Mathematics teachers' reflective practice within the ...

Page i of ii Volume 33 Number 3

Index December 2012

PYTHAGORAS

Journal of the Association for Mathematics Education of South Africa ISSN: 1012-2346 (print) ISSN: 2223-7895 (online)

ii ‘SierpinskiCuboctohedron’. Created by Phidelity. Available from http:// www.phidelity.com/photos/v/ Artwork/Xenodream/Sierpinski/ sierpinskiCuboctohedron.jpg

Information for Authors and Readers Original Research

1

Meeting the requirements of both classroom-based and systemic assessment of mathematics proficiency: The potential of Rasch measurement theory

Editor-in-Chief

Tim Dunne, Caroline Long, Tracy Craig, Elsie Venter

Alwyn Olivier

University of Stellenbosch, South Africa

Associate Editors

17

Michael de Villiers

Michael de Villiers

University of KwaZulu-Natal, South Africa

Anthony Essien

An illustration of the explanatory and discovery functions of proof

25

Coherence and connections in teachers’ mathematical discourses in instruction Hamsa Venkat, Jill Adler

University of the Witwatersrand, South Africa

Dirk Wessels

University of Stellenbosch, South Africa

33

choice

Editorial Board

Erica D. Spangenberg

Jill Adler

University of the Witwatersrand, South Africa

Bill Barton

University of Auckland, New Zealand

Thinking styles of Mathematics and Mathematical Literacy learners: Implications for subject

45

Pictorial pattern generalisation: Tension between local and global visualisation Duncan Samson

Doug Clarke

Australian Catholic University, Australia

Marcelo Borba

54

Barbara Posthuma

State University of Sao Paulo, Brazil

Jeremy Kilpatrick

University of Georgia, United States

Mathematics teachers’ reflective practice within the context of adapted lesson study

63

Alignment between South African mathematics assessment standards and the TIMSS assessment frameworks

Gilah Leder

La Trobe University, Australia

Mdutshekelwa Ndlovu, Andile Mji

Stephen Lerman

Southbank University, United Kingdom

Frederick Leung

73

Reviewer Acknowledgement

Liora Linchevski

75

Guidelines for Authors

University of Hong Kong, SAR, China Hebrew University of Jerusalem, Israel

John Malone

Curtin University, Australia

Andile Mji

Tshwane University of Technology, South Africa

Leonor Moreira

Universidade do Algarve, Portugal

Willy Mwakapenda

Tshwane University of Technology, South Africa

John Olive

University of Georgia, United States

David Reid

Acadia University, Canada

Paola Valero

Aalborg University, Denmark

Renuka Vithal

University of KwaZulu-Natal, South Africa

Anne Watson

Oxford University, United Kingdom

http://www.pythagoras.org.za

i

Pythagoras

Page ii of ii

Title Owner: The Assosiation for Mathematics Education of South Africa

The Association for Mathematics Education of South Africa

http://www.amesa.org.za

Index

INFORMATION FOR AUTHORS AND READERS OF PYTHAGORAS Focus and scope Pythagoras is a scholarly research journal that provides a forum for the presentation and critical discussion of current research and developments in mathematics education at both national and international level. Pythagoras publishes articles that significantly contribute to our understanding of mathematics teaching, learning and curriculum, including reports of research (experiments, case studies, surveys, philosophical and historical studies, etc.), critical analyses of school mathematics curricular and teacher development initiatives, literature reviews, theoretical analyses, exposition of mathematical thinking (mathematical practices) and commentaries on issues relating to the teaching and learning of mathematics at all levels of education. Pythagoras is a peer-reviewed journal, accredited and approved by the DoHET for inclusion in the subsidy system for being a research publication output for South Africa.

Indexed in: MathEduc (Gesellschaft für Didaktik der Mathematik) SA ePublications GALE, CENGAGE Learining ProQuest Google Scholar Directory of Open Access Journals Elsevier Scopus

Published by: AOSIS OpenJournals AOSIS (Pty) Ltd Postnet Suite #55 Private Bag X22 Tygervalley 7536 South Africa Tel: +27 (0)21 975 2602 Fax: +27 (0)21 975 4635 [email protected]

Title Operations Coordinator Rochelle Flint submissions@pythagoras. org.za

Subscriptions: [email protected] Tel: +27 (0)21 975 2602

Abbreviation and referencing This journal is marked and branded as Pythagoras. It is officially referenced as Pythagoras. Sponsors, endorsement, publication frequency and distribution In an endeavour to enhance its global support for mathematics education, AMESA has implemented a sponsorship system for all authors submitting to Pythagoras. Pythagoras publishes two issues per year and is also available at http://www.pythagoras.org.za The online version is an open access publication. Printed copies are distributed to subscribers (this includes AMESA members) – to subscribe contact orders@ openjournals.net or contact +27 (0)21 975 2602. Manuscript preparation and submission All articles must be submitted online at http://www.pythagoras.org.za Go to http://www.pythagoras.org.za. If you are already registered please proceed with the submission process by logging into the site. If not registered, please register with the website. Complete all the required fields and remember to tick the checkbox under ‘Register as Author’ at the bottom of the registration page. (You may also register here as a reviewer.) Login with your username and password and simply follow the easy five step process to upload your manuscript. Copyright Copyright on published articles is retained by the author(s). Licensee: AOSIS OpenJournals, AOSIS (Pty) Ltd. This work is licensed under the Creative Commons Attribution License. Every effort has been made to protect the interest of copyright holders. Should any infringement have occurred inadvertently, the publisher apologises and undertakes to amend the omission in the event of a reprint. Disclaimer The title owner and the publisher accept no responsibility for any statement made or opinion expressed in this publication. Consequently, the publishers and copyright holder will not be liable for any loss or damage sustained by any reader as a result of his or her action upon any statement or opinion in this issue. Correspondence Correspondence regarding manuscripts should be addressed to the Editor-in-Chief, Alwyn Olivier at [email protected]

AOSIS OpenJournals team: Rochelle Flint (TOC) Margo Martens (TOC) Duncan Hooker (TOC) Meghan McLachlan (Layout) Joanie Meintjes (Layout) Louwna van Heerden (Layout) Trudie Retief (POM)


ii

Pythagoras

Page 1 of 16

Original Research

Meeting the requirements of both classroom-based and systemic assessment of mathematics proficiency: The potential of Rasch measurement theory Authors: Tim Dunne1 Caroline Long2 Tracy Craig3 Elsie Venter4 Affiliations: 1 Department of Statistical Sciences, University of Cape Town, South Africa Centre for Evaluation and Assessment, University of Pretoria, South Africa

2

Academic Support Programme for Engineering in Cape Town, University of Cape Town, South Africa

3

The challenges inherent in assessing mathematical proficiency depend on a number of factors, amongst which are an explicit view of what constitutes mathematical proficiency, an understanding of how children learn and the purpose and function of teaching. All of these factors impact on the choice of approach to assessment. In this article we distinguish between two broad types of assessment, classroom-based and systemic assessment. We argue that the process of assessment informed by Rasch measurement theory (RMT) can potentially support the demands of both classroom-based and systemic assessment, particularly if a developmental approach to learning is adopted, and an underlying model of developing mathematical proficiency is explicit in the assessment instruments and their supporting material. An example of a mathematics instrument and its analysis which illustrates this approach, is presented. We note that the role of assessment in the 21st century is potentially powerful. This influential role can only be justified if the assessments are of high quality and can be selected to match suitable moments in learning progress and the teaching process. Users of assessment data must have sufficient knowledge and insight to interpret the resulting numbers validly, and have sufficient discernment to make considered educational inferences from the data for teaching and learning responses.

Independent researcher

4

Correspondence to: Tracy Craig Email address: [email protected] Postal address: Private Bag X3, Rondebosch 7701, South Africa Dates: Received: 18 July 2011 Accepted: 07 Oct. 2012 Published: 21 Nov. 2012 How to cite this article: Dunne, T., Long, C., Craig, T., & Venter, E. (2012). Meeting the requirements of both classroom-based and systemic assessment of mathematics proficiency: The potential of Rasch measurement theory. Pythagoras, 33(3), Art. #19, 16 pages. http://dx.doi.org/ 10.4102/pythagoras.v33i3.19

© 2012. The Authors. Licensee: AOSIS OpenJournals. This work is licensed under the Creative Commons Attribution License.

Introduction The assessment of mathematical proficiency is a complex task. The particular challenges inherent in this process depend on a number of factors, including the definition of what constitutes mathematical proficiency, an understanding of how children learn and the approach adopted as to the purpose and function of teaching. Besides these central questions in mathematics education, there are important questions to consider about the relationship between classroombased assessment and systemic assessment types. Whilst there is potential for positive information exchange between these two types of assessment, more often there is an unnecessary conflict or simply a lack of constructive communication. Classroom teachers are at times perplexed by the outcomes of systemic assessment, confused about what action to take as a result of the reported outcomes and, in the worst-case scenario, demoralised. The quest for positive information exchange demands that questions about quality at both classroom and systemic sites are addressed (see also Wyatt-Smith & Gunn, 2009, p. 83). In this article, we differentiate explicitly between the two broad types of assessment: classroombased and systemic (or external) assessment.1 The rationale for assessing, the demands of the stakeholders, the forms of the assessment instruments and the types of data produced can and do vary substantially. Having briefly explored the differences between the two assessment types, we discuss the broad distinction between two approaches to learning and teaching, one that may be termed a developmental approach and one that may be termed a deficit approach (Griffin, 2009). The particular approach adopted within a context will inevitably impact on the choice of and reasons for assessment. If systemic assessment is to be useful within the classroom the results need to be interpreted by teachers and found applicable in the classroom context. Underlying this requirement of applicability is the presence of a model of developing mathematical proficiency that includes both plausible conceptual development (from the mathematical perspective) and cognitive development (from the learner perspective). A model such as envisaged here should be somewhat loosely configured and address common issues so that it does not exclude different approaches to mathematical teaching and learning (see Usiskin, 2007). Such a degree of coherence (from broad consensus towards a developmental model, to a working curriculum document that outlines the broad ideas, to a more specified curriculum at school level and a school programme 1.For detailed descriptions of assessment types and a coherent framework, see Black (1998).


1

doi:10.4102/pythagoras.v33i3.19

Page 2 of 16

Original Research

providing more detail2) is at present a legitimate dream to work towards. Also envisaged in the dream is the idea that professional development, accountability testing and formative classroom experience are integrated around core core aspects of the discipline (Bennett & Gitomer, 2009). The theoretical insights informing a developmental model and the elaborated assessment programme are not the immediate concern of this article. We propose merely to show how applying Rasch measurement theory (RMT) may support such a project.

scenario, the forms of evidence used in classroom assessment may vary, from projects requiring extended planning to quick quizzes. Such variety embraces different learning styles and different facets of mathematical proficiency and adheres to cognitive science principles (Bennett & Gitomer, 2009, p. 49). The stakeholders in classroom-based assessment are the teacher and the learners. The data sets produced by the classroom assessment exercises are not necessarily designed to be expressly meaningful to anyone outside the classroom, although inevitably and importantly teachers within a school community may share ideas and discuss assessments and their results. The particularity and the immediacy of a test or assessment give it currency in the classroom context and for the classroom processes, at a specific period in time.

An essential part of that support is the facility of the Rasch model to yield measurement-like differences and changes. These quantities can enrich the evidence accessible from classroom-based assessment and satisfy the expectations of external stakeholders, in particular if one takes a developmental approach to learning (Griffin, 2009; Van Wyk & Andrich, 2006).

We may note that in any classroom test or assessment, the teacher is generally concerned with a current spectrum of learner skills and needs in the class, which invariably differs from the spectrum that confronts the educational decision-maker at a district or provincial level. The learners in a particular class may have test performances that are on average well above or well below the average performance associated with all learners of the corresponding grade in an entire school district or province, in the same or an equivalent test. Moreover, the variation of individual test performances within any particular classroom will generally be substantially less than the overall variation in performances on the same instrument across the school district or province.

An example is presented which illustrates the intervention potential of an assessment programme that adheres to RMT and within which the Rasch model is applied. We advocate that this model should be seriously considered for inclusion in the approach to national systemic and external assessment programmes, in particular for mathematics. In essence, we explore the question: What model of assessment may support teaching and learning in the classroom, and in addition enable broad-based monitoring of learning progression within districts and provinces? Reciprocally: How might systemic assessments not only serve their intrinsic purposes to inform decision-makers about performance levels in broad strokes, but simultaneously inform and enrich teaching and learning within the variety of classroom level challenges into which these single instruments intrude?

Systemic assessment Whilst classroom assessment is generally fine-grained and topic specific, external systemic is generally broadly banded, and attempts to ‘cover the curriculum’. From the perspective of the education departments, and in some cases other stakeholders such as funders of programmes, major purposes of systemic assessment are to assess the current performance and variability within a particular cohort of learners, according to some sort of external benchmark of desired proficiency, and to monitor progress, also according to some external standards for change and performance improvements over time. Overall averages (or percentage scores) and the associated pass rates (learner percentages at or above a specified pass criterion) may be deemed particular elements of interest, but their meaningfulness nonetheless has to be argued and established in a suitable robust exposition. These outcomes should be interpreted in relation to other assessment types, for example classroom-based assessment (see Andrich, 2009).

Classroom and systemic assessment The important distinctions between and commonalities within classroom-based assessment and systemic assessment types are discussed below. In addition, the complexities involved in reporting results at an individual level and monitoring change over time are noted.

Classroom-based assessment The teacher in the classroom is concerned with the learning processes and development of the learners in her class. Successful assessment is often of a formative nature and can emerge as continuous assessment, which helps to direct learning and teaching; the summative component, recording marks for the purpose of reporting, also plays a role.3 The rationale for a teacher to run assessment exercises is to determine whole-class and, particularly, individual levels of current development, to diagnose current obstacles to learning progress, and to provide subsequent targeted scaffolding to appropriate classroom segments. In the best

For systemic and external assessments, the sheer extent of the testing programmes, and the development time period and financial constraints, may impact resources and available turn-around time for testing, scoring and data capture. Systemic test designers may thus be obliged to limit the types of items to multiple choice or short-answer responses, to limit testing time to (say) a single period of a school day and, in consequence, to limit the maximum number of items that can reasonably be attempted.

2.See Thijs and Van den Akker (2009) for descriptions of curricula at the macro, meso and micro levels. 3.We consider the terms formative and summative assessment not as referring to discrete entities, but as depicting points on a continuum. Assessment moments may have elements of both kinds.


2


Page 3 of 16

Original Research

In an ideal situation, a systemic assessment is designed to produce, from a single short dipstick event, performance data about the current health of educational systems, which is meaningful to stakeholders, district officials and state educational bodies. This body of data and its interpretation may result in decisions requiring or offering intervention or other monitoring functions.

TABLE 1: Performance categories associated with percentage attained. %

The Department of Basic Education 2009 review claims that ‘externally set assessments are the only credible method of “demonstrating either to parents or the state whether learning is happening or not, or to what extent”’ (Dada et al., 2009, p. 36, citing Chisholm et al., 2000, p. 48, [emphasis added]). We contest that claim and, with Andrich (2009), maintain the view that the results of external assessment must be considered in conjunction with classroom assessment, rather than alone. In fact, one may argue that to invoke only external test results to convince stakeholders whether or not learning is happening at the individual or class level, and even perhaps at the grade level in a school, amounts to dereliction of duty and is a dangerous, unethical practice. The claim (of invoking only externally set assessment) itself is unethical, however, if it does not sufficiently address the complex issues of causation that lurk within the extensive variation of student performance on the test.

Performance categories

0−29

Not achieved

30−39

Elementary achievement

40−49

Moderate achievement

50−59

Adequate achievement

60−69

Substantial achievement

70−79

Meritorious achievement

80−100

Outstanding achievement

Reporting at an individual level A fairly recent expectation is that the results of systemic assessment be made available to parents. This new access to information may be well intentioned, but the form of the information is problematic, precisely because the data from a single and necessarily limited instrument are so fragmentary and imprecise. Systemic assessment is generally not finegrained enough to report to teachers, or parents, the results of individual learners, as if these single test performance results, ascertained from an instrument of about an hour’s duration, are on their own an adequate summative insight into a year’s progress in the classroom. Even bland descriptions limited to only pass versus fail criteria for a systemic test should be supported by some vigorous and robust debate amongst curriculum specialists, and result in an explicit consensus, before such pass or fail designations of test performance outcomes are communicated. These discussions may be most productive if they occur before a test is finalised for use, and again after the tentative results are available, with explicit minutes recorded at both stages.

Similar critique of inordinate emphasis on systemic tests, offered by Bennett and Gitomer (2009), rests primarily on two counts: firstly that systemic testing has unintended detrimental consequences for teachers, learners, administrators and policy makers, and secondly that this type of assessment generally offers limited educational value, as the assessment instrument is usually comprised largely of multiple choice or short answer questions (p. 45).

In some systemic tests administered under the auspices of the Department of Education (2005), designations of performance categories are assigned to the percentage of maximum scores attained on the instruments, as in Table 1.

A systemic assessment may in its totality give a valid overview of system-wide performance on the test instrument (through its constituent items) for the part of the subject and grade curriculum or domain which actually appears within a finite test. Possibly, by astute design and professional concurrence, the test may satisfy further criteria, so as to be viewed as a valid assessment of the whole curriculum at a system-wide level. The attainment of such all-encompassing curriculum validity would, however, require a complete revision of the current systemic test design, as noted and proposed by Bennett and Gitomer (2009).

On the basis of the systemic test score alone, a learner or parent is given a qualitative description that, however well intentioned, is simply arbitrary, invalid and possibly fraudulent, until other evidence justifies the descriptions offered. It is arguable that such descriptions are generally damaging, but especially when test design has not been informed at all by any criteria for item construction and selection that might relate to either the cut-points and the preferred 10% intervals or the adjectives chosen.

Whatever the virtues of a systemic test instrument, it simply cannot give the same level of precise inference about the performance of the individual, class or grade within a school as it does for aggregations at district or province levels. This comment applies even to the highly informative instrument4 we analyse further in this article. For that reason, any interpretation of classroom or grade performance data for a school has to be tempered with a deeper contextual understanding of those units of aggregation, for example, the particular class and the particular grade, and the school in its context and the history of its learners.

When systemic tests are designed5, there do not appear to be any explicit conditions or attempts made to warrant such achievement categorisations. Their valid use would suggest explicit design and the selection and inclusion of items precisely for the vindication of such verbal descriptions. For a 40-item test, the seven performance designators seem to imply a hierarchy of items, comprising 12 simple items

4.We distinguish here between a highly informative instrument and an instrument which through rigorous analysis and revision may be regarded as valid and fit for purpose.

5.These divisions may be the intentions of the test designers. In practice this balance is difficult to achieve.


Table 1 indicates instead a tortured avoidance of any verbal signals that learning is in distress, and of any recognition that some children are at precarious risk in the subject.

3


Page 4 of 16

Original Research

that all basically competent learners should have mastered, four items specifically associated with each of the five 10% interval categories, and eight items that address curriculum elements at the highest levels of cognitive demand for the particular grade cohort of the test.

A major purpose of equivalent tests is to legitimate comparisons. We may wish to examine progress within an individual over time, or to contrast the competencies elicited from two distinct cohorts of learners. Whilst such comparisons may and should admit and use profound qualitative insights and inferences, there is often an intention to seek numerical evidence to bolster those conclusions, and to argue their consequentiality. For that reason, inter alia, it will be of interest to obtain measurement-like outcomes of test instruments, in order to allow use of appropriate numerical differences and perhaps numerical ratios.

Such a table as a criterion-referenced end-product, using cut-points and descriptions, may be a laudable goal, but we suggest it cannot be achieved in the time horizons of planning and test design currently applicable in national and provincial tests. If we are correct, then it becomes important to redesign the systemic test construction agendas and timelines to ensure that the criterion-referenced outcomes are validly constructed within the instruments.

Inferences about systemic and classroom testing We argue that systemic testing is valuable as an external assessment technique at broader levels of aggregation, such as district or province, but is substantially less valuable where aggregation is narrower, such as at class and school level.

In general, current systemic tests include items that explore elements of the curriculum that warrant particular attention. For each of these elements, a test instrument will indicate how many learners exhibit the desired mastery in the associated responses. Thus the instrument can validly diagnose a series of current particular needs or inadequacies. Summarising the item performance of a class of moderate size in a grade will give an indication of those curriculum elements of which those learners as a group do not yet have mastery.

A well-functioning system of external assessment would involve teachers in the development of the test instruments. It would also feed the results and analysis back into constructive professional development, intended ultimately to impact on classroom practice. In reality, the current design cycles of systemic testing and most external assessments, with or without envisaged professional development support, are too short. The cycles do not encourage adequate engagement with teachers at either the design or analysis stages.

These indications, inferred from items that have elicited evidence of low proficiency, do not however identify what factors are contributing to the performances, whether good or poor. The class item scores report states, rather than relationships or processes. They may tell us where a problem is to be found, but not why it arose and what may be necessary to address it effectively.

Whilst engagement with teachers may not be a sufficient condition in itself to ensure subsequent effects in the classroom, it is certainly critical that assessment results make sense to teachers, and that the credibility and relevance of the outcomes are pursued. A systemic testing model proposed by Bennett and Gitomer (2009), provides an alternative model which avoids many of the pitfalls mentioned previously. This model includes three intersecting phases: an accountability phase, a formative phase and a professional development phase, wherein the engagement of teachers is a critical feature of the process.

Reporting change Any objective or intention to use systemic test performances to report on change between years, and possibly on trends over time, will involve an enormous amount of preparatory work to ensure the test performances for the various time periods are truly comparable. There needs to be demonstrable evidence that the associated tests are effectively equivalent. Where it is not possible to use the same instrument on two separate occasions, construction of equivalence is difficult and must be undertaken rigorously. Where the same instrument is used within too short a time frame, the problem of response dependence6 and appropriate targeting has to be addressed.

Summation and comparison Designers of any test instrument face the challenge of informing both the classroom and external stakeholders. We argue that, alongside this, traditional instruments assume validity of arithmetical functions, such as summation and comparison, which are not necessarily grounded in sound statistical theory.

Such preparatory work will involve subject and teaching expertise, but must necessarily impact on test construction and assessment. Without this work, and associated extensive piloting of all the test items or instruments in question as well as linking and equating processes, any apparent comparisons of individual test performances to measure change over time must be regarded as moot. It is safer to regard them as invalid until an equivalence relationship between performances over time or across tests has been explicitly argued and demonstrated.

We note that every assessment instrument will involve the summation of item scores. The validity of adding these scores underpins all assessment practice. Our current conventions of practice assume this operation is reasonable in every test, even though we may in contrast alert learners to the errors of adding apples to pears or grapes to watermelons. The unique role of Rasch measurement models in confirming the admissibility of summing test item scores to obtain a testperformance indicator, and in supporting interpretations of test results, will be outlined shortly.

6.Statistical techniques to resolve or account for issues of item dependence across replications of a single instrument for a particular cohort of learners are possible, and even necessary, to ensure validity of results (see Andrich & Marais, 2012).


4


Page 5 of 16

Original Research

many cases be counterproductive. Certainly opting for such post hoc ‘teaching to the test’ is something of a backward move, unless of course the ‘the test is worth teaching to’ (Bennett & Gitomer, 2009).

Comparison of assessment performances using numerical differences requires that there is some common scale against which the two sets of performances can be authentically captured as numbers of a common kind. Then we may compare by subtraction. In effect we mimic the way we compare 23 apples with 26 pears by obtaining a distinct currency values for each individual fruit of each set, and then use additions and a subtraction. We must assure ourselves that we can discern differences by use of a common inherent unit.

A developmental approach builds on and scaffolds from the existing knowledge base of individual learners. This approach, advocated by Steinbring (1998), requires that a teacher be attuned to the learner’s current understanding and hence current location on a developmental path. The teacher has to be able to diagnose and analyse the various students’ current constructions of mathematical knowledge within a curriculum. Then she has to compare these constructions with the mathematics knowledge required (informal assessment), and to adjust her teaching accordingly so as to facilitate the transition (Steinbring, 1998). This process happens against the background of a carefully constructed sequence of learning experiences, exhibiting a suitable sequence of logical and evolving mathematical concepts and theorems that are to be learnt.

Rasch approaches also allow evidence of change to emerge from the differences observed between two testing contexts whose comparability has been carefully constructed. The potential of the Rasch model to support use of informationenriched assessment for constructive classroom intervention in order to bring about changes in learning will be described shortly. Here we argue simply that educational objectives of assessing performance and monitoring for numerical evidence of change must rely on the admissibility of summing item scores and of subtracting test scores. Authorities need to explicitly establish and not simply assume that the conditions for using arithmetic operations are inherently defensible parts of the assessment instruments and their processes.

The developmental approach resonates with the work of Vergnaud (1988). He emphasises the important link between learners’ current intuitive knowledge and the targetted more formal knowledge, and where the teacher’s role assures scaffolding of the formal knowledge. The perceived ‘errors’ highlighted in a deficit model become the stepping stones to greater understanding and the construction of generalisable mathematical concepts.7

Deficit versus developmental approaches to learning and teaching Griffin (2009) makes a distinction between deficit and developmental learning approaches. A deficit approach may ‘focus on the things that people cannot do and hence develop a “fix-it approach” to education, and thereby focus on and emphasise “cures” for learning deficits’ (p. 187). The deficit approach is common practice where systemic assessment design processes take place in a short time period within the school year, with less than optimum engagement with any teachers and schools, and constrained to the use of a single instrument for a limited extent of class time.

Something of a paradigm shift is required in order to focus on a developmental trajectory which takes into account the network nature of mathematical concepts and considers that learners may learn different concepts at different rates and in different sequences. This shift may obviate a learning approach where the focus is only on those mathematical objects and skills which cannot yet be exhibited fluently. What is required is an assessment instrument which can more reliably inform teachers of the locations of learners along an intended trajectory of development. Such an assessment instrument may also more reliably inform the education departments, and stakeholders such as funders, of the current learning requirements of particular cohorts of learners, at least in the associated curriculum elements, through an explicit sequential rationale.

These practices are followed by a period of data scoring and capture, an extensive analysis being performed on the data, and some form of particular aggregated data provided to the schools, many months after the assessment was designed, and of no possible diagnostic value for the same classrooms from which the data arose.

When a test is well-designed for its purpose of distinguishing between different levels of learner performance, then we may simply order individual learner performances from lowest to highest, and order test items by their observed levels of difficulty. By partitioning learner scores into a range of ordinal categories, and similarly defining ranges of item difficulties, we may ascertain associations between these groupings that suggest educationally meaningful sequences within teaching and learning. Such a device is produced by

Invariably, the media are informed of the ‘research’ and information such as ‘x% of learners in Grade z cannot handle concept y’, thereby exemplifying a deficit approach. The Grade z teachers then, possibly as a result of a circular letter informing them that only x% of their learners have mastered concept y, change their teaching plans and focus an inordinate amount of energy on teaching concept y. The mathematical concept y may not singly be the problem, but may indicate a constellation of concepts that have not yet been mastered (see Long, 2011; Long, Wendt & Dunne, 2011). To focus on concept y without understanding the bigger picture may in http://www.pythagoras.org.za

7.The answers to constructed response items in a systemic test set are often found to be partly correct, thus supporting Vergnaud’s (1988) notion of ‘concepts-inaction’. The transition from localised concepts-in-action to formal and generalisable concepts is the challenge of mathematics education.

5


Page 6 of 16

Original Research

the Rasch measurement model and can be easily incorporated into systemic-testing design, so as to permit the provision of supplementary diagnostic information about items, inter alia for later communication after assessment results have been analysed.

In the social sciences, including education, the first step required towards measurement-like observations is to make explicit the construct to be tested. The operationalisation of the construct as various items, indicative of various levels of proficiency, makes up a test instrument whose overall purpose is to approximate measurement of a characteristic or an ability of persons. This ability is assumed to be plausibly described by a location on a continuum, rather than merely by membership of a discrete ordered category.

Rasch measurement theory We argue that the assessment opportunities provided by the application of the Rasch measurement model can resolve the potential conflicts between the contrasted viewpoints discussed above: classroom based assessment and systemic type assessment, and a developmental model and a deficit model. A well-designed assessment instrument, or sets of instruments, can provide detailed information on the individual development of each learner as well as simultaneously informing external stakeholders on the educational health of an education system. The requirements of the Rasch model resonate with the requirements of good educational practice.8

In designing a test instrument we are obliged to consider and specify both the construct of interest that we seek to measure, and the context or type of context within which the instrument is intended to be applicable. Having identified and described the construct of interest and designed a plausible test instrument, as a collection of items selected with an educational context in mind, the next step is administering the test to the intended study group (see Wright & Stone, 1979). There is no requirement that the items are all of equal or equivalent difficulty. They will generally be a collection with elements at various levels of difficulty.

Rasch measurement theory is explained in a number of publications (Andrich, 1988; Rasch, 1960/1980; Wilson, 2005; Wright & Stone, 1979, 1999). A comprehensive application of the Rasch model to a mathematical area, the multiplicative conceptual field, can be found in Long (2011), and an application pertaining to language assessment in Griffin (2007, 2009). In this article, the purpose is merely to illustrate the application of the Rasch model in one systemic test, through stipulating the requirements of the Rasch measurement model and through depicting the outcomes in a form that has the potential to inform both stakeholders and teachers, and to mitigate the misunderstandings that may arise when only aggregated data is used.

Of particular importance at this step is that the test instrument has been properly targeted to the cohort to be tested. This objective is a substantial challenge, because it involves hazarding judgements about how the overall study group will respond to the items, both individually and collectively, but doing so prior to having any corroborative information. In an educational context this challenge requires subject expertise, teaching experience and pedagogical insights into the learning journeys particular to the subject. It will preferably include cycles of engagement about suitability of items and item structures with specialists in the field, namely specialist teachers.

Whilst the example test was designed for a systemic application, it exhibits features which suggest areas of improvement in a subsequent design. The choice of setting happens to be mathematical, but the methodology is not tied to any single discipline.

Appropriate targeting is the requirement that the instrument will be able to distinguish effectively between various levels of performance across the spectrum of learner achievement arising in a specific context of assessment. We may be particularly interested in distinguishing between overall performances at precisely those levels most frequently observed in the assessments. The adequacy of our targeting will contribute to the precision which an assessment instrument can achieve, and hence validly discern differences in ability, in the specified context.

The Rasch measurement model is based on a requirement that measurement in the social sciences should aspire to the rigour that has been the hallmark of measurement in the physical sciences (Wright, 1997). A great deal of qualitative and theoretical work is required in order to construct a valid measurement instrument, as in the physical sciences (such as the thermometer, ruler, scale, or clock). In the natural sciences measurement devices are designed for specific contexts. Though notions such as length, mass and time have universal application, the selection of the specific instrument by which we choose to measure those characteristics is necessarily determined in part by the context in which we seek to comprehend and measure levels of extent and variation in extent.

We note that targeting a test instrument, in order to maximise its prospective use for distinguishing between performances in a specified study group, is a completely different issue from using the test as an instrument to decide which learners have performances exhibiting a desired level of competence in the curriculum of the test subject. We may order a set of student performances from best to worst, regardless of what subsequent judgment we may care to make about which of them attain a pass or attain a distinction on the basis of the ordered test scores.

8.The model was developed by Georg Rasch in the 1950s in order to solve an educational dilemma: that of measuring reading progress over time with different tests (Rasch, 1960/1980). Equating and linking of tests over time, initiated in the 1950s, are examples of the immense power of the Rasch model.


6


Page 7 of 16

Original Research

successfully on a dichotomous item i, with two ordered categories, lower and upper designated as 0 and 1, in the equation:

This difference between distinguishing and deciding arises from the contrasting nature of norm-based (internal ordering) and criterion-based (external notions of pass and distinction) inferences, Rasch models can admit the strengths of both norm and criterion approaches. More will be said on the matter of criterion-referencing in the illustrative example.

P { X ni = x } =

1 + e(

β n − iδ i )

=

x β − iδ e ( n i) λni

[Eqn 1]

Here P is the probability, Xni is the item score variable allocated to a response of person n on dichotomous item i, the number x is an observed score value (either 0 or 1), where βn is the ability of person n and δi is the difficulty or location of item i. Note that Equation 1 does not require any restrictions on either of the real numbers βn or δi, but it does require that the two values can be subtracted. The function of the denominator λni in Equation 1 is simply to ensure the (two) probabilities for the dichotomous item sum to 1.

The Rasch model and its item requirements 9

The essential idea underpinning the Rasch model for measuring ability by performance on a test, is that the whole test comprises a coherent set of appropriate items. Each item is conceptually relevant to the purpose of the test: it consistently gives partial information about the ability which we seek to measure (justifying a possible inclusion of the item), it enriches the information provided by all the other items collectively (contradicting possible redundancy and exclusion of the item), and it is substantially free of characteristics which might obscure the information obtainable from the instrument (contributing to the precision, rather than to uncertainty, of the instrument, and being free of bias).

The relationship of item to learner is such that if a learner labelled n is at the same location on the scale as an item labelled i, then βn = δi or (βn − δi) = 0. In consequence the two probabilities for the ordered categories are equal. Then substituting this zero difference for the bracketed terms into Equation 1 implies that the learner of any ability level will always have a 50% chance of achieving a correct response to any dichotomous item with a difficulty level equal to his or her ability level. If an item difficulty is above the ability location of any learner, then the learner has a less than 50% chance of achieving a correct response on that item, but if the item is located lower on the scale than the person location, the learner would have a greater than 50% chance of achieving a correct response.

Dichotomous items In educational settings, the Rasch model is a refutable hypothesis that measurement of an ability is being approximated by the test instrument outcomes in a specified context. It postulates that the ability level of a particular person can be represented by a single number βn. In its simplest form for dichotomous items (with outcomes success or failure, scored as 1 and 0), the model assumes that single numbers δi represent the difficulty levels of the items.

The graph of Equation 1 for a specified value of δi is obtained by setting the probability on the vertical axis, and person parameter βn on the horizontal axis (see Figure 1). The result is a symmetric s-shaped ogive curve, with a midpoint at (δi, 0.5). This curve is termed the item characteristic curve. The ascending curve (from low on the left to high on the right of the figure) indicates the probability of obtaining a correct response. The descending curve (from high left to low right) gives the complementary probability of obtaining an incorrect response.

Each outcome of an interaction between a person and an item is uncertain, but has a probability governed only by these two characteristics, that is by (βn, δi). The Rasch model avers that the arrays of numbers βn and δi are on the same linear scale, so that all differences between arbitrary pairs of these numbers such as (βn − δi), and hence also (βn − βm) and (δi − δj), are meaningful. Through these differences we may not only assign probabilities to item outcomes, but may also measure the contrasts between ability levels of persons and the contrasts between difficulty levels of items, and offer stochastic interpretations of those contrasts. The probability of any learner answering any item correctly is a function of only the difference between the locations of ability of the specified learner and the difficulty of the particular item (βn − δi). The model demands that no other person factor or item factor or other consideration intrudes into the probability of success on the item, and that the net joint interaction effect of the person ability and item difficulty is dominated entirely by that difference.

Equation 1 suggests that if we consider the subset of all persons whose common ability is precisely βn, then each of them will always have exactly the same probability of obtaining a score one, at each and every item whose difficulty is given by a

Probability

1.0

The logistic function with parameters (βn, δi), expresses the probability of a person n with ability βn responding

0.5

0.0

-3

-2

-1

0

1

2

3

4

Person Location (logits)

9.This section may be omitted on first reading, but readers are encouraged to become familiar with the underlying mathematical logic of the Rasch model.


x β − iδ e ( n i)

FIGURE 1: Item characteristic curve and complement: probability of 0/1 responses.

7


Page 8 of 16

Original Research

specific value δi. Similarly, these persons of common ability will all have the same probability of a zero score, for all items at the specified difficulty level. Moreover, this equivalence of probabilities will continue but with a revised common probability value, at any new set of items which are all at a distinct but common difficulty location δj, where δj ≠ δi.

Simultaneously, but distinctly, we also assume that higher levels of person ability βn will be associated with both higher item-score labels and marks x, for each polytomous item, and hence also with higher test total performance scores. These addition strategies are perfectly plausible and coherent, and have been common practice perhaps for many decades. But the issue of the conditions under which they can be defended as modes of obtaining objective and meaningful totals must still be addressed.

Equation 1 is a stringent requirement, but it is exactly as required for a dichotomous test item to be validly considered as unbiased and equally fair to all persons who take the test. It may appear to be only one equation in this format, but each version comprises two probability statements (for the values x = 0 and x = 1).Then there are 2 × N × K equations summarised within Equation 1 as Rasch models require that same stochastic structure for all possible N × K combinations of N persons in a study group, each interacting with each of K (dichotomous) items in a test instrument.

The levels of person ability can range over the entire set of real numbers (-∞, ∞). A consequence of the ordering of our categories in any polytomous item is that we also expect that each such item partitions the full ability range into a sequence of (m + 1) consecutive disjoint intervals, over which the corresponding most likely item category label or score will be 0, 1, 2, … m and in that ordering. If we wish to make inferences about the relative abilities of individual persons the Rasch measurement model is the only route by which to do so. All other models permit only vague general statements about the distribution of abilities for unspecified persons.

Multiple choice items Items offering a multiple choice amongst a closed set of response options are handled in the same way as dichotomous items. Some minor adaptations allow the analysis of test data to address the extent to which preference for the various false distractor items may exhibit patterns that vary over the ability range of the persons taking the test.

Software packages to perform Rasch analysis through stages of model checking, diagnostic processes and estimation procedures are available on the internet, and from development laboratories. This study made particular use of the RUMM2030 suite of programs. In the reported data (see Table 2), the five polytomous items are represented by the average thresholds.

Polytomous items Modifications of Equation 1 allow the probability relationships to be extended to polytomous test items that permit maximum score categories higher than one, for partially or completely correct responses. For polytomous items we permit each item response to be recorded as an ordinal category indicated as a single number within the set 0, 1, … m, where m > 1.

The Rasch model and consequences for test design Good test design seeks to have every item satisfying the design criteria outlined above. What Rasch methodology offers is the possibility of checking each of those item requirements, their collective functioning, and the various independence requirements. Constructing a valid instrument will require some arduous tasks at item level. When the item and independence requirements are each found to be reasonably satisfied by the test item data, the astonishing power of the Rasch model is harnessed.

It is important to note that we are making an ordinal set of categories, recorded primarily by numbers. Rules for allocating these number labels will be set out in a scoring memo for the polytomous item. Because we assume expert construction of each item and its scoring memo, we expect that higher item scores will be associated with higher abilities βn, and conversely that lower scores will be associated with lower abilities βn.

Statistical theory guarantees us that under these required conditions we can not only find a valid estimate of ability for each learner, but that for any person, the sum of his or her item scores is the key element in estimating that ability, and that all other detailed information from the data is neither needed nor helpful in the estimation process. We note that this sufficiency does not imply the total performance score itself is a suitable measure of the ability, but that the person ability measure is a mathematical function involving only that person’s total score.

We are only saying the labelled categories 0, 1, … m are distinct and uniquely ordered. We are not saying that unit differences between the scores x and x + 1 are the same, regardless of x. We are not considering any ratios to be valid. Here 2 is more than 1 but is not two times 1. Likewise 3 is higher than 2 and 1, but is not 3 times 1, nor 2 plus 1. This initial ordinal structure is therefore distinct from using the category labels x as marks. But we may go on to assume the labels to be marks, and also allow addition of these marks across all items. Then, for any particular item, as the marks x increase, we will expect higher total performance scores in general, and specifically, higher averaged total scores at each new higher observed label x. http://www.pythagoras.org.za

The same statistical theory also guarantees a similar result for items: counting how many of the N persons have been assigned into each of the (m + 1) score categories of an item, that is finding that item’s score frequencies, is sufficient to 8


Page 9 of 16

Original Research

obtain valid estimates for both the m thresholds of that item and for its average level of difficulty. No other information from the data is required, and no other information from the data set could possibly improve the estimation process. Again this sufficiency of the (m + 1) category frequencies for the m threshold estimates does not imply the frequencies themselves are suitable measures for the thresholds, but rather that threshold estimates are simply a mathematical function involving only those frequencies, whilst the person estimates are determined by the array of total scores.

stakeholders in a contextually meaningful way. We support our argument with an example drawn from recent practice in secondary school assessment.

An illustrative example A test instrument (K = 40 items) was designed for the purposes of measuring learner proficiency on Grade 8 mathematics. The test, as is common practice, combined several mathematical strands, such as data and probability, geometry, algebra, and number. The test was administered over a cohort of Grade 8 learners (N = 49 104) in one South African province. The study data was analysed applying the Rasch model, for the purposes of confirming appropriate difficulty level of the instrument as a whole for the learners and to identify and describe learner ability in relation to the test items (Long & Venter, 2009).

These two types of simple estimation structures are extraordinary. These simplicities do not hold for any other model than the Rasch measurement model. The Rasch model is essentially an hypothesis that an ability is measurable, indirectly, from test instrument data in a specified context. If the observed data do not fit the requirements of the Rasch model, then these measurement-like advantages, however desirable, do not arise. In consequence there is no way to coherently provide any statistical inferences relating to individual people or specific items, other than by frequency tables. Any long-term intention to make statistical comparisons between or within cohorts over time is irrevocably undermined.

The mean of all item locations is set at zero as a standard reference point in the Rasch measurement model11. The item difficulties are estimated and located on the scale. The learner ability values are then estimated. The learner proficiency estimates are located on the same scale in relation to the items. For the purposes of this analysis the scale was divided into bins of equal width. The left hand side of Figure 2 is a simplified histogram for the estimated ability values12. The chosen scale is the log-odds or logit scale, derived from

When the data fits a Rasch model, suitable transformation of the raw total scores for persons and raw frequencies of score categories of each item will enable calculation of estimates for both learner ability parameters and all item thresholds and average difficulty levels. All these estimates may then be legitimately represented and located on the same scale or linear dimension. All differences obtained from any pair of these N + M estimates have an explicit stochastic interpretation.

using the logarithm of odds (the ratio O =

Within this scale all the parameter estimates satisfy the required measurement-like properties, and have consistent stochastic interpretations.

We note that Figure 2 immediately provides decision-makers with an extensive but quick diagnostic summary of which items can be correctly answered by at least half (50%) of the learners at a set of specified ability levels, and which items are correctly answered by fewer than half of the tested persons at specified ability levels. The diagram provides a label in which the item number in the test is specified, and the item content is easily obtained by reference to that label.

The estimated item difficulties are calibrated to have a mean of zero10, and then the relative difficulties of the items are located accordingly. Thereafter the learner proficiencies are estimated in relation to the corresponding learner performance on each of the items. Figure 2 (in the illustrative example) displays a summary of item difficulty and person ability estimates in the same diagram. On the right side, all the items from the test instrument are located at their levels of relative difficulty. On the left side, all the learners are located at their individual levels of proficiency on the same vertical axis. Each learner is however only shown in the figure as hidden amongst the collective contributors to the cross (×) symbols at the particular interval in which their estimates appear. Note that the display gives valid insights into the test performance, but that no notions of fail, pass or distinction have been specified.

Here visual inspection of the proficiency histogram will suggest that the person (ability) mean is below the zero item mean, being located at approximately -1.0 logits. This negative location indicates that the test instrument is not appropriately targeted for the tested Grade 8 group as a whole. In consequence, somewhat less than optimum information for distinguishing between performance abilities on this test is obtainable for this cohort on this test. This graphical feature of the output indicates that the test could be improved to better match the variation in the study group. The data suggest that for this study group, more items of below the current average difficulty would improve the

The Rasch measurement model suggests an assessment system which provides statistically sound data and analysis which can inform classroom teaching as well as external

11.The software, RUMM2030 (Andrich, Sheridan & Luo, 2011), a programme designed to support the features and requirements of the Rasch measurement model, has been applied here.

10.There is a technical reason for setting the item mean equal to zero. A simple explanation is that there needs to be one arbitrary origin for all item difficulties because the data can only inform us about differences between item parameters in Equation 1, hence differences between person and item parameters.


Pr (X = 1) ). Pr (X = 0)

12.The terms ‘ability’ and ‘proficiency’ are both used to describe the location of persons. Proficiency is the preferred term as it denotes a current state rather than an innate condition.

9


Page 10 of 16

Original Research

TABLE 2: Items ordered from difficult to easy, with item location, standard error, item type, domain and item description. Item

Location

SE

Item type

Domain

Item description

I36

2.79

0.26

Poly

Data

Finds the mean of a data set

I40

2.42

0.23

Poly

Geometry

Calculates the coordinates reflected about the x-axis

I38

2.14

0.21

Poly

Number

Calculates rate problem

I37

1.81

0.19

Poly

Geometry

Calculates volume of a cylinder

I39

1.09

0.15

Poly

Number

Determines the exchange rate

I10

0.99

0.14

MC

Number

Calculates percentage increase

I23

0.98

0.14

MC

Geometry

Finds surface area of a prism

I26

0.75

0.14

MC

Geometry

Applies Pythagoras’ theorem

I34

0.70

0.13

MC

Data

Calculates total number in a stem and leaf plot

I35

0.55

0.13

MC

Data

Finds the mode of a data set

I33

0.46

0.13

MC

Data

Finds the median of a data set

I06

0.45

0.13

MC

Algebra

Manipulates algebraic fractions

I03

0.42

0.13

MC

Geometry

Estimates length measure in centimetres

I08

0.12

0.12

MC

Number

Calculates fractions

I21

0.08

0.12

MC

Algebra

Substitution of variables

I15

0.06

0.12

MC

Algebra

Addition and subtraction of algebraic terms

I13

0.03

0.12

MC

Geometry

Calculates angles of a triangle

I17

0.00

0.12

MC

Geometry

Applies horizontal translation

I19

-0.01

0.12

MC

Algebra

Solves problem applying multiplicative reasoning finds the range of a data set

I32

-0.08

0.12

MC

Data

I30

-0.08

0.12

MC

Data

Calculates theoretical probability

I25

-0.11

0.12

MC

Number

Applies knowledge of integers and square roots

I02

-0.20

0.12

MC

Number

Finds temperature difference, represents with integers

I31

-0.20

0.11

MC

Geometry

Determines exterior angle

I28

-0.23

0.12

MC

Algebra

Reasons about the square root of algebraic expression

I12

-0.25

0.11

MC

Algebra

Solves a linear equation

I01

-0.34

0.12

MC

Number

Identifies an irrational number

I11

-0.52

0.12

MC

Geometry

Reflects shape about the x-axis

I05

-0.60

0.11

MC

Geometry

Identifies coordinates of a linear function

I16

-0.69

0.11

MC

Number

Calculates fractions of time

I09

-0.70

0.11

MC

Number

Orders integers

I20

-0.70

0.11

MC

Algebra

Calculates arithmetical sequence

I22

-0.88

0.11

MC

Geometry

Identifies faces of a solid object

I07

-0.88

0.11

MC

Geometry

Knowledge of angles of a quadrilateral

I24

-1.03

0.11

MC

Data

Reads a pie chart

I27

-1.04

0.11

MC

Algebra

Converts additive problem into algebraic expression

I18

-1.05

0.11

MC

Number

Understands multiplication before addition convention

I04

-1.38

0.11

MC

Algebra

Recognises and predicts patterns

I14

-1.74

0.12

MC

Geometry

Identifies the net of a solid object

I29

-3.14

0.17

MC

Data

Interprets a bar chart

power of the test to distinguish between proficiencies at the lower segment of the person range, where most of the study group are located.

For learners clustered around the person mean, there are some items (below them) which are relatively easy, some items for which according to the model learners in this cluster have a 50% chance of answering correctly, but most items in the test (above them) are relatively difficult for this cluster of learners (fewer than 50% of them will answer correctly on any of the highest sets of items).

Augmenting the instrument with new items in the targeted range might make the instrument appear easier in the sense of possibly improved performances for all learners who performed well enough on the new items. That artefact of apparently increased scores and likely increased percentages, necessary in seeking better power to make finer comparisons between learner performances in the mid-range, will usually require a revised view of any corresponding criterionreferenced judgments such as pass-fail or distinction-pass applicable in a revised instrument.

Table 2 presents the same items from most to least difficult vertically down a table with brief descriptions of the K = 40 items in the associated levels. The easiest items therefore address the interpretation of a bar chart (I29) and the identification of a net (I14). The items, calculating rate (I38), coordinate geometry (I40) and calculating the mean (I36) emerge as the most difficult.

These revisions require precisely that same expert judgment which we hope originally contributes to the design of every systemic test, and to its educational interpretation, being exercised by the inclusion of new items and the interpretation of their consequences. http://www.pythagoras.org.za

For ease of analysis, some equally spaced levels, also denoted as proficiency zones, have been superimposed on the personitem map (see Figure 2). Items I15, I13, I17 and I19 are of average difficulty and therefore aligned with the item mean 10


Page 11 of 16

Original Research

--------------------------------------------------------------------LOCATION PERSONS ITEMS [locations] LOCATION PERSONS ITEMS [locations] --------------------------------------------------------------------4.0 4.0 | | | | | 3.0 3.0 | | I36(mean ofset) a data set) I36 (mean of a data Proficiency Zone 7 | | | 2.0 I40 (analytic geometry) geometry)I38(rate) I38 (rate) 2.0 × | I40(analytic × | I37(volume of a cylinder) I37 (volume of a cylinder) × | P-Zone 6 × | × | 1.0 I39 (exchange rates) rates) 1.0 ×× | I39(exchange I23 (surface area, prism) I26 (Pythagoras) prism)I26(Pythagoras) ××× | I23(surface area, I34 (total, data set) I10 (percent) ×× | I34(total, data set)I10(percent) P-Zone 5 I03 (length) I33 (median) I35 (mode, data set) ××× | I03(length)I33(median)I35(mode, data set) I08 I06 I06(fractions)I13(angles) (fractions) I13 (angles) ×××× | I08 0.0 I17 (trsfn) I25 (squ) I28 (intg) I30 (prob) I21 (alg) 0.0 ××× | I17(trsfn)I25(squ)I28(intg)I30(prob)I21(alg) ××××× | I02(Int)I32(range)I15 I19(algebra) I02 (Int) I32 (range) I15 I19 (algebra) ××××××× | I01(irrtnl)I12(Equations)I31(ext angles) I01 (irrtnl) I12 (Equations) I31 (ext angles) P-Zone 4 ×××××× | I09(Integers) I09 (Integers) 20 (sequ) I16 (fractn) I05 (anltc) I11 (trnsfmtn) ×××××××××××× | 20(sequ)I16(fractn)I05(anltc)I11(trnsfmtn) -1.0 I24 (pie chart) chart)I07(angles)I22(solids) I07 (angles) I22 (solids) -1.0 ×××××××××××××× | I24(pie ×××××××××××××××××××× | I18(calculation)I27(algebra) I18 (calculation) I27 (algebra) ×××××××××××××××× | P-Zone 3 I04 (geometric pattern) pattern) ×××××××××××××× | I04(geometric I14 (nets of a solid) ×××××××××× | I14(nets of a solid) -2.0 -2.0 ×××××× | ××× | × | Proficiency Zone 2 × | | -3.0 -3.0 | I29(bar I29 (bar chart) chart) | | Proficiency Zone 1 | | -4.0 -4.0 | | | | | -5.0 -5.0 | ---------------------------------------------------------------------× == 358358 Persons × Persons FIGURE 2: Person-Item map approximating person proficiency and item difficulty on a common scale.

Analysis of relative locations of learner proficiency and item difficulty in separate individual construct strands (for example Algebra) allows stakeholders in both classroombased and systemic assessment to further research and provide some appropriate intervention. For example, lesson sequences may be developed which attend to the increasing algebraic demands and the associated cognitive skills proximate to current levels of interpreted ability.

set at zero (see logits -0.01 to +0.06, see also Table 2). I29 and I14 are the easiest items, located at the lower end of the scale (logits -3.14, and -1.74), with I38, I40 and I36 the most difficult items, located at the top end of the vertical scale (logits 2.14−2.79). For the few learners at proficiency zones 8, 9 and above (not shown in Figure 2 due to the scale chosen), there are no items which challenge their mathematical proficiency. For learners at proficiency zones 6 and 7, there are five items located at a matching level, Items I39, I37, I38, I40 and I36, and for which the learners have around a 50% chance of being correct. http://www.pythagoras.org.za

Retrospectively, according to the model: when the amount [extent] of latent trait possessed by the candidate was equal to the amount [extent] needed to demonstrate the

11


Page 12 of 16

Original Research

fair reflection of learner proficiencies in algebra. See Items I04, I027, I020, I012, I019, I015, I015, I021, I006, arranged from least difficult algebra item (logit -1.74) to most difficult algebraic fractions item (logit 0.45).

criterion behavior, the probability that the person could demonstrate the behavior [in this instance] was 0.50. This [criterion] was an important idea in defining a person’s [current] ability, but it was crucial to the assessment being used to improve learning, identify appropriate teaching resources and to develop current policy. (Griffin, 2007, p. 90)

The potential is there, in the case of this systemic assessment, of identifying a hierarchy of competences within algebra through which learners could be guided in the small setting of a single classroom. The hierarchy of competences evident in Table 2, was derived from the responses collected from a very large sample of learners and not just from one classroom. This hierarchy could reflect increasing challenge in mastery of algebra as generally experienced by learners of that age. The development of a sequence of items, aligned with the theory of emerging proficiency in algebra, has the potential therefore to empower the researcher or professional teacher communities to structure learning opportunities in an informed manner, mapped to the needs of clusters of learners in her class whose proficiency has been mapped onto the same scale.

Systemic assessments and classroom intervention strategies We now make an educational assumption. We allow that the changing proficiencies between learners mapped against the static display of item difficulty as we move up Figure 2, will be very similar to the progression of proficiency on the corresponding curriculum elements particular to an individual learner. We assume that the learner is increasingly engaged in the teaching and learning classroom on tasks related to the test material and over time becomes better able to tackle items of greater difficulty up the vertical sequence. This assumption is debatable, since there is not necessarily only one pathway to mathematical sophistication in any grade. However its utility is that it allows us to interpret the static Figure 2 (with item descriptions) as part of a developmental model.

The efficacy of the instrument depends on the theoretical work that has informed the instrument and that also informs the analysis and the inferences to be made from the analysis. But given high quality theoretical work underpinning test construction and rigour in the refining of the instrument, we propose that the application of Rasch measurement theory may provide the means for meeting the needs of both the teacher or learners and the stakeholders interested in outcomes of large-scale assessment.

For each set of learners clustered at a level in Figure 2, we have some idea of the types of items which the cluster can currently manage (i.e. for which they have at least 50% chance of success). We also have some idea of the types of items just some small distance above the current cluster level, and hence located in what may be called the zone of proximal development (Vygotsky, 1962) for that cluster of persons:

Complementary strategies

The idea of ordering criteria and locating the criterion where the probability of success for each person is 0.50 can be linked to Vygotsky’s research which was driven by questions about the development of human beings and the role that formal education plays in the process. The challenge for educators was to identify students’ emerging skills and provide the right support at the right time at the right level. It was in this context that Vygotsky’s construct of Zone of Proximal Development (ZPD) – the zone in which an individual is able to achieve more with assistance than he or she can manage alone – was conceptualized. (Griffin, 2007, p. 90)

The advantage of identifying and targeting current need groups, emerging even from a non-optimal systemic test as reported here, arises if the results are known quickly. In large and complex educational structures where quick turn-around from data to results at a learner level is not easily achieved, it may be useful to consider an alternative complementary assessment strategy beyond systemic testing.

By specifying an assumed zone of proximal development for each cluster level, the teacher uses the test information to make teaching efforts more efficient. In this structure the teacher imposes temporary clusters within the class so as to more easily divide teaching efforts and time between groups with similar current needs, as reflected by the tested subject proficiency. For example, learners located in proficiency zone 3 have four items located within a similar zone. For these learners the model probability is 0.5 or 50% for answering correctly. For learners in proficiency zone 2, these same four items will be more difficult in general.

An external resource of a large collection of items, sufficient for several tests at any parts of the likely person ability range, along with associated already prepared diagnostic information, can be marshalled, and made available for devolved use by schools, grade leaders and teachers. There may be a need to provide facilitative scoring arrangements (e.g. electronic marking and outputs as provided for the example test in this article) so that the richness of the assessment resource feeds timeously into teaching. Given suitable systemic test and scoring resources, it will then be feasible for any classroom to be focused upon its own current needs, across all the very diverse ranges of classroom proficiency and school contexts.

From a conceptual development perspective, we see in Table 2, where the algebra items are in bold, that they are spread nicely over almost the whole range of item difficulties, and well aligned with learner proficiencies, therefore giving a

Making this option for selection and downloading of items feasible will require prior resource implications and processes. Many proposed items will need to be submitted, cleared for use, piloted and, where necessary, adapted. There


12


Page 13 of 16

Original Research

Where is the catch?

will be some attrition due to unsuitable proposals, and some necessity to ensure breadth of cover for the resource. All items will require grading and diagnostic ancillary information. The associated collaborations will generate teacher collegiality and contribute to professional development of classroom diagnostic skills and intervention initiatives.

In practice the validity of the output and analysis on which Figure 2 and Table 2 are based, is conditional on the adequacy of the fit of the test data to the Rasch model requirements. Checking the requirements of the model is an extensive and difficult task, precisely because this particular model embodies all the many requirements that permit measurement-like estimates. All these requirements should be checked. It may transpire that several iterations of design, analysis and identification of problems are required, before an instrument is deemed to be satisfactory for its intended measurement purposes. The checking of the fit is sketched here so as to obviate any impression that displays like Figure 2 are simply routine outputs of a test instrument and software which can be accepted without justification and analysis.

In this scenario, district and provincial decision-makers can usefully supplement external systemic-test results apparently signalling classrooms in current distress, with detailed analysis of the assessment initiatives and interventive strategies currently explored, or not yet explored, in those environments. Thus any systemic need to address incompetence or inexperience in the classroom can be informed in part by systemic tests, and give rise to other information or information processes that will be fairer to all teachers, affirming the dedicated and competent and alerting to incompetence or neglect.

The checking of model fit is the first of a set of cyclical processes, the purpose of which is to understand the data and where necessary to improve the functioning of the instrument. Here we distinguish between items that fit the model, items which are under-discriminating (often when learners are simply guessing), and over-discriminating items arising from item response dependence (e.g. where a correct response on a previous item increases the probability of a correct response on a current item).

Why Rasch The importance of requiring data to fit Rasch models is that fitting the model guarantees that scores arising from items which independently obey Equation 1, may always be summed together. These person totals and category counts will always permit separate estimation of each of the N person ability parameters and each of M item difficulty parameters.

A further possible violation of requirements to be considered when applying RMT is differential functioning of an item across distinct learner groups. For example, boys at an ascertained proficiency level may perform much better than girls at the same level on a particular item that involves bicycle gears. Checking for these group differences is important in the interest of assuring fairness of all items for all groups. Strategies for diminishing the effects of differential item functioning are to be found in the literature (Andrich & Hagquist, 2012; Andrich & Marais, 2012).

Only Rasch models have this property of guaranteeing the summation process to obtain a valid overall test score. All other methods (whether based on so-called traditional test theory or on so-called 2-parameter and 3-parameter structures for item responses) simply assume the summation is valid, even if there is demonstrable evidence that test items scores do not behave additively. In other words, all other models for summing of test item scores into a collective indicator will only assume the internal consistency within and between item scores as an incontestable truth, whereas the Rasch model allows the data to signal when such summation is dubious or false.

The Rasch model is essentially a single complex hypothesis built from several requirements about a context, about a test instrument and its constituent items, and about the way in which the context and instrument interact to produce special forms of measurement-like data. The whole purpose of the Rasch model might be characterised as seeking to make valid inferences at the level of an individual person and to avoid being limited only to inferences about the patterns within a totality of persons in a given context. It is inevitable that in demanding so much more detailed utility of an instrument of any kind, there will be more stringent properties required within its construction. In addition, we will require detailed description of the contexts within which such an instrument can be validly used.

This issue of permissible summation is not simply a mathematical nicety. It is an ethical imperative. If we claim we have an instrument that consistently accumulates scores from appropriate component parts, we are obliged to assess the extent to which both the accumulation and the behaviour of the parts are confirmed by the evidence in the data. We note that there is no requirement that the persons interacting with the items of an instrument are a random sample of any kind. The persons are simply part of the context, and not representative of any group other than themselves. We seek to make inferences about the relative abilities of any and all the persons tested.

Here we will take care to specify all the major requirements, and indicate some of the ways in which each of those requirements may be invalidated by evidence. Note that a single invalidation of any one requirement may be sufficient to send a test instrument back to a revised design stage, the beginning of a new cycle of iteration towards a data set with a validated Rasch measurement model.

Similarly, the items are not intended as a random sample from possible items. We seek to make valid inferences about the manner in which the selected items collectively discriminate between the persons who are the source the data. http://www.pythagoras.org.za

13


Page 14 of 16

Original Research

One such context may be the mathematical abilities of learners in a specified grade in all schools of a province. A test instrument is constructed with the purpose to measure the abilities of all the learners in the context, with sufficient precision. It will be impossible for the test instrument to yield exact measures, because it is composed of discrete item scores, subject to uncertainty. However we all recognise there is a point at which non-exact measures may be subject to such high levels of uncertainty that their utility is lost. In consequence all parameter estimates should be reported with an associated standard error of measurement, or by confidence intervals, as well as by point estimates. We may note that increasing numbers of persons will imply reduced standard errors for item parameters, and increasing numbers of items will imply reduced standard errors for person parameters.

responses for any particular item. The statistics identify items or persons for whom the interaction data does not conform to the required Rasch expectations. After identifying anomalous persons and anomalous items, the test designers have to explore what can be learnt from those elements. For the instrument, this process may involve changing or even dropping any anomalous item(s). The wording, structure and content of the item(s) will guide this choice. In general the final form of every item should enrich the collective power of the test instrument to distinguish between various persons on the basis of their ability alone. For the specified context, finding that any particular subset of persons responds anomalously, often warrants exploring their removal from the analysis. If a person’s item responses are random or incoherent, they do not address the construct which the items are intended to embody. Given that the vast majority of other learners are responding appropriately, we may eliminate the anomalous learners, precisely because their data are not contributing to an understanding of the relative difficulty of the items. In fact, including their anomalous item data will obscure the patterns in the data, and hence affect both the estimates obtained for the other learners and the estimates for the item parameters.

The test instrument and its items are expected to explore and reflect an underlying single dimension, rather than more than one dimension. One may argue that the complexity of mathematics implies more than one dimension. Detailed discussion on the topic of unidimensionality may be found in Andrich (2006). Here we note that unidimensionality implies all aspects of the test ‘pulling in the same direction’. Undue language difficulty for example, would be an example of an unwanted dimension.

We may eliminate such data, but must record the elimination and its rationale. This strategy still preserves a diagnostic value, for example identifying students who simply randomly guess for all or part of the instrument may have value for educational interventions.

On this single dimension we hypothesise that it is possible to meaningfully locate all N person abilities at particular numbers on a number line. We require that this arrangement must operate in such a way that all comparisons between person abilities would be consistently represented on the number line. We require that all K item average difficulties and all M item difficulty thresholds can be similarly organised on a single dimension, and that all comparisons of item parameters are consistently preserved. In addition, we require that the same straight line be used for both person and item arrangements, and that the two arrangements can be interwoven so that all differences of the type (βn − δi) will also be consistently preserved.

Only one ability-difficulty dimension is the intended construct of interest. However, it may be the case that an instrument taps into several dimensions, all inter-related in some way. Checking an instrument involves exploring if there is a suggestion that more than one dimension emerges from the data (Andrich & Marais, 2011). Having ascertained that the data largely manifest as a single scale for the person performances and the item difficulties, we check if each of the items suitably contributes to our objective of a measurement process. This process is lengthy and detailed (Andrich & Marais, 2011). It is also complicated, especially when by construction we seek to have an instrument with substantive validity, and that validity requires distinct aspects of the single dimension to be included. For example, we may in a mathematics test require items that tap into algebra, arithmetic, geometry and data handling.

Further, attention must be given to any extreme scores for persons and items. No test can usefully deal with estimating abilities for persons who score either 0% or 100% correct, except when further new assumptions are justified, or when new relevant information becomes available from beyond the current data set. Items on which 0% or 100% of persons are correct, tell us nothing about the distinct person abilities. These item data cannot contribute to a Rasch model for distinguishing either between persons, between 0% items or between 100% items, and are therefore eliminated from the analysis.

The data should be scrutinised for violations of the homogeneity of the learner responses over any features other than ability itself. Comparisons of the graphs produced by the Rasch analysis software for two or more groups may assist in determining whether various explanatory variables or factors give evidence for differences between groups.

Some violations of the required independence may arise only from specific persons or specific items. For each item and for each person we may calculate the corresponding Item fit and Person fit statistics. The values obtained for these statistics assess evidence for dependencies between item responses for any particular person, and dependencies between person http://www.pythagoras.org.za

Specifically we may check whether or not evidence exists for suspecting any items to be under-discriminating (as when learners are guessing rather than engaging with items), or over-discriminating (as when an item requires preknowledge or a threshold concept). 14


Page 15 of 16

Original Research

Discussion

in turn rests on an understanding of the central features of mathematics. The implicit beliefs underpinning current assessment practice may benefit from debate and explicit acknowledgement of any underpinning philosophy. For example, what view of learning and what view of evidence underpins the claim that ‘external’ assessment is the only credible method of demonstrating that learning is happening in schools (Dada et al., 2009)?

The example provided serves to illustrate the potential of an application of the Rasch model to an assessment instrument should the requirements be met. The potential of such an assessment model with its subsequent analysis is dependent on the quality of the instrument, and therefore on the prior theoretical work that has preceded the development and selection of items. Whilst in this example some worthwhile information is available for the stakeholders to observe, the potential for a more nuanced instrument may be envisaged. We note that the Rasch model is used routinely in TIMSS (Trends in Mathematics and Science Study) and PISA (Programme for International Student Assessment) to scale item difficulties and proficiency scores (see Wendt, Bos & Goy, 2011).

The recommendations resulting from the Department of Education review (Dada et al., 2009) are that continuous and broad-based assessment is limited and that external assessment at Grades 3, 6 and 9 be enshrined in policy. Given that this policy decision has been adopted, it is critical that the external assessments work in conjunction with classroom assessment. The relevant grade teachers, rather than being the objects of the testing policy, should be participants involved in the construction and analysis of tests. We aver that a collaborative strategy supporting regular use of formative assessments may impact more directly on their teaching, in ways that better address learner needs, and hence improve learning of the subject.

Given a well-targeted test instrument, informed by adequate theoretical investigation within the substantive discipline of the test, there is the potential for informing both the stakeholders and the educational officials. Well-targeted instruments may also require some type of pilot testing or external benchmarking. As it transpired, this well-intentioned test did not match the target population very well. Inferences can be explored to improve this aspect of the test instrument. Nonetheless, diagnostics relevant to the teaching of the material relating directly and indirectly to the test are readily available from the design work on the construction of the test. The design work permits the explicit statements in Table 2, and the ordering of items from the data, to suggest sequences of teaching and learning. It is readily conceded that further iterations with some altered or replaced items may produce revised Table 2 summaries that will conceivably be mildly or radically improved in usefulness.

In answer to the question: What model of assessment may support teaching and learning in the classroom, and in addition enable broad-based monitoring of learning progression within districts and provinces?, we advocate an approach which takes seriously the critical elements of mathematics, in the formulation of a developmental trajectory. Systemic provision of a large variety of test items and their diagnostic support material, together with informed and deliberative selection by committed teachers for classroom use, with facilities for electronic data capture and/or marking, are important strategies. Routine classroom tests drawn from such item bases can simultaneously support classroom innovations, whilst providing district structures with information about classroom efforts and needs. In such extended contexts, occasional systemic testing can be interpreted against a wider range of contextual information.

One may ask whether the information presented in this analysis is not already known to the stakeholders and education officials. We recognise the test design as somewhat typical of assessment instruments expected by current systemic assessment programmes; they should ‘cover the curriculum’. The issues may be well known, but the problem of coherence within such a test when analysed from a developmental learning approach is less explicitly recognised.

The role of assessment in the 21st century is ‘extremely powerful’ (Matters, 2009, p. 222). According to Matters, this role can only be justified on condition firstly that the assessment is ‘of sufficient strength and quality to support its use’, and secondly that the ‘users of assessment data have sufficient experience and imagination to see beyond the numbers’ (p. 222).

By its generality of coverage, the systemic instrument provides only scant or generic developmental information to the teacher. Perhaps it is time for cycles of systemic assessment of a more focused and limited nature, for example, an instrument with a focus only on algebra where the skills and concepts may be operationalised in a set of items requiring increasingly complex and critical skills that elaborate on the key areas identified in the literature. Associated specific developmental elements can be marshalled at the design stage, and modified in terms of the emerging patterns of the applied test context, to inform more specific target interventions for algebra in the classroom.

Assessment against this background of theoretical rigour fulfils a requirement of the Rasch measurement theory that the construct of interest be made explicit. The practical unfolding of the construct, in items that are realisations of the construct, is then formulated as a test instrument. The output from the Rasch model, provided the prior requirements are met, has the potential to inform current teaching practice, to orchestrate teacher insights into the challenges of their own classrooms and initiate two-way communication between classrooms and decision-makers.

Conclusions Any approach to mathematics assessment almost certainly follows a predicated view of teaching and learning, which http://www.pythagoras.org.za

15


Page 16 of 16

Original Research

Acknowledgements

Chisholm, L., Volmink, J., Ndhlovu, T., Potenza, E., Mahomed, H., Muller, et al. (2000). A South African curriculum for the 21st century. Report of the Review Committee on Curriculum 2005. Pretoria: DOE. Available from http://www.education.gov.za/ LinkClick.aspx?fileticket=Y%2bNXTtMZkOg%3d&tabid=358&mid=1301

Acknowledgement is due to the Schools Development Unit of the University of Cape Town for the development of the test instrument in collaboration with Caroline Long and Elsie Venter, and to the Western Cape Education Department for commissioning the work.

Dada, F., Dipholo, T., Hoadley, U., Khembo, E., Muller, S., & Volmink, J. (2009). Report of the task team for the review of the implementation of the National Curriculum Statement. Pretoria: DBE. Available from http://www.education.gov.za/LinkClick. aspx?fileticket=kYdmwOUHvps%3d&tabid=358&mid=1261 Department of Education. (2005). The national protocol on assessment for schools in the General and Further Education and Training band (Grades R to 12). Pretoria: DOE.

Competing interests

Griffin, P. (2007). The comfort of competence and the uncertainty of assessment. Studies in Educational Evaluation, 33, 87−99. http://dx.doi.org/10.1016/j. stueduc.2007.01.007

We declare that we have no financial or personal relationship(s) which might have inappropriately influenced our writing of this article.

Griffin, P. (2009). Teachers’ use of assessment data. In C. Wyatt-Smith, & J. J. Cumming (Eds.), Educational assessment in the 21st century: Connecting theory and practice (pp. 183−208). Dordrecht: Springer. http://dx.doi.org/10.1007/978-1-4020-99649_10 Long, C. (2011). Mathematical, cognitive and didactic elements of the multiplicative conceptual field investigated within a Rasch assessment and measurement framework. Unpublished doctoral dissertation. University of Cape Town, Cape Town, South Africa. Available from http://web.up.ac.za/sitefiles/file/43/314/ Long,_M__C__(2011)__The_mulitplcative_conceptual_field_investigated_ within_a_Rasch_measurement_framework_.PDF

Authors’ contributions T.D. (University of Cape Town) contributed to the conceptualisation of the article, and to the detailed explanations of the Rasch model. He wrote extensive sections of the article. He was also involved as an advisor to the original project where the data were collected. C.L. (University of Pretoria) was project leader for the original project where these data were collected. Together with E.V. (Independent Researcher) she was responsible for the initial analysis in the original project, and independently conducted the re-analysis using RUMM software for this article. She contributed to the conceptualisation of the article. She wrote the remaining elements of the article. T.C. (University of Cape Town) contributed to the conceptualisation of the article and thereafter assisted with critical revision of the manuscript. E.V. worked with C.L. on the pilot study analysis and the subsequent analysis of the data.

Long, C., & Venter, E. (2009). Report on the Western Cape Grade 8 Systemic Assessment Project. Pretoria: Centre for Evaluation and Assessment, University of Pretoria. Long, C., Wendt, H., & Dunne, T. (2011). Applying Rasch measurement in mathematics education research: Steps towards a triangulated investigation into proficiency in the multiplicative conceptual field. Educational Research and Evaluation, 17(5), 387−407. http://dx.doi.org/10.1080/13803611.2011.632661 Matters, G. (2009). A problematic leap in the use of test data: From performance to inference. In C. Wyatt-Smith, & J.J. Cumming (Eds.), Educational assessment in the 21st century: Connecting theory and practice (pp. 209−225). Dordrecht: Springer. http://dx.doi.org/10.1007/978-1-4020-9964-9_11 Rasch, G. (1960/1980). Probabilistic models for some intelligence and attainment tests (Expanded edition with foreword and afterword by B.D. Wright). Chicago, IL: University of Chicago Press. Steinbring, H. (1998). Elements of epistemological knowledge for mathematics teachers. Journal of Mathematics Teacher Education, 1, 157−189. http://dx.doi. org/10.1023/A:1009984621792 Thijs, A., & Van den Akker, J. (2009). Curriculum in development. Enschede: Netherlands Institute for Curriculum Development (SLO). Usiskin, Z. (2007). Would national curriculum standards with teeth benefit U.S. students and teachers? UCSMP Newsletter, 37, 5−7. Available from http:// d75gtjwn62jkj.cloudfront.net/37.pdf Van Wyk, J., & Andrich, D. (2006). A typology of polytomously scored items disclosed by the Rasch model: Implications for constructing a continuum of achievement. In D. Andrich, & G. Luo (Eds.), Report no. 2 ARC linkage grant LP0454080: Maintaining invariant scales in state, national and international assessments (n.p.). Perth: Murdoch University.

References

Vergnaud, G. (1988). Multiplicative structures. In J. Hiebert, & M. Behr (Eds.), Number concepts and operations in the middle grades (pp. 141−161). Hillsdale, NJ: National Council of Teachers of Mathematics.

Andrich, D. (1988). Rasch models for measurement. Beverly Hills, CA: Sage Publications. Andrich, D. (2006). On the fractal dimension of social measurements I. Perth: Pearson Psychometric Laboratory, University of Western Australia.

Vygotsky, L.S. (1962). Thought and language. Cambridge, MA: MIT Press. http:// dx.doi.org/10.1037/11193-000

Andrich, D. (2009). Review of the Curriculum Framework for curriculum, assessment and reporting purposes in Western Australian schools, with particular reference to years Kindergarten to Year 10. Perth: University of Western Australia.

Wendt, H., Bos, H., & Goy, M. (2011). On applications of Rasch models in international comparative large-scale assessments: A historical review. Educational Research and Evaluation, 17(6), 419−446. http://dx.doi.org/10.1080/13803611.2011.634582

Andrich, D.A., & Hagquist, K. (2012). Real and artificial differential item functioning. Journal of Educational and Behavioural Statistics, 37(3), 387−416. http://dx.doi. org/10.3102/1076998611411913

Wilson, M. (2005). Constructing measures: An item response modeling approach. London: Lawrence Erlbaum.

Andrich, D., & Marais, I. (2011). Introductory course notes: Instrument design with Rasch, IRT and data analysis. Perth: University of Western Australia. PMCid:3217813

Wright, B.D. (1997). A history of social science measurement. Educational Measurement: Issues and Practice, 16(4), 33−45. http://dx.doi.org/10.1111/j.1745-3992.1997. tb00606.x

Andrich, D., & Marais, I. (2012). Advanced course notes: Instrument design with Rasch, IRT and data analysis. Perth: University of Western Australia.

Wright, B.D., & Stone, M.H. (1979). The measurement model. In B.D. Wright, & M.H. Stone (Eds.), Best test design (pp. 1−17). Chicago, IL: MESA Press.

Andrich, D., Sheridan, B., & Luo, G. (2011). RUMM2030 software and manuals. Perth: University of Western Australia. Available from http://www.rummlab.com.au/

Wright, B.D., & Stone, M.H. (1999). Measurement essentials. Wilmington, DE: Wide Range, Inc.

Bennett, R.E., & Gitomer, G.H. (2009). Transforming K-12 assessment: Integrating accountability testing, formative assessment and professional development. In C. Wyatt-Smith, & J.J. Cumming (Eds.), Educational assessment in the 21st century (pp. 43−62). Dordrecht: Springer. http://dx.doi.org/10.1007/978-1-4020-9964-9_3

Wyatt-Smith, C., & Gunn, S. (2009). Towards theorising assessment as critical inquiry. In C. Wyatt-Smith, & J.J. Cumming (Eds.), Educational assessment in the 21st century: Connecting theory and practice (pp. 83−102). Dordrecht: Springer. http:// dx.doi.org/10.1007/978-1-4020-9964-9_5

Black, P.J. (1998). Testing: Friend or foe. London: Falmer Press.


16


Page 1 of 8

Original Research

An illustration of the explanatory and discovery functions of proof Author: Michael de Villiers1 Affiliation: 1 Mathematics Education, University of KwaZulu-Natal, Edgewood Campus, South Africa Correspondence to: Michael de Villiers Email: [email protected] Postal address: Private Bag X03, Ashwood 3605, South Africa Dates: Received: 15 Aug. 2012 Accepted: 27 Sept. 2012 Published: 30 Nov. 2012 How to cite this article: De Villiers, M. (2012). An illustration of the explanatory and discovery functions of proof. Pythagoras, 33(3), Art. #193, 8 pages. http://dx.doi. org/10.4102/pythagoras. v33i3.193 Note: This article is based on a presentation at the 12th International Congress on Mathematical Education, July 2012, COEX, Seoul, Korea. Workshops on Clough’s conjecture have also been conducted at the NCTM Annual Meeting, April 2004, Philadelphia, USA, as well as at the AMESA Congress, July 2004, North-West University, Potchefstroom, South Africa (De Villiers, 2004).


This article provides an illustration of the explanatory and discovery functions of proof with an original geometric conjecture made by a Grade 11 student. After logically explaining (proving) the result geometrically and algebraically, the result is generalised to other polygons by further reflection on the proof(s). Different proofs are given, each giving different insights that lead to further generalisations. The underlying heuristic reasoning is carefully described in order to provide an exemplar for designing learning trajectories to engage students with these functions of proof.

Introduction It seems that the human brain is designed, or has evolved over time, not only to recognise patterns, but also often to impose them on things we observe. Moreover, from a very young age, children naturally exhibit a need for an explanation of these patterns – a deep-seated curiosity about how or why things work the way they do. They ask questions about why the sky is blue, the sun rises in the East, or why more moss grows on the southern side of a tree (in the Southern Hemisphere). However, it sadly seems that once young children have entered the domain of mathematics in formal schooling, this natural inquisitiveness and quest for deeper understanding becomes severely repressed. Largely to blame is probably the traditional approach of focusing primarily on the teaching, learning and practising of standard algorithms. These are still presented in many classrooms as mystical chants to be memorised, rather than focusing on understanding why they work, as well as on the meaning of the basic operations underlying them. Lockhart (2002) laments this sorry state of affairs: By concentrating on what, and leaving out why, mathematics is reduced to an empty shell. The art is not in the ‘truth’ but in the explanation, the argument. It is the argument itself which gives the truth its context, and determines what is really being said and meant. Mathematics is the art of explanation. If you deny students the opportunity to engage in this activity — to pose their own problems, make their own conjectures and discoveries, to be wrong, to be creatively frustrated, to have an inspiration, and to cobble together their own explanations and proofs - you deny them mathematics itself. (p. 5, [emphasis in the original])

Extending the role of proof beyond verification Traditionally, the verification (justification or conviction) of the validity of conjectures has been seen as virtually the only function or purpose of proof. Most mathematics teachers probably see this as the main role of proof (Knuth, 2002) and this view, to the exclusion of a broader perspective, also still dominates much of curriculum design in the form of textbooks, lessons and material on teaching proof (French & Stripp, 2005). Even the majority of research conducted in the area of proof has been done from this perspective (Balacheff, 1988; Ball & Bass, 2003; Harel & Sowder, 2007; Stylianides & Ball, 2008). Harel and Sowder, for example, defines a ‘proof scheme’ as an argument that ‘eliminates doubt’, effectively restricting the role of reasoning and proof to only that of verification, although they acknowledge the explanatory role of proof in other places. In the past few decades, however, this narrow view of proof has been criticised by several authors (e.g. Bell, 1976; De Villiers, 1990, 1998; Hanna, 2000; Thurston, 1994; Rav, 1999; Reid, 2002, 2011). They have suggested that other functions of proof such as explanation, discovery, systematisation, intellectual challenge et cetera, have in some situations been of greater importance to mathematicians and can have important pedagogical value in the mathematics classroom as well. But these distinctions of other roles for proof are perhaps far older. For example, Arnauld and Nicole (1662) appear to be referring to the explanatory (illuminating or enlightening) role of proof by objecting to Euclid because they felt Euclid was ‘more concerned with convincing the mind than with enlightenment’ (cited by Barbin, 2010, p. 237).


17


Page 2 of 8

Original Research

Computing technology and the changing role of proof

1998, 2003), whilst the other functions of proof could be developed later or in other contexts. Furthermore, by initially referring to a deductive argument as a ‘logical explanation’ instead of a ‘proof, it may help to focus attention on its role as a means of deeper understanding of a dynamically verified result rather than of conviction or verification.

Mejía-Ramos (2002, p. 6) argues that the search for deeper understanding is what makes many mathematicians reject ‘mechanically-checked formal proofs and computational experiments as mathematical proofs’, for example, the famous use of computers by Appel and Haken in 1976 to prove the four-colour conjecture (Appel & Haken, 1977). Especially in the light of modern computing technology, such as dynamic geometry and symbolic algebraic processors, it is often the case that a very high level of conviction is already obtained before mathematicians embark on finding a proof. In fact, it can be argued that this ‘a priori’ conviction is more often a prerequisite and motivating factor (Polya, 1954, pp. 83−84) for looking for a proof than the mythical view that ‘eliminating doubt’ is the driving force.

Proof as a means of discovery Quite often, logically explaining (proving) why a result is true gives one deeper insight into its premises. On further reflection, one may then realise that it can be generalised or applied in other circumstances. Anderson (1996, p. 34) also clearly alludes to this aspect when writing, ‘Proof can bring understanding of why methods work and, consequently, of how these methods might be adapted to cope with new or altered circumstances.’ Rav (1999, p. 10) also describes this ‘productive’ role of proof when writing: ‘ ... logical inferences are definitely productive in extending knowledge by virtue of bringing to light otherwise unsuspected connections.’ More recently, Byers (2007, p. 337) has made a similar observation: ‘A “good” proof, one that brings out clearly the reason why the result is valid, can often lead to a whole chain of subsequent mathematical exploration and generalization.’

On the other hand, although such computing tools enable us to gain conviction through visualisation or empirical measurement, these generally provide no satisfactory insight into why the conjecture may be true. It merely confirms that it is true, and although considering more and more examples may increase our confidence to a greater extent, it gives no psychologically satisfactory sense of illumination (Bell, 1976) or enlightenment – for that, some form of proof is needed! In this regard, it is significant to note that young Grade 9 children still display a need for some form of further explanation (deeper understanding) of a result, which they had already become fully convinced of after empirical exploration on Sketchpad (Mudaly & De Villiers, 2000). Within the context of algebra, Healy and Hoyles (2000) also found that students preferred arguments that both convinced and explained, strongly suggesting that the need for explanation is perhaps an untapped resource in lesson design and implementation.

I have called this illuminating aspect of proof that often allows further generalisation, the discovery function (De Villiers, 1990), and it appears also to be the first explicit distinction of this function (Reid, 2011). For example, explaining (proving) Viviani’s theorem for an equilateral triangle by determining the area of the three triangles it is divided up into, and noticing the ‘common factor’ of the equal sides of these triangles as bases, may allow one to immediately see that the result generalises to any equilateral polygon, because exactly the same ‘common factor’ will appear (De Villiers, 2003, p. 26).

Appreciation of the verification (justification) function of proof is most easily developed in fields such as number theory, algebra, calculus, et cetera. In these fields one can give spectacular counter-examples to conjectures with massive empirical support (e.g. as in Stylianides, 2011). However, this is not quite the case with dynamic geometry. The difference is that one can transform geometric figures or graphs continuously (or at least closely) by dragging, as well as explore more deeply by zooming in to great levels of accuracy. With these facilities, one can usually find counter-examples to false conjectures fairly quickly and easily. It is possible to contrive didactical situations such as used in De Villiers (2003, pp. 73, 85) where students are given sketches with measurements preset to one decimal accuracy, therefore deliberately misleading them to make a false conjecture. However, genuinely authentic examples in dynamic geometry that are accessible to high school students are few and far between.

Nunokawa (2010, pp. 231−232) similarly claims that ‘explanations generate new objects of thought to be explored’. He gives an example of a problem involving two overlapping squares, and how explaining why the overlapping area remains constant as the one square remains fixed and the other is rotated, leads one to generalise to other regular polygons with the same feature. Two other ‘discovery via proof’ examples are discussed and presented in De Villiers (2007a, 2007b). Of course, for novices and less experienced students such generalisation (or specialisation) from a proof is not likely to be as automatic and immediate as with an experienced mathematician. Therefore, in didactically designing tasks to engage high school students, or even student teachers, with the discovery function of proof, sufficient scaffolding is often needed to provide adequate guidance for both the initial proof as well as for further reflection (Hemmi & Löfwall, 2011; Miyazaki, 2000).

It seems more natural and meaningful that within a dynamic geometry (mathematics) environment, potential use may be made of this cognitive need for explanation and understanding to design and implement alternative learning activities. Such learning activities could introduce novices for the first time to proof, not as a means of verification, but as a means of explanation and illumination (e.g. see De Villiers,


Jones and Herbst (2012, p. 267) reporting on a study on the instructional practices of a sample of expert teachers of geometry at Grade 8 level (pupils aged 13–14) in Shanghai, China, identify two important factors in developing an understanding of the discovery function of proof, namely,

18


Page 3 of 8

Original Research

(compare Movshovitz-Hadar, 1988); therefore creating a need for explanation, and provide an authentic mix of experimentation and proof of a possibly original result.

variation of the mathematical problems as well as the questions asked by teachers to guide their students. It is important to also point out here that with the ‘discovery’ function of proof is not only meant a discovery made after reflecting on a recently constructed proof. As illustrated in De Villiers (1990, p. 22, 2003, pp. 68−69), it also more broadly refers to situations where new results are discovered in a purely logical way by the application of known theorems or algorithms without resorting to any experimentation, construction or measurement. For example, using the tangents to a circle theorem, it is relatively easy to deduce logically (and proving at the same time) that the two sums of the opposite sides of a quadrilateral circumscribed around a circle are equal (and generalising to circumscribed 2n-gons).

Clough’s conjecture Although it is a rare occurrence, nothing gives greater pleasure to a teacher than when one of their students produces a conjecture of their own. The conjecture need not be entirely original, but the excitement created in the classroom when something goes ‘outside’ or ‘beyond’ the textbook gives a much more ‘real’ sense of genuine mathematical discovery and invention. Usually, students are also far more strongly motivated to want to solve such a problem because they perceive it as their own and not something old and boring from the textbook or the curriculum.

Another illustrative example is given in De Villiers (1999) involving the generalisation of a problem involving an area relationship between a square and a formed octagon. By dividing the sides into different ratios than the original, it was experimentally found with dynamic geometry that the area ratios remained constant. However, a purely inductive approach whereby the different ratios 0.1666; 0.3333; 0.4500 were looked at for the division of the sides into halves, thirds and quarters respectively, was not very helpful in finding a general formula and ultimately had to be derived logically.

By encouraging students, for example, to continually ask ‘what-if’ questions on their own until it becomes a regular occurrence, students are likely to more naturally start making more original conjectures of their own, providing an exciting injection to liven up the class. The availability of computing technology places at the disposal of students powerful new tools by which they can now easily make independent discoveries (Arzarello, Bartolini Bussi, Leung, Mariotti & Stevenson, 2012; Borwein, 2012). A useful overview and analysis for the task-design of activities for promoting conjecturing is given by Lin, Yang, Lee, Tabach and Stylianides (2012). Moreover, speaking from my own experience also, honouring students by attaching their names to discovered results is a powerful motivator to continue further mathematical studies (compare Leikin, 2011).

More generally, with the discovery function, it also means that a proof can reveal new, powerful methods of solving problems and creating new theories. Logical reasoning and proof can show that certain problems are unsolvable such as, for example, representing √2 as a rational (fraction), squaring the circle or solving a quintic (or higher order) polynomial equation with radicals. Hanna and Barbeau (2010, pp. 90−93) suggest a nice example for classroom use, showing how the problem of finding the quadratic formula naturally leads to an introduction to students of the strategy of completing the square. Grabiner (2012, p. 161) gives historical examples of how the distinction between pointwise and uniform convergence arose from counter-examples to Cauchy’s supposed theorem regarding infinite series, and of how Cantor’s theory of the infinite came about through trying to specify the structure of the sets of real numbers on which Fourier series converge. Similarly, the discovery (invention) of non-Euclidean geometry came about from attempts to use indirect proof (reductio ad absurdum) to prove Euclid’s 5th postulate. Grabiner (2012, p. 162) describes this as ‘another triumph of human reason and logic over intuition and experience’.

During 2003, a Grade 11 student from a high school in Cape Town was exploring Viviani’s theorem using dynamic geometry. The theorem says that the sum of distances of a point to the sides of an equilateral triangle is constant (i.e. in Figure 1 PPa + PPb + PPc is constant, irrespective of the position of point P inside triangle ABC). The student’s further exploration led him to measure the distances APc, BPa and CPb, and then add them. To his surprise, he noticed that APc + BPa + CPb also remained constant no matter how much he dragged P inside the triangle. However, he could not prove it. A

The main purpose of this article is to contribute further to the theoretical aspects of the role of proof by providing a heuristic description of some of my personal experiences of the explanatory and discovery functions of proof with a geometric conjecture made by a Grade 11 student. After logically explaining (proving) the result geometrically and algebraically, the result is generalised to other polygons by further reflection on the proof(s). This conjecture and its generalisations could easily be turned into a set of guided learning activities that elicit ‘surprise’ amongst students http://www.pythagoras.org.za

Pb Pc

B

P

Pa

FIGURE 1: Clough’s conjecture: APc + BPa + CPb is constant.

19


C

Page 4 of 8

Original Research

His teacher eventually wrote to me to ask whether I could perhaps produce a simple geometric proof, as he himself could only prove it algebraically by means of co-ordinate geometry. Below is the geometric proof I first produced, followed by further proofs, explorations and different generalisations of what has become known as Clough’s conjecture (De Villiers, 2004).

rectangle because all its angles are right angles. Therefore, A´P = APc, and similarly, B´P = BPa and C´P = CPb. Clearly the problem is now reduced to Viviani’s theorem in relation to ∆KLM. Considering that A´P + B´P + C´P is constant, it follows that APc + BPa + CPb is also constant. QED. The preceding proof is quite explanatory (Hanna, 1989) as one can almost immediately ‘visually see’ from the diagram in one ‘gestalt’, why the result is true and how it relates to Viviani’s theorem.

Geometric proof Problem solving heuristics are valuable in that they often direct the problem solver towards a successful solution of a problem. George Polya (1945) gives the following useful examples:

An alternative ‘algebraic’ proof In Polya’s final step of problem-solving, namely, looking back, he asks amongst other things whether one can derive or prove the result differently. In doing so, not only is one developing a variety of problem-solving (proving) skills, but one may also gain additional insight into the result. Recently, much has been written and researched about the value of posing such multi-proof tasks to students. Dreyfus, Nardi and Leikin (2012) provide a comprehensive survey and review of this particular field.

Have you seen it before? Or have you seen the same problem in a slightly different form? Do you know a related problem? Do you know a theorem that could be useful? Look at the unknown! And try to think of a familiar problem having the same or a similar unknown. Here is a problem related to yours and solved before. Could you use it? Could you use its result? Could you use its method? Should you introduce some auxiliary element in order to make its use possible? (p. xvii)

Following Polya’s heuristic, it seems natural to try and relate Clough’s conjecture to Viviani’s theorem and its proof. After several different attempts, I found by constructing perpendiculars to AB, BC and CA as ‘auxiliary elements’ respectively at A, B and C, that I obtained a triangle KLM as shown in Figure 2.

Considering that there are several right triangles, it seems reasonable to try the theorem of Pythagoras, and to apply it to each of these triangles and investigate where it leads. Let AB = a, APC = x, et cetera, as shown in Figure 3. We now need to show that x + y + z is constant. Applying Pythagoras to the right triangles adjacent to the hypotenuses AP, BP and CP, we obtain:

Considering that ∠ ABK = 30°, it follows that ∠ AKB = 60°. In the same way the other angles of ∆KLM can be shown to be equal to 60°; hence ∆KLM is equilateral. Next, drop perpendiculars from P to sides KM, KL and LM respectively. It then follows that quadrilateral APcPA’ is a

x 2 + PPc 2 = ( a − z ) 2 + PPb 2 y 2 + PPa 2 = ( a − x ) 2 + PPc 2 z 2 + PPb 2 = ( a − y ) 2 + PPa 2

K

It is often at this point, or even before reaching it, that a novice problem solver might lose hope of getting anywhere as it is not obvious from the start that this will lead somewhere useful. However, students should be encouraged to persist with such an exploration and not so easily give up and start asking for help. One might say that a distinctive characteristic of good mathematical problem solvers are that they are ‘stubborn’, and willing to spend a long time attacking a problem from

A A´

Pc

P

Pb

B´

A M a-z

B

x C

Pa

Pb Pc

C´

z a-x

B

L FIGURE 2: A geometric proof of Clough’s conjecture.


y

Pa

a-y

C

FIGURE 3: An alternative, algebraic proof of Clough’s conjecture.

20


Page 5 of 8

Original Research

(as it has all its sides equal), let us nonetheless see if we can use the same geometric approach with it as for the triangle, and whether it provides any new insights. By constructing perpendiculars as before to AD, DC, CB and BA respectively at A, D, C and B as shown in Figure 5, we find that the result is visually immediately obvious. For example, perpendiculars a and c are parallel to each other because they are respectively perpendicular to sides AD and BC. Because it is easy to show that FPH is a straight line, we see that AH + CF is simply equal to the constant distance between these two parallel lines. The same applies to the sum of the other two distances BE and DG between the parallel perpendiculars b and d. Therefore, AH + CF + BE + DG is the sum of two constants; hence constant. QED.

different vantage points and not easily surrendering. In this regard, Schoenfeld (1987, p. 190−191) also specifically refers to the importance of meta-cognition during problemsolving (i.e. maintaining a conscious awareness and control of a variety of possible approaches, and then monitoring how well things are going during the implementation of a possible approach). If we look at the set of three equations, however, an immediate observation is the cyclic fashion in which terms appear. This suggests that adding the left and right sides of the three equations, respectively, might lead to the quadratic terms cancelling out. Indeed, doing so, after simplification, 3 gives us the desired identity x + y + z = a . Considering that 2 a is constant for a fixed equilateral triangle, it completes the proof.

In many ways this proof is more explanatory than the preceding algebraic proof, which was more algorithmic, nonvisual and required quite a bit of manipulation. Moreover, following Polya, and looking back critically and examining this geometric proof, one should notice that we did not use the equality of the sides of the rhombus at all! We only used its property of opposite sides being parallel − it depends only on the parallel-ness of opposite sides. This implies that the result will immediately not only generalise to a parallelogram, but also in general to any parallel 2n-gon (n > 1); in other words to any even sided polygon with opposite sides parallel, as the same argument will apply!

3 a is half the perimeter of the 2 triangle, we also get the following bonus relationship: APc + BPa + CPb = PcB + PaC + PbA. Taking into account that

Although this algebraic proof appears less explanatory than the preceding geometric one, we have managed to find an additional property of the configuration that was not discovered experimentally, namely, that the sum of these distances is half the perimeter of the triangle. Nor was this clearly evident from the geometric proof at all, although one could now go back armed with this hindsight and use basic trigonometric ratios in Figure 2 to find that the side length 3 of ∆KLM is √3a; hence its height is a (which is equal to its 2 Viviani sum).

P5

A1

A5

x5

P4 x4

x1

However, more importantly, because of its cyclic nature, the algebraic proof suggests an immediate generalisation to equilateral polygons, giving a nice illustrative example of the discovery function of proof. It is not hard to see (at least for more experienced problem solvers) that from the structure of the proof, it will generalise as follows for an equilateral n-gon A1A2...An (refer to the notation in Figure 4, showing an equilateral pentagon):

P1

A2

x12 + PP12 = ( a − xn ) 2 + PPn 2

A4

P

x2

P2

A3

x3

P3

FIGURE 4: An equilateral pentagon.

x22 + PP2 2 = ( a − x1 ) 2 + PP12 .... xn2−1 + PPn −12 = ( a − xn − 2 ) 2 + PPn − 2 2

B

xn2 + PPn 2 = ( a − xn −1 ) 2 + PPn −12

C b

By again adding the left and the right sides as before, we get a collapsing ‘telescopic effect’ with all the squares of PPn and xi cancelling out, and all that remains is n 0 = na 2 – 2a(x1 + x2 + ... + x n) which simplifies to ∑ xi = a , 2 which as before, is also half the perimeter of the equilateral n-gon.

a

E P

c

d A

Revisiting the geometric proof Let us now revisit our explanatory geometric proof. Despite already knowing that Clough’s result is true for a rhombus http://www.pythagoras.org.za

F

G H

FIGURE 5: A geometric proof for a rhombus.

21


D

Page 6 of 8

Original Research

So here we have another excellent example of the discovery function of proof, leading us to a further generalisation, without any additional experimentation. As shown in Figure 6 for a hexagon with opposite sides parallel, exactly the same argument applies to the sum of the distances AH and DK respectively on opposite parallel sides, and lying between the two parallel perpendiculars a and d, et cetera.

produce another equi-angled polygon inside! Hence, the result generalises, and we have here another lovely example of the discovery function of proof. Another perhaps even easier way of logically explaining the theorem is shown in Figure 8. By translating the segments BG, CH, DI and EJ as shown, and then constructing perpendiculars at A, B´, C´, D´ and E´, we produce another

Generalising to equi-angled polygons Given that Viviani’s theorem generalises not only to equilateral polygons and 2n-gons with opposite sides parallel, but also to equi-angled polygons, it seemed reasonable to investigate whether Clough’s theorem is also true for polygons of this kind. Considering that it is true for parallelograms, it is true for the quadrilateral case (a rectangle), but what about an equi-angled pentagon?

A

M

H

F L

a P

E

A quick construction on Sketchpad showed me that the result was indeed also true for an equi-angled pentagon. Although I personally had no doubt about the equi-angled result from this experimental investigation, I was nonetheless motivated to look for a proof, because I wanted to know why it was true, as well as seeing it as an intellectual challenge (compare with Hofstadter, 1997, p. 10). It was therefore not about the ‘removal of doubt’ for me at all!

B

d I

K C

J

D

FIGURE 6: A hexagon with opposite sides parallel.

E

Once again, one can try the same strategy used before by constructing perpendiculars at the vertices and attempt to relate it to something we already know, namely Viviani’s generalisation to equi-angled polygons.

I K J

Given that ABCDE is a pentagon with equal angles as shown in Figure 7, draw perpendiculars to each side at the vertices A to E, and label as K the intersection of the perpendicular from A with that of the perpendicular from E. Similarly, as shown, label the other intersections of the perpendiculars as L, M, N and O. From Q draw perpendiculars QJ to AE and QX to EK (extended) to obtain rectangle EJQX. Therefore, QX = EJ.

O

L A

Q

H

M

F

C G

B

In the same way, construct rectangles to replace the other four segments AF, BG, CH and DI with the corresponding perpendiculars from Q to the sides of KLMNO as shown. Now note that ∠ EAB = 90° +∠ EAK, but ∠OKL = 90° +∠EAK, because ∠OKL is the exterior angle of ∆EAK. Hence, ∠OKL = ∠ EAB.

FIGURE 7: An explanatory proof for an equi-angled pentagon.

E I

Similarly, it can be shown that the other angles of the inner pentagon are correspondingly equal to that of the outer one; hence that KLMNO is also an equi-angled pentagon. But we know that the sum of the distances from a point to the sides of any equi-angled polygon is constant, and because all these five distances are correspondingly equal to the distances EJ, AF, BG, CH and DI by construction, the required result follows. QED.

D

J

E´

Q

A

B´

Looking back at this proof, we can also see that we did not use the angle size (108º) specific to the equi-angled pentagon to show that ∠OKL =∠EAB. This immediately implies that for any polygon with equal angles the same construction would http://www.pythagoras.org.za

D

X

H

F D´

C G

B C´ FIGURE 8: An alternative explanation (proof) for an equi-angled pentagon.

22


Page 7 of 8

Original Research

considering generalisations or pose new questions. Problem posing and generalisation through the utilisation of the ‘discovery’ function of proof is as important and creative as problem-solving itself, and ways of encouraging this kind of thinking in students need to be further explored.

equi-angled pentagon (left to the reader to prove) and the result follows as before.

Concluding comments Although it is probably not feasible to attempt to introduce complete novices to the ‘looking back’ discovery function of proof with the specific examples illustrated here, I believe it is possible to design learning activities for younger students in the junior secondary school and even in the primary school. This could at least acquaint students with the idea that a deductive argument can provide additional insight and some form of novel discovery.

Johnston-Wilder and Mason (2005, p. 93) and Mason, Burton and Stacey (1982, p. 9) have claimed that generalisation lies at the ‘heart’ of mathematics and is its ‘life-blood’, and give many instructive examples. It certainly is an important mathematical activity that students need to engage in far more than is perhaps currently the case in classroom practice. It is important to broadly distinguish between two kinds of generalisation, namely, inductive and deductive generalisation. With inductive generalisation is meant the generalisation from a number of specific cases by empirical induction or analogy, and is usually the meaning given to the word ‘generalisation’ in the literature. With deductive generalisation is meant the logical reflection (looking back on) and consequent generalisation of a critical idea to more general or different cases by means of deductive reasoning. In other words, generalising the essence of a deductive argument and applying it to more general or analogous cases. Three examples of this deductive kind of generalisation have been illustrated in this article.

For example, De Villiers (1993) shows that to algebraically explain why the sum of a two-digit number and its reverse is always divisible by 11 can lead students to see that the other factor is the sum of the digits of the original number, which they may not have noticed from considering only a few cases. This activity has been done many times with both high school students as well as pre-service and in-service teachers. It has been very seldom that any of them noticed this additional property in the empirical phase, and they would express appreciative surprise at finding this out later from the proof when their attention was directed towards it. Instead of defining proof in terms of its verification function (or any other function for that matter), it is suggested that proof should rather be defined simply as a deductive or logical argument that shows how a particular result can be derived from other proven or assumed results; nothing more, nothing less. It is not here suggested that fidelity to the verification function of proof is sacrificed at all, but that it should not be elevated to a defining characteristic of proof. Moreover, the verification function ought to be supplemented with other important functions of proof using genuine mathematical activities as described above. It is also not suggested that the preceding examples be directly implemented in a classroom as their success will depend largely on the past experience, expertise and ability of the audience, the classroom culture, as well as the skill of the teacher as a facilitator of learning. For example, Zack (1997, p. 1) contends that in her fifth grade classroom ‘for an argument to be considered a proof, the students need not only convince, but also to explain’. She then proceeds to give an example of how this broader ‘didactical contract’ with respect to proof motivated her students to actively engage in conjecturing, refuting and eventually developing a proof as a logical explanation through her continued insistence that they demonstrate why the pattern worked.

Schopenhauer (as quoted by Polya, 1954) aptly describes the educational value of the process of further generalisation to assist in the integration and synthesis of students’ knowledge as follows: Proper understanding is, finally, a grasping of relations (un saisir de rapports). But we understand a relation more distinctly and more purely when we recognize it as the same in widely different cases and between completely heterogeneous objects. (p. 30)

In terms of learning theory, the process of generalisation corresponds to some extent to ‘superordinate learning’ as distinguished by Ausubel, Novak and Hanesian (1978, p. 68), where an inclusive idea or concept is generalised or abstracted, under which already established ideas can be meaningfully subsumed. Finally, it is hoped that this article will stimulate some more design experiments in problem solving as suggested by Schoenfeld (2007), focussing not only on developing appreciation of the explanatory and discovery functions of proof, but also on other functions of proof such as systematisation, communication, intellectual challenge, et cetera. The aim is that ultimately, school curricula, textbooks and teachers can begin to present a more comprehensive, realistic and meaningful view of proof to students.

Leong, Toh, Tay, Quek and Dindyal (2012) similarly describe some success using a worksheet based on Polya’s model to guide a high achieving student to ‘look back’ at his solution and push him to further extend, adapt and generalise his solution. One could speculate, and it might be an interesting longitudinal study, that students who’ve been exposed to several such activities are more likely to spontaneously start ‘looking back’ at their solutions to problems and start


Acknowledgements Competing interests

I declare that I have no financial or personal relationships, which may have inappropriately influenced me in writing this article.

23


Page 8 of 8

Original Research

References

Hofstadter, D. (1997). Discovery and dissection of a geometric gem. In D. Schattschneider, & J. King (Eds.). Geometry turned on! (pp. 3−14). Washington, DC: MAA.

Anderson, J. (1996). The place of proof in school mathematics. Mathematics Teaching, 155, 33−39.

Horgan, J. (1993). The death of proof. Scientific American, 269(4), 92−103. http:// dx.doi.org/10.1038/scientificamerican1093-92

Appel, K., & Haken, W. (1977). Solution of the four color map problem. Scientific American, 237(4), 108–121. http://dx.doi.org/10.1038/ scientificamerican1077-108

Johnston-Wilder, S., & Mason, J. (Eds.). (2005). Developing thinking in geometry. London: Paul Chapman Publishing. Jones, K., & Herbst, P. (2012). Proof, proving, and teacher-student interaction: Theories and contexts. In G. Hanna, & M. de Villiers (Eds.), Proof and proving in mathematics education (pp. 261−277). Dordrecht: Springer.

Arzarello, F., Bartolini Bussi, M.G., Leung, A.Y.L., Mariotti, M.A., & Stevenson, I. (2012). Experimental approaches to theoretical thinking: Artefacts and proofs. In G. Hanna, & M. de Villiers (Eds.), Proof and proving in mathematics education (pp. 97−143). Dordrecht: Springer. Ausubel, D.P., Novak, D., & Hanesian, H. (1978). Educational Psychology: A cognitive view. New York, NY: Holt, Rinehart & Winston.

Knuth, E.J. (2002). Secondary school mathematics teachers’ conceptions of proof. Journal for Research in Mathematics Education, 33(5), 379−405. http://dx.doi. org/10.2307/4149959

Balacheff, N. (1998). Aspects of proof in pupils’ practice of school mathematics. In D. Pimm (Ed.), Mathematics, teachers and children (pp. 216−235). London: Hodder & Stoughton.

Leiken, R. (2011). Teaching the mathematically gifted: Featuring a teacher. Canadian Journal of Science, Mathematics and Technology Education, 11(1), 78−89. http:// dx.doi.org/10.1080/14926156.2011.548902

Ball, D.L., & Bass, H. (2003). Making mathematics reasonable in school. In J. Kilpatrick, W.G. Martin, & D. Schifter (Eds.), A research companion to Principles and Standards for School Mathematics (pp. 27−44). Reston, VA: National Council of Teachers of Mathematics.

Leong, Y.H., Toh, T.L., Tay, E.G., Quek, K.S., & Dindyal, J. (2012). Relooking ‘look back’: A student’s attempt at problem solving using Polya’s model. International Journal of Mathematical Education in Science and Technology, 43(3), 357−369. http:// dx.doi.org/10.1080/0020739X.2011.618558

Barbin, E. (2010). Evolving geometric proofs in the seventeenth century: From icons to symbols. In G. Hanna, H.N. Jahnke, & H. Pulte (Eds.), Explanation and proof in mathematics: Philosophical and educational perspectives (pp. 237−251). New York, NY: Springer. http://dx.doi.org/10.1007/978-1-4419-0576-5_16

Lin, F., Yang, K., Lee, K., Tabach, M., & Stylianides, G. (2012). Principles of task design for conjecturing and proving. In G. Hanna, & M. de Villiers (Eds.), Proof and proving in mathematics education (pp. 305−325). Dordrecht: Springer. Lockhart, P. (2002) A mathematician’s lament. In Devlin’s Angle (March 2008) at the Mathematical Association of America website. Available from http://www.maa. org/devlin/LockhartsLament.pdf

Bell, A.W. (1976). A study of pupils’proof-explanations in mathematical situations. Educational Studies in Mathematics, 7, 23−40. http://dx.doi.org/10.1007/ BF00144356

Mason, J., Burton, L., & Stacey, K. (1982). Thinking mathematically. London: AddisonWesley.

Borwein, J. (2012). Exploratory experimentation: Digitally-assisted discovery and proof. In G. Hanna, & M. de Villiers (Eds.), Proof and proving in mathematics education (pp. 69−96). Dordrecht: Springer.

Mejía-Ramos, J.P. (2005). Aspects of proof in mathematics. In D. Hewitt (Ed.), Proceedings of the British Society for Research into Learning Mathematics, 25(2). St. Martin’s College, Lancaster: Open University. Available from http://www. bsrlm.org.uk/IPs/ip25-2/BSRLM-IP-25-2-11.pdf

Byers, W. (2007). How mathematicians think. Princeton, NJ: Princeton University Press. De Villiers, M. (1990). The role and function of proof in mathematics. Pythagoras, 24, 7−24. Available from http://mzone.mweb.co.za/residents/profmd/proofa.pdf

Miyazaki, M. (2000). What are essential to apply the “discovery” function of proof in lower secondary mathematics? In T. Nakahara, & K. Mastaka (Ed.), Proceedings of the 24th Conference of the International Group for the Psychology of Mathematics Education, Vol. 4 (pp. 1−8). Hiroshima: Hiroshima University.

De Villiers, M. (1993). Computer verification vs. algebraic explanation. Pythagoras, 31, 46. Available from http://mzone.mweb.co.za/residents/profmd/alge.pdf De Villiers, M. (1998). An alternative approach to proof in dynamic geometry. In R. Lehrer, & D. Chazan (Eds.), New directions in teaching and learning geometry (pp. 369−415), Mahwah, NJ: Lawrence Erlbaum Associates.

Movshovitz-Hadar, N. (1988). School mathematics theorems: An endless source of surprise. For the Learning of Mathematics, 8(3), 34−40.

De Villiers, M. (1999). A generalization of an IMTS problem. KwaZulu-Natal AMESA Mathematics Journal, 4(1), 12−15. Available from http://mzone.mweb.co.za/ residents/profmd/imts.pdf

Mudaly, V., & De Villiers, M. (2000). Learners’ needs for conviction and explanation within the context of dynamic geometry. Pythagoras, 52, 20−23. Available from http://mysite.mweb.co.za/residents/profmd/vim.pdf

De Villiers, M. (2003). Rethinking proof with Geometer’s Sketchpad 4. Emeryville, CA: Key Curriculum Press.

Nunokawa, K. (2010). Proof, mathematical problem solving, and explanation in mathematics teaching. In G. Hanna, H.N. Jahnke, & H. Pulte (Eds.), Explanation and proof in mathematics: Philosophical and educational perspectives (pp. 223−236). New York, NY: Springer. http://dx.doi.org/10.1007/978-1-4419-05765_15

De Villiers, M. (2004). Clough’s conjecture: A Sketchpad investigation. In S. Nieuwoudt, S. Froneman, & P. Nkhoma (Eds.), Proceedings of the 10th Annual National Congress of the Association for Mathematics Education of South Africa, Vol. 2 (pp. 52−56). Potchefstroom: AMESA.

Polya, G. (1945). How to solve it. Princeton, NJ: Princeton University Press.

De Villiers, M. (2007a). A hexagon result and its generalization via proof. The Montana Mathematics Enthusiast, 4(2), 188−192.

Polya, G. (1954). Mathematics and plausible reasoning, (Vol. 1). Princeton, NJ: Princeton University Press.

De Villiers, M. (2007b). An example of the discovery function of proof. Mathematics in School, 36(4), 9−11. Available from http://frink.machighway.com/~dynamicm/ crossdiscovery.pdf

Rav, Y. (1999). Why do we prove theorems? Philosophia Mathematica, 7(3), 5−41. http://dx.doi.org/10.1093/philmat/7.1.5

Dreyfus, T., Nardi, E., & Leikin, R. (2012). Forms of proof and proving in the classroom. In G. Hanna, & M. de Villiers (Eds.), Proof and proving in mathematics education (pp. 191−213). Dordrecht: Springer.

Reid, D. (2002). What is Proof? International Newsletter on the Teaching and Learning of Mathematical Proof, June 2002. Available from http://www.lettredelapreuve. it/OldPreuve/Newsletter/02Ete/WhatIsProof.pdf

French, D., & Stripp, C. (Eds.), (reprinted 2005). Are you sure? Learning about proof. Leicester: The Mathematical Association.

Reid, D. (2011, October). Understanding proof and transforming teaching. Paper presented at the North-American Chapter of the International Group for the Psychology of Mathematics Education, Reno, NV: PME-NA. Available from http:// pmena.org/2011/presentations/PMENA_2011_Reid.pdf

Grabiner, J.V. (2012). Why Proof? A historian’s perspective. In G. Hanna, & M. de Villiers (Eds.), Proof and proving in mathematics education (pp. 147−168). Dordrecht: Springer.

Schoenfeld, A.H. (1987). What’s all the fuss about metacognition? In A.H. Schoenfeld (Ed.), Cognitive science and mathematics education (pp. 189−215). Hillsdale, NJ: Lawrence Erlbaum Associates.

Hanna, G. (1989). Proofs that prove and proofs that explain. In G. Vergnaud, J. Rogalski, & M. Artigue (Eds.), Proceedings of the 13th Conference of the International Group for the Psychology of Mathematics Education, Vol. 2 (pp. 45–51). Paris: CNRS.

Schoenfeld, A.H. (2007). Problem solving in the United States, 1970−2008: Research and theory, practice and politics. ZDM: The International Journal on Mathematics Education, 39, 537−551. http://dx.doi.org/10.1007/s11858-007-0038-z

Hanna, G. (2000). Proof, explanation and exploration: An overview. Educational Studies in Mathematics, 44, 5−23. http://dx.doi.org/10.1023/A:1012737223465 Hanna, G., & Barbeau, E. (2010). Proofs as bearers of mathematical knowledge. In G. Hanna, H.N. Jahnke, & H. Pulte (Eds.), Explanation and proof in mathematics: Philosophical and educational perspectives (pp. 85−100), New York, NY: Springer. http://dx.doi.org/10.1007/978-1-4419-0576-5_7

Stylianides, A.J., & Ball, D.L. (2008). Understanding and describing mathematical knowledge for teaching: Knowledge about proof for engaging students in the activity of proving. Journal of Mathematics Teacher Education, 11, 307−332. http://dx.doi.org/10.1007/s10857-008-9077-9

Harel, G., & Sowder, L. (2007). Towards comprehensive perspectives on the learning and teaching of proof. In F.K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 805–842). Greenwich, CT: Information Age Publishing.

Stylianides, A.J. (2011). Towards a comprehensive knowledge package for teaching proof: A focus on the misconception that empirical arguments are proofs. Pythagoras, 32(1), Art. #14, 10 pages. http://dx.doi.org/10.4102/pythagoras. v32i1.14

Healy, L., & Hoyles, C. (2000). A study of proof conceptions in algebra. Journal for Research in Mathematics Education, 31, 396−428. http://dx.doi. org/10.2307/749651

Thurston, W.P. (1994). On proof and progress in mathematics. Bulletin of the American Mathematical Society, 30(2), 161−177. http://dx.doi.org/10.1090/S0273-09791994-00502-6

Hemmi, K., & Löfwall, C. (2011, February). Making discovery function of proof visible for upper secondary school students. Paper presented at CERME 7: Working Group on Argumentation and Proof, Rzeszów, Poland. Available from http://www. cerme7.univ.rzeszow.pl/WG/1/CERME7_WG1_Lofwall.pdf

Zack, V. (1997). “You have to prove us wrong”: Proof at the elementary school level. In E. Pehkonen (Ed.), Proceedings of the 21st Conference of the International Group for the Psychology of Mathematics Education, Vol. 4 (pp. 291−298). Lahti, Finland: PME. Available from http://meru-urem.ca/articles/Zack1997(PME-Finland).pdf


24


Page 1 of 8

Original Research

Coherence and connections in teachers’ mathematical discourses in instruction Authors: Hamsa Venkat1 Jill Adler1 Affiliations: 1 School of Education, University of the Witwatersrand, South Africa Correspondence to: Hamsa Venkat Email: hamsa.venkatakrishnan@ wits.ac.za Postal address: Room 2, WMC Corridor, Wits School of Education, St Andrews Road, Parktown 2050, Johannesburg, South Africa Dates: Received: 13 Aug. 2012 Accepted: 21 Oct. 2012 Published: 11 Dec. 2012 Re-published: 21 Dec. 2012 Note: This article was re-published with a correction made on the last equation on this page. How to cite this article: Venkat, H., & Adler, J. (2012). Coherence and connections in teachers’ mathematical discourses in instruction. Pythagoras, 33(3), Art. #188, 8 pages. http://dx.doi. org/10.4102/pythagoras. v33i3.188

In this article, we share our combination of analytical concepts drawn from the literature with a set of grounded framing questions for thinking about differences in the nature of coherence and connections in teachers’ mathematical discourses in instruction (MDI). The literaturebased concepts that we use are drawn from writing focused on transformation activity as a fundamental feature of mathematical activity. Within this writing, the need for connections between stated problems and the representations introduced and subsequently produced through transformation steps are highlighted. Drawing from four empirical episodes located across primary and secondary mathematics teaching, we outline a set of framing questions that explore coherence and connections between these concepts, and the ways in which accompanying explanations work to establish these connections. This combination allows us to describe differences between the episodes in terms of the nature and degree of coherence and connection.

Introduction Teachers’ mathematical discourses in instruction (MDIs), essentially the mathematical aspects of what teachers say, do and write as they interact with learners in mathematics classrooms, are a key feature of classroom practice. Typically, these MDIs include a problem, a selected representation that is subsequently transformed, and explanations and justifications for the representations selected and transformations performed. Our interest in this article is in developing a language that can be used to describe a range of MDIs. This interest is driven by our need to understand MDIs that seem to us to disrupt coherence and connection in mathematical text in a range of ways, and thus impact on what mathematics is made available to learn. Transformation of representations, through manipulations within and across different representation forms, is a central feature of mathematical activity (Duval, 2006) and, therefore, of MDIs. Solving a problem in school mathematics often involves a set of steps through which one representation is transformed into another. For example, completing the square is comprised of a series of transformation steps that can act upon a quadratic function as input representation, if the stated problem is to find the turning point of the function. Consider the problem: Find the turning point of f (x) = x2 − 8x + 9 The first step to solving this stated problem could be to recognise that rewriting a quadratic expression as a perfect square, plus or minus some constant, allows us to ‘see’ vertical and horizontal shifts with respect to the parent function, and so the turning point, more easily. We would thus rewrite the function in the form f (x) = a(x − p)2 + q by completing the square: f (x) = x2 − 8x + 9 = x2 − 8x + 16 –16 + 9 = (x − 4)2 − 7 What is important for this transformation activity1 is that within the MDI, the input representation introduced, the representations produced through transformation activity, and the accompanying explanations connect with each other and cohere with the stated problem. Our observations, across primary and secondary classrooms within our respective projects, suggest that such coherence or connection is frequently, but varyingly, disrupted within MDIs.


In this article, we share our development of an empirically derived analytical language (elements of which are italicised above) that allows us to make visible a range of disruptions to connection 1.Our use of the term ‘transformation activity’ coheres with Duval’s (2006) use of the term, which refers to transformations within and across registers. It is more inclusive than Kieran’s (2004) notion of transformation, which is restricted to transformation of algebraic expressions whilst maintaining equivalence across these manipulations.


25


Page 2 of 8

Original Research

Describing transformation activity – a literature review

and coherence that come into play across four contrasting teaching episodes. We focus here on input objects, transformational activity and accompanying explanations in order to ‘see into’ the micro-level production of mathematics in classrooms through describing differences in the nature and degree of coherence and connection and to consider the consequences for what is made available to learn. This micro-level focus on specific episodes within lessons follows our observation of the occurrence of disruptions at this level, rather than at the broader level of lessons or lesson sequences that have been taken up in prior research (e.g. Sekiguchi, 2006).

Duval (2006), from a semiotic perspective, describes mathematical activity as comprised by the transformation of one semiotic representation into another. For Duval, mathematical transformations can happen within, or across, registers: encompassing natural language, numeric, symbolic and algebraic notations, graphical representations, geometric figures and tabular presentations. For our purposes, the focus is on the representations and transformations selected and produced within transformation activity, the turning of one representation into another, either within or across registers.

In order to present our thinking on making aspects of connection and coherence visible within transformation activity, we begin with a brief overview of the literature. We draw on writing focused on transformation activity and representations as these actions and objects are at the centre of all our episodes and, as noted already, at the heart of mathematical activity more generally. We also summarise evidence that points to the shortcomings that characterise practices in which transformation steps are emphasised at the expense of gaining understanding of the representations they act upon. From this review, we outline the key concepts that we found helpful in beginning to pull apart some of the range of procedural practices that we were working with. Centrally, we home in on the stated problem, the selected input representation, subsequent sequences of transformation steps, and the interim and final representations produced in this sequence. These concepts are all covered in the literature we review.

Paying attention to the representations selected and produced through transformation activity is described by Haapasalo and Kadijevich (2000), cited in Haapasalo (2003), as important within transformation activity underlain by strong procedural knowledge. Strong procedural knowledge, for them, involves: dynamic and successful utilization of particular rules, algorithms or procedures within relevant representation forms. This usually requires not only the knowledge of the object being utilized, but also the knowledge of format and syntax for the representational system(s) expressing them. (p. 98)

This synchronous attention to both representational objects and transformation techniques is often described as lacking in school mathematics. An important second thread in the mathematics education literature is highly critical of the ways in which transformation activity has come to be configured within classrooms. Artigue (2011), in her article for UNESCO on the challenges of extending basic mathematics and science education for all students, refers to international surveys to describe how schooling is very often unstimulating because the teaching of mathematics is framed by:

Somewhat absent in this literature is a focus on the MDIs that accompany transformation activity. Teaching involves the giving of accompanying explanations alongside transformation and so, unlike the (often predictably) piecemeal learner discourses that are in focus in much of the literature on transformation-oriented activity, one expects MDIs to be both coherent and connected, and to provide some of the rationales for the representations selected and transformation activity that is enacted. As noted above however, we see this expectation disrupted relatively frequently, and in a range of different ways. In order to consider the nature of these disruptions to coherence, we use a tentative set of framing questions, drawn from our grounded analysis of the episodes presented in this article, to analyse and differentiate the transformation activity in four selected teaching episodes. This could be criticised as somewhat circular: developing grounded framing questions from a dataset, and then using them to analyse the same dataset. Our aim in doing this is to share this set of literature-drawn concepts and grounded framing questions in order to start conversations across the mathematics teaching and teacher education communities that can help to build a more robust language for thinking about what constitutes coherence and connection within mathematics teaching. We have already been through several iterations of concepts and framing questions, and have seen that our current formulation can be applied to a significantly broader group of episodes that we have encountered. http://www.pythagoras.org.za

• formal teaching, centred on learning techniques and memorizing rules whose rationale is not evident to the pupils • pupils [who] do not know which needs are met by the mathematics topics introduced or how they are linked to known concepts. (p. 21) Implied within Artigue’s formulation is a situation within which representations tend to be backgrounded, whilst transformation techniques are foregrounded. The need for a sense of the ‘problem’ that drives the selection of representations and the transformations enacted on them (the ‘raison d’être’) in coherent ways is highlighted. Drawing from the analytical work of Duval, and the critiques presented here, we see that the concepts of stated problem, input and subsequent representations produced through transformation activity are highlighted as fundamental to mathematical activity. The international literature base is replete with evidence of the consequences of pedagogies based on these kinds of practices for mathematical learning (Thompson, Philipp, 26


Page 3 of 8

Original Research

statement, visual representation, et cetera. In each of our episodes, we refer to this introductory offering to be operated upon as the input representation (IR). Transformation activity (TA), constituted by the transformation steps enacted on this input representation, and the interim representations produced, then ensues. Given our focus on teaching, these are associated with accompanying explanations from the teacher.

Thompson & Boyd, 1994). De Lima and Tall (2008) provide evidence of learners enacting transformations of algebraic representations based on the ‘embodied actions they perform on the symbols, mentally picking them up and moving them around, with the added “magic” of rules’ (p. 3). Such actions indicate inattention to the representations being operated on and the syntax of the registers these representations are located in; they thus frequently produce incorrect answers. Learner performance in South Africa across all phases attests to similar problems (Department of Basic Education, 2011a, 2011b). More problematically, there is evidence in South Africa of some of these actions occurring in the context of teaching rather than at the level of learner working (Mhlolo, Venkat & Schafer, 2012).

We focus on the detail of transformation steps enacted and the interim representations produced through these transformations, noting, in the first instance, whether these representations retain connection to the input representation. It is worth noting here that algorithms in mathematics do sometimes break this connection at interim stages, and reinstate it at the final stage; the long division algorithm provides a well known example of this (see Long [2005], for an elaboration on the differences between procedures and algorithms, and the deep mathematical structures underlying algorithms). We note this point to emphasise that these breaks in connection need not be innately problematic if the scope of application to representations and the mathematics underlying the transformation sequence are considered within the accompanying explanation. We therefore look at teacher explanations for whether or how they establish connections between representations and transformations, coherence with the stated problem, and reference to the scope of application and mathematical structure of the transformation being dealt with.

This leads to the need to focus on MDIs. Instructional explanation has been described as a ‘commonplace’ of mathematics teaching (Leinhardt, 2001), and described in terms of the ‘orchestrations of demonstrations, analogical representations and examples’ (Leinhardt, Zaslavsky & Stein, 1990). The word ‘orchestration’ points to the coherence and connection between problems and representations that we are focusing on, but does not, in itself, provide descriptors of what might constitute ‘good’ orchestration. Similarly, whilst coherence was identified as a characteristic seen more frequently in some Asian countries within the TIMMS-video data (Hiebert et al., 2003), what constitutes this coherence within teaching is not detailed. Rowland (2012), in using the distinction made in Leinhardt’s (2001) work between disciplinary and instructional explanations, notes that deductive reasoning characterises the former, whilst the need to ‘help students learn, understand and use knowledge’ through the use of ‘carefully devised analogy’ that render explanations ‘more accessible and more palatable’ (p. 59) is key to instructional explanations. In this more learningfocused category, there is a need to ‘establish’ rather than ‘state’ deductive connections, in order to support learner understanding of critical links. Thus, we looked at the nature of both disciplinary and instructional explanations in MDIs through framing questions that would allow us to analyse differences in connection and coherence between our episodes.

The framing questions are presented below: • Does the input representation cohere with the stated problem by providing an appropriate representation to transform? • Does transformation activity produce representations that connect in mathematically defensible ways with the input representation? Does this happen (1) across all interim representations or (2) at final representation? • Does transformation activity, linked with the teacher’s accompanying explanation, serve to establish connections between its steps at each stage and the input representation? Our questions reflect more basic notions about transformation activity than have been dealt with in the existing literature that deals with both cognition and semiotics (Duval, 2008). In the literature we have summarised, the notion of coherence and connection in teachers’ selections and transformations of representations is largely assumed, and critique focuses on the absence of rationales for the transformations selected. Our episodes suggest the need to suspend this assumption and focus on the detail of problem-representation-transformation connections as they play out in teaching. Firstly, we look for whether the input representation presented to transform coheres with the stated problem. We then ask questions about how interim and final representations produced within transformation activity connect to the input representation. This often involves transformation activity that produces representations that maintain equivalence between the representation acted on

In the teaching episodes we present in this article, all drawn from previously published work in the South African landscape, practices are exemplified at a range of problematic levels, beginning in Artigue’s terrain (backgrounding of problems and representations and foregrounding of transformations), and moving to significantly deeper problems in relation to coherence and connection. Our framing questions allowed us to disaggregate episodes in ways that provided some windows into understanding what constitutes coherence and connection.

Concepts and framing questions Mathematical processing begins with a stated problem (SP). Stated problems have to be solved through the introduction of an initial input representation, which could be a symbolic http://www.pythagoras.org.za

27


Page 4 of 8

Original Research

and the representation produced. At a deeper level though, we can also ask whether this transformation activity, linked with the accompanying explanation, establishes connections with the input representation rather than simply assuming, or stating, the connection.

Learners: Nash:

[Chorus] Positive. It’s a positive gradient … we can see there’s our y-intercept, there’s our x-intercept [points to the points (0; -3) and (2; 0) respectively]. After a brief discussion on the labelling of points on a graph, Learner 2 and Learner 3 ask Nash: Learner 2: Learner 3:

In the next section, we present episodes drawn from previously published work.

Nash: Learner 2: Nash:

Episodes Episode 1 This episode was reported in Adler and Pillay (2007) and Adler (2012) and is drawn from a study by Pillay (2006). Nash (pseudonym), a secondary school teacher, is described as well respected in his school, at which student performance in Grade 12 national mathematics examinations was considered successful. The episode below is taken from the third in a unit of eight consecutive lessons on linear functions in a Grade 10 class. In Lesson 1 and Lesson 2, Nash had dealt with drawing the graph of a linear function first from a table of values, and then using the gradient and y-intercept method. All the linear function examples that were worked through in the first two lessons were in the standard form y = mx + c .

Learner 2: Nash: Learner 4: Nash:

In Lesson 3, he moves on to demonstrate how to draw the graphs of functions that are not expressed in standard form. He begins with a few examples (e.g.222yyy−−−444xxx===222), using the gradient-intercept method, and the manipulative work needed to get these into the standard form y = mx + c . This serves as motivation for the greater simplicity of the dual intercept method for drawing straight-line graphs. He returns to a function they had worked on,33xx−−22yy ==66, and begins a discussion of ‘dual’ meaning ‘two’, eliciting from learners that the two intercepts are where the graph cuts the x and y axes. He demonstrates how to find the coordinates of the y-intercept by calculating the value of y when x = 0 and, similarly, the coordinates of the x-intercept. He writes (0; -3) and (2; 0) on the chalkboard and proceeds to sketch the axes, explaining how you can ‘estimate’ where the points are on each of the axes. He plots the two points and continues: Nash:

Our interest here is in how the teaching of the dual intercept procedure unfolds through transformation activity and its accompanying explanation. Nash presents a function 33xx−−22yy ==66 (an input representation) where the stated problem is to draw the graph of this line. This representation does cohere with the stated problem, and the preceding activity also shows that producing a line using the transformation sequences that have already been taught can be done, but is long-winded, making an alternative method useful. Thus the need for an alternative transformation sequence is motivated in Nash’s accompanying explanation. He then presents the steps to follow to obtain the coordinates of each of the intercepts. All the interim representations and the final representation do connect appropriately and cohere with the stated problem.

… all I have to do join these important points and I got [states and labels the line] 333xxx−−−222yyy===666– wasn’t that much easier? – there’s less mathematics to do [points to the calculations from the previous question] [than] when you come to write it in y form. Simple, first make your x equal to zero – that gives me my y-intercept. Then the y equal to zero gives me my x-intercept. Put down the two points – we only need two points to draw the graph.

Whilst Nash’s advocacy of the dual intercept method in all instances blocks some of the flexibility that a more selective match between stated problem and input representation would allow, and also does not provide ways of dealing with special cases (horizontal, vertical lines and lines through the origin), important features of the input and subsequent representations and their connections do remain in focus across Nash’s lesson sequence. For example, the fact that the gradient can be derived from the application of the dual intercept transformation steps to a linear equation to produce the associated graph is explained in instructional terms, using the move between algebraic and graphical representations to emphasise how given information can be used to derive missing information.

Then a learner asks a question: Learner 1: Nash:

You don’t need all the other parts? … What’s important features of this graph? … we can work out … from here [points to the graph drawn] we can see what the gradient is … is this graph a positive or a negative?


Sir, is this the simplest method, sir? How do you identify which side must it go, whether it’s the right-hand side. [Nash interrupts] [Response to Learner 2] You just join the two dots. That’s it? Yeah … the dots will automatically … if it was a positive gradient it will automatically … if this was [refers to the line just drawn] negative … that means this dot [points the x-intercept] will be on that side [points to the negative x axis] … because if the gradient was negative, how could it cut on that side? [Points to the positive x axis] Is this the simplest method sir? The simplest method and the most accurate Compared to which one? Compared to that one [points to the calculation of the previous question where the gradient and y-intercept method was used] because here if you make an error trying to write it in y form … that means it now affects your graph … Whereas here [points to the calculations he has just done on the dual intercept method] you can go and check again … you can substitute … if I substitute for 2 in there [points to the x in333xxx−−−222yyy===666] I should end up with 0.

28


Page 5 of 8

Original Research

Episode 2

It could be argued that in transformation terms nothing has gone wrong here, but given that we see several learners filling gaps in the follow-up missing addend individual activity, with 11 at the centre, simply by adding the two numbers seen, there is evidence that neither the missing addend nor the subtraction routine have actually taken ‘hold’ for broad swathes of the class. Instead, the predominant interpretation of the problem involves ‘adding’ the two numbers that can be seen, a ‘putting together’ of the visual instruction to add with the numbers seen. Here, we see transformation activity which, whilst connected in mathematically coherent terms to the input object, does not work to establish this connection. The switch to subtraction is followed as an embodied imitation in the teacher’s presence, but no explanation for establishing this switch as valid is provided; at the same time, no interim representational supports or associated explanations that link more directly to the stated problem are enacted. Essentially, referring back to the literature, whilst the disciplinary explanation is coherent, an instructional explanation is lacking.

In this episode, drawn from Askew, Venkat and Mathews (2012), a Grade 2 teacher is working on missing addend problems using a wheel representation with three concentric circles: 7 written on the inner circle, and the numbers 0–7 placed in random order around the outermost circle in separate sectors. Askew et al. state: The task explained by the teacher was to fill in the intermediate circle with the numbers that needed to be added to the outer rim numbers of the wheel to make the number 7. (p. 29)

The stated problem of the lesson, indexed by the title on the board Hlanganisa (‘Addition’ in Zulu), is for the class to fill in the numbers that need to be added to the numbers on the edge to make 7. Initial answers from some of the children indicate that they are interpreting the task in terms of addition of the numbers shown. In one episode, the teacher is focused on the problem: ‘What number is added to 3 to make 7?’ She shows the class three open fingers on her hand as she asks this, pointing to the 3 on the circle rim, and then shows seven fingers as she indicates the need to make 7, pointing to the 7 in the centre. Some children are seen counting out seven on their fingers. When no correct answers are forthcoming from the class, the following exchange takes place: Teacher:

Learner:

Episode 3

Make 7 with your fingers. [Shows seven fingers on her hands and several learners seen showing seven fingers] Now hide three fingers. [Teacher closes three fingers on her hand and asks class to do the same] Which number can we add with this 3 to make 7? [Teacher’s hand shows four fingers remaining open] Now we made 7 and hide 3, and what is left? The number that when we will add with 3, we will get 7. [Teacher goes over and helps a child to close the same three fingers and asks her to count what is left] 4. [Accepting 4, the teacher then counts out three toothpicks at the front as 1, 2, 3, then another 4 toothpicks as 1, 2, 3, 4. She then counts them altogether from 1–7]

In this episode, drawn from Davis (2010), a Grade 10 teacher is working on integer addition sums, such as: -7 + 5. Davis describes the teacher’s instructions to the class as follows (p. 384): Teacher:

Learners: Teacher:

Prior to and following this episode, we see instances of some learners able to give correct answers. However, we also see several learners who appear unaware of how many fingers to open, and what to do once they have one of the given numbers showing. Here, a stated problem that is given in terms of missing addends comes to be ‘funnelled’ into a subtraction problem through a transformation step and associated explanation, and then verified by adding the two numbers as an addition problem. The teacher appears aware of the equivalence between missing addend problems and subtraction, but this equivalence is not established for learners; rather, the equivalence is simply assumed, and subsequently verified empirically. Thus, a problem stated in terms of missing addends is worked out in terms of subtraction-based transformation activity, and checked through addition. Essentially, the sum below is presented as the stated problem to be solved (though not in this form): 3+

Learners: Teacher:

As was the case in Episode 1, an ordered set of instructions is relayed to the class – ‘first you take ... and then you take ...’. Some conditions for the application of transformation sequences are established at the outset: essentially ways to distinguish the input representation in order to recognise which transformation must be selected. Davis (2010), discussing this episode in terms of operations (addition) and objects (integers), notes that: The regulative criteria required by the procedure indicate that the teacher and learners do not operate directly on the mathematical objects and relations being indexed (integer sums). They operate, instead on more familiar and intuitive objects and relations (‘whole number’ sums). (p. 385)

In terms of our analytical concepts, -7 + 5 is the input representation that is transformed through a series of TA steps that provide an algorithm for solving the problem. In the interim stages, following the instructions would produce these representations:

=7

whilst the transformation activity instead involves solving the following subtraction problem: 7−3 =


So if the signs are the same ... what do you do? ... You take the common sign ... and then ... you add. … If the signs are not the same ... what do you do? You subtract. [Chorus] Subtract. But first you take the sign of the what? The sign ... of the bigger number. You look at the bigger number between the two ... and then you take the sign ... of the bigger number. [Chorus] Yes. This should always be the case.

29


Page 6 of 8

Teacher:

Original Research

Calculate the answer to the sum [SP]: -7 + 5 = [IR] If the signs are not the same ... what do you do? You subtract. [TA step 1] [Following this TA step would result in the following interim representation]: 7 − 5 = 2 You look at the bigger number between the two ... and then you take the sign ... of the bigger number. [TA step 2] [Following this TA step would result in the following final representation]: -2

TABLE 1: Data table. No. of children in the family

8

1

14

2

20

3

17

4

10

5

11

Source: Venkat, H., & Mhlolo, M.K. (2011). Objects and operations in mathematics teaching – extending our understanding of breakdowns. In T. Mamiala, & F. Kwayisi (Eds.), Proceedings of the 19th Annual Meeting of the Southern African Association for Research in Mathematics, Science and Technology Education (pp. 246−259). Mafikeng: SAARMSTE

In this episode, the backgrounding of the input representation is more significant than in the previous two episodes. Within the transformation sequence that is presented, negative numbers simply do not figure. Further, whilst the transformation sequence is driven by the need to solve the given problem, the algorithm presented enacts steps that produce an interim representation that is not equivalent to the input object (-7 + 5), even though equivalence with the input representation returns at the final stage. Essentially, at the interim stage, transformation activity does not establish ‘reason-able’ connections between representations. As noted already, it could be argued that this is not problematic as a mnemonic device, given that correct answers across the range of input representations specified can be reliably produced. However, analyses of South African performance on TIMMS items (Howie, 2003) and our own data (Adler, 2011) point to multiple basic errors in the realm of both integer calculations and algebraic manipulations requiring knowledge of integer sum transformations. Further, Thompson et al. (1994) note that the orientation to ‘answer-getting’ tends to work most reliably only for the learners ‘who understood the problem in the first place, and understood it in such a way that the proposed sequence of operations fits their conceptualisation of the problem’ (p. 9).

first frequency value in her table. Then, pausing to ask the class if they have seen this (pointing to her tally) before, she explains further and demonstrates: OK so it’s one, it’s two, it’s three, it’s four and what happens to number five. [Indicates the diagonal line]. And then it’s one, it’s two and it’s three. OK. Your tally and your frequencies must be of the same number.

In this episode we note that whilst transformation activity (producing the tally) does connect to the stated problem, the stated problem does not cohere with the frequency table as the input representation presented. Thus, whilst equivalence is maintained, and indeed emphasised, in the teacher’s discourse between the input representation’s frequency values and the representation that is produced (the tallies), this equivalence in the absence of coherence between the stated problem and the input representation tends to nullify the equivalence that is produced. Table 2 summarises the application of the analytical concepts applied to the four episodes.

Discussion

In terms of our concepts, the input representation coheres with the stated problem, but the interim representation produced through the first transformation step does not retain equivalence with the input representation, even though this equivalence is recovered at the final stage.

Looking across Table 2, we can see that in relation to the analytical framing of transformation activity developed, differences in the detail of access to the representations and transformations involved emerge. Significantly, some episodes reflect more serious disruption of mathematical coherence and connection than others. We contend that having the discursive resources to see and talk about these differences is enabling for our work in teacher education.

Episode 4 Venkat and Mhlolo (2011) present an incident involving a Grade 11 teacher working on a data-handling problem based on the data table shown in Table 1, the input representation introduced in the lesson.

In Episode 1, all the criteria are answerable in the affirmative, suggesting that basic connections and coherence are in place. Nash’s practices, we believe, mirror the kind of teaching referred to in our earlier discussion of the literature. The fact that Nash is viewed as successful, with learners under him viewed as performing successfully, backs up our claim of basic connections and coherence. Thus, whilst concepts in the literature may well describe Nash’s practice as ‘procedural’, the connections and coherence we have identified appear to open up access to appropriate transformation activity.

In the process of asking generally about ways in which data can be presented, a student mentions the notion of a ‘tally table’. Venkat and Mhlolo note the subsequent return by the teacher, after several interim episodes focused on a range of other stated problems, to the notion of tallying. The teacher shifts attention from a focus on the meaning of the frequency values in Table 1 with the following question: OK before we move on, somebody talked about tally OK. Does anyone know how to tally the number 8? ... Do you know or you want to try?

In Episode 2, the key issue is that a missing addend problem is assumed to be solvable through manipulating the sum into subtraction format. Of course, this is mathematically correct, but the transformation from missing addend form to

Having asked the question, she then adds a further column to her frequency table and gives it the title ‘tallies’. She then shows the class how to tally the number ‘8’, this being the http://www.pythagoras.org.za

Frequency

0

30


Page 7 of 8

Original Research

TABLE 2: Applying the analytical concepts. Concepts and framing questions

Episode 1

Episode 2

Episode 3

Episode 4

Stated problem (SP)

Producing graphs of linear functions with equations in non-standard form

Calculating missing addends

Calculating integer sums involving one positive and one negative number

How to tally

Input representation (IR) 6

3x − 2y = 6

Clock diagram with missing addends for 7

-7 + 5

Frequency table with a range of frequency values

Transformation activity (TA)

Put x = 0 to get y-intercept; y = 0 to get x-intercept; plot points, join them; you get the graph

Take 7 fingers, then hide 3

Drop the signs, take the smaller from the larger and put the sign of the larger against the answer

Write one, one, one, one, then a cross through; check that tally and frequency show the same number

Does IR cohere with the SP by providing an appropriate representation to transform?

Yes 3x − 2y = 6

Yes Clock diagram allows link to missing addend

Yes -7 + 5

No A frequency number, rather than raw data, is given

Does TA produce representations that connect with the IR at: • interim stages • final stage in mathematical terms?

Yes Yes

Yes Yes

No Yes

No No

Does TA, with associated explanation Yes serve to establish connections between its steps and the IR?

No

No

No

Conclusions

subtraction form needs to be established for learners through explanation and mediating representations that allow for the equivalence between the two forms to be appropriated. Instead, an assertion or an assumption of equivalence is presented, rather than an establishment of the equivalence. An outcome of the assumed equivalence appears, in this lesson, to be ongoing difficulties with task completion. There is a notable absence of the analogical representations that Rowland (2012) describes as important within instructional explanations: in this instance, representations that provide ‘direct models’ (Carpenter, Fennema, Franke, Levi & Empson, 1999) of the stated missing addend problem. Thus, the MDI fails to provide representations that scaffold the connection between the missing addend situation and the abstract understanding of number relationships needed to ‘jump’ to subtraction as an appropriate transformation step to enact.

Several comments emerge from our analysis. Firstly, we note at the most basic level that if the input representation does not cohere with the stated problem, this appears to negate the possibilities for answering the other analytical questions in the affirmative. At the intermediate level, we suggest that two criteria allow for further disaggregation: • making transformation steps ‘reason-able’ by establishing connections between the representation and its transformation • producing transformation sequences that connect across representations. At the highest level, we have episodes that demonstrate coherence between the stated problem and the input representation, and connections between the representations produced through transformation activity where all three criteria are met.

In Episode 3, the problem seen in Episode 2 is further compounded by the fact that the first transformation step indicated by the teacher’s instructions produces an interim representation that does not connect with the input representation. Thus, the instruction that seeing one negative and one positive number means we have to subtract is arbitrary at this stage; it simply has to be remembered, and cannot be reasoned. Whilst at the final stage, equivalence with the input representation is resurrected, one needs to take on trust that this will happen through the interim working. As in Episode 2, the transformation activity does produce the correct answer with some efficiency. In this case though, connections between representations are broken at the interim stage.

Our sense is that the literature as it stands provides us with a discourse that can speak constructively to Nash’s practice, but offers few insights into the kinds of limitations seen in our other episodes. The analytic concepts and questions that we have presented in this article were derived from analysing problematic episodes of the teaching of procedures involving the transformation of representations. Whilst the framework still needs further testing, our application of these concepts and criteria to further episodes from our project data sets suggests that they may provide some general principles for basic coherence and connection within mathematics teaching. We therefore offer the concepts and questions developed and deployed in this article as a starting point that has some generality, and illuminating potential for the many classrooms in which the transformation activity that fundamentally underlies mathematical activity still appears to be problematic.

In Episode 4, given that the stated problem is tallying, the presentation of a frequency table as the input representation is inappropriate as an object for the process of tallying to act on. Thus, whilst equivalence between the tally graphic produced for each frequency is maintained, connections that could serve to establish the purpose of tallying processes in mathematics are not simply made invisible, but actively disrupted.


Acknowledgements This work is based upon the research of the Wits Mathematics Connect – Primary and Secondary Projects at the University

31


Page 8 of 8

Original Research

Department of Basic Education. (2011a). Report on the Annual National Assessments of 2011. Pretoria: DBE. Available from http://www.education.gov.za/LinkClick.asp x?fileticket=1U5igeVjiqg%3d&tabid=358&mid=1325

of the Witwatersrand, supported, respectively by FirstRand Foundation, Anglo American, Rand Merchant Bank, the Department of Science and Technology, and the FirstRand Foundation (FRF) Mathematics Education Chairs Initiative of the FirstRand Foundation, the Department of Science and Technology (DST). Both are administered by the National Research Foundation (NRF). Any opinion, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the institutions named above.

Department of Basic Education. (2011b). Report on the National Senior Certificate examination results: 2010. Pretoria: DBE. Available from http://www.education. gov.za/LinkClick.aspx?fileticket=ah59UwgTOHk%3d&tabid=358&mid=1325 Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61(1/2), 103−131. http:// dx.doi.org/10.1007/s10649-006-0400-z Duval, R. (2008). Eight problems for a semiotic approach. In L. Radford, G. Schubring, & F. Seeger (Eds.), Semiotics in mathematics education: Epistemology, history, classroom and culture (pp. 39−61). Rotterdam: Sense Publishers. Haapasalo, L. (2003). Linking procedural and conceptual mathematical knowledge in technology-based learning. In A. Rogerson (Ed.), Proceedings of the International Conference of the Mathematics Education into the 21st Century Project: The Decidable and the Undecidable in Mathematics Education (pp. 98−102). Brno, Czech Republic: Mathematics Education into the 21st Century Project. Available from http://math.unipa.it/~grim/21_project/21_brno03_Haapasalo.pdf

Competing interests

Haapasalo, L., & Kadijevich, D. (2000). Two types of mathematical knowledge and their relation. Journal für Mathematik-Didaktik, 21(2), 139−157.

We declare that we have no financial or personal relationship(s) which might have inappropriately influenced our writing of this article.

Hiebert, J., Gallimore, R., Garnier, H., Givvin, K.B., Hollingsworth, H., Jacobs, J., et al. (2003). Teaching mathematics in seven countries: Results from the TIMSS 1999 video study. Washington, DC: U.S. Department of Education, National Center for Education Statistics. Howie, S.J. (2003). Language and other background factors affecting secondary pupils’ performance in Mathematics in South Africa. African Journal of Research in Science, Mathematics and Technology Education, 7, 1−20.

Authors’ contributions H.V. (University of the Witwatersrand) began the writing of the manuscript in terms of its framing, but all ongoing work has been shared by H.V. and J.A. (University of the Witwatersrand).

Kieran, C. (2004). The core of algebra: Reflections on its main activities. In K. Stacey, H. Chick, & M. Kendal (Eds.), The future of the teaching and learning of algebra. The 12th ICMI Study (pp. 21−34). Boston, MA: Kluwer Academic Publishers. http:// dx.doi.org/10.1007/1-4020-8131-6_2 Leinhardt, G. (2001). Instructional explanations: A commonplace for teaching and location for contrast. In V. Richardson (Ed.), Handbook for research on teaching (pp. 333−357). Washington, DC: American Educational Research Association.

References

Leinhardt, G., Zaslavsky, O., & Stein, M.K. (1990). Functions, graphs and graphing: Tasks, learning and teaching. Review of Educational Research, 60(1), 1−64. Long, C. (2005). Maths concepts in teaching: Procedural and conceptual knowledge. Pythagoras, 62, 59−65. http://dx.doi.org/10.4102/pythagoras.v0i62.115

Adler, J. (2011). Research and Development Chairs: Annual report, 2010. Pretoria: National Research Foundation.

Mhlolo, M.K., Venkat, H., & Schafer, M. (2012). The nature and quality of the mathematical connections teachers make. Pythagoras, 33(1), 1−9. http://dx.doi. org/10.4102/pythagoras.v33i1.22

Adler, J. (2012). Knowledge resources in and for school mathematics teaching. In G. Gueudet, B. Pepin, & L. Trouche (Eds.), From text to ‘lived’ resources. Mathematics curriculum materials and teacher development (pp. 3−22). Dordrecht: Springer. http://dx.doi.org/2010.1007/978-94-007-1966-8_1

Pillay, V. (2006). An investigation into mathematics for teaching: The kind of mathematical problem-solving teachers do as they go about their work. Unpublished master’s thesis. University of the Witwatersrand, Johannesburg, South Africa. Available from http://wiredspace.wits.ac.za/handle/10539/2190

Adler, J., & Pillay, V. (2007). An investigation into mathematics for teaching: Insights from a case. African Journal of Research in Mathematics, Science and Technology Education, 11(2), 87−108.

Rowland, T. (2012). Explaining explaining. In S. Nieuwoudt, D. Loubscher, & H. Dreyer (Eds.), Proceedings of the 18th Annual National Congress of the Association for Mathematics Education of South Africa, Vol. 1 (pp. 54−66). Potchefstroom: AMESA. Available from http://www.amesa.org.za/AMESA2012/Volume1.pdf

Artigue, M. (2011). Challenges in basic mathematics education. Paris: UNESCO. Available from http://unesdoc.unesco.org/images/0019/001917/191776e.pdf Askew, M., Venkat, H., & Mathews, C. (2012). Coherence and consistency in South African primary mathematics lessons. In T-Y. Tso (Ed.), Proceedings of the 36th Conference of the International Group for the Psychology of Mathematics Education, Vol. 2 (pp. 27−34). Taipei, Taiwan: PME.

Sekiguchi, Y. (2006). Coherence of mathematics lessons in Japanese eighth-grade classrooms. In J. Novotná, H. Moraová, M. Krátká, & N.E. Stehlíková (Eds.), Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education, Vol. 5 (pp. 81−88). Prague: PME.

Carpenter, T.P., Fennema, E., Franke, M.L., Levi, L., & Empson, S.B. (1999). Children’s mathematics: Cognitively guided instruction. Portsmouth, NH: Heinemann.

Thompson, A.G., Philipp, R.A., Thompson, P.W., & Boyd, B.A. (1994). Calculational and conceptual orientations in teaching mathematics. In D. Aichele, & A. Coxford (Eds.), Professional development for teachers of mathematics (pp. 79−92). Reston, VA: NCTM. Available from http://www.patthompson.net/PDFversions/1994Calc& ConcOrientations.pdf

Davis, Z. (2010). Researching the constitution of mathematics in pedagogic contexts: From grounds to criteria to objects and operations. In V. Mudaly (Ed.), Proceedings of the 18th Annual Meeting of the Southern African Association for Research in Mathematics, Science and Technology Education, Vol. 1 (pp. 378−387). Durban: SAARMSTE. Available from http://www.sdu.uct.ac.za/usr/sdu/downloads/ conferences/saar_mste2010/longpapervol1.pdf

Venkat, H., & Mhlolo, M.K. (2011). Objects and operations in mathematics teaching – extending our understanding of breakdowns. In T. Mamiala, & F. Kwayisi (Eds.), Proceedings of the 19th Annual Meeting of the Southern African Association for Research in Mathematics, Science and Technology Education (pp. 246−259). Mafikeng: SAARMSTE.

De Lima, R.N., & Tall, D. (2008). Procedural embodiment and magic in linear equations. Educational Studies in Mathematics, 6(1), 3−18. http://dx.doi.org/10.1007/ s10649-007-9086-0


32


Page 1 of 12

Original Research

Thinking styles of Mathematics and Mathematical Literacy learners: Implications for subject choice Author: Erica D. Spangenberg1 Affiliation: 1 Department of Science and Technology Education, University of Johannesburg, South Africa Correspondence to: Erica Spangenberg Email: [email protected] Postal address: PO Box 5584, Krugersdorp West 1742, South Africa Dates: Received: 30 May 2012 Accepted: 04 Nov. 2012 Published: 06 Dec. 2012 How to cite this article: Spangenberg, E.D., (2012). Thinking styles of Mathematics and Mathematical Literacy learners: Implications for subject choice. Pythagoras, 33(3), Art. #179, 12 pages. http://dx.doi.org/10.4102/ pythagoras.v33i3.179

In this article I report on research intended to characterise and compare the thinking styles of Grade 10 learners studying Mathematics and those studying Mathematical Literacy in eight schools in the Gauteng West district in South Africa, so as to develop guidelines as to what contributes to their subject choice of either Mathematics or Mathematical Literacy in Grade 10. Both a qualitative and a quantitative design were used with three data collection methods, namely document analysis, interviews and questionnaires. Sixteen teachers participated in one-to-one interviews and 1046 Grade 10 learners completed questionnaires. The findings indicated the characteristics of learners selecting Mathematics and those selecting Mathematical Literacy as a subject and identified differences between the thinking styles of these learners. Both learners and teachers should be more aware of thinking styles in order that the learners are able to make the right subject choice. This article adds to research on the transition of Mathematics learners in the General Education and Training band to Mathematics and Mathematical Literacy in the Further Education and Training band in South Africa.

Introduction and background This article focuses on the characterisation of the thinking styles of Grade 10 Mathematics and Mathematical Literacy learners in eight schools in Gauteng West, South Africa. Since 2006, learners have had the choice to study either Mathematics or Mathematical Literacy in Grades 10–12 (Department of Education [DOE], 2003a). A subject is defined as ‘a specific body of academic knowledge’ where ‘knowledge integrates theory, skills and values’ (DOE, 2003c, p. 6). In this article I refer to a subject as a particular area of study that schools offer, for example Accounting, English, Mathematics or Mathematical Literacy. Learners study Mathematics from Grade 4 to Grade 9; Mathematical Literacy is a new subject which can only be studied in Grade 10 to Grade 12. Initially, parents and teachers guide learners in their subject choices, based on factors such as future career, language, socio-economic background, interests and achievements in the lower grades (Spangenberg, 2008). However, in the higher grades, learners’ own thinking styles influence their preference for the different subjects (Borromeo Ferri, 2004). Research reveals that thinking styles play an important role in teaching and learning (Borromeo Ferri, 2004; Cilliers & Sternberg, 2001; Grigorenko & Sternberg, 1997; Moutsios-Rentzos & Simpson, 2010; Sternberg, 1990; Sternberg & Wagner, 1992a; Zhang, 2006). In particular, Sternberg and Grigorenko (1993) found that certain thinking styles correlated positively to a learner’s success in a variety of academic tasks, whereas other thinking styles tended to correlate negatively to success in the same tasks. Van der Walt (2008) also noted that a learner’s thinking style is a factor that influences the effective learning and teaching of mathematics and could predict achievement of mathematics in school. Therefore, I argue that learners’ thinking styles could affect their choice to study either Mathematics or Mathematical Literacy. By establishing which thinking style is associated with learners in each of the two subjects, one should be able to guide them, their parents and teachers in making more informed decisions with regard to the choice between Mathematics or Mathematical Literacy as a subject.


A study that characterises and compares the thinking styles of Grade 10 learners taking Mathematics and those taking Mathematical Literacy is new to South Africa, although there have been investigations into thinking styles at both teacher and learner levels (Cilliers & Sternberg, 2001, De Boer & Bothma, 2003). In addition, Moutsios-Rentzos and Simpson (2010) conducted a study in Greece on the thinking styles of university students, Zhang (2006) asked ‘Does student–teacher thinking style match/mismatch matter in students’ achievement?’ in Hong Kong, and Borromeo Ferri (2004) conducted an empirical study on mathematical thinking styles of 15–16-year-old learners in Germany. All these researchers referred to the thinking styles


33


Page 2 of 12

Original Research

inventory of Sternberg and Wagner (1992b); I too have used the precepts of Sternberg’s theory in order to characterise and compare the thinking styles of learners taking Mathematics and those taking Mathematical Literacy in South Africa. This article reports on part of a broader study that I had previously conducted on the placement of Grade 10 learners to provide advice for learners, parents and teachers in terms of choosing between Mathematics and Mathematical Literacy (Spangenberg, 2008).

judging) and executive (preference for implementing rules and instructions) thinking styles. Forms are ‘general ways’ in which learners ‘approach their environments and the problems the environment presents’ including hierarchic (preference for having multiple prioritised objectives), anarchic (preference for flexibility), monarchic (preference for focusing on only one goal) and oligarchic (preference for having multiple equally important targets) thinking styles. Levels refer to the ‘amount of engagement individuals prefer in a given activity’ including local (preference for details and the concrete) and global (preference for general and the abstract) thinking styles. Scopes are ‘stylistic variables which divide learners into two basic personality types, including internal (preference for working alone) or external (preference for working in a group) thinking styles. Finally, learning explains ‘the methods and rules by which learners solve problems’ including liberal (preference for novelty and originality) and conservative (preference for conformity) thinking styles (Richmond, Krank & Cummings, 2006, p. 59). Each of these thinking styles has its own characteristics, as represented in Table 1.

The question arising from the above discussion is: Which thinking styles are associated with learners taking Mathematics and those taking Mathematical Literacy? Hence, I conducted a literature inquiry with regard to the thinking styles and the nature of Mathematics and Mathematical Literacy.

Thinking styles A style is a particular procedure or manner by which something is done or a specific way or tendency unique to a person (Soanes, 2002). In particular, Zhang and Sternberg (2000) define a thinking style as ‘a source of individual differences in academic performance that are related not to abilities but how people prefer to use their abilities’ (p. 469). Whereas a learning style refers to a way of approach to learning (Kolb, Boyatzis & Mainemelis, 1999), a thinking style refers to a particular act, idea, tendency or way of thinking about the execution of a task in the learning process (Sternberg, 1994). Thus, a learning style is how a learner receives information, whilst a thinking style is how a learner processes information and reflects on ideas in their mind. For Cilliers and Sternberg (2001, p. 14) a thinking style is a ‘preference’ for using abilities in certain ways during processing.

Sternberg (1990, p. 368) noted that a teacher may not appreciate a learner’s ability and may view him or her as ‘slow’ or ‘behind’ because of a difference in thinking style between the learner and the teacher. Conversely, Van der Walt (2008) argued that learners’ thinking styles may contribute to their inability to solve mathematical problems, even though they have the necessary knowledge. However, research conducted by Sternberg (Grigorenko & Sternberg, 1988, 1997; Sternberg, 1990, 1994, 1997) has revealed that teaching and learning can improve if teachers give more attention to thinking styles. Both learners and teachers bring their own individual characteristics and thinking styles to the learning environment (Zhu, 2011). These thinking style preferences lead to learning style preferences and in turn determine learners’ dominant cognitive modes, that is, the ways in which they communicate and receive information. More specifically, ‘cognitive functions are accommodated when teaching activities are constructed to comply with a learner’s preferred mode of thinking’ (De Boer & Bothma, 2003, p. 1).

The theoretical basis for this study is based on Sternberg’s theory of mental self-management. Sternberg and Wagner (1992b) developed a thinking style inventory consisting of 13 thinking style dimensions divided into five categories, namely functions, forms, levels, scopes and learning (Sternberg, 1990). Functions refer to the ‘basic types of thinking styles’, including legislative (preference for creativity), judicial (preference for TABLE 1: Thinking style dimensions. Category

Style

Characteristic

Functions

Legislative

Likes to create, discover, design; does things using own method; less structure.

Executive

Likes to follow instructions; does what is requested; structure must be given.

Judicial

Likes to criticise and evaluate people and things.

Monarchic

Likes to do one thing at a time; spends almost all the energy and resources on it.

Hierarchic

Likes to do many things at once; prioritises what and when to do a thing and how much time and energy to spend on it.

Oligarchic

Likes to do many things at once, but experiences problems with prioritising.

Anarchic

Likes to follow an extraordinary approach to problems; hates systems, guidelines and any restrictions.

Global

Likes to work with the bigger picture, generalisations and abstracts.

Local

Likes to work with detail, specifications and concrete examples.

Internal

Likes to work alone; focuses on the inside and is independent.

External

Likes to work with other people; focuses on the outside and is interdependent.

Liberal

Likes to do things in a new manner and deviates from traditions.

Conservative

Likes to do things in a proven and real manner and follows traditions.

Forms

Levels Scopes Learning

Source: Adapted from Sternberg, R.J. (1994). Allowing for thinking styles. Educational Leadership, 52(3), 36−40


34


Page 3 of 12

Original Research

The nature of Mathematics and Mathematical Literacy

express quantitative information in a verbal and visual form. Mathematical Literacy creates a consciousness about the role of mathematics in the modern world and is therefore driven by practical applications. The subject develops the ability and confidence of learners to think numerically in order to interpret daily situations (DOE, 2003b).

In South Africa, Mathematics and Mathematical Literacy relate to each other, but differ in terms of their nature and aims. In particular, Mathematics ‘enables creative and logical reasoning about problems in Mathematics itself’, which ‘leads to theories of abstract relations’ (DOE, 2003c, p. 9). On the other hand, Mathematical Literacy equips and sensitises learners with an understanding of the relevance of mathematics in real-life situations (DOE, 2003b, p. 9). Its purpose is to apply mathematics to make sense of the world. Mathematical Literacy was specifically introduced as an intervention to improve numeracy skills of South African citizens in response to poor performance in mathematics in the past (Bansilal, Mkhwanazi & Mahlabela, 2012).

According to Zhang (2002, p. 179) ‘students who reasoned at a higher cognitive developmental level tended to use a wider range of thinking styles than students who reasoned at a lower cognitive developmental level’. Therefore, learners taking Mathematics, which focuses on ‘creative and logical reasoning’ (DOE, 2003c, p. 9), likely utilise different styles of thinking when reasoning than learners taking Mathematical Literacy. Ideally, Mathematical Literacy learners should be able to reason by communicating, either verbally or in written form, because Mathematical Literacy uses everyday language for ‘practical relevance and applications’, which may be easier for learners to understand, whereas Mathematics uses highly technical mathematical language for ‘further math learning’ (Graven & Venkat, 2007, p. 69). In particular, Venkat, Graven, Lampen, Nalube and Chitera (2009) noted that Mathematical Literacy promotes ‘thinking as communication’ which ‘consists of acts such as asking questions, hypothesizing, finding counter-arguments and drawing conditional conclusions within a situation’ (p. 48). In contrast, Mathematics learners should use higher levels of visual-spatial reasoning and abstract thinking. Hence, to achieve in Mathematics, the ‘use of symbols and notations’ and ‘mental processes that enhance logical and critical thinking, accuracy and problem solving’ and ‘mathematical problem solving’ (Department of Basic Education [DBE], 2011b, p. 8) should be emphasised as the content is in an abstract and generalisable form.

Learners who can think in terms of ‘symbolic representation or abstract conceptualization – thinking about, analyzing, or systematically planning’ and in terms of the concrete reality, including mathematical modelling and more applied mathematics, should achieve in Mathematics, whereas learners who can only think ‘through experiencing the concrete, tangible, felt qualities of the world, relying on our senses and immersing ourselves in concrete reality’, and not in terms of abstraction, should achieve in Mathematical Literacy (Kolb, Boyatzis & Mainemelis, 1999, p. 3). Moreover, the Grade 10–12 Mathematics syllabus, as set out in the National Curriculum Statement (DOE, 2003c) is a purely academic subject which focuses more on content that the learners have to deal with, memorise and reflect on, as opposed to Mathematical Literacy which is a practical subject where learners learn practical skills that will enable them to find concrete solutions to numeric, spatial and statistical problems associated with the everyday challenges of life. In Mathematics attention has to be paid to specific details and, as Bohlmann and Pretorius (2008, p. 43) claim, ‘the conceptual complexity and problem-solving nature of Mathematics make extensive demands on the reasoning, interpretive and strategic skills of learners.’ Mathematics is an abstract, deductive discipline that is required in the scientific, technological and engineering world. According to Venkat (2007):

Mathematics deals with concepts as ideas or abstractions which learners have to bring together to solve a mathematical problem to enable them ‘to understand the world’ (DBE, 2011b, p. 8). In contrast, Mathematical Literacy deals with ‘making sense of real-life contexts and scenarios’ and ‘mathematical content should not be taught in the absence of context’ (DBE, 2011a, p. 8). Table 2 presents the differences between Grade 10 Mathematics and Grade 10 Mathematical Literacy with regard to the content as prescribed by the National Curriculum Statement (DOE, 2003b, 2003c).

Emphasis is laid on abstract rather than concrete concepts, on intra-mathematical connections rather than mathematics-realworld connections, on rigour and logic rather than interpretation and critique, and on knowledge itself, as well as applications of knowledge. (p. 77)

Due to the differences in the natures of Mathematics and Mathematical Literacy, it is expected that learners require different thinking styles to achieve in these subjects. The identification of the different thinking styles will contribute towards informing teachers, parents and learners, in an objective manner, about the choice of either Mathematics or Mathematical Literacy as a subject.

In comparison to Mathematics, which is more abstract in nature, most definitions of Mathematical Literacy focus on the concrete dimension of mathematics with the context determining the content to be learned. Learners use reallife situations to gain new knowledge; thus, a learner who tends to process information in a concrete way should achieve in Mathematical Literacy. Gal (2009) pointed out that Mathematical Literacy focuses on the relevance of learned knowledge to everyday life and linked it to diverse realworld contexts, whilst Frith and Prince (2006) stated that people are mathematically literate if they have the ability to http://www.pythagoras.org.za

Research design The research question is intended to characterise and to compare the thinking styles of learners taking Mathematics and of learners taking Mathematical Literacy. As noted 35


Page 4 of 12

Original Research

TABLE 2: Differences in content between Grade 10 Mathematics and Mathematical Literacy. Grade 10 Mathematics

Grade 10 Mathematical Literacy

Number and number relationships: • convert between terminating or recurring decimals • fluctuating foreign exchange rates.

Number and operations in context: • percentage • ratio • direct and inverse proportion • scientific notation.

Functions and algebra: • graphs to make and test conjectures and to generalise the effects of the parameters a and q on the graphs • algebraic fractions with monomial denominators • linear inequalities in one variable • linear equations in two variables simultaneously.

Functional relationships: • numerical data and formula in a variety of real-life situations, in order to establish relationships between variables by finding the dependent variable and the independent variable.

Space, shape and measurement: • volume and surface area of cylinders • co-ordinate geometry • the trigonometric functions sin θ, cos θ and tan θ; and solve problems in two dimensions by using the trigonometric functions in right-angled triangles.

Space, shape and measurement: • international time zones • circles • draw and interpret scale drawings of plans to represent and identify views.

Data handling and probability: • measures of dispersion (range, percentiles, quartiles, interquartile and semiinterquartile range) • frequency polygons • Venn diagrams.

Data handling: • investigate situations in own life by formulating questions on issues such as those related to social, environmental and political factors, people’s opinions, human rights and inclusivity • collect or find data by appropriate methods (e.g. interviews, questionnaires, the use of databases) suited to the purpose of drawing conclusions to the questions • representative samples from populations.

• There are significant differences between a learner’s thinking style dimensions and the subject they choose, either Mathematics or Mathematical Literacy.

above, learners’ thinking styles can influence their subject choices. The lack of guidelines from the DOE with regard to the placement of Grade 10 learners in either Mathematics or Mathematical Literacy convinced me to research for a practical solution to guide learners, parents and teachers to make informed subject choices and, thus, to adopt a pragmatic philosophy that is concerned with ‘what works’ and ‘what provides solutions’ in an authentic situation (Creswell, 2003, p. 11). I utilised both quantitative and qualitative techniques in the study because a combination of the two techniques provides a more in-depth knowledge of the theory and practice (Creswell, 2003).

Research methods and procedures Sample A convenience stratified sampling technique (Creswell, 2003) was used to select teachers and learners from secondary schools in a single district in South Africa, namely Gauteng West. The area was chosen because I worked in the area and had easy access to schools. Before sampling, the population of 32 secondary schools in the district was divided into types: there were nine Afrikaans-medium suburban schools, three English-medium suburban schools, three rural schools and 17 township schools, all of which were heterogeneous in respect of learners studying Mathematics and Mathematical Literacy. Thereafter, the population was sampled within each stratum; I chose eight schools: two Afrikaans-medium suburban schools, one English-medium suburban school, one rural school and four township schools. These schools were selected to ensure that all types of school in the district were represented proportionally and because I had good working relations with them. The sample also included eight Mathematics teachers and eight Mathematical Literacy teachers, one teacher from each subject (Mathematics and Mathematical Literacy) from each participating school. The teachers were selected on a voluntary basis and they granted me the right to interview them. All the Grade 10 Mathematics and Mathematical Literacy learners from each selected school were included in the sample, a total of 1046 Grade 10 learners. Moreover, I could only utilise a naturally formed group, namely learners in a classroom setup, for this research, which justifies a convenience sample (Creswell, 2003, p. 162).

Purpose of the study The purpose of this article is to establish which thinking styles are associated with learners studying either Mathematics or Mathematical Literacy, so as to develop guidelines that will contribute to the subject choice of either Mathematics or Mathematical Literacy by Grade 10 learners, and eventually to better performance in the two subjects. I established teachers’ perceptions regarding the differences in thinking styles between learners selecting Mathematics and those selecting Mathematical Literacy as a subject through a qualitative technique whilst I compared the thinking styles of learners quantitatively. The following research questions were addressed in the qualitative approach: • Which thinking styles of learners are you using to advise learners on their choice between Mathematics and Mathematical Literacy? • Which characteristics would you attribute to learners who have chosen Mathematics? • Which characteristics would you attribute to learners who have chosen Mathematical Literacy?

The 16 teachers who were interviewed were selected on the basis that they had at least one year’s teaching experience in Grade 10 Mathematics and/or Mathematical Literacy, thus ensuring that they had the necessary knowledge and

In order to compare the thinking styles of Grade 10 learners taking Mathematics and those taking Mathematical Literacy, the following hypothesis was interrogated in the quantitative approach: http://www.pythagoras.org.za

36


Page 5 of 12

Original Research

experience to teach these subjects. The teachers differed in age and both men and women were included in the sample. The teachers participated on a voluntary basis.

TABLE 3: Themes, sub-themes and codes.

In terms of the quantitative phase of the study, 1046 learners completed the questionnaire on thinking style dimensions, indicating their choice of either Mathematics or Mathematical Literacy. Of these, 56.2% (588) selected Mathematics.

Sub-theme

Codes

Characteristics of a Mathematics learner and a Mathematical Literacy learner

Mathematics learner

Interest, motivation and perseverance Hard-working, sense of duty, reliable and punctual Logical and critical thinker Basic knowledge and insight Can focus and concentrate Independent and self-discipline Cannot function in a group

Data collection: Document analysis, interviews and questionnaires

Mathematical Literacy learner

Good general knowledge and social skills Enjoy reading and research Entrepreneurial skills

Both a qualitative and a quantitative design were used to collect the data through document analysis, interviews and questionnaires (Creswell, 2003). Document analysis of the content for Grade 10 Mathematics in comparison with the content for Grade 10 Mathematical Literacy, as prescribed by the National Curriculum Statements of South Africa, was conducted to establish whether different thinking styles would be demanded in Mathematics and Mathematical Literacy. This document analysis was used to supplement the data obtained from the other methods (Bell, 1995). Unfortunately, access to learners’ written work and assessment documents was denied due to the integrated quality management system at the schools.

Poor self-image and short attention focus Lack of basic knowledge and cannot comprehend Lack of perseverance, undisciplined, non-participation and no interest in Mathematics Unorganised, untidy and not reliable

(2009) method of coding was used. According to this method, coding is a ‘heuristic exploratory problem-solving technique without specific formulas to follow’ (Saldana, 2009, p. 8) where a code in qualitative inquiry refers to a ‘word or short phrase that symbolically assigns a summative, salient, essence-capturing, and/or evocative attribute for a portion of language-based or visual data’ (p. 3). After coding was applied to the data, codes sharing the same characteristics were grouped into sub-themes. Similar sub-themes were then grouped together to form concepts or themes (Saldana, 2009). Table 3 summarises the themes, sub-themes and codes emerging from the data qualitative analysis.

I followed a qualitative approach during the first phase of the research by conducting one-to-one interviews with teachers through a semi-structured questionnaire (see Appendix 1). The aim was to ascertain their perceptions regarding the characteristics and differences in thinking styles between learners selecting Mathematics and those selecting Mathematical Literacy as a subject.

The Statistical Package for the Social Sciences, edition 15 (SPSS 15) was used in collaboration with the Statistical Consultation Service at the university concerned, to conduct the quantitative data analyses. The quantitative data analysis included univariate descriptive measures, namely frequencies and percentages and cross-tabulations of the categorical variables and descriptive statistics of the scale variables (specifically the thinking style dimensions). Inferential statistics, namely independent sample t-tests (for the scales variables) and chi-squared tests of independence (for the categorical variables), were used in order to identify significant differences between the thinking styles of learners taking Mathematics and those taking Mathematical Literacy. A significance level of 0.05 was assumed throughout. The internal reliability of each of the thinking style dimensions was determined by the Cronbach α coefficient.

During the second phase of the research, I used a quantitative research method: a survey (structured questionnaires) amongst learners. The aim of this phase was to compare thinking styles of a Mathematics learner compared to those of a Mathematical Literacy learner. The questionnaire (see Appendix 2) was based on an existing standardised instrument, the thinking style inventory of Sternberg and Wagner (1992b), which aimed to determine the different strategies used by learners to solve problems, execute tasks or projects and make decisions. The questionnaire consisted of 13 thinking style dimensions divided into five categories. For each characteristic (e.g. selfmanagement function: legislative) there were eight questions on a 1–7 point Likert scale, with 1 = Not at all well and 7 = Extremely well. Thus, for 13 thinking style dimensions, there were 104 questions. Scores were then averaged over each characteristic. The characteristic associated with each of these appears in Table 1 (Sternberg, 1994).

To analyse the research question, descriptive statistics for each of the group variables (the independent variable being Mathematics or Mathematical Literacy and the dependent variable thinking style) were used. For the purposes of classifying participants into thinking style categories, each participant’s highest score for a given category was chosen to represent the category. For example, if a learner’s scores on the 7-point Likert scale on average were for Legislative = 3.2, Judicial = 6.2 and Executive = 4.7, they would be categorised as judicial thinkers. This process was performed on all five thinking style categories.

Data analyses Tesch’s protocol of data analysis (Creswell, 1994) was used to analyse the data from the interviews for the qualitative inquiry. Firstly, each interview was audio recorded and transcribed. Secondly, the transcriptions were read to obtain a holistic perspective, after which relevant answers were separated from irrelevant answers. Thereafter, Saldana’s http://www.pythagoras.org.za

Theme

37


Page 6 of 12

Original Research

Ethical considerations

Some of the items on the original thinking style inventory could be ignored, namely the judicial, oligarchic, anarchic, global, local, internal and conservative thinking styles, because of the findings from the reliability analyses. However, due to the standardisation of the instrument, no item was deleted and the dimensions, as identified by Sternberg and Wagner (1992b), were calculated.

The ethical committee of the education department at the university concerned granted ethical clearance for the study and permission was obtained from the Gauteng DOE, Gauteng West district and the schools in Gauteng West to conduct the research. All the participants’ contributions were recognised by proper referencing. The rights and interests of the participants were protected and sensitivity was shown towards them based on common trust (Mouton, 2001). Furthermore, all information supplied was treated with confidentiality and the outcomes of the research made available on request. Tape recordings and data were kept under lock and key and were destroyed after completion of the research study.

Validity The characteristics and differences of the thinking styles of the Mathematics and Mathematical Literacy learners were measured by means of the thinking style inventory of Sternberg and Wagner (1992b), which is an existing standardised instrument that had already complied with all validity aspects. Therefore, no items were omitted and the dimensions as identified by Sternberg and Wagner were calculated.

Sternberg granted permission for the usage of the thinking style inventory of Sternberg and Wagner (1992b) and the intellectual property rights were recognised. Furthermore, data obtained was personally analysed by means of statistical verified methods and procedures (Eiselen, 2006).

Main findings

Findings from the document analysis The basic principles of numeracy laid out in the General Education band develop in Grade 10 Mathematics, using more symbolic methods, such as numeric sequences and series (DOE, 2003c). In comparison, Grade 10 Mathematical Literacy does not include number systems, numeric or geometric patterns, but focuses on using numbers within contexts relevant to daily life, such as profits and losses, budgets, loans, commission and banking (DOE, 2003b). Furthermore, Grade 10 Mathematics includes mathematical modelling, linear, exponential and quadratic equations, linear inequalities, products and factorisation, trigonometry, coordinate geometry and Euclidean geometry, which do not appear in the Grade 10 Mathematical Literacy curriculum (DOE, 2003c), as displayed in Table 2.

Reliability The questionnaires were shown to colleagues for comments and responses, to ensure that the constructs were clearly conceptualised. Consequently, the questionnaires were amended with regard to timeframes, language, terminology, readability and clarity and piloted with one class group of 30 learners at a school that was not part of the sample before they were administered to the eight schools in the sample. The purpose was to ensure coherency and consistency of the questions. The questionnaires were administered under examination conditions. The internal reliability of each thinking style dimension was determined by using the Cronbach α coefficient, after which descriptive statistics, namely averages and standard deviations, of each dimension were used. The Cronbach α coefficient is recommended for large samples where items are not scored right or wrong and was thus suitable for this study of 1046 learners. A score of 0.7 and higher was assumed as reliable for this study. The internal consistency of the five thinking style categories is presented in Table 4.

From the above, it is evident that there is more mathematical content in the Grade 10 Mathematics curriculum than in the Grade 10 Mathematical Literacy curriculum. Hence, learners who choose Mathematics are likely to do many things with a ‘hierarchic thinking style’ (Sternberg, 1990, p. 369). Mathematics requires that learners be able to think in terms of ‘symbolic representation or abstract conceptualization’ (Kolb, Boyatzis & Mainemelis, 1999, p. 3). Thus, learners should have a preference to create, discover and design (‘legislative thinking style’) (Sternberg, 1990, p. 38). This viewpoint is also supported by Bohlmann and Pretorius (2008, p. 43), who claimed that ‘the conceptual complexity and problem-solving nature of Mathematics make extensive demands on the reasoning, interpretive and strategic skills of learners’.

TABLE 4: Internal reliability of the thinking styles dimensions. Category

Style

No. of items

Cronbach α

Functions

Legislative

8

0.736

Executive

8

0.786

Judicial

8

0.672

Monarchic

8

0.787

Hierarchic

8

0.780

Oligarchic

8

0.596

Anarchic

8

0.675

Global

8

0.617

Local

7

0.695

Levels

Internal

8

0.666

External

8

0.767

Learning

Liberal

8

0.739

Conservative

8

0.684

Forms

Scope


Grade 10 Mathematical Literacy focuses more on contexts relevant to daily life. Thus, learners who prefer to work with other people, focus on the outside and are interdependent, with an ‘external thinking style’ (Sternberg, 1990, p. 38) should rather choose Mathematical Literacy. 38


Page 7 of 12

Original Research

Findings from the interviews

Mathematics, firstly, as being interested in the subject. Sam explained that

From the 16 personal interviews conducted with teachers, I could find no evidence that teachers consider the thinking styles of learners when they advise learners on their choice between Mathematics and Mathematical Literacy. Rather, teachers indicated that they use three other methods. In the following protocols the names of teachers are pseudonyms to protect their identity. All the protocols are from Spangenberg (2008, pp. 229–242).

it will depend … on the interest of the learner. Maybe if he likes working with numbers, he can choose the Mathematics … but the learners that choose Maths … you see that these learners are interested in the subject.

These learners are perceived as self-disciplined and diligent. Santa mentioned that ‘the learner must be dedicated’, supported by Shisha that ‘they tend to be the more conscientious student’. Christine argues further that Mathematics learners are motivated and focused. She noted that ‘I would choose a learner who’s able to focus … to concentrate’.

Firstly, learners’ marks obtained in Grade 9 are used as an indication of which subject to take. Matle mentioned that he is guided by ‘the mark that the learner obtained in Grade 9’. Bana noted that ‘if you performing poor in natural sciences you can see that you are not going to do Maths. You are going to do Maths Literacy’.

Zane added that these learners are hardworking by commending that ‘most of them are quite conscientious workers, enthusiastic workers’ and Carmen supported that ‘those that choose Mathematics tend to be those that are very hard workers’.

Samuel added: [I]f he gets 60% in Grade 9 and above then basically you are allowed to choose Maths. Anything less than 60% you wouldn’t have a choice … That is like a policy in our school at the moment in time. (Spangenberg, 2008, p. 229)

Teachers also described Mathematics learners as having the ability to memorise in a logical manner. Geoff claimed that Mathematics is for the brainy … that there are those that are very intelligent and as a result they need to do Mathematics, because they can think far to their ability and then Mathematical Literacy is like it’s made for those who are less able to do Literacy … but those who are very intelligent, they have to do Mathematics.

Secondly, tests guide teachers when advising learners in their subject choices. Jack stated that ‘we give them aptitude test in terms of Mathematics … If he pass … we just place him or her’ and Mary alluded to ‘some test that they do to test their ability to do Maths as it test their ability to do Maths Lit … we set an internal Maths paper and we use that as a guide’.

Shisha add that ‘they are able to work logically’ and noted that ‘they understand theorems and them they are able to apply them immediately’. Lastly, teachers described learners taking Mathematics as independent workers, able to work on their own, thus displaying an ‘internal thinking style’ (Sternberg, 1990, p. 38). Sam stated that Mathematics learners are ‘very independent … they can work on their own … they are very disciplined learners who are taking Maths seriously’.

Lastly, subject combination packages and future careers are also indicators of the subjects learners should take. Rosen claimed that ‘if a learner chooses a subject package, then the package makes provision for him in a certain direction’, adding that for the science then we have included the pure maths in that package and if it has accounting we have include the pure maths in the accounting package and in all the other courses whereby the learner has the commerce fields or learning areas we have included the Maths Literacy in that package.

On the other hand, teachers characterised Mathematical Literacy learners as having good general knowledge. Samuel noted that for Mathematical Literacy learners ‘the most important thing is the knowledge of the outside environment that they are in’. He further continued that these learners are socially adaptable, by arguing that

Jack referred to four streams: The first steam is for Maths and Science. The second one is Maths and Accounting. The third one is Maths Literacy and Economics. The fourth one is Maths Literacy and History … this learners who are doing Maths Literacy, most of them, they must consider this career opportunities of law, human resources, those that are not attached with Mathematics.

how you can be able to adapt in your everyday life and how you adapt in the environment you in … You don’t have to be a good academic learner to be a good social person being … you just need to be well equipped to handle everyday experiences, have the grasp or basically knowing what’s happening around you and being interesting.

Bana explained that:

Ilze described Mathematical Literacy learners as entrepreneurial, perceiving a Mathematical Literacy learner as ‘a child who can stand on his own feet, a child who wants to start his own business’. She further added that these learners have an interested in life and people, noting that ‘you learn him about the life’ and ‘more interested in the human being’. Shisha mentioned that Mathematical Literacy learners are able to express themselves, thus displaying an ‘external thinking style’ (Sternberg, 1990, p. 38) by claiming that ‘a learner must be open-minded. Usually we’re using the open-minded, especially the history learners, because they

I look at his or her ambitions, whether which career does he or she want to follow. If she wants to be a scientist, then I say okay Maths is good for you. If you want to be a lawyer I say Maths Literacy is good for you.

In addition, Sam argued that ‘some of the learners choose it because of the career that they want to go in’. Teachers could clearly distinguish between the characteristics of each type of learner. From their observations and perceptions, teachers described learners who choose http://www.pythagoras.org.za

39


Page 8 of 12

Original Research

and Matle agreed: ‘fear … many learners they’ve got this mentality that Mathematics is a difficult subject … there is a possibility that I will fail.’

are used to expressing themselves’. In contrast, however, other teachers described these learners as lacking discipline. Hannah perceived a Mathematical Literacy learner as ‘a guy who does not have discipline’ and Bana added that ‘they are not willing to learn. They are not willing to participating in classes’.

It is, however, important to note that the above-mentioned findings were based on teachers’ perceptions regarding the characteristics and differences in thinking styles between learners selecting Mathematics and those selecting Mathematical Literacy as a subject. The findings should thus not be generalised to all Grade 10 Mathematics learners or Grade 10 Mathematical Literacy learners.

Hannah continued to describe a Mathematical Literacy learner as having a short attention span, by noting that ‘he disappears in class, he loses concentration. He does also not have the ability to concentrate’ and Carmen argued that Mathematical Literacy learners lack an interest in Mathematics, by stating that Mathematical Literacy learners are ‘those that for any reason don’t like Mathematics. They haven’t enjoyed Mathematics’. Zane agreed that a Mathematical Literacy learner is

Findings from questionnaires Statistical analysis was undertaken to investigate whether differences between learners’ thinking style dimensions and which subject they are studying (Mathematics or Mathematical Literacy) were significant. The precise means (X) and standard deviations (SD) pertaining to the thinking style dimensions (dependant variable) of learners taking Mathematics and Mathematical Literacy (independent variable) (according to the questionnaires constructed for this study) are indicated in Table 5.

one who just only doesn’t want to choose Maths, but he is forced to do something in the Maths field now, because it is compulsory. So, he chooses Maths Literacy.

Lastly, teachers described Mathematical Literacy learners as having a fear of Mathematics. Rosen mentioned that ‘some learners will choose it because of their fear for pure Maths’

The findings support the hypothesis about the comparisons between learners’ thinking style dimensions and which subject they are studying (Mathematics or Mathematical Literacy). Table 6 shows the value of the test statistics in terms of the null hypothesis (t), the p-value for each case and the degree of freedom (df) where the variances were accepted.

TABLE 5: Descriptive statistics pertaining to the thinking style dimensions of learners taking Mathematics and Mathematical Literacy. Category

Functions

Forms


Thinking style

Mathematics (N = 588) X SD

Mathematical Literacy (N = 460) X SD

Legislative

5.3095

0.88738

5.0713

0.95301

Executive

5.1949

0.95755

4.8973

0.98888

Judicial

4.6395

0.82793

4.4194

0.88665

Monarchic

4.6256

1.10803

4.6441

0.98406

Hierarchic

5.1173

0.93658

4.8001

0.95987

Oligarchic

4.6951

0.85636

4.5975

0.89014

Anarchic

4.9031

0.88603

4.7869

0.94682

Global

4.3345

0.86228

4.2997

0.85161

Local

4.7901

0.87187

4.5342

0.96353

Internal

4.7010

0.93962

4.4961

0.98286

External

4.9173

1.05198

4.8156

1.02982

Liberal

5.5013

0.94377

4.8691

0.97416

Conservative

4.4627

0.96726

4.3951

0.97027

Mean score

4.86092

0.93051

4.66357

0.95244

Learners studying Mathematics and those studying Mathematical Literacy differed significantly at a 95% level in terms of eight of the 13 Sternberg thinking style dimensions, namely legislative (p = 0.000 < 0.05), executive (p = 0.000 < 0.05), judicial (p = 0.000 < 0.05), hierarchic (p = 0.000 < 0.05), anarchic (p = 0.043 < 0.05), local (p = 0.000 < 0.05), internal (p = 0.001 < 0.05 and liberal (p = 0.002 < 0.05). It was found that Mathematics learners (average = 5.1949) are more likely than Mathematical Literacy learners (average = 4.8973) to like following instructions, to do whatever they are told to do and to prefer fixed structures (executive thinking style). Mathematics learners (average = 5.3095) are also more likely than Mathematical Literacy learners (average = 5.0713) to design and do things in their own ways (legislative thinking style). Furthermore, Mathematics learners (average = 4.6395) are also more likely than Mathematical Literacy learners (average = 4.4194) to be critical, in the sense that they like to judge people and to evaluate things (judicial thinking style).

N, total number of learners; X, precise means.

TABLE 6: The independent sample test of learners taking Mathematics and Mathematical Literacy and their different thinking styles. Category

Thinking style

t

p

df

Functions

Legislative

4.175

0.000

1046

Executive

4.921

0.000

1046

Judicial

4.139

0.000

1046

Monarchic

-0.267

0.789

1046

Hierarchic

5.382

0.000

1 046

Oligarchic

1.799

0.072

1046

Anarchic

2.044

0.041

1046

Global

0.651

0.515

1046

Local

4.501

0.000

1046

Forms


Internal

3.433

0.001

1046

External

1.567

0.117

1046

Liberal

3.058

0.002

1046

Conservative

1.121

0.262

1046

Although the Mathematics learners on average obtained a slightly higher score than the Mathematical Literacy learners in terms of each of the three above styles, it was found that the legislative thinking style (average = 5.3095) on average measured the highest for the Mathematics group, followed by the executive thinking style (average = 5.1949), and lastly by the judicial thinking style (average = 4.6395). Even though

p, probability value. Correlation is significant at the 0.05 level.


40


Page 9 of 12

Original Research

Both the interviews with teachers and the survey of learners revealed differences between the two groups as far as characteristics and thinking styles are concerned. Even though the teachers’ reflections on the difference between Mathematics learners’ and Mathematical Literacy learners’ characteristics suggest broad distinctions, these are not clearly mirrored in the learners’ responses. Furthermore, the differences between Mathematics and Mathematical Literacy, as identified from the curriculum analysis, may not directly link to pedagogy and thinking styles. However, learners’ thinking styles could be taken into consideration, as a guideline, when advising learners regarding their subject choices.

these findings appear contradictory, it is important to note that, according to Sternberg (1990), the mind performs each of the legislative, executive and judicial functions, but one of these tends to be more dominant in a person. In terms of the four forms of cognitive self-management distinguished by Sternberg (1990), namely hierarchic, anarchic, monarchic and oligarchic, the two groups only differed significantly in terms of the first two: Mathematics learners (average = 5.1173) are more likely than Mathematical Literacy learners (average = 4.8001) to do many things at the same time and to set priorities pertaining to what to do, at what time to do it and how much time and energy to spend on it (hierarchic thinking style). In contrast, Mathematical Literacy learners (average = 4.6441) are more likely than Mathematics learners (average = 4.6256) to do one thing at a time and spend almost all their energy and resources on it (monarchic thinking style).

Both learners and teachers should be more aware of thinking styles if they are to make the right subject choice and thus minimise switching between subjects. If learners understand their thinking styles and how these match either with Mathematics or Mathematical Literacy, they are more likely to select the appropriate subject. Also, the pressure on teachers who have to deal with larger classes due to subject changes later during a year will decrease and they will not have to re-teach subject content to learners which switch subjects.

In terms of the two levels of self-management, namely local and global, it was found that Mathematics learners (average = 4.7901) are more likely than Mathematical Literacy learners (average = 4.5342) to find detail, specifications and concrete examples to be important (local thinking style). As far as the scope of self-management is concerned, described by Sternberg (1990) as being either internal or external, it was found that a Mathematics learner (average = 4.7010) is more likely than a Mathematical Literacy learner (average = 4.4961) to prefer to work alone, to focus inward and to be independent (internal thinking style).

Based on the findings of this research, further research is required to develop a quantitative instrument to capture the backgrounds and thinking styles of Grade 9 learners to enable schools to provide learners with the necessary information to make an informed choice. In particular, the following information should be included in an instrument of this nature:

Lastly, in terms of the distinction made by Grigorenko and Sternberg (1997) between two ways of learning, namely liberal and conservative, it was found that a Mathematics learner (average = 5.5013) is more likely than a Mathematical Literacy learner (average = 4.8691) to do things in new ways and to deviate from traditions (liberal thinking style).

• age • number of times retained in a grade • perceptions of the quality of tuition they received in Mathematics in the past • Grade 9 marks in Mathematics • subject choice in Grades 10−12 (excluding Mathematics of Mathematical Literacy). The findings also suggest that a shorter edition of the thinking style inventory (Sternberg & Wagner, 1992b) should be used. In particular, only the legislative, executive, judicial, hierarchic, anarchic, local, internal and liberal dimensions of the inventory need to be measured, as this study found differences in these thinking styles between learners taking Mathematics and those taking Mathematical Literacy.

The findings did not correspond with those of Sternberg and Grigorenko (1993), who found that the judicial and legislative styles correlated positively to academic achievement, whereas the executive, legislative, oligarchic and liberal styles tended to correlate negatively to academic success. It is, however, important to note that thinking styles are interrelated (Garcia & Hughes, 2000, p. 413). One must take into account that thinking style interrelationship is complex, since it is influenced by many variables, such as education, subject, age and gender (Sternberg & Wagner, 1992a), which should be researched in depth.

Given the localised nature of this study, namely a single district in a single province in South Africa, the findings obtained should be confirmed through similar studies of this nature in other provinces and districts. In this way, a better understanding of the differences, both cognitive and noncognitive, between Mathematics learners and Mathematical Literacy learners can be obtained.

Discussion This study characterised and compared the thinking styles of Grade 10 learners taking Mathematics and those taking Mathematical Literacy. It could not, however, find any evidence that teachers use thinking styles of learners to advise learners on their choice between Mathematics and Mathematical Literacy in the Gauteng West district. http://www.pythagoras.org.za

The ways in which empirical realities manifest are much more complex than the broad groupings pointed to in the literature and curriculum analysis in this article. Further research in this regard should also be conducted. 41


Page 10 of 12

Original Research

Conclusion

References

The article focused on the characterisation and comparison of thinking styles of learners studying Mathematics and those studying Mathematical Literacy. This was an extract from a broader study on the placement of Grade 10 learners in either subject (Spangenberg, 2008). The aim was to establish which thinking styles are associated with learners studying these subjects, so as to develop guidelines that will contribute to the subject choice of either Mathematics or Mathematical Literacy by Grade 10 learners, and eventually to better performance in the two subjects. It was found that there is a relationship between learners’ thinking style dimensions and which subject they are studying. Mathematics learners are more likely than Mathematical Literacy learners to execute instructions, to design and do things in their own way and to be critical. Where Mathematics learners are more likely to do many things at the same time and to set priorities pertaining to what to do, at what time to do it and how much time and energy to spend on it, Mathematical Literacy learners are more likely to do one thing at a time and spend almost all their energy and resources on it. This information about the thinking styles of learners could be used to help place learners more appropriately and possibly reduce the number of learners who make inappropriate choices. Also, this will ease pressure on teachers who have to deal with larger classes due to subject changes later during a year and may have to re-teach subject content. More learners will gain university exemption. In support, Borromeo Ferri (2004, p. 2) argued that thinking styles should ‘not be viewed as being unchangeable, but they may change depending on time, environment and life demands’.

Bansilal, S., Mkhwanazi, T., & Mahlabela, P. (2012). Mathematical Literacy teachers’ engagement with contextual tasks based on personal finance. Perspectives in Education, 30(3), 98−109. Bell, J. (1995). Doing your research project: A guide for first-time researchers in education and social science. Buckingham: Open University Press. Bohlmann, C., & Pretorius, E. (2008). Relationships between Mathematics and Literacy: Exploring some underlying factors. Pythagoras, 67, 42−44. http://dx.doi.org/ 10.4102/pythagoras.v0i67.73 Borromeo Ferri, R. (2004). Mathematical thinking styles – An empirical study. Available from http://www.erzwiss.uni-hamburg.de/Personal/Gkaiser/pdf-dok/borrom2.pdf Cilliers, C.D., & Sternberg, R.J. (2001). Thinking styles: Implications for optimising learning and teaching in university education. South African Journal of Higher Education, 15(1), 13−24. http://dx.doi.org/10.4314/sajhe.v15i1.25375 Creswell, J.W. (1994). Research design: Qualitative and quantitative approaches. Thousand Oaks, CA: Sage. Creswell, J.W. (2003). Research design: Qualitative, quantitative and mixed methods approaches. (2nd edn.). Thousand Oaks, CA: Sage. De Boer, A, & Bothma, T.J.D. (2003, June). Thinking styles and their role in teaching Library and Information Science. Paper presented at the 24th Annual Conference of the International Association of Technological University Libraries, Ankara, Turkey. Department of Education (2003a). Qualifications and assessment policy framework Grades 10–12 (General). Pretoria: DOE. Department of Education (2003b). National curriculum statement. Grade 10–12 (General) Mathematical Literacy. Pretoria: DOE. Department of Education (2003c). National curriculum statement. Grade 10–12 (General) Mathematics. Pretoria: DOE. Department of Basic Education. (2011a). Curriculum and assessment policy statement. Grades 10–12. Mathematical Literacy. Pretoria: DBE. Available from http://www. education.gov.za/LinkClick.aspx?fileticket=jB%2fjGQ35UgI%3d&tabid=420&m id=1216 Department of Basic Education. (2011b). Curriculum and assessment policy statement. Grades 10–12. Mathematics. Pretoria: DBE. Available from http://www.education. gov.za/LinkClick.aspx?fileticket=QPqC7QbX75w%3d&tabid=420&mid=1216 Eiselen, R.J. (2006). Predicting achievement in Mathematics at tertiary level. Unpublished doctoral dissertation. University of Johannesburg, Johannesburg, South Africa. Available from http://hdl.handle.net/10210/547 Frith, V., & Prince, R. (2006). Reflections on the role of a research task for teacher education in data handling in a Mathematical Literacy education course. Pythagoras, 64, 52−61. http://dx.doi.org/10.4102/pythagoras.v0i64.99 Gal, I. (2009). South Africa’s Mathematical Literacy and Mathematics curricula: Is probability literacy given a fair chance? African Journal of Research in Mathematics, Science and Technology Education, 13(1), 50−61.

In conclusion, now that access to education and the right to learning have been established for most learners in South Africa, the time is ripe to set key priorities for the country’s future. There is an urgent need to increase the number of learners with sufficient and well-established mathematical knowledge and skills, and so enable them to progress in the short, medium and long term to higher education, the business world and industry. There is a great demand for teachers in Mathematics and Mathematical Literacy to equip learners with the necessary knowledge and skills.

Garcia, F.C., & Hughes, E.H. (2000). Learning and thinking style: An analysis of their interrelationship and influence on academic achievement. Education Psychology, 20(4), 413−430. http://dx.doi.org/10.1080/713663755 Graven, M., & Venkat, H. (2007). Emerging pedagogic agendas in the teaching of Mathematical Literacy. African Journal of Research in Mathematics, Science and Technology Education, 11(2), 67−84. Grigorenko, E.L., & Sternberg, R.J. (1988). Mental self-government: A theory of intellectual style and their development. Human Development, 31, 197−224. http://dx.doi.org/10.1159/000275810 Grigorenko, E.L., & Sternberg, R.J. (1997). Are cognitive styles still in style? American Psychologist, 52(7), 700−710. http://dx.doi.org/10.1037/0003-066X.52.7.700 Kolb, D.A., Boyatzis, R.E., & Mainemelis, C. (1999). Experiential learning theory: Previous research and new directions. Cleveland, OH: Case Western Reserve University.

Acknowledgements

Mouton, J. (2001). How to succeed in your master’s & doctoral studies. Pretoria: Van Schaik Publishers.

I am grateful to the participants and the authorities of the participating schools for their cooperation. A special word of thanks to my supervisors for their professional assistance, guidance and support, STATKON at the University of Johannesburg for professional consultation and analyses of the data, and the Gauteng Department of Education for permitting the research in the schools. Also, many thanks to my colleagues who read the draft manuscript and made comments. The study was funded by the University of Johannesburg.

Moutsios-Rentzos, A., & Simpson, A. (2010). The thinking styles of university mathematics students. Acta Didactica Napocensia, 3(4), 1−10. Available from http://dppd. ubbcluj.ro/adn/article_3_4_1.pdf Richmond, A.S., Krank, H.M., & Cummings, R. (2006). A brief research report: Thinking styles of online distance education students. International Journal of Technology in Teaching and Learning, 2(1), 58−64. Saldana, J. (2009). The coding manual for qualitative researchers. London: Sage. Soanes, C. (Ed.). (2002). Pocket Oxford English Dictionary. (9th edn.). New York, NY: Oxford University Press. Spangenberg, E.D. (2008). Riglyne vir die plasing van leerders in Wiskunde of Wiskundige Geletterdheid [Guidelines for placing learners in Mathematics or Mathematical Literacy]. Unpublished doctoral dissertation. University of Johannesburg, Johannesburg, South Africa. Available from http://hdl.handle. net/10210/3332

Competing interest

Sternberg, R.J. (1990). Thinking styles: Keys to understanding student performance. Phi Delta Kappan, 71(5), 366−371. Available from http://www.jstor.org/stable/20404156

I declare that I have no financial or personal relationship(s) which may have inappropriately influenced me in writing this article.

Sternberg, R.J. (1994). Allowing for thinking styles. Educational Leadership, 52(3), 36−40.


Sternberg, R.J. (1997). Thinking styles. London: Cambridge University Press. http:// dx.doi.org/10.1017/CBO9780511584152

42


Page 11 of 12

Original Research

Sternberg, R.J., & Grigorenko, E.L. (1993). Thinking styles and the gifted. Roeper Review, 16(2), 122–130. http://dx.doi.org/10.1080/02783199309553555

Venkat, H., Graven, M., Lampen, E., Nalube, P., & Chitera, N. (2009). Reasoning and reflecting in Mathematical Literacy. Learning and Teaching Mathematics, 7, 47−53. Available from http://www.amesa.org.za/amesal_n7_a13.pdf

Sternberg, R.J., & Wagner, R.K. (1992a). Tacit knowledge: An unspoken key to managerial success. Creativity and Innovation Management, 1, 5–13. http:// dx.doi.org/10.1111/j.1467-8691.1992.tb00016.x

Zhang, L. (2002). Thinking styles and cognitive development. Journal of Genetic Psychology, 162(2), 179−195. http://dx.doi.org/10.1080/00221320209598676, PMid:12095088

Sternberg, R.J., & Wagner, R.K. (1992b). Thinking styles inventory. Unpublished test. New Haven, CT: Yale University.

Zhang, L. (2006). Does student-teacher thinking style match/mismatch matter in students’ achievement? Education Psychology, 26(3), 395−409. http://dx.doi. org/10.1080/01443410500341262

Van der Walt, M.S. (2008). Aanpassing van die studie-oriëntasievraelys in Wiskunde vir gebruik in die intermediêre fase [Adaptation of the study orientation questionnaire in Mathematics for use in the Intermediate Phase]. Unpublished doctoral dissertation. North-West University, Potchefstroom, South Africa. Available from http://hdl.handle.net/10394/2069

Zhang, L., & Sternberg, R.J. (2000). Are learning approaches and thinking styles related? A study in two Chinese populations. Journal of Psychology, 137, 469−489. http://dx.doi.org/10.1080/00223980009598230, PMid:11034129 Zhu, C. (2011). Thinking styles and conceptions of creativity among university students. Educational Psychology, 31(3), 361−375. http://dx.doi.org/10.1080/01 443410.2011.557044

Venkat, H. (2007). Mathematical Literacy – mathematics and/or literacy: What is being sought? Pythagoras, 66, 76−84. http://dx.doi.org/10.4102/pythagoras.v0i66.82

Appendix starts on next page →


43


Page 12 of 12

Original Research

APPENDIX 1

Teachers’ interview questions 1. Which criteria do you use to advise a learner in choosing between Mathematics and Mathematical Literacy? 2. Which method(s) or criteria does your school use to place Grade 10 learners in either Mathematics or Mathematical Literacy? 3.1 Which other factors influence the placement of learners in Mathematics or Mathematical Literacy? 3.2 Can you motivate why you made that statement? 4. Which thinking styles of learners do you use to advise learners on their choice between Mathematics and Mathematical Literacy? 5. Which characteristics would you attribute to learners that have chosen Mathematics? 6. Which characteristics would you attribute to learners that have chosen Mathematical Literacy? 7. Is there anything that you wish to add with regard to the placement oflearners in Mathematics or Mathematical Literacy?

APPENDIX 2

Sample of learners’ thinking style questionnaire Circle the number that best describes the way you do things. Use the following code: 1

2

3

4

5

6

7

Not at all well

Not very well

Slightly well

Somewhat well

Well

Very well

Extremely well

1. When discussing or writing down ideas, I like criticising others’ way of doing things.

1

2

3

4

5

6

7

2. I prefer to deal with specific problems rather than with general questions.

1

2

3

4

5

6

7

3. I enjoy working on projects that allow me to try novel ways of doing things.

1

2

3

4

5

6

7

4. When making decisions, I tend to rely on my own ideas and ways of doing things.

1

2

3

4

5

6

7

5. When discussing or writing down ideas, I follow formal rules of presentation.

1

2

3

4

5

6

7

6. When talking or writing about ideas, I stick to one main idea.

1

2

3

4

5

6

7

7. When starting a task, I like to brainstorm ideas with friends or peers.

1

2

3

4

5

6

7

8. I tend to base my decisions only on concerns important to my group or peers.

1

2

3

4

5

6

7

10. I like to set priorities for the things I need to do before I start doing them.

1

2

3

4

5

6

7

11. I like situations or tasks in which I am not concerned with details.

1

2

3

4

5

6

7

17. I like to control all phases of a project, without having to consult with others.

1

2

3

4

5

6

7

20. I like to do things in ways that have been used in the past.

1

2

3

4

5

6

7

27. I stick to standard rules or ways of doing things.

1

2

3

4

5

6

7

Source: Sternberg, R.J., & Wagner, R.K. (1992b). Thinking styles inventory. Unpublished test. New Haven, CT: Yale University


44


Page 1 of 9

Original Research

Pictorial pattern generalisation: Tension between local and global visualisation Author: Duncan Samson1 Affiliation: 1 Education Department, Rhodes University, South Africa Correspondence to: Duncan Samson Email: [email protected] Postal address: PO Box 94, Grahamstown 6140, South Africa Dates: Received: 16 Apr. 2012 Accepted: 26 Aug. 2012 Published: 26 Oct. 2012 How to cite this article: Samson, D. (2012). Pictorial pattern generalisation: Tension between local and global visualisation. Pythagoras, 33(3), Art. #172, 9 pages. http://dx.doi. org/10.4102/pythagoras. v33i3.172

This article engages with the notion of local and global visualisation within the context of figural pattern generalisation. The study centred on an analysis of pupils’ lived experience whilst engaged in the generalisation of linear sequences presented in a pictorial context. The study was anchored within the interpretive paradigm of qualitative research and made use of the complementary theoretical perspectives of enactivism and knowledge objectification. A crucial aspect of the analytical framework used was the sensitivity it showed to the visual, phenomenological and semiotic aspects of figural pattern generalisation. A microanalysis of a vignette is presented to illustrate the subtle underlying tensions that can exist as pupils engage with pictorial pattern generalisation tasks. It is the central thesis of this article that the process of objectifying and articulating an appropriate algebraic expression for the general term of a pictorial sequence is complicated when tension exists between local and global visualisation.

Introduction The use of number patterns, specifically pictorial or figural number patterns, has been advocated by numerous mathematics educators as a didactic approach to the introduction of algebra and as a means of promoting algebraic reasoning and supporting the fundamental mathematical processes of generalisation and justification (e.g. De Jager, 2004; Mason, Graham, Pimm & Gowar, 1985; Pegg & Redden, 1990; Walkowiak, 2010). Number patterns presented as a sequence of pictorial terms have the potential to open up meaningful spaces for classroom exploration and discussion. However, despite the potential richness of such pictorial contexts, potentially meaningful pattern generalisation activities carried out in the classroom often become degraded to simple rote exercises in which the given pictorial sequence is simply reduced to an equivalent numeric sequence. As such, the generalisation process becomes a somewhat superficial or mechanistic exercise using set algorithmic methods. Whilst such an approach may well be successful in arriving at the correct general formula, the potential for genuine mathematical exploration offered by the context of the question is largely lost, with the generalisation process becoming ‘an activity in its own right and not a means through which insights are gained into the original mathematical situation’ (Hewitt, 1992, p. 7). As Thornton (2001) remarks, the danger with such an approach is that the focus becomes ‘… the development of an algebraic relationship, rather than the development of a sense of generality’ (p. 252), the result being little more than disconnected algebraic formulation. This disconnection from the original context becomes particularly problematic when importance is placed on the justification or validation of the general rule, since algebraic expressions arrived at through this process become ‘statements about the results rather than the mathematical situation from which they came’ (Hewitt, 1992, p. 7). This is unfortunate since not only can generalisation of pictorial patterns lead to different but algebraically equivalent expressions of generality, thereby opening up excellent opportunities to engage with the notion of algebraic equivalence, but, as Orton (2004, p. 114) points out, justifying pattern generalisations provides pupils with legitimate and valuable experiences of proof en route to more formal mathematical proofs. Number patterns presented in the form of a sequence of pictorial terms are thus far more than simply a visual representation of a given numeric pattern. In essence, the use of a pictorial context aims to exploit the visual decoding of the pictorial sequence to give meaning to the symbolic expressions constructed. Two critical aspects of this process are the ability not only to grasp in a meaningful way the perceived underlying structure of the pictorial context, but also the ability to use this structure to articulate a direct expression for the general term of the sequence.


Theoretical background

Visualisation and figural apprehension As Duval (2006, p. 116) succinctly notes, there are many different ways of seeing. Consider the simple geometrical figure composed of a number of line segments (Figure 1).


45


Page 2 of 9

Original Research

intrinsically contained within the image, simply waiting to be extracted or noticed by an observer. Rather, such features are seen to co-emerge from the interaction of a perceiver and the given figural context. A further dimension is added to the visual image shown in Figure 2 when the pictorial sequence is used as a referential context for finding an algebraic expression for the number of lines in the nth diagram (or term) of the sequence. Although the context remains the same as that represented in Figure 2, the process of generalisation provides a further layer of complexity as it necessitates not only the perception of the figure within the context of a sequence of similar figures, but it requires the perception of generality, the notion that the figures in the sequence have a related structure. Finally, there is a requirement that this perceived generality must be articulated in such a way that it can be written in the form of an algebraic expression. A critical aspect of the visualisation process thus relates to the usefulness or meaningfulness of the perceived structure of the image in terms of the extent to which this perceived structure supports or hinders the process of generalisation.

FIGURE 1: A simple geometrical figure.

FIGURE 1: A simple geometrical figure.

Pupils visually interacting with the figure have the potential to perceive it in a number of different ways. In the same vein, a single pupil may be able to perceive the figure in multiple ways. If we take Figure 1 as an example, the figure could be perceived as comprising two overlapping Hs. Alternatively, it could be seen to comprise four overlapping squares, where the ‘lids’ of the top two squares and the ‘bases’ of the bottom two squares are missing. A third possibility is for the figure to be perceived as comprising three vertical lines with two shorter interconnecting horizontal lines. Yet another possibility is for the figure to be seen as a ‘+’ symbol contained within two sets of vertical lines, one on either side.

Drawing on the nomenclature used by Fischbein (1993), figures such as the image shown in Figure 1 could be said to contain figural properties. What one sees in the image is a result of the Gestalt laws of figural organisation (Helson, 1933; Katz, 1951; Wertheimer, 1938; Zusne, 1970, pp. 111−135). Images such as that shown in Figure 2 could be said to contain both figural properties and conceptual qualities. What one sees in the image is still based on the Gestalt laws of figural organisation, but this is further influenced by the additional conceptual qualities of the image that have been added by virtue of the image being contextualised, in this case within a sequence of similar images. The critical point here is that an underlying tension is likely to pervade visual strategies applied to pictorial pattern generalisation tasks as a result of the relationship between the figural properties and conceptual qualities of the given images.

If we now take the same image but place it in the context of a pictorial sequence, then the visual scenario is profoundly altered (Figure 2). The middle diagram in Figure 2 is identical to the image shown in Figure 1. However, the contextual setting is no longer simply that of a geometrical figure. There is now a tacit suggestion that the image is part of a larger context, that it is part of a sequence of visually or structurally similar images. Given this new context, and the associated yet implicit sense of sequential growth, the middle diagram in Figure 2 could be perceived as a vertical line on the left followed by two sideways T-shapes. Alternatively, it could possibly be perceived in terms of a horizontal ‘backbone’ with vertical lines extending off it in two directions, upwards and downwards. The previous four visualisations are of course still possible, but the added context provides additional/ alternative perceptual features to be made apparent. Thus, by modifying the context, different ways of perceiving the figure are brought forth. On a note of clarity, use of the word ‘feature’ is not meant to imply that such features (or structures) are

Local versus global visualisation Visual approaches to pictorial pattern generalisation can be divided into two broad categories. The first category, which I previously termed local visualisation (Samson, 2011a), incorporates those visual strategies that are characterised by the foregrounding of the local additive unit – that is the structural unit which is added to (or inserted into) a given pictorial term in order to form the next term in the sequence.

FIGURE 2: A simple pictorial sequence.

FIGURE 2: A simple pictorial sequence. http://www.pythagoras.org.za

46


Page 3 of 9

Original Research

Shape 3

Shape 5

FIGURE 3: Term 3 and Term 5 of a typical pictorial sequence.

FIGURE 3: Term 3 and Term 5 of a typical pictorial sequence.

Tn  1 3n

Shape 5 Shape 5

Tn  1 3n

and Term 5 of a typical pictorial sequence.

Tn T 4  (n  1)  3  1 3n n

Shape 5 FIGURE 4: Local visualisation of Term 5.

FIGURE 4: Local visualisation of Term 5. Shape 5

horizontal TnT 4 1  (n3n 1)  3 matches n

FIGURE 4: Local visualisation of Term 5.

Tn  4  (nn +11)vertical  3 matches

Shape 5

FIGURE 4: Local visualisation of Term 5. horizontal Tn  4  (n  1)  3 matches

Tn  2n  (n  1)

Tn  1 3n

FIGURE 4: Local visualisation of Term 5. Shape Shape 33

Shape 5

Tn  1 3n Shape 5

FIGURE 3: Term 3 and Term 5 of a typical pictorial sequence. Shape 5

4-match Tn  4  (n  1)  3 squares

FIGURE 4: Local visualisation of Term 5. Tn  4  (n - 11)overlaps 3

Tn  4n  (n  1)

FIGURE 5: Global visualisation of Term 3.

FIGURE 4:FIGURE Local visualisation of visualisation Term 5. 5: Global of Term 3.

in the form of a backwards C-shape, the visual additive unit. The visual deconstruction of the pictorial context based on this local visualisation could be based on either (1) an initial starting match and n multiples of three matches in the form of a backwards C-shape (the additive unit), yielding the general formula Tn = 1 + 3n, or (2) a 4-match constant followed by (n − 1) multiples of the visual additive unit, yielding the general formula Tn = 4 + (n − 1) × 3. Both of these scenarios are illustrated in Figure 4.

This focus on the structural additive unit represents an iterative or recursive process of visual reasoning. The second category, which I previously termed global visualisation (Samson, 2011a), incorporates those visual strategies characterised by a more holistic or global view, where each term of the pictorial context is seen in terms of a generalised structure that does not make use of the iterative addition of the additive unit. By way of example, consider Figure 3, which shows Term 3 and Term 5 of a pictorial sequence1.

By contrast, global visualisation is characterised by a more holistic view that does not make use of the iterative addition of the additive unit. Rather, each pictorial shape is visualised in terms of a generalised structure.

Using a local visualisation, one could reason that to get from one term to the next requires the addition of three matches

By way of example (see Figure 5), one could subdivide each term of the given pictorial context into an upper row of n

1.On a note of clarity, the expressions ‘shape’ and ‘term’ are used synonymously and interchangeably throughout the text. Both expressions refer to the independent variable (i.e. the position of the term) in a sequence.


47


Page 4 of 9

Original Research

horizontal matches, a lower row of n horizontal matches, and a central row of n + 1 vertical matches, thus yielding the general formula Tn = 2n + (n + 1). Alternatively, each term could be seen to contain n overlapping squares, each made up of four matches, thus giving a count of 4n matches. However, this would result an overcount because there are n − 1 overlaps. Correcting for this overcount gives the final formula Tn = 4n − (n − 1). These two different global visualisations are illustrated in Figure 5.

fundamental role in the formation of knowledge (Radford, 2005a, p. 142). Secondly, in order to study the process of knowledge production one needs to pay close scrutiny to multiple means of objectification, for example words, linguistic devices, gestures, rhythm, graphics and the use of artefacts, where ‘…meaning is forged out of the interplay of various semiotic systems’ (Radford, 2005b, p. 144). It is through this multi-systemic, semiotic-mediated activity that the objects of perception, or rather the objects of knowledge, progressively emerge – a process of ‘concept-noticing and sense-making’ (Radford, 2006, p. 15). Importantly, from an enactivist stance, use of the word ‘object’ by no means suggests that these ‘objects’ are pre-existing properties inherent in the environment. Rather, the ‘objects’ of perception are brought forth through the co-determination of knower and known, the co-evolution of individual pupils and their surroundings.

Duval (1998, p. 41) makes the pertinent point that most diagrams contain a great variety of constituent gestalts and subconfigurations. Critically, this surplus constitutes the heuristic power of a geometrical figure, since specific subconfigurations may well trigger different visual generalisations. Thus, within the context of figural pattern generalisation, the processes of visualisation and generalisation are deeply interwoven, and a complex relationship is likely to exist between different modes of visualisation. One of these underlying tensions is that between local and global visualisation, and it is this particular tension that forms the focus of this article.

Methodology The broader study (Samson, 2011b) of which this article forms part is oriented within the conceptual framework of qualitative research, and is anchored within an interpretive paradigm. The study aims ultimately to gain insights into the embodied processes of pupils’ visualisation activity when engaged in figural pattern generalisation tasks through an in-depth analysis of each pupil’s lived experience. A mixedgender, high-ability Grade 9 class of 23 pupils constituted the research participants for the broader study (Samson, 2011b). From this group of 23 pupils, seven research participants were identified as preferring a visual mode when solving pattern generalisation tasks. These seven research participants were individually provided with a linear pattern presented in a pictorial context and were required to provide, in the space of one hour, multiple expressions for the nth term of the sequence. Tools such as paper, pencils and highlighters as well as appropriate manipulatives such as matchsticks were provided. Participants were asked to think aloud whilst engaged with their particular pattern generalisation task, and the researcher also prompted the participants to keep talking or provide further explication as and when necessary. Each session was audio-visually recorded and field notes were taken. Audio-visual recordings were analysed with specific reference to how participants made use of multiple means of objectification en route to a stable form of awareness. These means of objectification included the use of words, linguistic devices, metaphor, gestures, rhythm, graphics and physical artefacts. These processes of ‘coming to know’ were carefully scrutinised through multiple viewings of the audio-visual recordings of each research participant.

Enactivism and knowledge objectification The broader study (Samson, 2011b) of which this article forms a part centres on two key theoretical ideas, enactivism (Maturana & Varela, 1998; Varela, Thompson & Rosch, 1991) and knowledge objectification (Radford, 2003, 2008). The manner in which these complementary theoretical lenses combine to provide a rich tool for analysis is described elsewhere (Samson & Schäfer, 2011), but a brief overview is presented here. The basic tenet of enactivism is that there is no division between mind and body, and thus no separation between cognition and any other kind of activity. Within an enactivist framework, there is a purposeful blurring of the line between thought and behaviour (Davis, 1997, p. 370), and cognition is thus viewed as an embodied and co-emergent interactive process. From this theoretical stance, as Davis (1995, p. 4) points out, language and action are not merely outward manifestations of internal workings, but should rather be seen as ‘visible aspects of … embodied (enacted) understandings’. Radford’s (2008) theoretical construct of knowledge objectification resonates strongly with an enactivist theoretical framework since it foregrounds the phenomenological and semiotic aspects of figural pattern generalisation. Knowledge objectification thus represents an ideal theoretical construct to critically engage with pupils’ whole-body experience and expression whilst they explore the potentialities afforded by a given pictorial pattern generalisation task.

The data analysis was guided by an enactivist methodological framework in which the researcher and research environment are seen to co-emerge (Reid, 2002). This interdependence of researcher and context was characterised by a flexible and dynamic process of investigation (Trigueros & Lozano, 2007). The iterative and reflexive process of co-emergence was built up over time through the use of multiple perspectives and the continuous refinement of methods and data analysis protocols. Audio-visual data was examined repeatedly in different forms (e.g. video and transcript) and in conjunction with additional data retrieved from field-notes and participants’ worksheets. In

Knowledge objectification is a theoretical construct to account for the manner in which learners engage or interact with a given scenario or context in order to make sense of it en route to a stable form of awareness (Radford, 2006, p. 7). Knowledge objectification is premised on two notions. Firstly, semiotic means such as gestures, rhythm and speech are not simply epiphenomena, but are seen as playing a http://www.pythagoras.org.za

48


Page 5 of 9

Original Research

addition, nodes of activity that seemed particularly interesting were identified and meticulously characterised with reference to the various semiotic means of objectification in the form of descriptive vignettes.

of a perceiver and the given figural context. Since not all patterning tasks could be presented unambiguously using a single term, it was decided to use two non-consecutive terms for all questions. This purposeful decision was thus not intended to encourage global visualisation per se, but rather to ensure that the potential for global engagement with the pictorial context was not discouraged by virtue of terms being presented consecutively.

Ethical considerations Before the broader study started, formal permission to conduct the research was obtained from the principal of the school in question. Anonymity of both the school and the research participants was assured, and appropriate pseudonyms are used throughout the text when referring to research participants. In addition, only those pupils who agreed to participate in the study through voluntary informed consent formed part of the research sample, and participants had the freedom to withdraw from the study at any stage without explanation. In the case of participants who were audio-visually recorded, written consent was obtained from each research participant as well as from each participant’s parents or legal guardians.

In addition, data collection and analysis protocols were sensitive to the enactivist underpinnings of the study, and thus made use of multiple data sources and approaches to data handling (as previously outlined). This process acted as a form of triangulation, which sought to ensure validity.

Findings and discussion Participants were presented with two non-consecutive terms of a linear pictorial sequence rather than a series of consecutive terms. Nonetheless, during their engagement with the presented pictorial terms, all pupils created physical instantiations of specific terms of their pictorial sequence through the process of drawing. This drawing process was in many ways a two-edged sword. In some instances, the physical process of drawing led to the emergence of structural commonalities or regularities. This supported the generalisation process where these regularities were algebraically useful (i.e. where the generality of what was noticed in the phenomenological realm could be readily expressed using algebraic symbolism). However, the physical process of drawing often led to attention being focused on the recursive nature of the step-by-step process of construction, thereby foregrounding local considerations rather than allowing for a more holistic or global apprehension. This often resulted in an underlying tension between these two different modes of visualisation.

From a more philosophical standpoint, there is also an important ethical consideration stemming from the enactivist theoretical underpinnings of this study. In enactivist terms we need to be sensitive to the notion that ‘… our actions have the potential to alter the worlds and possibilities of others’ (Simmt, 2000, p. 158). Furthermore, an enactivist stance compels us to see each person’s certainty as being ‘… as legitimate and valid as our own’ (Maturana & Varela, 1998, p. 245). Sensitivity to both of these ethical considerations was maintained throughout the study.

Reliability and validity In terms of reliability and validity considerations, not only the appropriate choice of the figural pattern generalisation questions themselves, guided by pertinent literature, but also the nature of their presentation were of critical importance. A literature review and previous research experience (Samson, 2007) suggested that linear sequences of the form ax ± c (c ≠ 0) would be most appropriate in terms of eliciting rich data.

An illustrative vignette The following vignette attempts to capture and characterise the tension between local and global visualisations as evidenced by the generalisation activity of one of the research participants (Terry, a high-ability Grade 9 pupil).

It has been shown that patterning tasks presented with consecutive terms encourage a recursive strategy (Hershkowitz et al., 2002; Samson, 2007) and thus tend to draw attention away from global structural features that could potentially co-emerge through the interaction

Part 1 Terry was presented with two non-consecutive terms of a typical pictorial sequence (see Figure 6).

Shape 5

Shape 3 FIGURE 6: Pictorial pattern presented to Terry.


49


Page 6 of 9

Original Research

After staring at the two terms for a few seconds, he remarked:

so it’s +4; and then however many things in between, just work out how many it is for that [indicating the 7-match additive unit] (…) well how many it is for the top triangle, bottom triangle and then plus that one [indicating the vertical match connecting the top and bottom triangles in the 7-match additive unit]. (Samson, 2011b, p. 132)

It seems like it’s basically just adding on the same sort of thing again, every time, and then just finishing it off with that [indicating the rightmost > shape]. (Samson, 2011b, p. 131)

After making this remark he carefully drew Term 4 in a very structured manner. He began by drawing the three matches of the leftmost triangle. Thereafter he drew the middle section of the structure in a very rhythmic fashion: 4,5,6 … 7,8,9 … 10,11,12 … 13,14,15. Interestingly, instead of drawing each group of three matches in the flowing form of a backward C-shape (top match, then vertical match, then bottom match) which would have been slightly more economical, he instead methodically drew each group of three matches by first drawing the top horizontal match, then the bottom horizontal match, and then finally the vertical match. To check that he had drawn the correct number of matches for Term 4, he then carefully counted the four ‘squares’ in the diagram before adding on the two oblique matches on the far right. After completing the middle section of the diagram, he drew a series of inverted V-shapes across the top of the structure (rhythmically drawing them in pairs from left to right) and finished the diagram by drawing a series of V-shapes along the bottom of the diagram, once again in rhythmic pairs from left to right (Figure 7).

At this point Terry wrote down the formula 7n + 4. However he quickly realised that he had not taken into account the first vertical match from the left: Shape 3 is Term 3, it’s got 3 of these little squares like that [pointing to the three central squares of Term 3] and Shape 5 has 5. So then you’ve got these 2 [indicating the leftmost < shape] so you start off, there’s +2, there’s another 2 [indicating the rightmost > shape] plus 4, then you’ve already got [points to the leftmost vertical match and realises he has missed it out] – oi! (…) With the front triangle [indicating the triangle formed from the leftmost < shape and the leftmost vertical match] it’s the full triangle that you’re starting off with, so it’s 1, 2, 3 matchsticks. With the end one you’ve already got the (…) base of the triangle coming from the previous square. (Samson, 2011b, pp. 132−133)

Terry thus gave his final formula as 7n + 5 which he subsequently altered to 3 + n(3 + 3 + 1) + 2 as being more representative of how he was visualising the pictorial context (Figure 8). Both expressions represent a local visualisation since they are based on a recursive addition/insertion of the 7-match additive unit. In the altered version of the formula, the 7-match unit is further subdivided into a ‘top triangle’, a ‘bottom triangle’ and a vertical line. An important distinction here is that Terry is not seeing each term as being holistically

Based on this drawing procedure, Terry was able to arrive at the formula 7n + 5: So you started off with your little triangle [indicating the leftmost < shape] so that’s obviously +2, then you finish it off with a little triangle again [indicating the rightmost > shape], plus another 2,

18 1

19 4

3

5

2 26

20

27

21 7

6

29

24

23

10

12 11

9

8 28

22

30

31

32

25

13 14

16 15 17

33

FIGURE 7: Terry’s drawing procedure for Term 4.

→ → 3  n(3  3  1)  2 3  n(3  3  1)  2

7n  5 7n  5

FIGURE 8: Two different formats of Terry’s initial visualisation.


50


Page 7 of 9

Original Research

composed of multiple 7-match units enclosed between a triangle on the left and a > shape on the right. Rather, the 7-match unit is seen as an additive structural unit in that it is recursively or iteratively added to (or inserted into) an existing term in order to construct the next term in the sequence, as evidenced by Terry’s remark ‘it’s basically just adding on the same sort of thing again, every time.’

signify existing physical structures in the particular diagram he was looking at. His second set of gestures, accompanying the words ‘and gets connected to another square’ mark a transition from existential signification to what Sabena, Radford and Bardini (2005, p. 134) refer to as imaginative signification. This second set of gestures moves from indicating materially instantiated aspects of the pictorial term to miming an ongoing sequence of connected squares, squares that are not yet materially present. We thus see a progressive distancing from the physical referent. Another important aspect of Terry’s objectification process is his use of the words ‘another square’. These words serve an important generative action function in terms of objectifying the generality of the interconnecting squares through an imaginative conception of iterative potential action. This linguistic device supports the process of objectification by allowing the recursive addition of squares to be ‘… repeatedly undertaken in thought’ (Radford, 2000, p. 248). However, and critically important in terms of the local-global visual tension, the words ’and gets connected to another square’ foreground an iterative process. Thus, whilst supporting an important generative action function in terms of objectifying the generality of the structural unit of the square, they also tend to focus attention on the recursive nature of step-bystep construction, thus drawing attention away from a more holistic view of the overall general structure.

An interesting aspect of Terry’s discussion is his frequent reference to squares in the pictorial terms. He makes express reference to the fact that the nth term in the sequence would contain n ‘little squares’. In addition, when he initially drew Term 4 he did so by drawing the central structure first and then checking that he had drawn the correct number of matches by quickly counting the four squares. However, these squares do not feature anywhere in either of the two versions of his initial formula. In fact, the horizontal matches from these squares are seen to form part of an upper and lower triangle. Thus, after visually deconstructing the diagram into triangles, the squares become negative space as the matches that originally formed them have been apportioned to different component parts. Nonetheless, Terry continued to refer to them as a helpful structural unit. There are two possible reasons for this that are worth considering. Firstly, the ultimate aim of the patterning task from Terry’s perspective is to arrive at an algebraic expression for the general term through a process of visualisation. It is possible that this goal had an unconscious influence on the visualisation process since some visualisations would be algebraically more useful than others – for example, squares would overlap and a correction would thus be necessary for the resulting overcount. A second possibility is that there is a tension between local and global aspects of the pictorial context. Local considerations focus on the additive unit by virtue of attention being focused on the step-by-step process of constructing the next term from the previous one. It is possible that this local focus obscured a more global outlook where the structural unit of a square could be properly incorporated into the general expression.

Terry then went on to draw Term 4. Interestingly, the order in which he drew the various lines did not seem to correlate with his description of overlapping squares. Instead, his drawing process seemed to suggest a subdivision into a triangle at either end, two rows of horizontal matches, a row of vertical matches, and V-shapes at the top and bottom. After completing his drawing of Term 4, Terry sat staring at it for just over a minute before commenting: ‘I had something and now I’ve, I had something else but now I’ve lost it.’ It thus seems that his initial idea of using overlapping squares came from a flash of insight that has since receded. It is possible that this may have been at least partially precipitated by Terry’s drawing of Term 4 in a manner which did not mimic his initial visualisation of overlapping squares. In this instance, it is possible that the drawing process itself obfuscated the visual apprehension. However, another interpretation of the data could suggest that the drawing process actually reflects a competing, albeit unconscious, visualisation of the pictorial terms, thus suggesting an underlying visual tension between two different apprehensions of the pictorial context.

Part 2 After silently and motionlessly staring at the two printed pictorial terms for a few minutes he made the following comment: What I’m trying to do now is almost use the squares. So now instead of having that sort of backwards C, actually have a fullon square (…) that gets connected to another square [gesturing to the right with his pencil], that then, just got to take out that one [indicating the overlapping match between two squares], and gets connected to another square [making multiple gestures further to the right with his pencil]. (Samson, 2011b, p. 134)

Terry then came up with the formula 3 + n(4 − 1) + 4n + 2. The ‘3’ at the beginning of the formula represents the starting triangle on the far left whilst the ‘+2’ at the end of the formula represents the > shape at the extreme right of each term. Terry described the n(4 − 1) portion of his formula as representing ‘each square minus the one that’s being taken up by either the previous one or the next one’ whilst the 4n is required for ‘the triangles above and below it’. Although he

Whilst Terry was explaining his strategy, he made use of a number of crucial semiotic means of objectification. The first of these was his gesturing to the right whilst saying the words ‘that gets connected to another square’. This indexical or deictic gesturing was specifically related to Term 3, which Terry had in front of him. He thus used the gestures to


51


Page 8 of 9

Original Research

→ FIGURE 9: Terry’s change in visual apprehension to a local visualisation.

specifically refers to ‘squares’, these structural features are not reflected in his present visualisation. Although his initial visualisation was suggestive of overlapping squares, he has in essence reverted to a previous visualisation in which the central structure is seen not in terms of overlapping squares but rather in terms of a series of backward C-shapes (Figure 9). A possible explanation for this reversion to an earlier visualisation is that Terry’s focus on the recursive nature of the construction process supported a local generalisation, but not a global one. A global generalisation or visualisation would entail seeing the structure in a holistic manner as being composed of a series of n overlapping squares. Since four matches are needed for each square, the n squares would require a total of 4n matches. However, this would lead to an overcount since overlapping would mean that some matches would in effect have been counted twice. To correct for this, we would need to subtract n − 1 matches from the tally since n overlapping squares would have n − 1 overlaps. However, Terry’s constant focus on a recursive, step-by-step process of construction is incompatible with this global view. To proceed from one term to the next would require the addition of a square and the removal of the overlapping match each time. The addition of a whole square each time thus becomes a redundant process if the overlapping match is immediately removed, since the process could be accomplished in a far simpler manner by just adding on three matches in the form of a backward C-shape each time, thereby avoiding the unnecessary removal of the overlapping match. It is this focus on a stepwise process of construction that is likely to have contributed to the initial visualisation of overlapping squares being transformed into a visualisation of backward C-shapes.

FIGURE 10: Terry’s final global visualisation of overlapping squares.

overlapping squares with n − 1 overlaps. At my suggestion he continued to pursue his initial idea. After staring at the diagrams for about half a minute he commented: I think I might have found it … So what I’m trying to do is now, is almost separate it so you’ve got, you just put all the squares together (…) and then take out this extra match right at the end [pointing in turn to each of the three overlaps between the four squares in Term 4]. (Samson, 2011b, p. 137)

This marks the crucial moment when Terry changes from a local to a global visualisation (Figure 10) and is thus able to make sense of, and articulate, his initial fleeting visualisation. After trying to incorporate (n − (n − 1)) into his general expression, he eventually abandoned it and came up with the final formula 2 + 4n − (n − 1) + 4n + 2: That works, then you’ve got your 2 that starts it off [indicating the leftmost < shape], your 2 that finishes it off [indicating the rightmost > shape], you’ve got your four for each square, then the − (n − 1) (…) for each square there’s an extra line except for the first (…) then + 4n for each triangle above and below it. (Samson, 2011b, p. 137)

Concluding comments The vignette serves to illustrate the subtle underlying tensions that can exist as pupils engage with pictorial pattern generalisation tasks. Within the context of figural pattern generalisation, the processes of visualisation and generalisation are deeply interwoven. Pattern generalisation rests on an ability to grasp a commonality from a few elements of a sequence, an awareness that this commonality is applicable to all the terms of the sequence, and finally being able to use it to articulate a direct expression for the general term. There are thus two important aspects of this notion of generalisation, namely, (1) a phenomenological element related to grasping the generality, and (2) a semiotic element related

At this point I asked Terry what had happened to his initial idea of focusing on the squares: I don’t know, I had something … I was busy looking at it and something hit me and then I lost it. I noticed something to do with n minus, open brackets n minus 1, and then that in brackets [i.e. (n − (n − 1))], that had something to do with it, but I cannot for the life of me remember what it was. (Samson, 2011b, p. 137)

Terry’s reference to his noticing something to do with (n − (n − 1)) does not initially seem to make any sense as the expression simplifies to +1. However, it retains an interesting remnant of his initial visualisation in which there are n http://www.pythagoras.org.za

52


Page 9 of 9

Original Research

to the sign-mediated articulation of what is noticed in the phenomenological realm (Radford, 2006, p. 5).

Hershkowitz, R., Dreyfus, T., Ben-Zvi, D., Friedlander, A., Hadas, N., Resnick, T., et al. (2002). Mathematics curriculum development for computerised environments: A designer-researcher-teacher-learner activity. In L.D. English (Ed.), Handbook of international research in mathematics education (pp. 657−694). Mahwah, NJ: Lawrence Erlbaum Associates.

Although competing visualisations have been shown to cause tension, the crucial aspect relates to the process of coming to realise how the visualisation is regular, and how this regularity can be expressed in an algebraically useful manner. Thus, although both local and global visualisations can be useful in their own particular way, it is likely that the process of objectifying and articulating an appropriate algebraic expression for the general term is complicated when tension exists between these two modes of visualisation. An awareness of and appreciation for these subtle tensions has the potential to provide an added depth of engagement for a sensitive practitioner.

Radford, L. (2005a). The semiotics of the schema: Kant, Piaget, and the calculator. In M.H.G. Hoffmann, J. Lenhard, & F. Seeger (Eds.), Activity and sign – grounding mathematics education (pp. 137−152). New York, NY: Springer.

Acknowledgements

Radford, L. (2005b). Why do gestures matter? Gestures as semiotic means of objectification. In H.L. Chick, & J.L. Vincent (Eds.), Proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education, Vol. 1 (pp. 143−145). Melbourne: PME.

Hewitt, D. (1992). Train spotters’ paradise. Mathematics Teaching, 140, 6−8. Katz, D. (1951). Gestalt psychology: Its nature and significance. London: Methuen. Mason, J., Graham, A., Pimm, D., & Gowar, N. (1985). Routes to/Roots of algebra. Milton Keynes: Open University Press. Maturana, H.R., & Varela, F.J. (1998). The tree of knowledge: The biological roots of human understanding (rev. edn.). Boston, MA: Shambhala. Pegg, J., & Redden, E. (1990). From number patterns to algebra: The important link. The Australian Mathematics Teacher, 46(2), 19−22. Radford, L. (2000). Signs and meanings in students’ emergent algebraic thinking: A semiotic analysis. Educational Studies in Mathematics, 42(3), 237−268. http:// dx.doi.org/10.1023/A:1017530828058 Radford, L. (2003). Gestures, speech, and the sprouting of signs: A semiotic-cultural approach to students’ types of generalization. Mathematical Thinking and Learning, 5(1), 37−70. http://dx.doi.org/10.1207/S15327833MTL0501_02

The financial assistance of the National Research Foundation (NRF) towards this research is hereby acknowledged. Opinions expressed and conclusions arrived at, are those of the author and are not necessarily to be attributed to the NRF. This work is based on research supported by the FirstRand Foundation Mathematics Education Chairs Initiative of the FirstRand Foundation, Rand Merchant Bank and the Department of Science and Technology. Any opinion, findings, conclusions or recommendations expressed in this material are those of the author and therefore the FirstRand Foundation, Rand Merchant Bank and the Department of Science and Technology do not accept any liability with regard thereto.

Radford, L. (2006). Algebraic thinking and the generalization of patterns: A semiotic perspective. In S. Alatorre, J.L. Cortina, M. Sáiz, & A. Méndez (Eds.), Proceedings of the 28th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Vol. 1 (pp. 2−21). Mérida, Mexico: Universidad Pedagógica Nacional.

Competing interests

Samson, D. (2011b). The heuristic significance of enacted visualisation. Unpublished doctoral dissertation. Rhodes University, Grahamstown, South Africa. Available from http://eprints.ru.ac.za/2881/

Radford, L. (2008). Iconicity and contraction: a semiotic investigation of forms of algebraic generalizations of patterns in different contexts. ZDM: The International Journal on Mathematics Education, 40(1), 83−96. http://dx.doi.org/10.1007/ s11858-007-0061-0 Reid, D.A. (2002). Conjectures and refutations in Grade 5 mathematics. Journal for Research in Mathematics Education, 33(1), 5−29. http://dx.doi. org/10.2307/749867 Sabena, C., Radford, L., & Bardini, C. (2005). Synchronizing gestures, words and actions in pattern generalizations. In H.L. Chick, & J.L. Vincent (Eds.), Proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education, Vol. 4 (pp. 129−136). Melbourne: PME. Samson, D.A. (2007). An analysis of the influence of question design on pupils’ approaches to number pattern generalisation tasks. Unpublished master’s thesis. Rhodes University, Grahamstown, South Africa. Available from http://eprints. ru.ac.za/1121/ Samson, D. (2011a). Capitalising on inherent ambiguities in symbolic expressions of generality. The Australian Mathematics Teacher, 67(1), 28−32.

The author declares that he has no financial or personal relationships which may have inappropriately influenced him in writing this article.

Samson, D., & Schäfer, M. (2011). Enactivism, figural apprehension and knowledge objectification: An exploration of figural pattern generalisation. For the Learning of Mathematics, 31(1), 37−43. Simmt, E. (2000). Mathematics knowing in action: A fully embodied interpretation. In E. Simmt, B. Davis, & J.G. McLoughlin (Eds.), Proceedings of the 2000 Annual Meeting of the Canadian Mathematics Education Study Group (pp. 153−159). Montreal: Université du Québec à Montréal.

References

Thornton, S. (2001). A picture is worth a thousand words. In A. Rogerson (Ed.), New Ideas in Mathematics Education: Proceedings of the International Conference of the Mathematics Education into the 21st Century Project (pp. 251−256), Palm Cove, Queensland, Australia.

Davis, B. (1995). Why teach mathematics? Mathematics education and enactivist theory. For the Learning of Mathematics, 15(2), 2−9. Available from http://www. jstor.org/stable/40248172

Trigueros, M., & Lozano, M.D. (2007). Developing resources for teaching and learning mathematics with digital technologies: An enactivist approach. For the Learning of Mathematics, 27(2), 45−51. Available from http://www.jstor.org/ stable/40248571

Davis, B. (1997). Listening for differences: An evolving conception of mathematics teaching. Journal for Research in Mathematics Education, 28(3), 355−376. http:// dx.doi.org/10.2307/749785 De Jager, T. (2004). Introducing and teaching first-degree equations. Learning and Teaching Mathematics, 1, 8−11.

Varela, F.J., Thompson, E., & Rosch, E. (1991). The embodied mind: Cognitive science and human experience. Cambridge, MA: MIT Press.

Duval, R. (1998). Geometry from a cognitive point of view. In C. Mammana, & V. Villani (Eds.), Perspectives in the teaching of geometry for the 21st century (pp. 37−52). Boston, MA: Kluwer.

Walkowiak, T.A. (2010). An examination of algebraic reasoning: Elementary and middle school students’ analyses of pictorial growth patterns. In P. Brosnan, D.B. Erchick, & L. Flevares (Eds.), Proceedings of the 32nd Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Vol. VI (pp. 194−201). Columbus, OH: The Ohio State University.

Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61(1/2), 103−131. http:// dx.doi.org/10.1007/s10649-006-0400-z

Wertheimer, M. (1938). Laws of organization in perceptual forms. Summary of Wertheimer, M. (1923) “Untersuchungen zur lehre von der gestalt,” II, Psychol. Forsch., 4: 301−350. In W.D. Ellis (Ed.), A source book of gestalt psychology (pp. 71−88). London: Kegan Paul.

Fischbein, E. (1993). The theory of figural concepts. Educational Studies in Mathematics, 24(2), 139−162. http://dx.doi.org.za/10.1007/BF01273689 Helson, H. (1933). The fundamental propositions of Gestalt psychology. Psychological Review, 40, 13−32. http://dx.doi.org/10.1037/h0074375


Zusne, L. (1970). Visual perception of form. New York, NY: Academic Press.

53


Page 1 of 9

Original Research

Mathematics teachers’ reflective practice within the context of adapted lesson study Author: Barbara Posthuma1 Affiliation: 1 Faculty of Education Sciences, North-West University, South Africa Correspondence to: Barbara Posthuma Email: barbara.posthuma@gmail. com Postal address: Private Bag X1290, Potchefstroom 2520, South Africa Dates: Received: 17 Nov. 2011 Accepted: 21 Oct. 2012 Published: 19 Nov. 2012 How to cite this article: Posthuma, A.B. (2012). Mathematics teachers’ reflective practice within the context of adapted lesson study. Pythagoras, 33(3), Art. #140, 9 pages. http://dx.doi. org/10.4102/pythagoras. v33i3.140

There seems to be paucity of research in South Africa on mathematics teachers’ reflective practice. In order to study this phenomenon, the context of lesson study (in an adapted form) was introduced to five mathematics teachers in a rural school in the Free State. The purpose was to investigate their reflective practice whilst they collaboratively planned mathematics lessons and reflected on the teaching of the lessons. Data were obtained through interviews, video-recorded lesson observations, field notes taken during the lesson study group meetings and document analyses (lesson plans and reflective writings). The adapted lesson study context provided a safe space for teachers to reflect on their teaching and they reported an increase in self-knowledge and finding new ways of teaching mathematics to learners. This finding has some potential value for planning professional learning programmes in which teachers are encouraged to talk about their classroom experiences, share their joys and challenges with one another and strive to build a community of reflective practitioners to enhance their learners’ understanding of mathematics.

Introduction The ability to reflect on practice is considered a necessity for effective instruction (Sowder, 2007). When teachers reflect on their classroom practice they carefully consider the problems in their own teaching and think about how those problems are related to their learners’ understanding of concepts. They are aware of the consequences of their teaching and how their own assumptions or beliefs can influence their teaching. However, Day (1999) argues that other teachers are needed in the reflective process in order to achieve deep levels of reflection. It seems that systematic investigation of practice with the help of a critical colleague enhances the reflective process. Teachers may for instance find it beneficial to come together in groups or teams to discuss their teaching in a supportive atmosphere (Farrell, 2004). This view mirrors Pollard’s (2002) finding that the value of engaging in reflective activity is almost always enhanced if carried out in association with other colleagues. York-Barr, Sommers, Ghere and Montie (2006) concur with Pollard and maintain that reflecting on practice with another person has the potential to greatly enrich understanding and support improvements in practice. They believe that reflecting with a colleague can assist in gaining awareness of some of the fixed assumptions a teacher might have and as such help the teacher to view events from another perspective (York-Barr et al., 2006). Lesson study is a professional development model that provides an opportunity for teachers to reflect collaboratively on their planning and teaching of a lesson. Murata (2011, p. 10), for instance, claims that lesson study is centred around teachers’ interests, is learner focused, provides opportunities for teachers to be researchers, provides plenty of time and opportunities for teachers to reflect on their teaching practice and learner learning, and is collaborative.

Problem statement


In spite of what has been said in the introduction, there seems to be paucity of research in South Africa on mathematics teachers’ reflective practice, which strengthened the rationale to research this issue. Furthermore, lesson study has not been used widely as a professional development model in South Africa, despite evidence that the lesson study structure enables teachers to engage in significant professional development with a minimum of resources and offer possibilities for rural teachers to use the resources they already have to improve the teaching and learning of mathematics (Taylor, Anderson, Meyer, Wagner & West, 2005). Based on evidence regarding the importance of teachers reflecting collaboratively on their teaching, lesson study (in an adapted form) was used as the context of an investigation of the nature of mathematics teachers’ reflective practice (Posthuma, 2011). This article reports on how


54


Page 2 of 9

Original Research

both inwardly at their own practice and outwardly at the social conditions wherein these practices are situated (Carr & Kemmis, 1986). The second advocates a democratic and emancipatory force as teachers focus their reflections on issues related to inequality and injustice, in the context of their own classroom, school and society. The third demonstrates a commitment to reflection as a communal activity and seeks to create communities of learning where teachers can support and sustain one another (Zeichner & Tabachnik, 1991, p. 9).

the adapted lesson study context influenced mathematics teachers’ reflections before, during and after teaching a lesson. The article is structured as follows. Firstly, a conceptual theoretical framework is outlined. This is followed by an exposition of the research method that was applied. The findings are listed thereafter, followed by a discussion against the backdrop of the conceptual theoretical framework. The article concludes with suggestions for further research.

The main concepts that this research addressed were mathematics teachers’ reflective practice and lesson study (in an adapted form). In this section these concepts are linked theoretically to establish a conceptual theoretical framework for the study that was done. Two theories, symbolic interaction and social reconstructionism, informed the conceptualisation of the study.

According to Mewborn (1999), ‘reflection and action together are seen as a bridge across the chasm between educational theory and practice’ (p. 317). Lesson study formed the practice part of this study where the teachers tried to construct meaning about their classroom practice socially through their interaction with one another. In the adapted lesson study process the teachers were constantly reflecting on their practice and both theories (theory of symbolic interactionism and social constructionist theory) embrace the social interaction and reflection concepts.

Symbolic interaction theory

Mathematics teachers’ reflective practice

Conceptual theoretical framework

Symbolic interaction theory examines the ways in which people make sense of their life situations and the ways in which they go about their activities, in conjunction with others, on a day-to-day basis (Prus, 1996). As a theoretical framework, it suited the interpretive nature of this study which is centrally concerned with meanings teachers construct whilst reflecting collaboratively on their planning and teaching of a lesson. The four basic propositions of symbolic interaction are:

Thoughts on reflection and reflective practice have evolved over many decades, if not centuries, through carefully constructed theory and research applications (York-Barr et al., 2006). Dewey is frequently recognised as the eminent 20th-century influence on reflection in education (Ottesen, 2007; Pollard, 2002; Rodgers, 2002; York-Barr et al., 2006; Zeichner & Liston, 1996). For Dewey (1933), true reflective practice takes place only when the individual is faced with a real problem that they need to resolve and seek to resolve in a rational manner.

• individuals act and interact within larger networks of individuals and groups that have an impact on them • human beings are active in shaping their own behaviour • individuals can engage in thought and change their behaviour as they interact with others • and to understand human conduct requires a study of their covert behaviour (Blumer, 1969).

The seminal work of Schön (1983) has also inspired a renewed interest in reflective practice in the field of education (Lee & Tan, 2004; Valli, 1997). Although Schӧn clearly relates reflection to action through using the terms ‘reflection-inaction’ and ‘reflection-on-action’, other researchers seem to view reflection as a special form of thought (Artzt, ArmourThomas & Curcio, 2008; Sparks-Langer & Colton, 1991). For the purpose of this investigation, reflection was considered to involve questioning the effectiveness of practice before (reflection-for-practice), during (reflection-in-practice) and after the act of teaching (reflection-on-practice).

This final proposition has major methodological implications, namely that the procedures used should allow sympathetic introspection as a part of the methodology (Pedro, 2001). The adapted lesson study context was explored not only against the backdrop of symbolic interaction, but also within the milieu of the social reconstructionist theory, discussed in the following section.

Research indicates that the content of teachers’ reflection in their classrooms relates to their learners’ prior knowledge (Ward & McCotter, 2004), their own instructional strategies (Lee, 2005; Ward & McCotter, 2004), discipline (Lee, 2005), the teacher-learner relationship (Lee, 2005), mathematical content (McKeny, 2006) and disturbing aspects of teaching experiences (Ottesen, 2007).

Social reconstructionist theory Zeichner and Tabachnick (1991) outlined four major theoretical traditions in the reflective teaching literature. Each of these is concerned with thoughts and practices connected to particular educational aims and values. These traditions are the academic, social efficiency, developmentalist and social reconstructionist traditions. The latter views schools and teachers as agencies of change for the creation of a more just and humane society. This tradition has three central strands. The first encourages teachers to focus their attention http://www.pythagoras.org.za

The quality of teachers’ reflection can be measured at different levels as identified by a number of researchers (e.g. Lee, 2005; Valli, 1992; Van Manen, 1977). For the purpose of this study the teachers’ quality of reflection was rated according to Lee’s (2005) levels of reflection. According to Lee, a teacher who 55


Page 3 of 9

Original Research

on one’s teaching that occurs during the lesson study process is a habit that remains with teachers long after the lesson study process is over.

reflects on Level 1 (R1: ‘Recall level’) is mainly concerned with mastery and/or application of technical means for achieving given educational ends, and includes a simple description of observation or a focus on behaviours or skills from past experience. Reflection on Level 2 (R2: ‘Rationalisation level’) is directed at an interpretive understanding of the meanings of educational experiences and choices of action within a particular social and institutional context (Lee, 2005). Reflection on Level 3 (R3: ‘Reflective level’) links classroom practice to the broader arena of political, moral and ethical forces (Lee, 2005) and this type of reflection is considered to be critical reflection (Jay & Johnson, 2002).

Research on lesson study and mathematics teaching Meyer and Wilkerson (2011) investigated the impact of lesson study on teachers’ knowledge for teaching mathematics and found that lesson study provided opportunities for improved teacher knowledge in three of the five groups that participated. They conclude that more research is needed to examine the impact of lesson study on teachers’ knowledge. Olson, White and Sparrow (2011) studied the influence of lesson study on mathematics teachers’ pedagogy and found that three of the five teachers who completed the lesson study process changed their pedagogies over 18 months. Before lesson study, the three teachers maintained control of the classroom discourse and frequently asked ‘why?’ without knowing what to do with the information that they gained. After completing the lesson study process, these three teachers asked purposeful questions and used the learners’ responses to guide instructional decisions.

In order to reflect on a critical level about the teaching and learning of mathematics, the teacher has to question moral, ethical, and other types of normative criteria related directly and indirectly to the classroom (Van Manen, 1977). Within the social reconstructionist tradition, teaching and teacher education are important determinants for the creation of a more just and humane society. This perspective prioritises critical reflection in terms of how teachers’ actions influence the stability within both the school and society (ZwodiakMyers, 2009).

Lesson study in the South African context

Lesson study

In South Africa, a school-based in-service education intervention programme, modelled along the lines of the Japanese lesson study, was launched in 2000 in Mpumalanga (Jita, Maree & Ndlalane, 2006). According to Jita et al. (2006), the lesson study approach has established a system in which teachers have grown accustomed to relying on one another, coaching, leading discussions and exploring alternative solutions to problems experienced in their teaching of mathematics.

Lesson study is a collaboration-based teacher professional development approach that originated in Japan (Fernandez & Yoshida, 2004). According to Takahashi, Watanabe and Yoshida (2006), lesson study has played an important role in professional development in Japan since the public education system was introduced more than a hundred years ago. One reason for the popularity of lesson study in Japan might be that: lesson study provides Japanese teachers with opportunities to make sense of educational ideas within their practices; change their perspectives on teaching and learning; learn to see their practices from a learner’s perspective; and enjoy collaborative support among colleagues. (Takahashi, Watanabe & Yoshida, 2006, p. 201)

A study by Coe, Carl and Frick (2010), in a rural primary school in the Western Cape, sought to determine the value that a group of teachers placed on the process of lesson study as a model for their own learning and instructional improvement. Their findings highlight the following benefits of lesson study:

The lesson study process is cyclical and has the following basic stages (Murata, 2011, p. 6):

1. Lesson study offers an effective strategy to bring teachers out of isolation, allowing them to experience meaningful collaboration with fellow teachers. 2. The process of lesson study is embedded within the classroom context by setting goals and then planning instruction with the purpose of moving the learners closer to the goals. A connection between the content of the research lesson and the remainder of the curriculum is established. Furthermore, lesson study provides an opportunity to observe the learners during the research lesson. The post-lesson discussion is also valuable in terms of validating and developing the perceptions of learners in relation to the prescribed goal. 3. Lesson study can be a catalyst for transforming new instructional strategies into routine classroom practice. 4. Continuous support (by fellow teachers) is embedded within the model of lesson study.

1. Consider goals for student learning and development, plan a research lesson based on these goals. 2. Observe the research lesson and collect data on student learning and development. 3. Use these data to reflect on the lesson and on instruction more broadly. 4. If desired, revise and re-teach the research lesson to a new group of students. Research has shown that lesson study impacts on teachers’ understanding of learner thinking, it enhances teachers’ content knowledge and awareness of new approaches to teaching, it helps teachers to connect their practices to school goals and broader goals, and it creates a demand for improved instruction and allows competing views to be heard during the reflection-stage of the lesson study cycle (Lewis, 2000). Friedman (2005) argues that the habit of personally reflecting http://www.pythagoras.org.za

56


Page 4 of 9

Original Research

The biographical information of the participants is provided in Table 1. Pseudonyms are used to protect the identities of the participants.

The lesson study process involves reflection on the planning and teaching of a lesson as a communal activity, and hence fits into the framework of the social reconstructionist tradition. Teacher reflection within the lesson study model of professional development focuses on social conditions of practice (a group of teachers trying to make sense of the teaching and learning of mathematics) in a democratic way.

Table 1 indicates that all the participants were experienced teachers with basic teaching qualifications and, except for Dianne, had all been teaching for 14 years or longer. The pass rate for mathematics at the rural township school was 19% in 2010. The medium of instruction at the school was English. Setati, Adler, Reed and Bapoo (2002) argue that in the remote rural areas of South Africa where access to English outside the classroom is severely limited, the classroom context is more appropriately described as a learning environment in which English is a foreign language. This notion has implications for the present study, where the participants had to reflect on their practice through the medium of English. In addition, they all taught mathematics to learners who did not have textbooks.

The conceptual theoretical framework (discussed previously) links mathematics teaching to teachers’ reflective practice, within the context of adapted lesson study. The lesson study cycles of planning, teaching and evaluation correspond with teachers’ reflection-for-practice, reflection-in-practice and reflection-on-practice. The theory of symbolic interaction and the social reconstructionist tradition both embrace critical reflection and social interaction and therefore together form the backdrop of this study.

Procedure

Empirical investigation

The data-gathering procedure followed is outlined in Table 2.

Research design

Although the group provided feedback on how to improve each of the lesson plans, the final lesson plan for each teaching phase remained the responsibility of the teacher who would teach the lesson. This is a deviation from the Japanese lesson study, although the same procedure was followed as suggested by Lewis (2000):

A case study design suitable for a qualitative approach was chosen. According to Cohen, Manion and Morrison (2005) a case study provides a unique example of real people in real situations. Case studies offer a multi-perspective analysis in which the researcher considers not just the voice and perspective of one or two participants in a situation, but also the views of other relevant groups and the interaction between them (Nieuwenhuis, 2010).

the research lesson should be observed by other teachers; the research lesson should be planned collaboratively; there should be an overarching goal for the research lesson; and the research lesson should be recorded and discussed. (pp. 4–6)

The lesson study cycle is illustrated in Figure 1.

Participants The participants in this study were five mathematics teachers from one school in the Thabo Mofutsanyana district in the Free State. The criteria for selection included the factors of convenience, access and willingness to participate. Meetings with the teachers took place in the teachers’ school environment.

Data collection The lesson study cycle involves three phases: • planning • teaching • evaluation (reflection).

TABLE 1: Biographical information of participants. Category

Dianne

Mary

Morgan

Sipho

Vicky

Age

32

44

39

48

44

Highest qualification

ACE

FDE

BSc Ed

FDE

BEd Hons

Teaching experience (years)

8

19

14

22

17

Grades teaching

11,12 ML

8

10, 12

8, 9

9, 10, 11

Home language

Sesotho

Sesotho

Sesotho

Sesotho

Sesotho

Source: Posthuma, A.B. (2011). The nature of mathematics teachers’ reflective practice. Unpublished doctoral dissertation.University of Pretoria, Pretoria, South Africa. Available from http://upetd. up.ac.za/thesis/available/etd-04252012-164207/ ACE, Advanced Certificate in Education; FDE, Further Diploma in Education; ML, Mathematical Literacy.

TABLE 2: Procedure for data gathering. Lesson study cycle

Description of cycle

Planning phase

The participants in the lesson study group cooperatively planned an initial lesson for Grade 8 on solving linear equations. The lesson study goal was to improve learners’ understanding of related mathematics concepts.

Teaching phase

During the teaching phase of the first lesson study cycle the planned lesson was taught by the Grade 8 teacher (Mary). This lesson was observed by the researcher together with an assistant who managed the video recorder. The same procedure was followed during the next four lesson study cycles.

Reflection (evaluation) phase

During the feedback (evaluation/reflection) phase of each lesson study cycle the participants of the lesson study group viewed the video-recorded lesson the same afternoon in a post-conference. (This is another deviation from the original Japanese lesson study model, where the lesson study group observes the lesson whilst it is taught.) The focus was not only on the teacher’s presentation but also on the learners’ understanding of the concepts that were taught. A consensus was reached on whether the lesson study goal was achieved. During each of these sessions the group suggested improvements to the lesson plan for the next teaching cycle.


57


Page 5 of 9

Original Research

Ethical considerations Permission to access a school in the Thabo Mofutsanyana district was obtained from the Free State Education Department. Potential participants were verbally briefed and presented with details of the research in writing. Each teacher signed a letter of consent. Participation was voluntary and the teachers had the option to withdraw at any stage. They were assured that all data would be kept confidential and pseudonyms would be used in the report. In addition, the principles of respect for personal autonomy, benevolence and justice guided the research.

Planning lesson

Improving lesson plan for the next teaching phase

Teaching lesson

Findings

Findings from the analyses of the lesson plans The first lesson was planned collaboratively by the group (the ‘research lesson’) and subsequent lesson plans were refined by the lesson study group and presented by the different teachers. The participants used a lesson plan template that made provision for reflection on their learners’ current understanding of the concepts. The template also made provision for reflection after the lesson was taught. The lesson plans were analysed to establish whether the teachers reflected on their learners’ thinking and understanding of the concepts to be taught, whether provision was made for learners who might have struggled with the concepts or learners who needed to be challenged, and whether there was any evidence of linking the current lesson to the lesson study goal determined by the lesson study group.

Lesson study group reflection

FIGURE 1: Lesson study cycle.

Data were collected to coincide with each of these phases. During the planning phase, data were collected through document analyses (lesson plans); during the teaching phase, data were collected through lesson observation (video recorded so that the lesson study group could observe the lesson the same afternoon); during the reflection or evaluation phase, data were collected through semistructured interviews, reflective writings and field notes. A final group interview with all the participants was conducted after the last lesson study cycle to establish how the reflective processes of lesson study influenced their classroom practice.

The analysis of the lesson plans revealed no reflection on learners’ current mathematical knowledge or on how to address any misconceptions that might have occurred when teaching the concepts. There was also no evidence in these lesson plans of teaching mathematical concepts with a longterm goal in mind. For example, during the post-observation interview Mary admitted that she had not planned the lesson with her learners in mind: ‘you know, I did not now think of them … grasping gradually …’ and Dianne said: ‘… previously we did not spend much time on lesson plans.’ In addition, the level of the content the participants selected to teach differed. For example, the content that the lesson study group selected for Mary’s Grade 8 class (the first lesson plan) was on a very low cognitive level (the examples used were x + 3 = 7, x – 10 = 3 and 32 + x = 34) in contrast to the content that Morgan selected for his Grade 10 class (which was arranged from basic examples to more complicated examples, including fractional algebraic equations such as 10 y + 1 4 ( 7 y + 1 ) ). The selection of tasks on a low cognitive = 2 5 level might reveal the individual teacher’s perception of their learners’ ability to cope with the mathematics content.

Data analysis The data gathered during the lesson study cycles were analysed during and after the data-gathering process, in line with Creswell’s (2003, p. 18) qualitative case study approach in which the researcher collects open-ended, emerging data with the primary intent of developing themes from the data. All interviews were transcribed verbatim and the computer software program Atlas.ti 6 (ATLAS.ti Scientific Software Development, 2010) was used to assist with the data management, coding, categorisation, abstracting and conceptualising stages of the analysis.

Trustworthiness The qualitative data collected were in the form of observations, interviews and document analyses. The observations and interviews were electronically recorded and transcribed. Participants had the opportunity to review these transcriptions at the end of the data collection period to ensure accuracy and provide additional research data. Data from multiple sources were used to help verify findings: data collected through interviews were verified, for instance, with information gathered from the observations and the document analyses.


Findings of lesson study group reflection after observing lessons The lesson study group observed the video recording of each lesson and discussed their observations. According to Taylor et al. (2005), observing a lesson enables teachers to

58


Page 6 of 9

Original Research

Table 3 summarises the content of reflection, as well as the level of the group reflections on each lesson.

shift their thinking from a teaching focus to a learning focus whilst puzzling over their learners’ mathematical thinking. As observers, they are free to focus on the actual work the learners are doing, as well as the learners’ thought processes.

From Table 3, it appears that the participants reflected on their teaching styles, on their learners’ understanding of concepts, and on matters such as class size, textbooks, class arrangement and language. Within the framework of the symbolic interaction theory, the teachers were making sense of their own and each other’s classroom practice collaboratively. During the group reflections on the lessons they observed, the lesson study group reflected critically (R3) on Mary and Vicky’s lessons, considering the implications of each teacher’s actions on their learners’ understanding of mathematics. The video recordings revealed Mary’s impatience with the learners’ inability to solve basic linear equations as well as Vicky’s impatience with her learners’ lack of basic mathematical knowledge (such as the product of multiplying two negative numbers will be a positive number) and Vicky admitted: ‘Ai, ... I lose my temper sometimes … (laughs).’ Morgan replied: ‘We need to be very patient with the learners. When we teach them, we need to make sure you motivate them.’ Against the backdrop of the social reconstructionist theory, the teachers who participated in this study reflected communally, supported each other during the feedback sessions, but also critically considered the effects of their own and their colleagues’ classroom practice on their learners’ mathematical growth and well-being.

During the lesson observations the focus was on the teacher’s presentation as well as on the learners’ content knowledge. Mary’s learners struggled to transpose the constant term to the other side of the equation (using the additive inverse) and as a result the volume of work done during the 45-minute period was very low. It seemed to me that Mary did not understand her learners’ thinking and failed to encourage them to clearly explain and justify their reasoning. For example, she asked one learner to come to the board to solve the linear equation 32 + x = 34. The learner struggled and Mary did not help her, but instead told her to ask learners from the class to help, which resulted in chaos. She moved between the desks to mark the learners’ work and if a learner wrote a wrong answer, she sent the learner to the board to look at the examples, not helping the learner to understand the method or asking them to explain their solution method to her. According to Warfield, Wood and Lehman (2005) such teachers do not reflect deeply about their learners’ mathematics or about their own teaching. However, Morgan allowed learners to solve the equations using their own methods and expected them to explain and justify their reasoning and to listen to and question the reasoning of other learners. In this way he actively involved all learners in the lesson:

Findings of final group reflection on the influence of lesson study on teaching mathematics

I invite the learners to come and show me what they are doing on the board, so that I can exactly know what they know and what they don’t know …

All the participants reported that they had gained personally and professionally from being part of the lesson study group. A summary of their reflections appears in Figure 2.

The lesson study group reflected critically on Mary’s lesson and expressed concern about treating the learners fairly and catering for all learners’ needs (reflecting on Lee’s R3: Reflective level). During the group reflection, Morgan said:

The benefits of lesson study reported by the participants are summarised in Figure 2, a circular diagram that illustrates the influence of the lesson study cycle on the participants’ reflective journeys. They reported that they had improved their lesson planning as a result of the lesson study group planning sessions. They were more confident about their teaching after seeing themselves on video. They expressed a deeper awareness of their learners’ needs. They learned from watching their fellow participants on video to change their teaching to become more learner-centred, and they felt as if

We should cater for all learners ... sometimes you plan a worksheet for your class, and after one or two examples you see they don’t understand, and then only a few sums are done ...

The lesson study group seemed to reflect more openly and talk more freely as the lesson study process continued. They constantly reflected on their learners’ lack of understanding of basic concepts, for example, learners struggling to add 3x to 6x.

TABLE 3: Summary of content and level of reflection during the lesson study group reflections. Lesson study group reflections

Mary’s lesson on 24 February 2011

Morgan’s lesson on 04 March 2012

Vicky’s lesson on 10 March 2011

Sipho’s lesson on 05 May 2012

Dianne’s lesson on 05 May 2012

Content of reflection

The introduction; her teaching style; learners’ understanding; class size and lack of textbooks; her lesson plan

Teaching of the content; teaching style (learnercentred); lesson plan

Her methodology; learners’ understanding of concepts; her expectations of her learners; teaching style; class arrangement and management; learners’ needs

His introduction; his teaching style; learners’ understanding of concepts

Her teaching style; the topic; language

Level of lesson study group reflection

R3 (Critical reflection level): Considering the implications of her teaching for learners

R2 (Rationalisation level): Thinking about his teaching from different perspectives

R3 (Critical reflection level): Considering the implications of her teaching for learners

R1 (Recall level): Describing his actions in class

R2 (Rationalisation level): Thinking about her teaching from different perspectives

Source: Posthuma, A.B. (2011). The nature of mathematics teachers’ reflective practice. Unpublished doctoral dissertation. University of Pretoria, Pretoria, South Africa. Available from http:// upetd.up.ac.za/thesis/available/etd-04252012-164207/


59


Page 7 of 9

Original Research

As expected in terms of the symbolic interaction theory, the adapted lesson study provided a context for the teachers to derive meaning about their teaching actions through social interaction with their fellow teachers. However, they were also actively shaping their own behaviour and reflecting on how to change their teaching actions to improve their learners’ understanding of mathematics concepts. In line with the social reconstructionist tradition, which prioritises reflection-in-practice and -on-practice, the teacher becomes an agent of change to improve social conditions of practice as a result of critical reflection. This issue is especially applicable in South African society, where most teachers teach in rural areas, using a language that is not their mother tongue, in schools with limited resources.

Improved lesson planning

Selfresearch

Influence of lesson study on teaching

Learning from colleagues

Teaching with more confidence

A deeper awareness of learners’ needs

Limitations of the study

Source: Posthuma, A.B. (2011). The nature of mathematics teachers’ reflective practice. Unpublished doctoral dissertation. University of Pretoria, Pretoria, South Africa. Available from http://upetd.up.ac.za/thesis/available/etd-04252012-164207/

The five participants were of the same cultural and language group, and taught at the same school. A more diverse sample that excludes ethnic and geographical biases could arguably have cast more light on the research problem.

FIGURE 2: Influence of lesson study on teaching as reported by the participants.

they were doing self-research by being part of this research study. Against the framework of the symbolic interaction theory, these teachers reflected collaboratively and tried to make sense of their classroom practice and their teaching of mathematics. They reported positively on the lesson study process, but whether they will adapt their classroom behaviour as revealed by the lesson observations remains to be researched.

Another limitation of this research study pertains to the researcher’s inability to speak the home language of the participants. Language emerged as a contextual factor that possibly influences participants’ reflective practice (Posthuma, 2011). Although allowance was made for an interpreter during the last group reflection, the participants’ verbal and written reflections showed that they struggled to express themselves in English during the lesson study group meetings. One has to allow for the fact that their reflections might have been misinterpreted.

Discussion Lesson study as reported by this research supports Lewis’s (2000) view that lessons planned in the lesson study context impact on teachers’ individual professional development and their view of learners (they ‘learn to see children’). New content and approaches are acquired, competing views of teaching emerge and a demand for improvement is created. It also confirms Hix’s (2008) contention that the sum of planning collaboratively, anticipating learner responses, creating evaluation questions for observers, observing the public teaching, and discussing and reflecting on the observations are beneficial to teachers’ reflective practice. The results of this study mirror Friedman’s (2005) report that the major advantage of lesson study for teachers is the collaboration factor. Taylor et al. (2005) report on the following benefits of the lesson study professional development model:

Further research It would be worthwhile to undertake a follow-up study with the participants of this study to understand the longterm effects of reflective processes in a lesson study context. Furthermore, there might be other contextual factors that influence mathematics teachers’ reflective practice that need to be researched further, for example, gender, personality characteristics and culture. The reason gender is considered to be a possible influence on the reflective process is because the male participants in this study were very reluctant to write about their reflections in a reflective diary. Personality characteristics might also play a role in teachers’ reflective practice, based on Dewey’s (1933) three attitudes that he considered to be integral to reflection, namely, openmindedness, responsibility and wholeheartedness. Reflection might also be culturally bound. For example, Lee and Tan (2004) investigated student teachers’ reflective practice in Malaysia and found that their private reflections were on a deeper level than their public reflections. An intercultural study, for example, comparing South African mathematics teachers’ reflective practice to that of a different culture, will provide for a more comprehensive body of knowledge on reflective practice.

• an effective detailed lesson plan achieves the goal of more effective learning by learners • the lesson study model provides a highly motivated structure for planning and teaching a lesson • reflecting and thinking in the company of other teachers allow for sharing, interacting questioning assumptions, and reassessing common practices • observing a lesson enables a shift in thinking from a teaching focus to a learning focus • focusing on learner thinking provides opportunities for feedback to support changes in teaching mathematics • lesson study transforms working relationships and conversations between teachers. http://www.pythagoras.org.za

The results of this study could furthermore be used in the planning of future continuing professional teacher 60


Page 8 of 9

Original Research

development programmes. The positive feedback of the participants on the lesson study process suggests that lesson study should be initiated in other settings. Further research should explore lesson study as a model in South Africa for successful continuing professional teacher development and teacher education programmes.

Creswell, J.D. (2003). Research design: Qualitative, quantitative and mixed methods approaches. (2nd edn.). Thousand Oaks, CA: Sage Publications.

Conclusion

Fernandez, C., & Yoshida, M. (2004). Lesson study: A Japanese approach to improving mathematics teaching and learning. Mahwah, NY: Lawrence Erlbaum.

The teachers who participated in this study were teaching mathematics to learners in a rural area with few or no resources, in a language that was not their mother tongue. The symbolic interaction theory provided a backdrop for the adapted lesson study as a way for these mathematics teachers to derive meaning about their teaching actions through social interaction with their fellow teachers. The adapted lesson study experience allowed them an escape from their isolated practices, providing them an opportunity to reflect on their problems cooperatively in a safe space. However, they were also actively shaping their own behaviour and some were critically reflecting on how to change their teaching actions to improve their learners’ understanding of mathematics concepts. The social reconstructionist approach prioritises reflection-in-practice and reflection-on-practice that enables the teacher to become an agent of change to improve social conditions of practice. This issue is especially applicable in South African society where teachers’ histories need to be taken into account when planning and implementing professional development programmes.

Friedman, R.E. (2005). An examination of lesson study as a teaching tool in U.S. public schools. Unpublished doctoral dissertation. Ashland University, Ashland, OH, United States. Available from http://etd.ohiolink.edu/send-pdf.cgi/Friedman%20 Ruth%20E.pdf?ashland1116871771

Day, C. (1999). Professional development and reflective practice: Purposes, processes and partnerships. Pedagogy, Culture & Society, 7(2), 221−233. http://dx.doi. org/10.1080/14681369900200057 Dewey, J. (1933). How we think: A restatement of reflective thinking to the educative Process. Boston: Heath. Farrell, T.S.C. (2004). Reflective practice in action. Thousand Oaks, CA: Corwin Press, Inc.

Hix, S.L. (2008). Learning in lesson study: A professional development model for middle school mathematics teachers. Unpublished doctoral dissertation. University of Georgia, Athens, GA, United States. Available from http://jwilson. coe.uga.edu/pers/hix_sherry_l_200808_phd.pdf Jay, J.K., & Johnson, K.L. (2002). Capturing complexity: A typology of reflective practice for teacher education. Teaching and Teacher Education, 18, 73−85. http://dx.doi. org/10.1016/S0742-051X(01)00051-8 Jita, L.C., Maree, J.G., & Ndlalane, T.C. (2006). Lesson study (Jyugyo Kenkyu) from Japan to South Africa: A science and mathematics intervention program for secondary school teachers. In B. Atweh, A.C. Barton, M. Borba, N. Gough, C. Keitel, C. VistroYu, et al. (Eds.), Internationalisation and globalisation in mathematics and science education (pp. 465−486). Dordrecht: Springer. Lee, H-J. (2005). Understanding and assessing preservice teachers’ reflective thinking. Teaching and Techer Education, 21, 699–715. Lee, W.H., & Tan, S.K. (2004). Reflective practice in Malaysian teacher education. Singapore: Marshall Cavendish Academic. Lewis, C.C. (2000, April). Lesson study: The core of Japanese professional development. Paper presented at the Annual Meeting of the American Educational Research Association, New Orleans, LA. Available from http://www.lessonresearch.net/ aera2000.pdf McKeny, T.S. (2006). A case-study analysis of the critical features within field experiences that affect the reflective development of preservice secondary mathematics teachers. Unpublished doctoral thesis. Ohio State University, Columbus, OH, United States. Mewborn, D.S. (1999). Reflective thinking among preservice elementary mathematics teachers. Journal for Research in Mathematics Education, 30, 316−341. http:// dx.doi.org/10.2307/749838

Acknowledgements

Meyer, R.D., & Wilkerson, T.L. (2011). Lesson study: The impact on teachers’ knowledge for teaching mathematics. In L.C. Hart, A. Alston, & A. Murata (Eds.), Lesson study research and practice in mathematics education (pp. 15−26). Dordrecht: Springer. http://dx.doi.org/10.1007/978-90-481-9941-9_2

I would like to acknowledge the support of my supervisor, Prof. Kobus Maree and co-supervisor, Dr Gerrit Stols in the completion of this research. In addition, I thank Prof. Hannes van der Walt from North-West University as well as SANPAD (South Africa Netherlands Research Programme on Alternatives in Development) for supporting me in writing this article.

Murata, A. (2011). Introduction: Conceptual overview of lesson study. In L.C. Hart, A. Alston, & A. Murata (Eds.), Lesson study research and practice in mathematics education (pp. 1−12). Dordrecht: Springer. http://dx.doi.org/10.1007/978-90481-9941-9_1 Nieuwenhuis, J. (2010). Qualitative research designs and data gathering techniques. In K. Maree (Ed.). First steps in research (pp. 69−97). Pretoria: Van Schaik Publishers. Olsen, J.C., White, P., & Sparrow, L. (2011). Influence of lesson study on teachers’ mathematics pedagogy. In L.C. Hart, A. Alston, & A. Murata (Eds.), Lesson study research and practice in mathematics education (pp. 39−57). Dordrecht: Springer. http://dx.doi.org/10.1007/978-90-481-9941-9_4

Competing interests

Ottesen, E. (2007). Reflection in teacher education. Reflective Practice, 8(1), 31−46. http://dx.doi.org/10.1080/14623940601138899

I declare that I have no financial or personal relationship(s) which may have inappropriately influenced the writing of this article.

Pedro, J.Y. (2001). Reflection in teacher education: Exploring preservice teachers’ meanings of reflective practice. Unpublished doctoral dissertation. Virginia Polytechnic Institute and State University, Blacksburg, VA, United States. Pollard, A. (2002). Reflective teaching: Effective and evidence-informed professional practice. London: Continuum.

References

Posthuma, A.B. (2011). The nature of mathematics teachers’ reflective practice. Unpublished doctoral dissertation. University of Pretoria, Pretoria, South Africa. Available from http://upetd.up.ac.za/thesis/available/etd-04252012-164207/

Artzt, A.F., Armour-Thomas, E., & Curcio, F.R. (2008). Becoming a reflective mathematics teacher. New York, NY: Lawrence Erlbaum Associates.

Prus, R. (1996). Symbolic interaction and ethonographic research: Intersubjectivity and the study of human lived experience. New York, NY: State University of New York Press.

ATLAS.ti Scientific Software Development GmbH. (2010). ATLAS.ti 6 Qualitative Data Analysis Software. Berlin: ATLAS.ti Scientific Software Development GmbH. Available from http://www.atlasti.com

Rodgers, C. (2002). Defining reflection: Another look at John Dewey and reflective thinking. Teachers College Record, 104(4), 842−86. http://dx.doi. org/10.1111/1467-9620.00181

Blumer, H. (1969). Symbolic interactionism: Perspective and method. Englewood Cliffs, NJ: Prentice-Hall, Inc.

Schön, D.A. (1983). The reflective practitioner. San Francisco, CA: Jossey-Bass.

Carr, W., & Kemmis, S. (1986). Becoming critical: Education, knowledge and action research. Sussex: Falmer Press.

Setati, M., Adler, J., Reed, Y., & Bapoo, A. (2002). Incomplete journeys: Code-switching and other language practices in Mathematics, Science and English language classrooms in South Africa. Language and Education, 16, 128−149. http://dx.doi. org/10.1080/09500780208666824

Coe, K., Carl, A., & Frick, L. (2010). Lesson study in continuing professional teacher development: A South African case study. Acta Academica, 42(4), 206−230. Available from http://journals.sabinet.co.za/WebZ/images/ejour/academ/academ_ v42_n4_a8.pdf?sessionid=01-59893-1679460615&format=F

Sowder, J.T. (2007). The mathematical education and development of teachers. In F.K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 157−223). Charlotte, NC: Information Age.

Cohen, L., Manion, L., & Morrison, K. (2005). Research methods in education. (5th edn.). London: Routledge Falmer.


61


Page 9 of 9

Original Research

Sparks-Langer, G., & Colton, A. (1991). Synthesis of research on teachers’ reflective thinking. Educational Leadership, 48(6), 37–44.

Ward, J.R., & McCotter, S.S. (2004). Reflection as a visible outcome for preservice teachers. Teaching and Teacher Education, 20(3), 243–257.

Takahashi, A., Watanabe, T., & Yoshida, M. (2006). Developing good mathematics teaching practice through lesson study: A U.S. perspective. Tsukuba Journal of Educational Study in Mathematics, 25, 197−204. Available from http://www. human.tsukuba.ac.jp/~mathedu/2516.pdf

Warfield, J., Wood, T., & Lehman, J.D. (2005). Autonomy, beliefs and the learning of elementary mathematics teachers. Teaching and Teacher Education, 21, 439−456. http://dx.doi.org/10.1016/j.tate.2005.01.011 York-Barr, J., Sommers, W.A., Ghere, G.S., & Montie, J. (2006). Reflective practice to improve schools. Thousand Oaks, CA: Corwin Press.

Taylor, A.R., Anderson, S., Meyer, K., Wagner, M.K., & West, C. (2005). Lesson study: A professional development model for mathematics reform. The Rural Educator, Winter 2005, 17−22. Available from http://www.ruraleducator.net/ archive/26-2/26-2_Taylor.pdf

Zeichner, K.M., & Liston, D.P. (1996). Reflective teaching: An introduction. Mahwah, NJ: Lawrence Erlbaum Associates, Inc.

Valli, L. (1992). Reflective teacher education: Cases and critiques. Albany, NY: State University of New York Press.

Zeichner, K., & Tabachnick, B.R. (1991). Reflections on reflective thinking. In B.R. Tabachnick, & K. Zeichner (Eds.), Issues and practices in inquiry-oriented teacher education (pp. 1−21). London: Falmer Press.

Valli, L. (1997). Listening to other voices: A description of teacher reflection in the United States. Peabody Journal of Education, 72(1), 67−88. http://dx.doi. org/10.1207/s15327930pje7201_4

Zwodiak-Myers, P. (2009). An analysis of the concept reflective practice and an investigation into the development of student teachers’ reflective practice within the context of action research. Unpublished doctoral dissertation. Brunel University, Middlesex, United Kingdom.

Van Manen, M. (1977). Linking ways of knowing with ways of being practical. Curriculum Inquiry, 6, 205−228. http://dx.doi.org/10.2307/1179579


62


Page 1 of 9

Original Research

Alignment between South African mathematics assessment standards and the TIMSS assessment frameworks Authors: Mdutshekelwa Ndlovu1,2 Andile Mji2 Affiliations: 1 Institute for Mathematics and Science Teaching (IMSTUS), Department of Curriculum Studies, University of Stellenbosch, South Africa Faculty of Humanities, Tshwane University of Technology, South Africa

2

Correspondence to: Mdutshekelwa Ndlovu Email: [email protected] Postal address: Private Bag X1, Matieland 7602, South Africa Dates: Received: 28 June 2012 Accepted: 24 Oct. 2012 Published: 10 Dec. 2012 How to cite this article: Ndlovu, M., & Mji, A. (2012). Alignment between South African mathematics assessment standards and the TIMSS assessment frameworks. Pythagoras, 33(3), Art. #182, 9 pages. http://dx.doi.org/10.4102/ pythagoras.v33i3.182


South Africa’s performance in international benchmark tests is a major cause for concern amongst educators and policymakers, raising questions about the effectiveness of the curriculum reform efforts of the democratic era. The purpose of the study reported in this article was to investigate the degree of alignment between the TIMSS 2003 Grade 8 Mathematics assessment frameworks and the Revised National Curriculum Statements (RNCS) assessment standards for Grade 8 Mathematics, later revised to become the Curriculum and Assessment Policy Statements (CAPS). Such an investigation could help to partly shed light on why South African learners do not perform well and point out discrepancies that need to be attended to. The methodology of document analysis was adopted for the study, with the RNCS and the TIMSS 2003 Grade 8 Mathematics frameworks forming the principal documents. Porter’s moderately complex index of alignment was adopted for its simplicity. The computed index of 0.751 for the alignment between the RNCS assessment standards and the TIMSS assessment objectives was found to be significantly statistically low, at the alpha level of 0.05, according to Fulmer’s critical values for 20 cells and 90 or 120 standard points. The study suggests that inadequate attention has been paid to the alignment of the South African mathematics curriculum to the successive TIMSS assessment frameworks in terms of the cognitive level descriptions. The study recommends that participation in TIMSS should rigorously and critically inform ongoing curriculum reform efforts.

Introduction Hencke, Rutkowski, Neuschmidt and Gonzalez (2009) make the important remark that the Trends in International Mathematics and Science Study (TIMSS) examines the effectiveness of curriculum and instruction in relation to student achievement. There is increasing global interest in and attention paid to the resultant rankings of participating countries, making the very participation in TIMSS a high-stake local decision. As a consequence of the heightened (political and educational) stakes, the relevance of the tests to local curricula has come under sharp scrutiny, which makes the issue of alignment of the South African (SA) curriculum with TIMSS important for educators, curriculum workers, test developers and policymakers. Hencke et al. (2009) concede upfront that whilst TIMSS assessments were developed to represent an agreed-upon framework with as much in common across countries as possible, it was inevitable that the match between test and curriculum would not be identical in all countries. However, the more aligned a national curriculum is to what is common across countries the greater the chance of that country’s students performing well. In other words, rather than reject the common core assessments as irrelevant it might be beneficial to investigate in depth what discrepancies exist between SA’s curricula and TIMSS, with special focus on the overlapping content. Mullis, Martin, Ruddock, O’Sullivan and Preuschoff (2009) refer to the TIMSS curriculum model as consisting of an intended curriculum, an implemented curriculum and an attained curriculum, all of which are familiar terms in curriculum theory. For instance, Porter (2004, p. 1) suggests that a curriculum can be divided into four aspects: the intended, enacted, assessed and learned curriculum. The enacted curriculum refers to instructional events in the classroom whereas the assessed curriculum refers to student achievement tests. Mullis et al.’s (2009) attained curriculum refers to student achievement in those tests. For cross-national tests such as TIMSS to be valid, it is critical that their assessed curricula correspond with the intended national curricula. Moreover, assessments aligned with the assessment standards can guide instruction and raise achievement (Martone & Sireci, 2009; Polikoff, Porter & Smithson, 2011). In view of the foregoing it is expected that, in order to be relevant, cross-national studies or tests should provide curriculum information that can help countries to improve the quality of their education systems on the basis of benchmarking performance (Reddy, 2006). This makes curriculum matching analysis a logical starting point.


63


Page 2 of 9

Original Research

term ‘outcomes’ has been used widely to frame statements about both subject matter content and anticipated learning behaviours.

In bemoaning the absence of extensive use of alignment research in the classroom, Martone and Sireci (2009) point out lost opportunities to help policymakers, assessment developers and educators to make refinements so curriculum, assessment and instruction support each other in achieving what is expected of students. In an attempt to bridge this gap the aim of the study was to analyse the alignment between SA’s Grade 8 Mathematics curriculum and TIMSS by means of the Porter (2002) procedure. To achieve this goal the remainder of this article gives the theoretical background to alignment studies in general and shows why the Porter index was chosen. Thereafter we spell out the research questions guiding the study and outline the procedure for determining the index before presenting and discussing the results. The article concludes with summary observations and recommendations.

Porter’s model for evaluating alignment From the three commonly used primary models of evaluating alignment, that is Webb’s (1997, 2005) Depth of Knowledge Procedure, Rothman, Slattery, Vranek, and Resnick’s (2002) Achieve Procedure model and Porter’s (2002) Surveys of Enacted Curriculum index, we opted for the last one. Unlike the other two approaches, the Surveys of Enacted Curriculum index does not rely on direct comparison of assessments or assessment items with objectives or standards. Instead, content analysts first code the standards and assessments onto a common framework, a content taxonomy, developed by subject matter experts. The taxonomy defines content in terms of two variables: topics or sub-topics and levels of cognitive demand. The two variables compare favourably with Webb’s Categorical Concurrence and Performance Centrality. Analysts place assessment items and objectives from standards documents into the taxonomy and the documents are then represented as matrices of proportions, where the proportion in each cell (topic and cognitive demand) indicates the proportion of total content in the document that emphasises that particular combination of topic and cognitive demand. The matrices for standards and assessments are then compared, cell by cell, and an alignment index is calculated. We believe that the Porter procedure achieves in two dimensions what the Webb and Achieve procedures do in four measures. More importantly, the Porter alignment model ‘can be applied to analyse the match between any two of curriculum, instruction and assessment’ (Liu et al., 2009, p. 795). It was therefore appropriate for our purpose since we wanted to compare two curriculum documents: the TIMSS frameworks and RNCS.

Theoretical framework of alignment studies Definition of alignment and scope of alignment studies

For purposes of comparing the Grade 8 Revised National Curriculum Statements (RNCS) for Mathematics and the TIMSS assessment we analyse measures of curricula and assessment alignment based on research that has developed methods for judging the extent and nature of alignment (e.g. Porter, 2002; Porter & Smithson, 2001; Webb, 2005). Alignment can be defined as the degree of agreement, match or measure of consistency between curriculum content (content standards) for a specific subject area and the assessment(s) used to measure student achievement of these standards (Bhola, Impara & Buckendahl, 2003; Näsström, 2008; Näsström & Henricksson, 2008). A major feature of alignment studies is the development of common languages of topics and categories of cognitive demand for describing content in different subject areas such mathematics, reading and science (Berends, Stein & Smithson, 2009, p. 4). The underlying logic is that if standards specify what and how well students should be learning and tests measure what they know and can do, then the two ought to be synchronised (Herman & Webb, 2007, p. 1). In other words, the language of the assessment items must match the language of the outcomes stated in the RNCS or its successor, Curriculum and Assessment Policy Statements (CAPS) (Department of Basic Education, 2011a). Similarly, the content and cognitive domain language of the CAPS should match that of the TIMSS assessment frameworks as closely as possible. Alignment, thus, has both content and consequential validity in terms of the knowledge and skills prescribed and tested (Bhola et al., 2003, p. 21) Although the alignment between standards and assessment has been most commonly studied (e.g. Bhola et al., 2003; Herman & Webb, 2007), the alignments between standards and instruction as well as between instruction and assessment have also been studied (e.g. Porter, 2002). In curriculum theory and practice, standards have lately come to refer to ‘descriptions of what students are expected to know and be able to do’ (Näsström, 2008, p. 16), which makes them synonymous with the intended relationship between educational objectives and subject matter content. In SA, the http://www.pythagoras.org.za

The calculated Porter alignment index ranges from 0 to 1 with 0.5 as its centre since it uses absolute differences, a characteristic that has to be taken into account when interpreting the computed values. Fulmer (2011) has recently provided critical values for the strength of the Porter index of alignment based on the number of cells and number of standard points used. Furthermore, the Porter procedure agrees with Bloom’s Taxonomy of educational objectives which has also been used by TIMSS and the RNCS.

Purpose of the study The purpose of this study was to determine the degree of alignment between the Grade 8 RNCS for Mathematics and the 2003 TIMSS assessment frameworks by means of the Porter index. TIMSS regularly assesses learners at the Grade 4 and 8 levels and SA has previously participated (in 1995, 1999, 2003 and 2011). The Grade 12 level has not been consistently assessed. We chose the Grade 8 curriculum because it is a transitional grade between primary and secondary phases. The 2003 results were the latest available of South Africa’s participation in TIMSS because the 2011 results were still pending at the time of this article. The following research questions guided the study: 64


Page 3 of 9

Original Research

• What is the structure of the content and cognitive domain matrices for the components of the 2003 TIMSS Grade 8 Mathematics assessment frameworks? • What is the structure of the content and cognitive domain matrix for the RNCS Grade 8 Mathematics assessment standards? • What are the computed Porter indices of alignment within and between the components of the 2003 TIMSS assessment frameworks and the RNCS assessment standards for Grade 8 Mathematics? • What is the structure of discrepancies in emphasis between the RNCS assessment standards and the 2003 TIMSS assessment objectives? • How do the discrepancies assessment objectives compare with SA’s performance in TIMSS 2003?

points, 5 integer points, 10 fractions and decimals points, 2 irrational number and financial mathematics points, and 7 ratio, proportion and percentage points); Algebra: 26 (3 patterns, 7 algebraic expressions, 7 equations and formulas, 9 relationships/functions); Measurement: 16 (3 attributes and units, 13 tools, techniques and formulae); Geometry: 18 (3 lines and angles, 6 two-dimensional and three-dimensional shapes, 3 congruence and similarity, 4 location and spatial relationships, 3 symmetry and transformations); and Data: 17 (4 data collection and organisation, 4 data representation, 5 data interpretation, 4 uncertainty and probability). In the common template, 88% of the RNCS assessment standards were covered whilst 89% of the TIMSS assessment objectives were covered. The facilitators were introduced to the mathematics cognitive domain categories used in TIMSS 2003 (Mullis et al., 2003, pp. 27−33) and asked to code the cognitive domain levels elicited by each standard point in the template according to the verbs used (see Table 1). Following the level descriptors, the two facilitators independently coded the standards and the author allocated marks according to their coding, totalling 1 score point per standard point. Table 1 shows the content and cognitive domain categories, the weightings and the verbs or descriptors that characterise the cognitive levels.

To help answer these questions we adopted the document analysis methodology in this study.

Methodology

The document analysis methodology The methodology of document analysis was adopted for this study as it entails systematic and critical examination rather than mere description of instructional or curriculum documents (Center for Teaching and Learning, 2007). Document analysis is also referred to as qualitative content analysis (Daymon & Holloway, 2011), an analytical method used in qualitative research to gain an understanding of trends and patterns that emerge from data. The aim of qualitative document analysis is to discover new or emergent patterns, including overlooked categories (Daymon & Holloway, 2011, p. 321). Statistical reports within a qualitative study should reveal ways in which the data and statistics have been organised and presented to convey the key messages and meanings intended. The qualitative document analysis in this study is organised and presented statistically by means of the Porter alignment procedure to convey messages and meanings about the strength of the alignment between the TIMSS (2003) mathematics assessment frameworks and the RNCS for Grade 8 Mathematics.

Table 2 shows an example of five selected RNCS standard points coded following Airasian and Miranda’s (2002, pp. 251−253) procedure of coding objectives according to the Revised Bloom’s Taxonomy of educational objectives. Assessment standard point 1.1, for example, uses three verbs, the first of which is in the ‘knowing’ category and the other TABLE 1: Content and cognitive domains for mathematics used in TIMSS 2003. TIMSS 2003 content domains

TIMSS 2003 cognitive domain levels

Number (30%) Algebra (25%) Measurement (15%) Geometry (15%) Data (15%)†

Knowing facts and procedures (15%) (recall, recognise/identify, compute, use tools) Using concepts (20%) (know, classify, represent, formulate, distinguish) Solving routine problems (40%)† (select, model, interpret, apply, verify/check)‡ Reasoning (25%) (hypothesise/conjecture/predict, analyse, evaluate, generalise, connect, synthesise/ integrate, solve non-routine problems, justify/ prove)‡

For the empirical work the first author worked with two experienced mathematics in-service facilitators for the Senior Phase and the Further Education and Training phase. The Grade 8 RNCS Mathematics assessment standards were compared with the TIMSS assessment objectives and a common template consisting of 110 standard points or fragments as follows: Number: 32 (8 whole number

†, See Martin, M.O., Mullis, I.V., & Chrostowski, S.J. (2004). TIMSS 2003 technical report. Chestnut Hill, MA: International Association for the Evaluation of Educational Achievement. Available from http://timss.bc.edu/PDF/t03_download/T03TECHRPT.pdf ‡, See Mullis, I.V., Martin, M.O., Ruddock, G.J.,O’Sullivan, C.Y., & Preuschoff, C. (2009). TIMSS 2011 assessment frameworks. Chestnut Hill, MA: International Association for the Evaluation of Educational Achievement. Available from http://timssandpirls.bc.edu/timss2011/ downloads/TIMSS2011_Frameworks.pdf for more detailed examples of these verbs.

TABLE 2: Coding of the RNCS assessment standards according to TIMSS 2003 cognitive levels. Standard point examples

Sample common template standard points: The student …

Knowing facts and procedures

Using concepts

Solving routine problems

Reasoning

Score points

1.1

Recognises, classifies, and represents to describe and compare integers

X

XX

3 × 0.333 = 1

1.2

Recognises, classifies, and represents to describe and compare decimal fractions

X

XX

3 × 0.333 = 1

2.1

Solves problems that involve ratio

3.1

Designs and uses questionnaires with a variety of possible responses to collect data

4.1

Investigates and extends numeric and geometric patterns, relationships or rules

Score points

X

0.667


65

1.833

X

1

X

2 × 0.5 = 1

1.5


X

1

1

-

Page 4 of 9

Original Research

two are both in the ‘using concepts’ category. Standard point 1.2 uses the same three verbs but for a different content sub-topic (fractions instead of integers); standard point 2.1 solves problems and standard point 4.1 investigates and extends at the reasoning level. Reliability was assured by the independent coding of the experts who were given copies of the relevant pages for the classification of TIMSS assessment objectives as they appear in Mullis et al. (2003, pp. 27−33) and implored to adhere to these as closely as possible. (The interrater kappa reliability index could not be computed because it applies to items falling in mutually exclusive categories.)

1. Frequency matrices Matrix X

Matrix Y

4

6

6

4

7

3

3

7

2. Ratio matrices Matrix x

Matrix y

0.2

0.3

0.3

0.2

0.35

0.15

0.15

0.35

3. Absolute discrepancies |xjk − yjk|

Computation of the Porter index

∑ k =1

1−

K

J

k =1

j

jk

− b jk

2 0.70

FIGURE 1: Porter alignment index example calculation for 2 × 2 matrices. TABLE 3: Sample mean alignment indices by number of cells and standard points. Cells 10 20 30 50 70 90 100

J

30 0.9464 0.8674 0.8054 0.6587 0.5451 0.4614 0.4309

Standards points 60 90 0.9782 0.9875 0.9428 0.9635 0.8974 0.9291 0.7958 0.8384 0.7039 0.7478 0.6258 0.6633 0.5908 0.6276

120 0.9916 0.9737 0.9438 0.8553 0.7600 0.6716 0.6337

Source: Adapted from Fulmer, G.W. (2011). Estimating critical values for strength of alignment among curriculum, assessments and instruction. Journal of Educational and Behavioural Statistics, 36(3), 381−402. http://dx.doi.org/10.3102/1076998610381397

∑ a jk − b jk j

, where J is the number of rows, K is 2 the number of columns in each of matrices X and Y, and xjk and yjk are ratios of points in the cells at row j and column k for each of ratio matrices x and y respectively.

values as determined by the number of cells and standard points. Table 4 shows sample reference (or critical) value estimates from the corresponding number of cells and standards points.

Critical values for the strength of the alignment index

From results presented by Porter (2002), for instance, the alignment between the standards of four US states (and the NCTM) and their own assessments ranged from 0.30 to 0.47 for 30 standards points. Six content areas and five cognitive levels were used, which meant 30 squares made up matrices A and B. Table 4 gives a critical value of 0.7372 for 0.05 = 0.025 ) if a two-tailed test is used the lower quantile ( 2 at the alpha level of 0.05 (i.e. lower than might be expected by chance). Therefore one can conclude that alignment amongst assessment and standards was very low. Liu et al. (2009) used a coding structure with five content categories and six cognitive levels (hence 30 squares again) to compare the alignment of physics curriculum and assessments for China, Singapore and New York state, China and Singapore had alignments of 0.67, which were significantly lower than the 0.7372 + 0.8667 = 0.80195 ) at the 0.05 level (below the mean ( 2 critical value of 0.7372); New York’s alignment index of 0.80 was equivalent to the mean.

A greater number of cells in the matrices will yield a range of likely values that is lower than for matrices with fewer cells. Hence the total number of cells in the A and B matrices can have an effect on the significance of the alignment index. When we also consider that the centre of the distribution of indices is not zero, as noted earlier, we need to assess how far an observed alignment index is from 0.5. Fulmer (2011) generated a matrix of means and critical values for alignment indices with results also demonstrating the expected (mean) distribution of pattern of alignment indices (see sample entries in Table 3). In addition to matrix-size dependence, the alignment index also depends on the number of curriculum or standards statements or test items being coded. If the total number of cells in the matrix is N (= J × K) then for matrices A and B, J = 2 and K = 2 yields N = 4. In this study we used Fulmer’s (2011) estimates of the critical


0.2

∑ ∑a

1. Create matrices of frequencies for the two documents being compared and label these as X and Y. 2. For each cell in matrices X and Y, compute the ratio of points in the cell with the total number of points in the respective matrix. Label the matrices of ratios as x and y. 3. For every row j and column k in matrices X and Y (the matrices of ratios), calculate the absolute value of the discrepancy between the ratios in cells xjk and yjk. 4. Compute the alignment index using the formula

P =1−

0.1

0.2

4. Alignment index

As already noted, the Porter procedure analyses the extent of alignment between two matrices or matrices of frequencies (Fulmer, 2011, p. 384). It produces a single alignment index, ranging from 0 to 1 to indicate how closely the distribution of points in the first matrix (of standards) aligns with the second matrix (of assessment). The alignment index P is arrived at in four steps as shown in Figure 1:

K

0.1

66


Page 5 of 9

Original Research

TABLE 4: Sample reference values for indices of alignment by number of cells and standard points. Cells 10 30 60 90

0.025 0.9167 0.7372 0.5211 0.3684

0.050 0.9250 0.7500 0.5342 0.3836

20 50 70 100

0.9100 0.7225 0.6137 0.4833

0.9200 0.7412 0.6333 0.5021

10 20 50 90

0.9811 0.9302 0.7537 0.5556

0.9822 0.9404 0.7696 0.5773

20 60 90 114

0.9333 0.7083 0.5667 0.4833

0.9417 0.7250 0.5833 0.5000

Quantiles for 30 standards points 0.100 0.900 0.9250 0.9667 0.7643 0.8500 0.5479 0.6438 0.4026 0.5176 Quantiles for 60 standards points 0.9288 0.9583 0.7577 0.8313 0.6521 0.7548 0.5230 0.6549 Quantiles for 90 standards points 0.9833 0.9911 0.9535 0.9727 0.7894 0.8786 0.5889 0.7273 Quantiles for 120 standards points 0.9620 0.9821 0.7495 0.8616 0.6000 0.7391 0.5167 0.6500

0.950 0.9667 0.8565 0.6571 0.5316

0.975 0.9750 0.8667 0.6667 0.5429

0.9610 0.8394 0.7657 0.6722

0.9625 0.8455 0.7758 0.6859

0.9922 0.9747 0.8869 0.7421

0.9933 0.9758 0.8931 0.7556

0.9833 0.8720 0.7551 0.6728

0.9845 0.8801 0.7312 0.6877

Source: Adapted from Fulmer, G.W. (2011). Estimating critical values for strength of alignment among curriculum, assessments and instruction. Journal of Educational and Behavioural Statistics, 36(3), 381−402. http://dx.doi.org/10.3102/1076998610381397

Results and discussion

TABLE 5: Results of coding the TIMSS 2003 assessment objectives by content and cognitive domain.

The structure of the content and cognitive domain matrices for the components of the 2003 Grade 8 TIMSS Mathematics assessment frameworks

Content domain

Three matrices were derived in respect of the three components of the TIMSS assessment frameworks: the TIMSS 2003 assessment objectives, the TIMSS 2003 target percentages and the released TIMSS 2003 test items. The content and cognitive domain matrix for the TIMSS 2003 assessment objectives was derived from the list of objectives given in the TIMSS assessment frameworks document (Mullis et al., 2003, pp. 27−33). Table 5 shows the results of the coding of the 98 (fine-grained) TIMSS objectives. The numerical values form the required matrix. The 98 fine-grained objectives were accorded equal weight guided by the estimated time to be devoted to each of them. Of the 110 fine-grained standard points (assessment objectives or standards) in the template, 12 were not amongst the TIMSS assessment objectives.

Using concepts

Routine problem solving

Reasoning

Number

0.141

0.073

0.068

0.000

Algebra

0.096

0.044

0.081

0.018

Measurement

0.016

0.000

0.141

0.000

Geometry

0.117

0.044

0.013

0.008

Data

0.019

0.055

0.043

0.040

TABLE 6: Derived target percentages of TIMSS 2003 mathematics assessment devoted to content and cognitive domain by grade level. Content domain


Using concepts


Reasoning

Number

0.045

0.060

0.120

0.075

Algebra

0.075

0.050

0.100

0.063

Measurement

0.023

0.030

0.060

0.038

Geometry

0.023

0.030

0.060

0.038

Data

0.023

0.030

0.060

0.038

TABLE 7: TIMSS 2003 Grade 8 Mathematics content and cognitive domain matrix for test items. Content domain

The content and cognitive domain matrix for the TIMSS 2003 target percentages (Table 6) was computed by extrapolation from Exhibit 2 (Mullis et al., 2003, p. 9) showing the target percentages of TIMSS 2003 mathematics assessment time devoted to content and cognitive domain for the Grade 8 level. Time devoted was assumed to be equivalent to the importance attached to the respective categories in the frameworks as underpinned by the respective objectives.


Using concepts


Reasoning

Number

0.070

0.051

0.125

0.032

Algebra

0.060

0.056

0.047

0.084

Measurement

0.042

0.009

0.070

0.037

Geometry

0.032

0.037

0.047

0.042

Data

0.005

0.028

0.065

0.060

Mullis & Chrostowski, 2004, p. 35). The results of the tallying (Table 7) agreed with Exhibit 2.24 in Martin et al. (2004, p. 60) and were converted to proportions.

The content and cognitive domain matrix (Table 7) for the released TIMSS 2003 test items was derived from tallying the coding of all the released TIMSS 2003 test items. The assumption was that the items were accurately coded and accurately reported on. In the coding process multiple-choice items were allocated one score point each whilst constructed response items were allocated marks, depending on the amount of work to be done, so that at least one third of the assessment came from constructed response items (Martin, http://www.pythagoras.org.za


The structure of the content and cognitive domain matrix for the Grade 8 RNCS for Mathematics Table 8 shows the content and cognitive domain matrix obtained for SA’s Grade 8 Mathematics curriculum. All of the 98 RNCS assessment standards whose content was covered by TIMSS were coded. The resultant score points were 67


Page 6 of 9

Original Research

totalled for each content and cognitive domain category and converted to the proportions shown.

TABLE 8: Grade 8 RNCS Mathematics content and cognitive domain matrix.

Computed Porter indices of alignment for this study Table 9 shows the calculated raw cell-by-cell differences between RNCS assessment standards and TIMSS assessment objectives. These raw differences were converted to absolute differences, from which the Porter index of alignment k =1

j

∑ ∑a was computed using the formula 1 −

jk


Reasoning

Number

0.141

0.073

0.068

0.000

Algebra

0.096

0.044

0.081

0.018

Measurement

0.016

0.000

0.141

0.000

Geometry

0.117

0.044

0.013

0.008

Data

0.019

0.055

0.043

0.040

TABLE 9: Raw cell-by-cell differences between the RNCS assessment standards and the TIMSS 2003 assessment objectives.

− b jk

Content domain

. The 2 computed index was 0.735. The mean-simulated alignment index for a 5 × 4 comparison with 20 cells is 0.9635 (see Table 3). Using a two-tailed test, at the 0.05 alpha level, we looked to the 0.025 and 0.975 quantiles in Table 4. Close to 100 standards points and matrices of 20 squares each were used, so the critical values for 90 standard points are 0.9302 and 0.9758 respectively whilst those for 120 standard points are 0.9333 and 0.9843 respectively. The computed alignment value is well below 0.9302 and 0.9333 in the 0.025 quantiles 0.05 ( = 0.025 ). The alignment was therefore significantly 2 lower than would be expected by chance at the 0.05 level.


Using concepts


Number

-0.050

-0.003

0.012

0.002

Algebra

-0.077

-0.002

0.002

0.071

Measurement

Reasoning

0.009

0.012

-0.031

0.000

Geometry

-0.049

-0.008

-0.008

0.048

Data

-0.012

-0.047

0.042

0.006

TABLE 10: Porter indices of alignment amongst the TIMSS components and the RNCS. No.

Further iterations were conducted to determine the pair-wise alignment amongst the RNCS assessment standards and the three components of the TIMSS 2003 assessment frameworks. The indices obtained are as shown in Table 10. Surprisingly the pair-wise alignment is significantly lower in all instances, without exception, suggesting a low internal consistency even amongst the TIMSS components themselves.

1

2

3

4

1

RNCS assessment standards

Curriculum Component

1.000

0.751

0.758

0.698

2

TIMSS 2003 assessment objectives

0.751

1.000

0.647

0.647

3

TIMSS 2003 test items

0.758

0.647

1.000

0.845

4

TIMSS 2003 target percentages

0.698

0.647

0.845

1.000

Knowing facts and procedures Discrepancies between RNCS and TIMSS 2003 assessment objectives 2. 1.

0.080 0.080

Using concepts Routine problem solving 4. Reasoning 3.

4

Cell-by-cell differences

0.060 0.060

The structure of discrepancies between TIMSS 2003 framework components and RNCS assessment standards

4

3

0.040 0.040 0.020 0.020

3

-0.020 -0.020 -0.040 -0.040 -0.060 -0.060

1

4

0.000 0.000

2

4

3 2

2

Number

2

Algebra

3

1

Geometry

Measurement

Data

3 2

1

1

-0.080 -0.080

The first structure of discrepancies investigated was between RNCS assessments standards and the TIMSS assessment objectives. Figure 2 shows the structure of (mis)alignment by content and cognitive domain.

1

-0.100 -0.100

and cognitive domain categories ContentContent and cognitive domain categories

FIGURE 2: Discrepancies between RNCS assessment standards and the 2003 TIMSS assessment objectives.

From the graph it is evident that the RNCS assessment standards were stronger than TIMSS assessment objectives in all cases where the bars extend upwards above zero but weaker in those cases where the bars extend downwards below zero. Whilst there is a common criticism of teachers concentrating on knowledge of facts and procedures, the tables shows that the RNCS was weaker than TIMSS objectives in this cognitive level in four of the content domains, namely Number, Algebra, Geometry and Data. The RNCS was stronger with respect to routine problem solving in Number and Data but weaker in Measurement and Geometry. A similarly mixed picture emerged in respect of Reasoning.

1.


Discrepancies between RNCS and TIMSSUsing 2003concepts targets 2. 3.

0.060 0.060


0.040 0.040 0.020 0.020

4.

3

1

-0.040 -0.040

-0.080 -0.080

1

3 4

2

Number

2

Algebra 3

4

2

1

-0.000 0.000 -0.020 -0.020

Routine problem solving Reasoning

4

Measurement 2

Geometry

1

Data 2

4

3

-0.060 -0.060

The second structure to be investigated was between the RNCS objectives and the TIMSS target percentages. Figure 3 summarises the structure. Surprisingly, the RNCS was stronger on knowledge of facts and procedures in http://www.pythagoras.org.za

Using concepts

Cell by cell diffferences

J


Cell by cell differences

K

Content domain

1

3

4

Content andandcognitive domain Content cognitive domains

FIGURE 3: Discrepancies between RNCS assessment standards and TIMSS target percentages.

68


Page 7 of 9

Original Research

above in Algebra, Measurement and Data, but below in Number and Geometry.

Number, Measurement and Geometry but weaker in Algebra and Data. Taken together, these discrepancies were significant.

The next logical question is whether there was any relationship between the structure of discrepancies and South African students’ performance in TIMSS 2003. Given the discrepancies within the TIMSS components themselves, the ultimate question is whether the discrepancies between the RNCS and the TIMSS 2003 test items had any correlation with student performance as they were partially in force at the time of the 2003 TIMSS assessments.

A third and final structure to be investigated was that between the RNCS and TIMSS test items. Figure 4 summarises the discrepancies. A marked shift in this comparison is that the RNCS was stronger with respect to knowledge of facts and procedures in Number, Geometry and Data. The RNCS was, however, weaker in respect of Reasoning in all content categories. Routine problem solving was almost evenly split,

A comparison of the RNCS-TIMSS test discrepancies with South African students’ performance in TIMSS 2003

Discrepancies between RNCS and TIMSSKnowing 2003 test items facts and procedures 1. 2.

0.060 0.060

3. 4.

3

3

Cell by cell differences


0.040 0.040 1

0.020 0.020

Using concepts Routine problem solving Reasoning

Figure 5 was compiled after extracting South African students’ performance in each of the released test categories by content and cognitive domain relative to the international average. It is already well known that South African students performed below the international average across the board (e.g. Reddy, 2006).

1

3

2 4 4

2

0.000 -0.000 Number

-0.020 -0.020

Algebra 2

1

Measurement

Geometry

Data

1

4

2

Beyond that, however, the intention in this study was to additionally investigate if the pattern of discrepancies between RNCS assessment standards and TIMSS assessment objectives was in any way related to South African students’ performance (i.e. the achieved TIMSS curriculum).

4

-0.040 -0.040

4

1

3

-0.060 -0.060

3

Content cognitive domain Contentand and cognitive domains

FIGURE 4: Discrepancies between RNCS and assessment standards and TIMSS 2003 test items.

Table 11 and Figure 6 attempt to answer that question. It is evident that student performance correlated negatively with discrepancies in Number but positively with discrepancies in Algebra, Measurement, Geometry and Data. That is, the narrower (or positive) the discrepancy was, the closer the performance was to the international average in all content domains except Number.

Figure 5: SA learners' performanceKnowing inTIMSSfacts2003 and procedures Using concepts problem solving compared to the internationalRoutine average Reasoning 1. 2. 3. 4.

0.00 0.00 Cell-by-cell differences

-5.00 -5.00

Number

Measurement

Algebra

Geometry

Data

-10.00 -10.00 -15.00 -15.00 -20.00 -20.00 -25.00 -25.00

4

-35.00 -35.00

2 4

1

1 4

3 2

-30.00 -30.00

In Number, SA students performed worst in items on Using Concepts even though this was not the weakest cognitive domain representation in the RNCS assessment standards. In Algebra, SA students performed worst in the Knowledge of Facts and Procedures and this was the weakest category. In Measurement they performed the worst in Routine Problem Solving which was the weakest category of RNCS. In Geometry they performed worst in the Routine Problem Solving category which was the second weakest in the RNCS curriculum. In Data Handling they performed worst

3

4

3

2

3

1

2

4

3 2

1

Content and cognitive domain Knowing facts and procedrues Using concepts Routine problem solving

Reasoning

FIGURE 5: South African students’ performance by content and cognitive domains in TIMSS 2003 compared to the international average. TABLE 11: Comparison of RNCS-TIMSS discrepancy with SA performance in TIMSS 2003. Cognitive domain Number Algebra Measurement Geometry Data

Discrepancy

Knowledge of facts and procedures

RNCS-TIMSS discrepancy Deviation from international average RNCS-TIMSS discrepancy Deviation from international average RNCS-TIMSS discrepancy Deviation from international average RNCS-TIMSS discrepancy Deviation from international average

Using concepts


Reasoning

-0.044

0.001

0.017

0.003

-22.000

-27.429

-23.714

-18.667

-0.076

0.000

0.007

0.076

-32.000

-18.833

-29.833

-21.500

0.010

0.013

-0.025

0.000

-21.286

-28.500

-28.667

-14.000

-0.045

-0.006

-0.008

0.051

-27.250

-28.000

-28.667

-23.000

RNCS-TIMSS discrepancy

-0.011

-0.047

0.048

0.009

Deviation from international average

0.000

-31.500

-13.250

-24.167


69


Page 8 of 9

6a -0.050

-0.040

-0.030

-0.020

Original Research

in Using Concepts, which was also the weakest point in the RNCS. Overall, SA learners performed worst in Using Concepts, suggesting little conceptual understanding being achieved by the curriculum. Routine Problem Solving was second worst. This pattern has implications for the intended curriculum which determines what curriculum materials should emphasise and ultimately what teachers should teach in the classroom. Brijlall (2008) notes that the lack of problem-solving skills in SA may be a result of the way it has been taught in schools: individual solution by learners, presentation of abstract problems foreign to learners. There is little doubt that the ultimate answer lies in the implemented curriculum but what feeds into the implemented curriculum is the intended curriculum.

a

0.000 -0.010 0.000 -5.000

0.010

0.020

0.030

-10.000

6b

-15.000

-0.100

-0.050

-22.000

0.000 -20.000 0.000 -5.000 -25.000 -10.000 -30.000 -15.000

-18.667 0.050

0.100 -23.714

-27.429

6b -18.833

-20.000 -0.100

-0.050

-32.000

0.000 -25.000 0.000 -5.000 -30.000

-21.500 0.050

b

0.100

-29.833

Conclusions

-10.000 -35.000 -15.000 -18.833

-20.000

The study reported in this article set out to investigate the alignment of South Africa’s RNCS for Grade 8 Mathematics with the TIMSS 2003 Grade 8 Mathematics assessment frameworks. From the results we conclude that the computed Porter index of 0.751 suggests that the misalignment was low enough to warrant urgent attention, from curriculum designers, assessment practitioners, educators, teacher educators and policymakers alike, in order to enhance prospects of improved performance in future participations. In particular there is need to pay attention to the observed discrepancies between the content and cognitive domain emphases. The fact that, even where the RNCS curriculum was stronger than TIMSS, performance was still generally poor suggests the likelihood of a gap between the intended curriculum and the implemented curriculum. Such a gap further suggests a possible mismatch in emphasis between the intended curriculum and the curriculum support materials that actualise it. However, this conjecture requires further investigation. The study also points to the likelihood of a consequential gap between the implemented (SA) curriculum and the attained (TIMSS) curriculum reported by Reddy (2006, p. xiv). From a developing country perspective, what is even more disconcerting is that the three components of TIMSS do not appear to be aligned. That the misalignment is statistically significant calls into question the value-neutrality of TIMSS which currently appears to be a constantly shifting target that only well-resourced, developed countries can cope with.

-21.500

-25.000 -32.000

-29.833

-30.000 -35.000

6c -0.030

-0.020

c

0.000 0.000 -5.000

-0.010

0.010

0.020

-10.000 -14.000

-15.000

6c -20.000

-0.030

-0.060

-0.060

-0.020 -28.667

-0.040

-0.040 -28.667

-21.286 0.000 -0.010 -25.000 0.000 0.010 0.020 -5.000 -28.500 -30.000 -10.000 -35.000 0.000 -14.000 -15.000

6d

6d

-0.020 0.000 -5.000 -20.000 0.000 -0.020 0.000 -5.000 -15.000 -30.000 -10.000 -35.000 -20.000 -15.000

-10.000 -25.000

-25.000

-27.250

-20.000

0.040

0.060

-21.286

0.020

d

0.040 0.060 -28.500

-23.000 -23.000

-28.000 -28.667

-30.000

-25.000

-27.250

0.020

-35.000 -28.000

-30.000

-28.667

-35.000

-0.060

-0.040

-0.060

-0.040

0.000 0.000 -0.020 0.000 0.000 -5.000 0.000 -0.020 0.000 -5.000 -10.000

e 0.020 0.020

-10.000 -15.000

-25.000

-25.000

-31.500 -31.500

-30.000

0.040

0.060 0.060

-13.250

-15.000 -20.000 -20.000

0.040

Finally, participation in TIMSS should not be another bureaucratic ritual. Rather, it should rigorously and reflexively inform curriculum reform and innovation. In an increasingly globalised knowledge economy the school system needs to be globally competitive in the gateway fields of mathematics and science education. A simple illustration of the current disconnect is that, despite SA’s participation in previous TIMSS studies, the recently published Curriculum and Assessment Policy Statements (CAPS), which are largely a refinement of the RNCS, proclaim to have been influenced by the cognitive domain levels used in TIMSS 1999 (Department of Basic Education, 2011a, p. 55, 2011b, p. 59). This is so in spite of changes in 2003 (when the country participated in TIMSS for the third time) and further changes

-13.250

-24.167 -24.167

-30.000

-35.000

-35.000

FIGURE 6: Comparison of the RNCS-TIMSS discrepancy in assessment objectives and SA students’ performance in TIMSS 2003: (a) Number, (b) Algebra, (c) Measurement, (d) Geometry and (e) Data.


70


Page 9 of 9

Original Research

References

in 2007 (when the country did not participate), which were carried over to 2011. Accordingly, the influence of the Revised Bloom’s Taxonomy on TIMSS as evident in the 2007 and 2011 frameworks was apparently not taken into account in the latest curriculum revisions. This suggests that even the newly introduced Annual National Assessments (ANA) for Grades 1−6 and 9, together with the Senior National Certificate examinations, will continue to be guided by outof-sync domain categories.

Airasian, P.W., & Miranda, H. (2002). The role of assessment in the revised taxonomy. Theory into Practice, 41(4), 249−254. http://dx.doi.org/10.1207/ s15430421tip4104_8 Bansilal, S. (2011). Assessment reform in South Africa: Opening up or closing spaces for teachers? Educational Studies in Mathematics, 78, 91−107. http://dx.doi. org/10.1007/s10649-011-9311-8 Berends, M., Stein, M., & Smithson, J. (2009). Charter public schools and mathematics instruction: How aligned are charters to state standards and assessments? Nashville, TN: National Center on School Choice, Vanderbilt University. Bhola, D., Impara, J.C., & Buckendahl, C.W. (2003). Aligning tests with states’ content standards: Methods and issues. Educational Measurement: Issues and Practice, 22(3), 21−29. http://dx.doi.org/10.1111/j.1745-3992.2003.tb00134.x Brijlall, D. (2008). Collaborative learning in a multilingual class. Pythagoras, 68, 52−61. http://dx.doi.org/10.4102/pythagoras.v0i68.67 Center for Teaching and Learning. (2007). Instructional assessment resources: Document analysis. Austin, TX: The University of Texas at Austin. Available from http://www.utexas.edu/academic/ctl/assessment/iar/teaching/plan/method/ Daymon, C., & Holloway, I. (2011). Qualitative research methods in public relations and marketing communications. New York, NY: Routledge. Department of Basic Education. (2011a). Curriculumm and assessment policy statement: Mathematics Grades 7−9. Pretoria: DBE. Available from http://www.education.gov. za/LinkClick.aspx?fileticket=7AByaJ8dUrc%3d&tabid=672&mid=1885 Department of Basic Education. (2011b). Curriculum and assessment policy statement: Mathematics Grades 10−12. Pretoria: DBE. Available from http://www.education. gov.za/LinkClick.aspx?fileticket=QPqC7QbX75w%3d&tabid=420&mid=1216 Fulmer, G.W. (2011). Estimating critical values for strength of alignment among curriculum, assessments and instruction. Journal of Educational and Behavioural Statistics, 36(3), 381−402. http://dx.doi.org/10.3102/1076998610381397 Hencke, J., Rutkowski, L., Neuschmidt, O., & Gonzalez, E.J. (2009). Curriculum coverage and scale correlation on TIMSS 2003. In D. Hastedt, & M. von Davier (Eds.), IERI monograph series: Issues and methodologies in large-scale assessments, Vol. 2 (pp. 85−112). Princeton, NJ: IEA-ETS Research Institute. Herman, J.L., & Webb, N.M. (2007). Alignment methodologies. Applied Measurement in Education, 20, 1−5. Liu, X., Zhang, B.H., Liang, L.L., Fulmer, G.W., Kim, B., & Yuan, H.Q. (2009). Alignment between the physics content standards and standardized tests: A comparison among US-NY, Singapore and China-Jiangsu. Science Educaton, 93, 777−797. http://dx.doi.org/10.1002/sce.20330 Martin, M.O., Mullis, I.V., & Chrostowski, S.J. (2004). TIMSS 2003 technical report. Chestnut Hill, MA: International Association for the Evaluation of Educational Achievement. Available from http://timss.bc.edu/PDF/t03_download/T03TECHRPT.pdf Martone, A., & Sireci, S.G. (2009). Evaluating alignment between curriculum, assessment, and instruction. Review of Educational Research, 19(9), 11−16. http://dx.doi.org/10.3102/0034654309341375 Mullis, I.V., Martin, M.O., Ruddock, G.J., O’Sullivan, C.Y., & Preuschoff, C. (2009). TIMSS 2011 assessment frameworks. Chestnut Hill, MA: International Association for the Evaluation of Educational Achievement. Available from http://timssandpirls. bc.edu/timss2011/downloads/TIMSS2011_Frameworks.pdf Mullis, I.V., Martin, M.O., Smith, T.A., Garden, R.A., Gregory, K.D., Gonzalez, E.J., et al. (2003). TIMSS assessment frameworks and specifications 2003. (2nd edn.). Chestnust Hill, MA: International Association for the Evaluation of Educational Achievement. Available from http://timss.bc.edu/timss2003i/PDF/t03_af_book.pdf Näsström, G. (2008). Measurement of alignment between standards and assessment. Umeå, Sweden: Umeå universitet. Available from http://umu.diva-portal.org/ smash/get/diva2:142244/FULLTEXT01 Näsström, G., & Henricksson, W. (2008). Alignment of standards and assessment: A theoretical and empirical study of methods for alignment. Electronic Journal of Research in Educational Psychology, 6(3), 667−690. Polikoff, M.S., Porter, A.C., & Smithson, J. (2011). How well aligned are state assessments of student achievement with state content standards? American Educational Research Journal, 48(4), 965−995. http://dx.doi.org/10.3102/0002831211410684 Porter, A.C. (2002). Measuring the content of instruction: Uses in research and practice. Educational Researcher, 31(7), 3−14. http://dx.doi.org/10.3102/ 0013189X031007003 Porter, A.C. (2004). Curriculum assessment. Nashville, TN: Vanderbilt University. Available from http://datacenter.spps.org/uploads/curricassess.pdf Porter, A., & Smithson, J. (2001). Defining, developing, and using curriculum indicators. Philadelphia, PA: Consortium for Policy Research in Education, University of Pennsylvania. Available from http://www.cpre.org/sites/default/ files/researchreport/788_rr48.pdf Reddy, V. (2006). Mathematics and Science achievement at South African schools in TIMSS 2003. Cape Town: Human Sciences Research Council. Rothman, R., Slattery, J.B., Vranek, J.L., & Resnick, L.B. (2002). Benchmarking and alignment of standards and testing. Los Angeles, CA: Center for the Study of Evaluation. Schwab, K. (2012). The global competitiveness report 2012−2013. Geneva: World Economic Forum. Available from http://www3.weforum.org/docs/WEF_ GlobalCompetitivenessReport_2012-13.pdf Spaull, N. (2011). A preliminary analysis of SACMEQ III South Africa. Stellenbosch: Department of Economics, University of Stellenbosch. Available from http://www. ekon.sun.ac.za/wpapers/2011/wp112011/wp-11-2011.pdf Webb, N.L. (1997). Criteria for alignment of expectations and assessments in mathematics and science education. Madison, WI: National Institue for Science Education. Webb, N.L. (2005). Alignment analysis of mathematics standards and assessments, Michigan high schools. Madison, WI: Author.

By implication, curriculum and assessment will continue to be out of step with international trends resulting in mixed messages for teaching and learning. Bansilal (2011), for example, calls for a closer alignment of curriculum implementation plans with classroom realities. It is ironic that although SA teachers and educationists have complained of rapid curricula changes, the National Curriculum Statement has not changed at the same pace as TIMSS. Accordingly, educators, teacher educators and education researchers should be engaged more constructively in the curriculum and assessment reform processes for sustainable curricula coherence to be achieved. Reddy (2006, p. xiv) reports that during the period of TIMSS 2003, SA teachers consulted disparate curricula documents to determine what and how they taught. As affirmed by Airasian and Miranda (2002, p. 253), severe misalignment of assessment, standards and instruction can cause numerous difficulties. Given the extent to which the misalignment of the SA curriculum has gone relatively unchecked, the school system will continue to buckle for some time to come when subjected to international scrutiny. The latest of such scrutiny is the Global Competitiveness report (Schwab, 2012) and the Southern and East African Consortium for Monitoring Educational Quality report (Spaull, 2011), in which SA ranks very unfavourably.

Acknowledgements This research was funded by the Tshwane University of Technology through its postdoctoral fellowship placement of the first author. However, the opinions expressed do not necessarily reflect the views of the university. The authors are also grateful to Jeram Ramesh and Cerenus Pfeiffer for their time and expertise in the coding of the RNCS Grade 8 Mathematics assessment standards and the TIMSS assessment objectives. We are further grateful to Iben Christiansen’s comments on an earlier draft of this manuscript.

Competing interests We declare that we have no financial or personal relationship(s) which might have inappropriately influenced our writing of this article.

Authors’ contributions M.N. (University of Stellenbosch) was the researcher responsible for the empirical study as well as the writing of the manuscript. A.M. (Tshwane University of Technology), as postdoctoral supervisor, shared his expertise during the main study and the development of the manuscript. http://www.pythagoras.org.za

71


Page 1 of 2


Acknowledgement to reviewers The quality of the articles in Pythagoras crucially depends on the expertise and commitment of our peer reviewers.

In an effort to facilitate the selection of appropriate peer reviewers for manuscripts for Pythagoras, we ask that you take a moment to update your electronic portfolio on www. pythagoras.org.za, allowing us better access to your areas of interest and expertise, in order to match reviewers with submitted manuscripts. If you would like to become a reviewer, please visit the Pythagoras website and register as a reviewer. To access your details on the website, follow these steps: 1. Log into Pythagoras online at http://www. pythagoras.org.za 2. In your ‘user home’ select ‘edit my profile’ under the heading ‘my account’ and insert all relevant details, bio statement and reviewing interest. It is good practice as a reviewer to update your personal details regularly to ensure contact with you throughout your professional term as reviewer to Pythagoras. Please do not hesitate to contact me if you require assistance in performing this task. Rochelle Flint submissions@pythagoras. org.za Tel: +27 (0)21 975 2602 Fax: +27 (0)21 975 4635

Reviewing is an important part of scholarly work, making a substantial contribution to the field. Reviewers’ comments serve two purposes, guided by two inter-dependent objectives: • Pythagoras wishes to publish only original papers of the highest possible quality, making a meaningful contribution to the field. Reviewers advise the Editor on the scholarly merits of the manuscript to help him evaluate the manuscript and to decide whether or not to publish it. Reviewers are encouraged to reject a manuscript if it is scientifically flawed, merely sets out observations with no analysis, provides no new insights, or is of insufficient interest to warrant publication. • Pythagoras is committed to support authors in the mathematics education community. Reviewers help the author to improve the quality of their manuscript. Reviewers are encouraged to write their comments in a constructive and supportive manner and to be sufficiently detailed to enable the author to improve the paper and make the changes that may eventually lead to acceptance. The following summary of outcomes of the reviewing process in 2012 shows that our reviewers do well in achieving both objectives: No. of manuscripts processed in 2012 (outcome complete) Accepted without changes Accepted with minor changes (to the satisfaction of the Editor)1 Accepted after major revisions (re-submit, then re-review)2 Rejected after review – not acceptable to be published in Pythagoras3 Rejected without review – not acceptable to be published in Pythagoras4

0 11 11 12 13

47 (0.0%) (23.4%) (23.4%) (25.5%) (27.7%)

We sincerely thank the following people who have reviewed these manuscripts for Pythagoras in 2012. We very much appreciate their time, expertise and support of Pythagoras amidst pressures of work. David Mtetwa Deborah Moore-Russo Deonarain Brijlall Dirk Wessels Duncan Samson Ednei Becher Elizabeth Pretorius Eric Gold Erica Spangenberg Eugenia Vomvoridi-Ivanovic Faaiz Gierdien Gawie du Toit Gelsa Knijnik Gilah Leder Hamsa Venkatakrishnan Hannatjie Vorster Hans Niels Jahnke Hennie Boshoff Herbert Khuzwayo

Aneshkumar Maharaj Anne Watson Ansie Harding Anthony Essien Antonia Makina Arthur Powell Belinda Huntley Benard Okelo Bill Atweh Brian Greer Busisiwe Goba Calvin Jongsma Caroline Long Clement Dlamini Craig Pournara Cyril Julie Danie Strauss Daniela Coetzee-Manning David Mogari

1.Accepted after one round of review, with ‘minor’ changes as specified by reviewers and Editor. 2.Accepted after two or more rounds of review, with major changes specified by reviewers and Editor. 3.Includes three cases where authors did not resubmit after required to make major changes. 4.All submissions undergo a preliminary review by the Editor (and Associate Editors) to ascertain if it falls within the aims and scope of Pythagoras and is of an acceptable standard. Includes six cases where authors did not resubmit after extensive feedback prior to reviewing.


73

Pythagoras

Page 2 of 2


Reviewers (Continued):

If you would like to become a reviewer, please visit the Pythagoras website and register as a reviewer.

Iben Christiansen Ida Marais Ingrid Mostert Jacob Jaftha Jacques du Plessis Jill Adler Johann Engelbrecht Jurie Conradie Kakoma Luneta Karen Coe Karin Brodie Kate le Roux Kosie Smit Leila Goosen Leonard Mudau Lindiwe Tshabalala Lorna Holtman Lovemore Nyaumwe Luckson Kaino Lyn Webb Lynn Bowie Marc Schäfer Marcus Bizony Margot Berger Marietjie Potgieter Marit Johnsen-Høines Marthie van der Walt Mellony Graven Michael de Villiers Michael Mhlolo Michael Samuel Mogege Mosimege Murad Jurdak Neil Eddy Nelis Vermeulen Nick Taylor


Norman Webb Nyna Amin Paola Valero Paula Ensor Percy Sepeng Peter Gates Piet Human Pragashni Padayachee Retha van Niekerk Richard Barwell Ronel Paulsen Sandra Heldsinger Sarah Bansilal Shaheeda Jaffer Sheena Rughubar-Reddy Sibawu Siyepu Sizwe Mabizela Stefan Haesen Stephen Lerman Stuart Rowlands Sudan Hansraj Susan van Rensburg Tad Watanabe Temesgen Zewotir Thomas Morman Thulisile Nkambule Tim Dunne Tony Cotton Tracy Craig Umesh Ramnarain Vasuthavan Govender Vera Frith Washiela Fish Yusef Waghid Yusuf Johnson Zain Davis

74

Pythagoras

Page 1 of 1

Guidelines

Guidelines for Authors Pythagoras publishes research that explores the concerns, the research methods employed and the variety of didactical, methodological and pedagogical issues related to teaching and learning mathematics at all levels of education.

• Findings: What is the answer? The main findings (as a result of completing the above procedure/study what did you learn/invent/create?). Identify trends, relative change or differences on answers to questions. • Conclusion: What are the implications of your answer? Brief summary and potential implications (what are the larger implications of your findings, espcially for the problem/gap identified in your motivation).

All submissions must be made online at the Pythagoras website at http:// www.pythagoras.org.za. Pythagoras is a peer-reviewed, fully accredited academic journal, publishing only original research and scholarly work of a high quality. During the submission process authors must indicate that the research is their own original work, and the manuscript, or similar work, was not simultaneously submitted for review to any other journal, or previously accepted for publication or published elsewhere, including congress proceedings. A paper published in congress proceedings will only be considered if it is a substantial extension and revision of the previous paper. Authors should provide the Editor with all the relevant information (including copies of recent related papers) to enable the Editor to make the judgement whether such papers are sufficiently different to justify two separate primary papers.

• Acknowledgements: If, during your study, you received any significant help in conceiving, designing, or carrying out the work, or received materials from someone who did you a favour by supplying them, you must acknowledge their assistance and the service or material provided. Authors always acknowledge outside readers (but not the Editor and formal reviewers) of their drafts and any sources of funding that supported the research.

The guidelines for authors which is available on the online Pythagoras website at www.pythagoras.org.za includes information about the criteria for publication, preparing a manuscript for submission to Pythagoras and the online submission process. Information about the journal’s policies and the review process can also be found online.

General specifications for manuscripts • Layout: Start each paragraph at the margin (no tabs to indent first line). Include a line space between paragraphs to separate paragraphs. • Heading styles: First level headings: (Boldface, upper case, centred, on a separate line, 14pt). Second level headings: (Boldface, normal case, justified at left margin, on a separate line, 14pt). Third level headings: (Boldface, normal case, justified at left margin, on a separate line, 12pt). • Quotations in the text: Single quotation marks are used for all quotations; for quotes within a quote, use double quotation marks. If citations are longer than 30 words, indent the citation without quotation marks and do not use italics. • Tables and figures: Upload all tables, figures, images, and supplementary files in Step 4 of the online submission process. Tables should be saved and uploaded as separate Excel (.xls) files, no more than 10 figures and tables in total per article. Ensure all personal identifying information is removed from the supplementary files as per the provided instructions. All captions should be provided together on a separate page. Tables and figures should use numerical numbers. Figures and images should be saved and uploaded using high quality image formats: BMP, TIFF, EPS, JPEG (uncompressed); not GIF or compressed JPEG. Audio and video files should be saved and uploaded using the MPEG format (MP3 for audio and MPEG for video). At least two large originals of each figure in greyscale and colour at 600 dpi format should be submitted. Black and white line drawings should ideally be supplied in 1200 dpi (but no less than 800 dpi). • Acronyms: If a phrase with an established acronym or abbreviation is used, and appears more than five times in your article, please include the acronym/abbreviation in brackets after first mention of the phrase, then use the acronym/abbreviation only after that. Please note that you should not define acronyms or abbreviations in any of your headings. If either have been used in your abstract, you need to define them again on their first use within the main text. • Units: The use of units should conform to the SI convention and be abbreviated accordingly. Metric units and their international symbols are used throughout, as is the decimal point (not the decimal comma), and the 24-hour notation. • Spacing and punctuation: There should be one space (not two) between sentences; one space before unit terms (e.g. 5 kg, 5 cm, 5 days, 5 °C); but no space before %. Thousand/millions are marked with a space, not a comma (e.g. 12 345, 1 000 000). Ranges are expressed with an extended hyphen, not with a short hyphen (e.g. 1990–2000, pp. 34–42). • Dates: Dates are written in the following style: 12 July 2012. • Permission: Permission should be obtained from the author and publisher for the use of quotations, illustrations, tables and other materials taken from previously published works that are not in the public domain. The author is responsible for the payment of any copyright fee(s) if these have not been waived. The letters of permission should accompany the manuscript and can be downloaded from the journal website. The original source(s) should be mentioned in the figure legend or as a footnote to a table. • Proofs: Authors can provide feedback on the publication process of their manuscript, at two stages: 1. Copy-editing queries in a Word Document 2. Final PDF proof approval

• References: Begin the reference list on a separate page, with no more than 60 references. Pythagoras uses the APA referencing style. Note: No other style will be permitted. The Pythagoras APA referencing style guide can be downloaded from the website.

Submission of manuscripts • Editorial process: To read more about the editorial process, editorial criteria for publication, reviewing policy and how editors handle papers after submissions, visit the Pythagoras website. • Pre-submission enquiries: If you wish to enquire whether your submission might be suitable for consideration by Pythagoras, please use our online enquiry service, or email the Editor at [email protected]. • Readability: Contributions should be written clearly and simply so that they are accessible to readers for whom English is not a first language. Note: Please use UK spelling and not US spelling. If in doubt, consult the Oxford English Dictionary. Presentation of content • Format of articles: One and a half line spacing, between 4000–8000 words with a maximum of 60 references in Palatino or Times New Roman font, 12 point size. If other non-Latin/foreign alphabets are used, a covering note should specify which Unicode and HTML fonts have been used in the electronic copy submission. All specialised fonts must be supplied by the author in TrueType or on PostScript Type 1 format and uploaded as supplementary files duringthe online submission process. • Compulsory cover letter (page 1): The compulsory cover letter forms part of your submission and is located on the first page of your manuscript. You should provide all of the following elements: • Article title: Provide a short title of 50 characters or less. • Significance of work: Briefly state the significance of the work being reported. • Full author details: Title(s), Full name(s), Position(s), Affiliation(s) and contact details (postal address, email, telephone and cellular number) of each author. • Corresponding author: Identify to whom all correspondence should be addressed. Note: For manuscripts with multiple authors, proof of approval to submit the manuscript to the journal, signed by all authors, should be attached and uploaded as a supplementary file – this document can be downloaded from the journal website. • Authors’ contributions: Briefly summarise the nature of the contribution made by each of the authors, along the lines of the following: M.K. (Rhodes University) conducted all the research and wrote the manuscript. H.V. (University of Witwatersrand) was the project leader and made some final conceptual contributions to the manuscript. M.S. (Rhodes University) made initial conceptual contributions to the manuscript. • Possible reviewers: Authors are encouraged to provide the names and full contact details (including email address) of two or three potential referees who might be available to evaluate the work (referees should not be people with whom the researcher has recently collaborated or published). • Title, abstract and keywords page (page 2): The article’s full title should contain a maximum of 95 characters. The abstract should be no longer than 300 words and must be written in the past tense. Remember – the abstract is a summary of the article, not the introduction. The structured abstract should consist of five unlabelled paragraphs describing the background, objectives, method, findings and conclusion of your research: • Background: Why do we care about the problem? The context and purpose of the study (what practical, scientific or theoretical gap is your research filling). • Objectives: What problem are you trying to solve? What is the scope of your work (a generalised approach, or for specific situation)? Be careful not to use too much jargon. • Method: How did you go about solving or making progress on the problem? How the study was performed and statistical tests used (what did you actually do to get the results). Clearly express the basic design of the study, name or briefly describe the basic methodology used without going into excessive detail. Be sure to indicate the key techniques used.


These must be returned promptly to the journal Coordinator within 48 hours to avoid delays in publication. Substantial changes made at PDF proof stage will be charged to the author. Submissions and correspondence: All submissions must be made online at http://www.pythagoras.org.za, where authors can track the progress of the review and editing of their manuscripts online. Other enquiries regarding manuscripts should be addressed to: The Editor, Pythagoras, email: editor@ pythagoras.org.za. Ensure that the article ID [reference] number is included in the subject of your email correspondence. Reprints: The journal publishes on an open-access model and authors can download their material from the journal website freely and distribute it under the Creative Commons Attribution License. Strict adherence to these guidelines will expedite the publication process. This set of guidelines and supplementary documentation is available online at http://www.pythagoras.org.za.

75

Pythagoras

RSA Tel 086 1000 381 International Tel +27 21 975 2602 Email [email protected]

ONLINE CPD

Healthcare HPCSA & HPCNA accredited

E-LEARNING Train students online with Moodle®. Used by educational institutions & companies.

PUBLISHING Scholarly journal publisher. Free access to peer-reviewed articles.

Accountancy Partner: The Ethics Institute of South Africa & Compass Ethics Financial Services Partner: Financial Planning Institute (FPI)

S C H O L A R LY J O U R N A L S P U B L I S H E D B Y A O S I S Engineering International Journal of Machine Learning and Applications w w w. i j m l a . n e t Veterinary Onderstepoort Journal of Veterinary Research w w w. o j v r. o rg Journal of the South African Veterinary Association w w w. j s ava . co. z a

Education Pythagoras w w w.pythagoras.org.za Reading & Writing w w w.r w.org.za Social Sciences Jàmbá: Journal of Disaster Risk Studies w w w.jamba.org.za SA Journal of Industrial Psychology w w w.sajip.co.za

Sciences Koedoe w w w. ko e d o e. co.za

SA Journal of Human Resource Management w w w.sajhrm.co.za

Suid-Afrikaanse Tydskrif vir Natuurwetenskap en Tegnologie w w w. s at nt. a c. z a

South African Journal of Information Management w w w.sajim.co.za Koers w w w.koersjournal.org.za Literator w w w.literator.org.za

w w w. a o s i s g r o u p . c o m

Religion HTS Teologiese Studies/Theological Studies w w w.hts.org.za Verbum et Ecclesia w w w.ve.org.za In die Skriflig / In Luce Verbi w w w.inluceverbi.org.za Health Health SA Gesondheid w w w.hsag.co.za Curationis w w w.curationis.org.za African Journal of Disability w w w.ajod.org African Journal of Laboratory Medicine w w w.ajlmonlin e.org Open Journal of Implant Dentistry w w w.ojid.org African Journal of Primary Health Care & Family Medicine w w w.phc fm.org