Teacher Integration of Technology into Mathematics Learning By Michael O. J. Thomas and Ye Yoon Hong The University of Auckland, Department of Mathematics, PB 92019, Auckland, New Zealand [email protected] Yonsei University, Mathematics Department, Korea [email protected] Received: i^"'July 2012

Revised: 1" November 2012

Graphic calculators (GC) have been widely used in teaching for around 20 years, and yet many teachers still appear to be either unconvinced, or unaware, of how to release the potential of such technology to assist students to learn mathematics. This research describes a study addressing aspects ofthe teacher's role in integrating technology into their pedagogy. It considers issues related to the didactical contract, pedagogical technology knowledge (PTK), and procedural and conceptual knowledge in GC integration in teaching by two groups of secondary teachers. The results describe a number of factors that identify teacher progress toward pedagogical integration of digital technology and from these some tentative inferences for teacher professional development may be drawn.

1

BACKGROUND

Learning and teaching mathematics with technology is a complex process requiring a teacher to marshal a number of key competencies. In this paper we outline what we believe seme of these to be and illustrate how teachers can make progress in gaining these by considering cases of specific tîachers. It seems clear that the key factor in whether, £ind if so how, technology is used in the learning of mathematics is the teacher (or lecturer). In turn, there are many factors that influence a teacher's attitude to, and practice of, using a graphic calculator (GC), as with any technology. These include the affordances and constraints ofthe environment they work in (see e.g., Greeno, 1994) and their attitude to, and beliefs about, the mathematics and the technology, as well as their confidence and ability in using it to teach mathematics. Teacher affective variables, or orientations (a broad term that includes attitudes, dispositions, beliefs, values, tastes and preferences - see Schoenfeld, 2011, p. 29), their perceptions of the nature of mathematical knowledg(; and how it should be learned, their mathematical knowledge for teaching (Ball, Hill and Bass, 2005; Hill and Ball, 2004), which includes pedagogical content knowledge (Shulman, 1986), all influence their teaching. When we add the potentially subversive element of technology to this mix (Thomas, Tyrrell and Bullock, 1996) then the situation becomes even more complex. 1.1

Pedagogical Technology Knowledge

by a teacher of what we have previously called pedagogical technology knowledge (PTK) (Thomas and Hong, 2005; Hong and Thomas, 2006; Thomas and Chinnappan, 2008). A teacher's PTK applied to mathematics includes the principles, conventions and techniques required to teach mathematics through the technology. PTK is an emerging framework that can be used by researchers to generate explanations about teachers' use of technology to scaffold mathematical learning; it comprises partly the teacher's perspective on the technology and partly their familiarity with it (see Figure 1). They need to appreciate that technological tools, such as GCs, may be employed in teaching in qualitatively different ways, such as for: property investigation; computation; transformation; data collection and analysis; visualising; and checking (Doerr and Zangor, 2000) and that students may form a variety of different relationships with them (Goos, Galbraith, Renshaw and Geiger, 2000) depending on teacher direction. This directional emphasis in the use of technology, which has been called teacher privileging (Kendall and Stacey, 1999, 2001), has been shown to shape student preferences, so that they often follow the teacher's lead. We argue that teachers who have built up a sound basis of PTK are more likely to feel comfortable in accessing tools of technology in designing mathematical learning experiences. This privileging can have a positive impact on students' uptake of technology in exploring mathematics. Another key aspect of PTK surrounds the transforming of a technological tool into one or more instruments (Rabardel, 1995). This engages the teacher in actions and decisions to produce the personal schemes required to adapt the tool to a particular mathematical task, considering what it can do and how. PTK grows as teachers progress through the stages of instrumentalisation and instrumentation of the tool, gaining a personal appreciation of its role in learning mathematics, and importantly, of ways in which students may be assisted through various teaching approaches to emulate their instrumental genesis of the technological tool. This concerns more than just the mechanics of getting the syntax and semantics of the input/output, developing algebraic insight and expectation (Pierce and Stacey, 2004a), and coping with the difficulties of navigating between screens and between menu operations. It means seeing opportunities for epistemic mediation by the technology, between the user and the mathematics.

Due to this complexity of teaching mathematics with technology we hypothesise that it requires the development

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Pedagogical Knowledge

Mathematical Content Knowledge

Mathematical Knowledge for Teaching Personal orientations

Technology instrumental genesis

Pedagogical Technology Knowledge Figure 1 An outline ofthe construction of Pedagogical Technology Knowledge (PTK). In this regard. Artigue (2002) describes pragmatic value as a "productive potential (efficiency, cost, field of validity)" (p. 248) and epistemic value as a contribution "to the understanding ofthe objects they involve" (p. 248). For example, technology may have pragmatic value in providing a means (an instrument with associated techniques) for solving quadratic equations, but it can also have epistemic value if it is used to promote mathematical thinking about the kind of objects that quadratic functions and equations are and the nature of the transformations involved in the solution process. Discussing instrumentation and conceptualisation, Guin and Trouehe (1999) stress the need for them to occur concurrently in the classroom, and hence teachers need to focus technological activity on specific conceptions. In work somewhat parallel with the general concept of PTK, Pierce and Stacey (2004b) outline four key aspects of knowledge and skills for teaching effectively with Computer Algebra System (CAS) technology: mathematical, machine, technical and personal, and maintain that the technical and personal need the most attention by new teachers. In order to assist development ofthe above range of skills, they have proposed a framework for teaching to promote effective use of CAS by monitoring progress. In summary, PTK involves a teacher understanding the principles and techniques required to incorporate technology into mathematics classrooms so that mathematical learning through the technology, in the form of epistemie rather than simply pragmatic knowledge (Artigue, 2002), is the outcome. The factors that combine to produce PTK involve instrumental genesis, strong mathematical knowledge for teaching, and positive teacher orientations and goals (Schoenfeld, 2011), especially beliefs about the value of technology and the nature of learning mathematical knowledge, along with affective aspects, such as confidence (see Figure 1 ). 1.2

Affordances Situations

and

Constraints

in

Didactical

A useful framework for examining technology use in the classroom is the Theory of Didactical Situations (TDS) of © 2013 Research Information Ltd. All rights reserved.

Brousseau (1997). Here, Brousseau describes three situations, action, communication and validation that have as their endpoint student acquisition of specific knowledge. These didactical situations arise from the students' relationship with a didactical milieu established by the teacher. The teacher's role is to orchestrate the elements of this milieu so that the specified knowledge is attained. In doing so she takes into account the dynamic didactical contract between the students and the teacher, which is the mutual recognition of the roles that the student and teacher play in the acquisition of the knowledge. This recognises, through the tacit acceptance of each party, that in any didactical situation there are reciprocal obligations in the relationship. Not least among these is the expectation by students that they will gain knowledge, and by the teacher that the students will want to learn. Of course this social contract is a dynamic entity, changing and adapting to new circumstances that arise in the classroom situation. The provision by a teacher of GCs in the milieu of a didactical situation can alter the dynamics of how students might gain the expected knowledge. Even when they have a high level of PTK teachers may still faee difficulties in implementing technology use in didactical situations. One useful way of analysing these has been presented by Gibson ( 1977), who has initiated a theory that presents relationships between agents, in our case teachers, and the environment they seek to interact with. A key feature of this interaction is the presence of affordances (attributes of the environment that contribute to the potential for an interaction to occur) and constraints (characteristics of the affordance that provide structure and guidance for the interaction - see Kennewell, 2001). In this mix, according to Greeno (1994, p. 338), "An affordance relates attributes of something in the environment to an interactive activity by an agent who has some ability". In short, affordances speak about the potential for action, while constraints impose the structure for that action. Thus, an example of an affordance, given by Gibson is the provision of mailboxes for posting letters; its location and the size of its opening would be constraints. In a technology classroom setting, the presence of technology is an affordance, with student or teacher www.technologyinm atheducation.com

Teacher Integration of Technology into Mathematics Learning instrumentation, time available to use the technology and the content of curriculum as examples of consfraints. Similarly, in terms of the present research, a graphic calculator is an affordance and the size and resolution of its screen display a consfraint. Other researchers have used different words to describe the absence of an affordance with regard to technology use in mathematics learning. For example, Forgasz (2006) talks about encouraging and inhibiting factors, vk^hile Thomas (2006) and Thomas, Hong, Bosley and delos Santos (2007) use the terminology of obstacles to technology use. In this paper we delineate between a consfraint, which implies the presence of an affordance, and an obstacle, something that inhibits or prevents the presence of an affordance-producing entity in the classroom situation. A classroom didactical situation with GCs in the milieu requires changes in the didactical confract. Factors inñuencirig the didactical contract include the teacher's affective variables (beliefs and attitudes), their perceptions of the nature; of mathematical knowledge and how it should be learned, as well as their mathematical and pedagogical content knowledge (Shulman, 1986). One may assume that if a teacher possesses a limited knowledge of a concept and its related subconcepts (Chinnappan and Thomas, 2003) then they will find it more difficult to provide the kind of environment and experiences that will assist students in the construction of rich conceptual thinking. Instead they may regress to a process-oriented approach (Thomas, 1994), presenting; students with a toolbox selection of procedures that may be applied to each problem that arises. While such procedures and skills are important, mathematical thinking is clearly much wider than this, and requires procedural and conceptual interactions with the various representational forms of mathematics (Thomas and Hong, 2001). However, teaching is not mediated simply by the mathematical understanding ofthe teacher (Cooney, 1999), but it is also influenced by the teacher's mathematical knowledge for teaching (Hill and Ball, 2004; Ball, Hill and Bass, 2005). This refers to understanding the mathematical content, the constructs and ideas involved in a particular topic, and how these relate to the principles and techniques required to teach and learn it, including appropriate structuring of content and relevant classroom discourse and activities (Simon, 1995; Cooney, 1999; Chinnappan and Thomas, 2003).

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multiple representational approach to teaching. In this sense they act as a physical medium for the purpose of reexpression and translation (re-presentation), and for mathematical communication (Noss and Hoyles, 1996; Yackel, 2000). However, when teachers use graphic calculators they may sfress different teaching goals, since these are anchored in their personal orientations, including their beliefs about the potential of the calculator to support the goals (Doerr and Zangor, 1999; Simmt, 1997; Harskamp, Suhre and van Sfreun, 1998). Such goals can include student-cenfred initiatives, interactive-inquiry teaching styles and student-cenfred views of learning, as well as contentoriented goals and a view of learning as listening. Teacher progression in their belief in the value ofthe technology in teaching mathematics leads to a key issue they have to face, namely the level of integration of the technology in learning that they will practice. This may range from using it at prescribed moments as a teacherdirected add-on, to an ever-present insfrument that is an extension of cognition; as an instrument for pragmatic mediation to one for epistemic mediation. The research described in this paper followed a group of teachers as they began, or continued, their instrumental genesis of the GC tool, thus extending their PTK. It attempted to understand teacher practice in relation to the congruency of content knowledge, mathematical knowledge for teaching, instrumentation and instrumentalisation, PTK and the didactical contract. 2

METHOD

This research comprised two case studies. In the first 22 secondary school teachers, from 22 different schools were using GCs in their teaching in Auckland, New Zealand. Of these 22 teachers who were using GCs, 17 were experienced in their use and 5 were not. The second-named researcher carried out the fieldwork and had an initial meeting with each ofthe teachers. At this initial meeting the topics to be taught, the time-lines and the technology to be used were discussed, and they were given a Likert-style attitude test (see Figure 2) with five subscales, comprising attitude to: mathematics, technology in general, personal learning, technology and GC in learning mathematics. Following this meeting a further, more detailed, discussion took place by email.

The introduction of new technology into the classroom has been shown to be capable of a subversive effect (Thomas, Tyrrell and Bullock, 1996) radically altering the didact cal confract. Thomas, Tyrrell, and Bullock (ibid. p. 49) suggest that the introduction of technology requires a new mindset on the part of teachers, a 'shift of mathematical focus ', to a broader perspective of the implications of the technology for the learning of the mathematics. As teacher and student insfrumental genesis of the technological tool proceeds teacher beliefs and attitudes are shaped, changing their teaching emphasis and the didactical confract to give increased emphasis to the insfrument. There is some evidence that if the use of technological tools, such as computers and graphic calculators, is made an integral part of a didactical contract, they can provide valuable support for a

The teachers in this first case study were also given a diary and encouraged to complete it for all their GC lessons, including their lesson aims and expected outcomes, teaching method, classroom organisation, details of GC use, and their reflection on student learning. Three of these group one teachers, whose teaching practice is featured in this paper, chosen to illustrate their comparatively positive work with the GCs, and to cover a wide range of obstacles and consfraints. These three teachers, we call A2, A6 and A8. Teacher A2 had 9 yeats' teaching experience, had made some use of technology in teaching, and taught in a middle socio-economic school (decile 5), while A8 was a relatively new teacher, with 4 years' experience, who had made only a little use of GCs in teaching, and who taught in a low socioeconomic school (decile 2). Teacher A6 was a deputy head of department (HOD) with 20 years' teaching experience.

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including a number of years implementing technology in her teaching, who worked in a high socio-economic school (decile 10). Following discussion, they chose to teach topics including families of curves, linear programming, limit, drawing graphs and derivatives. We made a total of 8 lesson observations; two in Year 13 (aged 18 years) for each of teachers A2 and A6 and three for Year 13 for teacher A8. In addition, one Year 12 (aged 17 years) lesson for teacher A6

was observed. The classroom practice of these three teachers is considered in more detail below (along with three group two teachers). The lesson videotapes were transcribed for analysis along with the data from the questionnaire, the lessons and the diaries. After these observations were completed the teachers were interviewed for about 40 minutes, and these were tape-recorded and later transcribed for analysis.

Mathematics Attitude Questionnaire Name: School: Levels of Teaching (Circle): Y7 Y8 Y9 YIO Yll Y12 Y13 Years of Teaching: Please circle the numbers on the right below corresponding to which of the following indicates your level of agreement with each statement. 5 - 1 STRONGLY AGREE (SA) with the statement 4 - 1 AGREE (A) with the statement 3 -- NEUTRAL (N) 2 - I DISAGREE (D) with the statement 1 - I STRONGLY DISAGREE (SD) with the statement

1. More interesting mathematics problems can be done when students have access to technology. 2. Students understand mathematics better if they solve problems using paper and pencil. 3.1 have lots of ideas about how I can make use of technology in mathematics. 4. Students should not be allowed to use technology during mathematics tests or examinations. 5.1 think technology is a very important tool for learning mathematics. 6. Technology can be used as a tool to solve problems students could not solve without it. 7. Technology is only a tool for doing calculations more quickly. 8. Technology can make mathematics more fun. 9. Students should use technology less often in mathematics. 10. Using technology will cause students to lose basic computational skills. 11.1 want to improve my ability to teach with technology. 12. Students rely on technology too much when solving problems. 13. Technology should only be used to check work once the problem has been worked out on paper. 14. Mathematics students need to know how to use technology. 15. Students should not be allowed to use technology until they have mastered the idea or the method. 16. Mathematics is easier if technology is used to solve problems. 17. Learning how to use technology is difficult for me. 18. Using technology makes students better problem solvers. 19.1 lack the confidence to use technology to solve mathematical problems. 20. Learning mathematics is mostly memorising a set of facts and rules. 21. When doing mathematics it is more important to know how to do a process than to understand why it works. 22. Learning mathematics means exploring problems to discover patterns and make generalisations. 23. Students would be better motivated in maths if they could use a graphic calculator. 24. Using a graphic calculator removes some learning opportunities for students. 25. Students would understand maths better if they had a graphic calculator. 26. Using a graphic calculator would make the management of data easier. 27. Students would be more confident in maths if they had a graphic calculator. 28. Since students can use a graphic calculator, they do not need to leam to draw graphs by hand. 29. I feel that computer algebra system calculators should be allowed in mathematics tests and examinations. 30. Using a graphic calculator to solve statistics makes the problems easier to understand.

S A 5

A N D S D 2 1 4

5 5 5

4 3 2 4 3 2 4 3 2

1 1 1

5 5 5 5 5 5 5 5 5

4 4 4 4 4 4 4 4 4

2 2 2 2 2 2 2 2 2

1 1 1 1 1 1 1 1 1

5 5

4 3 2 4 3 2

1 1

5 5 5 5 5 5

4 4 4 4 4 4

2 2 2 2 2 2

1 1 1 1 1 1

5

4 3 2

1

5 5 5 5 5 5

4 4 4 4 4 4

2 2 2 2 2 2

1 1 1 1 1 1

5

4 3 2

1

5

4 3 2

1

3 3 3 3 3 3 3 3 3

3 3 3 3 3 3

3 3 3 3 3 3

Figure 2 The attitude questionnaire.

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Teacher Integration of Technology into Mathematics Learning A second group of seven Auckland teachers completed a workshop covering both content and pedagogy for algebra and calculus, using the TI-83Plus GC and several downloadable memory-based Flash App[lication]s. None of these teachers had much experience of using the GC to teach mathematics, although two of the seven had previously made some use of them. Apart from one teacher trainee, all the teachers were experienced, with between 10 and 24 years teaching. Each teacher was given their own GC and a class set that they kept these for six months after the course. Once the teach(;rs were familiar with the GC they began to ask questions related to their teaching, and discussed ideas with one another. In the month following the workshop the teachers v/ere given a brief questionnaire on their perspective of the value of the TI-83P]us GC for teaching mathematics, and six teachers, four from one school and two from a second, agreed to take part in the classroom-based phase of the research. During this three-week phase we observed their teaching with the GC for Years 10-13 (age 15-18 years) recording observations and video taping their practice. Three of these t(;achers, F1, F2 and F3 are used below, along with A2, A6 and A8, to exemplify the differing rates of progress Teacher

Teacher GC Experience/Confidence

Al A2 A7 A8 AID B4 C6 É4 E5 E6

New user/Weak New user/Weak Experienced/Weak New user/Weak Experienced/Weak New user/Weak Some experience/Weak Some experience/Weak Some experience/Weak Some experience/Weak

A3 A4 A5 A6 A9 B3 B5 C3 D2 D5 El E2

Experienced/Strong Experienced/Strong Experienced/Strong Experienced/Strong Experienced/Strong Experienced/Strong Experienced/Strong New/Strong Experienced/Strong Experienced/Strong Experienced/Strong Experienced/Strong

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in constructing PTK, and its subsequent infiuence on their technology use. 3

RESULTS

Potential affordances for teachers using GCs (or other technology) arise in a number of domains. For example, there may be a supportive environment, comprising a school that supplies GCs and time for professional development, a mathematics department with a head of department who encourages GC use, and other mathematics teachers of a like mind. The level of confidence in GC use in mathematics of the 22 teachers in case study one was deduced through discussion with them and from classroom observation, and we concluded that 12 had strong, and 10 weak confidence. The teachers were from 15 different schools, and our aim was to investigate the factors influencing the different levels of confidence and to see if confidence was linked to style of teaching with the GCs. Table 1 lists some ofthe school and department factors we identified, along with the personal experience and confidence of the teachers in the study, divided into groups based on confidence.

Department HOD/Teacher Support Weak Confidence Group Weak/None Weak/None Strong/Others Weak/None Weak/None Strong/Others Weak/None Strong/Others Weak/None Moderate/Some Strong Confidence Group Strong/Others Strong/Others Strong/Others Strong/Others Weak/None Strong/Others Strong/Others Strong/Others Strong/Others Weak/None Strong/Others Strong/Others

School Use of GC/Teacher Support New/Strong New/Weak Experienced/Strong New/Weak New/Weak Experienced/Strong Experienced/Weak Experienced/Strong Experienced/Moderate Experienced/Moderate New/Strong New/Strong Experienced/Strong Experienced/Strong New/Strong Experienced/Strong Experienced/Strong New/Strong Experienced/Strong Experienced/Weak Experienced/Strong Experi enced/Strong

Table 1 Background Factors Infiuencing the 22 Teachers' Use of GCs From Table 1 we see that of the 12 teachers with strong confidence in their ability to teach with the GC, 11 were experienced users, 11 had strong school support, 10 had strong HOD support, and 11 had the support of other teachers in their department. In contrast, of the 10 teachers with weak confidence in their GC use, 8 were inexperienced users, and seven had little HOD or other teacher support in their schools. Interestingly the level of school support was split, with five being supportive and five not.

This initial analysis seems to suggest that among the key variables in producing confident users of GCs are the teacher's own experience, and the immediate affordance of support from others in the mathematics department. However, these may be neither all necessary nor sufficient, since there are exceptions to the general trend. On the one hand teacher C3, who was new to using the GC was confident, but was in a school where there was strong support from the HOD and the school, even though the school was new to using the technology, and teachers D5 and

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A9 were confident in spite of no support (D5), or no support from the department (A9). On the other hand, teacher A7 was experienced, had strong support from both department and school, and yet lacked confidence in his ability to teach with the GC; teachers A10, C6, E5 and E6 all had some (limited) experience with the GC but had little or no support from their department or school, and teacher E4 had some experience and strong support, yet they were all weak in confidence. It seems support from others in the school, and especially the mathematics department may be a primary environmental factor influencing teacher confidence. In addition to the above examination of the personal and environmental factors influencing GC use, the teachers were given an attitude questionnaire, to probe for other possible influences on teaching practice. The scale, given in Figure 1, comprised 5 subscales, on attitude to: mathematics (Q's 2, 20, 2 1 , 22); technology (Q's 7, 13, 15, 18); using technology to learn mathematics (Q's 1, 4, 5, 6, 8, 9, 10, 12,

Sfrong Confidence (JV= 12) Non-users (A'= 10) p-value

14, 16); using the GC to learn mathematics ( Q ' s 23-30); and personal learning (Q's 3,11, 17, 19). The internal consistency of the scale, measured using Cronbach's Alpha reliability coefficient, was 0.90, suggesting that it was a reliable scale. For the teachers in case study one we managed to collect data based on the scales in Figure 1 from another group often teachers in their schools who did not use GCs at all in their teaching. Comparing the 22 teachers who used the GC with the 10 who did not we found that there was no difference between these two groups on the subscales except, not surprisingly, weak evidence of a significantly more positive attitude {p' = 5. So I'll just give you an example to try yourself That should say 1 and 5. Anyway I'll give you an example today. So following the insfructions and directions on the sheet try doing this question. In summary we could say that for these two teachers insfrumental genesis was very much in progress, but since they were primarily working on their insfrumentation, as shown by their emphasis on which keys to press, this may have delayed progress on the instrumentalisation phase. They also emphasised the ability of the GCs to produce solutions rather than improve mathematical thinking. As a consequence their practice was focussed on the use of the technology rather than on the mathematics, and how understanding of it could be enhanced though the technology. Further, they both had low levels of confidence in using the technology to teach mathematics. These atfributes seem to be a feature of what we see as the early stage of progress in the acquisition of PTK. 3.3

More progress

Teacher A2, from the first case study group, worked in a medium socio-economic level school (decile 5). She had 9 years' teaching experience and had been using GCs for 3 years. We made 2 observation visits to her classroom, both involving Year 13 students (age 18 years) studying firstly frigonomefric graphs and their fransformations, and then the solution of trigonomefric equations. The school allowed GC use in examinations and encouraged students to buy their own calculator, but the financial situation at the school was not considered good enough to support technology; according to the head of the department "the budget doesn't allow for it." Hence the mathematics department did not have a class set of GCs or a viewscreen, so she had an obstacle to overcome when she wanted to demonsfrate working with a GC. When we visited teacher her class only seven of the 14 students had their own scientific or graphic calculator, and so the students shared with each other or worked without a calculator. Thus she worked under the constraint that the calculators were not all the same, and so she had to explain how to work with each model. Teacher A2 used a CASIO fx-9750G GC and an overhead projector (OHP) on which she wrote to demonsfrate and explain key points. In her interview she spoke about the visual value of the GC, how "when they have a graphics calculator, it's very useful for them to see how the graphs... what the graphs look like, and you can change numbers". She also mentioned the time-saving aspects of its use "It's much faster, quicker and easier". However, she also thinks ofthe conceptual value of the GC in helping students make connections. I think it's important for them to understand concepts in mathematics and there's got to be a balance between the skills they do and the word www.technologyinmatheducation.com

Teacher Integration of Technology into Mathematics Learning problems...Basically we want them to be able to see what the concept is, and instead of sketching it every time... so if you want them to check for '>^=3cos(4x)', they have independently seen what is happening to 'cos (4x)' and what happens when they [do] '2cos(x)', and they can put the two together and get it from the graphics calculator quicker and they can see the changes much faster. Teacher A2 did not put a lot of emphasis on student instrumentation, getting them to think about what buttons to press, etc. She says that once they have the basics students are quick to pick up what they need in each lesson: "I don't put it all up at the beginning of the lesson or they'll get confused with it. I do it as and when I feel it's necessary. But generally, switching it on, feeling the menu and how to use the cursor, most of the students know. That's why I put the instructions up and most of them caught on to it very quickly". Her emphasis on conceptual understanding was seen in aii interview comment "I think it's important for them to understand concepts in mathematics and there's got to be a balance between the skills they do and the word problems they work out." In her first lesson with the Year 13 students (age 18 years) she used the GC to allow students to investigate graphs of the form y = AsmB(x-i-C)-\-D, etc. First she concentrated on the effect of a single parameter, using >'=yísin;í:, with ^ = 2, 3 and 0.5, and asking what these numbers signified, and then moved on to graphs of the form y = cos(Bx), with ß = 2 and 3. Students were encouraged to work together, "Discuss with the person sitting next to you the effect of y = sin(.ßx).", and she also got students to come out and sketch graphs on the white board. In each case she tried to geit students to focus on the concepts of domain, range, period, amplitude and frequency, asking questions such as "V/liat does the number 2 signify?". She explained the conceptual approach, and the role of the GC this way: Instead of just sketching y = cos(2x)' and then after it doing 'v = cos(3x)', they've got to see the connection. If they see... keep changing the variables, and they see the effect ofthat, that's conceptual understanding and that's v/hat we should be getting at and the graphics calculator is really useftil for that."

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Moving on she focused on the concept of translation, asking "What sort of translation is y = sim- -I- 2? What is [sic] the domain, range and period? What does [the] graph look like?" Figure 3 shows her use of some of the key ideas on the overhead projector. In the second lesson was spent solving equations such as sinx = 0.5. To do this she got the students to draw the graphs of>' = sinx and y = 0.5 on the GC (see Figure 4(a)) and consider their intersection. Afterwards she used this same concept to get them to solve 7 - 3x = 6cosx. Since she did not want her students to be procedural users of the GC, but to think about the mathematics, she used this example as an opportunity to get them to look through what they saw on the screen. She pointed to the apparent intersection of the two graphs near the y-axis (see Figure 4(b)) and asked the students to use the GC to zoom in on that area. They could then see that the line does not actually intersect the curve (see Figure 4(c)) and that was why there was no solution given by the GC, instead "When you press [G-Solve], it always give you first intersection from left to right, then x = 3.85,

(a) y = sinx and j ; = 0.5 for sinx = 0.5

(b) y = 7 — 3x and y = 6cosx for 7 — 3x = 6cosx

(c) Zooming in Figure 3 Teacher A2 stresses the concepts of range and period for j ; = sinx + 2.

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Figure 4 Teacher A2 shows that trig equations can be solved with intersecting graphs

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Teacher A8, also in the first group, worked in a low socio-economic level school (decile 2). He was less experienced than the others, with 4 years' teaching, was new to using GCs in his teaching and had not attended any GC professional development course. He worked in the same school as teacher A2, and the school allowed GC use in examinations and encouraged students to buy them, even though the majority of the students could not afford one. However, the school was trying to support technology use so the mathematics department had purchased one class set of CASIO fx-9750G calculators and had one classroom with a computer set up for PowerPoint use. Teacher A8 commented on this constraint that "Very few of the students can afford their own graphics calculator therefore, the only chance they get to practise with them is in the lesson when we hand them out and draw them back again, so they don't get the familiarity ...they haven't had the practice in using the technology." We made 3 observations of his classroom. All three lessons involved year 13 students (age 18 years) using the class set of GCs, one studying the binomial distribution in statistics, and the others the use of power and exponential functions in statistical modelling. Once again he had to work around the obstacles ofthe lack of an overhead projector and a viewscreen to project his calculator screen, but managed this by using the affordance of a poster of the GC to show students the right key presses. However, this clearly has constraints, such as not being able to show the result of the key presses.

In his interview teacher A8 showed that he did not want students simply to use the GC in a procedural manner, but saw the value of an inter-representational approach. The graphical side of the GC was important to him since "it can also provide a visual help to understanding the overall idea.", and he talked about how "students have come to me and said, 'I now understand what you're saying', by having a little presentation of the graph." He also confirmed his desire for a non-procedural approach, saying that his "prime aim would be for them to understand the method and be able to apply it rather than to arrive at the right answer." Actually, his first lesson was rather procedural, using the computing power of the GC to calculate Pix = 6) = ^C^ 0.4* (1 - 0.4)^ and finding other probabilities, P(x > k) or P{x < k). It appeared from his interview that the purpose of this lesson was revision, since "Basically the type that they're going to get in the exam." However, for the second lesson on power and exponential functions he tried to integrate the GC into students' mathematical thinking. He gave students the ftinction y = 2e""'*^*, asking them to sketch the graph for x values from 0 to 3, by completing a table he gave them (see Figure 5(a)) and then plotting the function. Students completed the table by putting x = 0, 0.5, 1, 2, 3 into the ftinction y = 2é~°''*^' on their GC and then plotting by hand (Figure 5(b)). He used this method rather than getting the GC to draw the final graph so that the students could see how the graph was constructed.

Í 5

Figure 5(a)

Figure 5(b)

Figure 5.Teacher A8 integrates the GC into graph plotting. A similar process was then followed for graphs ofthe form y = x , for k = -2, 1.5 and -0.5. Finally a contextual problem involving the volume of water in a lake at time /, with the function V = 1275/"^^^, was considered. Here the ñmction was drawn using the GC. Teacher A8 asked the students "At what point does the amount of water in the lake drop below 200 m"? When y = 200, what is x-value? Use [Trace] key, approach to y = 200, you can find the x-value." (variables had been changed on the GC). Here the students had a choice of method. They could either use [G-solve], which gave them the x-value directly, or the [Trace] key on the GC; they preferred the former. This work led into the third lesson on using power and exponential functions to model two variable regression. In summary we might say that the instrumentation of these teachers was improved but they were still making progress on instrumentalisation of the tool, with classroom constraints a factor in this. However, rather than simply © 2013 Research Information Ltd. All rights reserved.

focussing on the instrument, such as menu operations and key presses, they were beginning to use the technology, for example through graphs, to assist students with conceptual understanding ofthe mathematics. Examples of this include getting students to think about generalising transformations of the form of the form ^ = ^ sin B(x + C) + Z) and graphs ofthe form j ; = jc* by linking algebraic, tabular and graphical representations. They were also growing in confidence in use ofthe technology to the point where they were willing to experiment with their approach. 3.4

Greater strides forward

Teacher F3 from the second group exemplifies a teacher who had made greater strides forward. She had over six years' experience of using GCs in her teaching and had been involved in a previous research study with them. In spite of her experience, in the questionnaire she admitted "Sometimes it's hard to see how to use it effectively so I www.technologyinm atheducation.com

Teacher Integration of Technology into Mathematics Learning don't use it as continuously as I should." Her motive was a rather pragmatic "We should move with the times" and she had a small reservation about the GC that "It is OK. By now expected better resolution though." Due to her relative experience she appeared confident in her use of the GC and spoke at length in her interview, describing how "In the past I have also done some exploratory graphs lessons where students get more freedom to input functions and observe the plots." Further, she explained that she was happy to loosen control of the students and let them explore the GC and help one another: "Students learn a lot by their own exploration...In past lessons I have never had a student get lost while using a graphics calculator. Sometimes friends around will assist someone." However, she acknowledged her need to progress in her types of GC use, "1 would like to see them used more frequently and beneficially in class with structured lessons." Wliile she could perceive imaginative GC usage, part of her difficulty involved the pressure of day-to-day teaching, since she admitted that "I was not relaxed enough with the term coming to an end and other aspects in this year's teaching to be inspired to use the calculator with imagination for the students." Responding to what she wanted her students to learn she replied in terms ofthe challenge ofthe depth of mathematics, "The success for me as a teacher is when they want to learn more and students show a joy either in what they are doing or in challenging themselves and their teacher with more deeper or self posed mathematical problems." She was convinced that the novel and challenging nature of :he GC could motivate students - "The calculator Appears parallel

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puts a radiant light in the class... With a graphics calculator lesson no one notices the time and no one packed up." - and that her perspective on learning mathematics influenced her PTK came out in the comment that "Today we find a lot of Maths does not need underlying understanding... I feel as teachers what we need to really be aware of is what the basics are that students must know manually... when we sit down to work with graphics calculators we need to consider carefully what still should be understood manually." In this way she addressed an important factor of the integration of technology use into mathematics, namely learning what is better done by hand and what could be done better with the technology (Thomas, Monaghan and Pierce, 2004). One of her lessons with the GC was with Year 12 students and she considered families of functions with the aim of exploring exponential and hyperbolic graphs and noting some of their features, "we're going to utilise the calculator to show that main graph and then we're going to go through families of >' = 2^". She was comfortable enough to direct them to link a second representation "Another feature of the calculator I want you to be aware of .[pause] you've got also a list of x and y values already done for you in a table." Teacher F3 had moved away from giving explicit key press instructions, instead declaring "I want you to put these functions in and graph them and see what's going on.", and "You can change the window if you want to see more detail, and if you want to see where it cuts the x-axis, you can use the "trace" function." Figure 6 shows a copy of her whiteboard working.

y=2''+l y=2'' +2 Original graph of y=2'' shifted up Cuts J'axis different places

Figure 6 Teacher F3's whiteboard working: Viewscreen projection and overwriting. She was also able to move towards an investigative mode of teaching "if you're not sure where the intercepts are, you can use the "trace" key, remember, and I want you to observe what is happening.", encouraging students to use the GC in a predictive manner, to investigate a different family. We want to do some predictions... Looking at the screen try to predict where 3 x 2' will go then press "'y=..." and see if it went where you expected it to go. You may get a shock... Can you predict where "y = 4 X 2*" will be? Now you learned from that, so can you predict where it'll lie. The gap between them gets smaller. If you're interested put in 'y = 100 x 2^". Does it go where you expect? There was also some discussion of mathematical concepts and how this could help with interpretation of the www.technology inmatheducation. com

GC graph. She linked 2 x 2' with 2'^' and then during examination of the family of equations y = 2\ y = 2^*', y = l''^- said of >'=2''^' "We expect this to shift 1 unit to the left [compared with 2']. Did it?" In this way she made a link with previous knowledge of translations of graphs parallel to the X-axis, and then reinforced this with the comment that "With this family, when you look at the graph can you see that the distance between them stays the same because it's sliding along 1 unit at a time. The whole graph shifts along 1 unit at a time." In addition, there was a discussion of the relationship between the graphs in the family of y =2' + k, and the relative sizes of 2^ and k. ... as the exponential value gets larger, because we're adding a constant term that is quite small, it lands up becoming almost negligible. So, when...all they're differing by is the constant part, you'll find that they appear to come together. Do they actually equal the International Joumal of Technology in Mathematics Education Vol 20. No 2

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Michael O J Thomas and Ye Yoon Hong same values ever? Do they ever meet at a point? No, because of the difference by a constant, but because ofthe scaling we have, they appear to merge.

The discussion on the relative size of terms in the function continued with "How significant is "+1" or "+2"? We know that 2^ is 32." and again the use of prediction was evident "I want you to predict where y = 2'' + 3" would be." Teacher A6, from the first group, worked in a high socio-economic school (decile 10), had 20 years' teaching experience and had been using GCs for three years. She had also made very good progress in her GC use. We made 3 visits to observe her classroom. The first involved year 12 students (age 17 years) studying probability simulations in statistics, and the second and third observations were of year 13 students (age 18 years) working on calculus, stationary points and trigonometry. In each lesson the teacher and all her students had access to T1-83+ calculators. While in her interview she sfressed the value of the technology for covering more ground "...the number of things that they can do in any one lesson is far greater so it's better than drawing. Usually it takes you all lesson to draw three (graphs). You can draw ten with a graphics calculator and they can really understand it for themselves...", she mentions student understanding, which was a focus for her. She also stressed the visual benefits ofthe GCs in this: ...it gives the students a visual interpretation, a handson approach; it's not all just writing, they can see things happening, particularly with the probability simulations yesterday. 1 felt they had a much better comprehension of what was actually happening so

The lesson's concepts

they got a visual picture but also every time they did it ...they could see it happening differently every time. These two ideas converge in her mind to give students the ability to generalise "I think it's the amount of visual information they can get, and the amount of examples they can get through, so they really feel they understood it because they've seen so many that they can actually accept and they can generalize for themselves and it gives a better understanding." Rather than focussing simply on getting her students to perform procedures she talked about how "they had a much better concept of what was actually happening". She does not put the emphasis in class on the instrumentation of the GC, instead "Basically if I'm organized I'll make a worksheet with the 5 keys we'll use that day or put the 5 keys we'll use on the blackboard, go over those and just spend 5 minutes on what we're going to day for the day". This approach was confirmed by the observation of her second lesson, on concavity. Her teaching was focused on using the technology to improve conceptual understanding, with an emphasis on visual representations of functions. She explained the concept of concavity on the whiteboard (see Figure 7), followed by turning point, stationary point, local maximum and minimum points. Teacher A6 also gave the definition of a point of inflection, showing that it may have a non-zero gradient. The students then worked on the function y = x'^- 2x^, to find its key features, such as concavity, being encouraged to work by-hand and on the GC (with, eg, [2"''] [Trace] to find turning points) in parallel.

The point of inflection

Figure 7 Teacher A6's conceptual emphasis in differentiation. In her third lesson she considered how the GC could help her students understand and recognise key concepts of functions y = AúnB{x+C)+D, such as amplitude, period, maximum and minimum values. This led to a question using trig functions to model temperature: A patient in the hospital had an illness in which his temperature (in degrees census) varied front it low of 37° to a high of 40.4°. The length of time between successive highs is 16 days. Determine the formula for the temperature, T, of the patient at time in days since the beginning of the illness. Assume that the function describing the temperature can be modelled with a sine function, with no phase shift.

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Teacher Integration of Technology into Mathematics Learning

a) Preliminary data

b) Beginning the curve

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c) Using the GC

Figure 8 Teacher A6 integrates the GC in modelling. In this she got the students to work by-hand on the conceptual structure of the problem, and then integrated the GC into the solution process. The students found A as \.l from (40.4 - 37)/2 (Figure 8a), D as 38.7 (37 + 1.7), and B as

of the GC and are thus more able focus on other important aspects, such as the linking of representations such as graphs, tables, and algebra, and to use other features of GCs. In turn, this better instrumentation ofthe GC produces a higher level of confidence in classroom use. Considering their PTK they — from B = 27i/period, to give y = 1.7siní — + 38.7begin to see the GC in a wider way than simply as a 8 \8 j calculator. They feel free to loosen confrol and encourage students to engage with conceptual ideas of mathematics In summary, these two teachers seem to have reached through individual and group exploration of the technology, a point where they have both strong insfrumentation and investigation of mathematical ideas, and the use of prediction insfrumentalisation of the technological tool. They are able and test methodology. For these teachers the mathematics to use the affordances of the technology (within acceptable rather than the technology has again been thrust into the consfraints) to provide an epistemic focus on the foreground, and the GC has been better integrated into the mathematics through the technology. Examples of this lessons and the didactical contract. If we think that the include the idea of testing concepts against definitions and a approach of this second group is preferable, then we must sfrong emphasis on generalisation, with the use of ask how we assist teachers to progress towards it. One investigation to form conjectures about graphs of the forms crucial answer involves the provision of pedagogicallyy — 2^^ + ic and y = fc • 2^. This focus has as its aim the focussed professional development, relevant resources and important aspect of generalisation (Mason, Graham and good lines of communication. Johnston-Wilder, 2005), with attendant modelling, of mathematii;al processes and constructs. In addition, these Although mathematics teachers often claim to be teachers had a high level of confidence in using the supportive ofthe use of technology in their teaching (Forgasz, technology to teach mathematics. 2006; Thomas, 2006) the degree and type of use in the classroom often does not correlate with this (Becker, 2000). 4 DISCUSSION Research into the uptake and implementation of technology in mathematics teaching has identified a range of factors that In summary we may describe the differences in the influence it. Goos (2005) lists some of these as: skill and progress of the teachers we have observed in terms of a previous experience in using technology; time and number of variables that delineate two different groups, with opportunities to learn (pre-service education, guidance a third proj^essing between the two. The first group may be during practicum and beginning teaching, professional identified, in terms of their insfrumental genesis (Lagrange, development); access to hardware (computers and 1999), as teachers who are still coming to grips with basic calculators), software, and computer laboratories; availability operational aspects of the technology, such as key presses of appropriate teaching materials; technical support; support and menu operations. This appears to be related to a low from colleagues and school adminisfration; curriculum and level of confidence in terms of teaching with the GC in the assessment requirements and how teachers interpret these for classroom. In terms of their PTK, this group is characterised students perceived to have different mathematical abilities; by an over-emphasis on passing on operational matters to knowledge of how to integrate technology into mathematics students, such as key presses and menu operations, to the teaching; and beliefs about mathematics and how it is learned. detriment c i a focus on the mathematical ideas. Furthermore, Forgasz (2006) agrees, with her computer survey listing the mathematics approached through the technology has an access to computers and/or computer laboratories as the most emphasis on technology, and work tends to be very processprevalent inhibiting factor (constraint), with lack of oriented; based on procedures and calculating specific professional development and technical problems, including answers to standard problems. There is little or no freedom lack of technical support next. Thomas's (2006) survey of given to students to explore with the GC, and it tends to be computer use in New Zealand secondary schools found that seen as an add-on to the lesson rather than an integral part of teachers make similar statements, citing availability of it. These features then become part of the teacher-initiated computers as the major issue, followed by a lack of software, expectations in the didactical confract. training and confidence. The first two may be described as obstacles, while the last two are consfraints. In contrast to this, the second group have advanced to the point where they are competent in basic instrumentation An analysis of possible affordances for the action of wwTv.technologyinmatheducation.com

International Joumal of Technology in Mathematics Education Vol 20, No 2

Michael O J Thomas and Ye Yoon Hong

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teaching with technology led us to consider a number of domains where the affordances arise, including the school's physical infrastructure, the school's personnel structure and relationships, and the technology itself. Hence, when considering the obstacles and constraints that might be infiuencing the teachers' use of GCs (and other technology) we considered three areas, departmental, school, and personal factors. It appears that teachers' confidence in using technology is related to the support of colleagues, including the head of department, and the school. However, we note that two teachers who did well did not have this support. Similarly, although two of the teachers had relatively weak personal instrumentation of the GC they did not allow this to prevent them from trying to introduce it in a conceptual way. In fact only teacher A6 had the profile that we would consider ideal for a successful outcome. She also was more experienced with GCs and was the only teacher to have had professional development with them, as reflected in her lessons. This research suggests that the affordances produced by supportive colleagues in a mathematics department constitute a strong infiuence on personal confidence, and that this confidence is ñirther linked to a teacher's attitude to

using technology in teaching mathematics and to personal leaming, as well as to epistemic teaching with technology. In view of this we suggest a tentative model (see Figure 9), with teacher confidence as a pivotal variable in producing conceptual teaching with technology. There has been some attempt to substantiate this (Thomas and Palmer, In Press) with positive results, but we suggest that further research to confirm or reftite this relationship would be valuable. There are some possible implications of these findings for both preand in-service professional development of teachers with regard to technology. It may be beneficial to building teacher confidence to enable them to be part of a supportive group sharing knowledge of instrumentation, practical classroom activities and ideas about the calculator use, especially in the initial stages of leaming about the technology. Hence, constructing professional development around a supportive community of practice in a manner that gives teachers the opportunity to use technology in a classroom might lead to positive changes in the perception of its value in leaming mathematics, to increased confidence in using the technology to teach mathematics, and improved classroom practice based on the epistemic value of the technology.

Attitude to: teaching maths with technology; personal learning

^ Instrumentation technology

of

1

4

Confidence in teaching with technology

i

Improved PTK

Conceptual teaching with technology

Support of school and mathematics department

Figure 9 A model of factors influencing conceptual teaching with technology. Placing confidence in a key role like this is in agreement with some research that identifies the infiuence of teacher beliefs and attitudes on teaching practice. For example, in her study, Forgasz (2006) found that teacher confidence, experience, skills or enjoyment of computers was the third highest factor encouraging computer use. While we cannot come to any definite conclusions from our small-scale studies it is interesting to note that teachers who have few school resources, are not well supported by their head of department, and who do not have strong personal GC skills can do quite well in implementing technology use. Thus, our research suggests that the teacher's personal orientations (Schoenfeld, 2011), especially their attitudes and beliefs, if strong enough, can override other negative constraints and obstacles. In particular, our results indicate that a strong belief in the value of technology in leaming mathematics coupled with a strong willingness to be open to personal leaming could be crucial factors. Further research will be necessary to test these claims.

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ACKNOWLEDGEMENT We would like to acknowledge the support of Texas Instruments and a Ministry of Education of New Zealand, Teaching, Leaming and Research Initiative (TLRl) grant for this research.

REFERENCES Artigue, M. (2002) Leaming mathematics in a CAS environment: The genesis of a reflection about instrumentation and the dialectics between technical and conceptual work. International Journal of Computers for Mathematical Learning, 1, 245-274. Ball, D. L., Hill, H. C. and Bass, H. (2005) Who knows mathematics well enough to teach third grade, and how can we decide? American Educator, Fall Issue, 14-22 and 43-46.

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Teacher Integration of Technology into Mathematics Learning Becker, H. J. (2000) Findings from the teaching, learning and computing survey: Is Larry Cuban right? paper presented at the 2000 School Technology Leadership Conference of the Council of Chief State Officers, Washington, DC. Brousseau, G. (1997) Theory of didactical situations in mathematics: Didactique des mathématiques. 1970-1990, Balacheff, N., Cooper, M., Sutherland, R. and Warfield, V. (trans, and eds), Dordrecht: Kluwer Academic Publishers. Chinnappan, M. and Thomas, M. O. J. (2003) Teachers' function schémas and their role in modelling. Mathematics Education Research Journal, 15(2). 151-170.

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Education, 35(5), 330-351. Harskamp, E. G., Suhre, C. J. M. and van Struen, A. (1998) The graphics calculator in mathematics education: An experiment in the Netherlands, Hiroshima Journal of Mathematics Education, 6, 13-31. Hong, Y. Y. and Thomas, M. O. J. (2006) Factors infiuencing teacher integration of graphic calculators in teaching. Proceedings of the 11''' Asian Technology Conference in Mathematics, Hong Kong, 234-243.

Cooney, T. J. (1999) Conceptualizing teachers' way of g. Educational Studies in Mathematics, 38, 163-187.

Kendal, M. and Stacey, K. (1999) Varieties of teacher privileging for teaching calculus with computer algebra systems. The International Journal of Computer Algebra in Mathematics Education, 6(4), 233-247.

Doerr, H. and Zangor, R. (1999) The teacher, the task and the tool: The emergence of classroom norms. International Journal of Computer Algebra in Mathematics Education, 6, 267-280.

Kendal, M. and Stacey, K. (2001) The impact of teacher privileging on learning differentiation with technology. International Journal of Computers for Mathematical Learning, 6, 143-165.

Doerr, H. and Zangor, R. (2000) Creating meaning for and with the graphing calculator. Educational Studies in Mathematics, 41, 143-163.

Kennewell, S. (2001) Using affordances and constraints to evaluate the use of informal and communications technology in teaching and learning. Journal of Information Technology for Teacher Education, 10, 101-114.

Forgasz, H. (2006) Factors that encourage and inhibit computer use for secondary mathematics teaching. Journal of Computers in Mathematics and Science Teaching, 25(1), 7793. Gibson, J. I. (1977) The theory of affordances, in Shaw, R. and Bransford, J. (eds) Perceiving, acting and knowing: Towards an ecological psychology, Hillsdale, NJ: Erlbaum, 67-82. Goos, M. (2005) A sociocultural analysis ofthe development of pre-service and beginning teachers' pedagogical identities as users of technology. Journal of Mathematics Teacher Education, 8, 35-59. Goos, M., Galbraith, P., Renshaw, P. and Geiger, V. (2000) Reshaping teacher and student roles in technology-enriched classrooms. Mathematics Education Research Journal, 12(3), 303-320. Goos, M., Galbraith, P., Renshaw, P. and Geiger, V. (2000) Reshaping teacher and student roles in technology-enriched classrooms. Mathematics Education Research Journal, 12(3), 303-320.

Lagrange, J.-B. (1999) Complex calculators in the classroom: Theoretical and practical reflections on teaching pre-calculus. International Journal of Computers for Mathematical Learning, 4(1), 51-81. Mason, J., Graham, A. and Johnston-Wilder, S. (2005), Developing thinking in algebra, London, UK: Paul Chapman Publishing and The Open University. Nosss, R. and Hoyles, C. (1996) Windows on mathematical meanings: Learning culture and computers, Dordrecht: Kluwer. Pierce, R. and Stacey, K. (2004) Monitoring progress in algebra in a CAS active context: Symbol sense, algebraic insight and algebraic expectation. International Journal for Technology in Mathematics Education, 11(1), 3-11. Pierce, R. and Stacey, K. (2004b) A framework for monitoring progress and planning teaching towards the effective use of computer algebra systems. International Journal of Computers for Mathematical Learning, 9, 59-93.

Greeno, J. G.(1994) Gibson's affordances. Psychological Review, 101 (2), 336-342.

Rabardel, P. (1995) Les hommeset les technologies, approche cognitive des instruments contemporains, Paris: Armand Colin.

Guin, D. and Trouche, L. (1999) The complex process of converting tools into mathematical instruments: The case of calculators. International Journal of Computers for Mathematical Learning, 3, 195-227.

Schoenfeld, A. H. (2011) How we think. A theory of goaloriented decision making and its educational applications, Routledge: New York.

Hill, H., and Ball, D. L. (2004) Learning mathemafics for teaching: Results from California's mathematics professional development institutes. Journal for Research in Mathematics www.technologyinm atheducation.com

Shulman, L. C. (1986) Those who understand: Knowledge growth in teaching. Educational Researcher, 15, 4-41. Simmt, E. (1997) Graphing calculators in high school International Joumal of Technology in Mathematics Education Vol 20, No 2

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mathematics. Journal of Computers in Mathematics and Science Teaching, 16(2/3), 269-289. Simon, M. (1995) Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, 26(2), 114-145.

Yackel, E. (2000) Creating a mathematics classroom environment that fosters the development of mathematical argumentation, paper presented at the Ninth International Congress of Mathematical Education, Tokyo. BIOGRAPHICAL NOTES

Thomas, M. O. J. (1994) A process-oriented preference in the writing of algebraic equations, Proceedings of the 17th Mathematics Education Research Group of Australasia Conference, Lismore, Australia, 599-606. Thomas, M. O. J. (2006) Teachers using computers in the mathematics classroom: A longitudinal study. Proceedings of the 30''' Conference of the International Group for the Psychology of Mathematics Education, Prague, Czech Republic, 5, 265-272. Thomas, M. O. J. and Chinnappan, M. (2008) Teaching and learning with technology: Realising the potential, in Forgasz, H., Barkatsas, A., Bishop, A., Clarke, B., Keast, S., Seah, WT., Sullivan, P. and Willis, S. (eds). Research in Mathematics Education in Australasia 2004-2007, Sydney: Sense Publishers, 167-194. Thomas, M. O. J. and Hong, Y. Y. (2001) Representations as Conceptual Tools: Process and Structural Perspectives, Proceedings of The 25''' Conference of the International Group for the Psychology of Mathematics Education, Utrecht, The Netherlands, 257-264.

Mike Thomas is a full Professor in the Mathematics Department at The University of Auckland, New Zealand. His research interests are in using technology to improve learning, developing theories of advanced mathematical thinking, the learning and teaching of calculus and undergraduate mathematics, school and university teaching, and connections between mathematics education and cognitive neuroscience. He has given invited research seminars in a number of countries and is on the editorial boards of the international journals. Mathematics Education Research Journal and the International Journal of Mathematical Education in Science and Technology. He recently led an international survey team for the 2012 International Congress on Mathematical Education (ICME), on the mathematical difficulties inherent in the transition from school to university. Ye Yoon Hong is a lecturer in the Mathematics Department at The University of Yonsei, Korea. Her research interests are in using technology to improve learning and teaching of calculus and undergraduate mathematics, school and university teaching.

Thomas, M. O. J. and Hong, Y. Y. (2005) Teacher factors in integration of graphic calculators into mathematics learning, in Chick, . L. and Vincent, J. L. (eds). Proceedings ofthe 29th Conference of the International Group for the Psychology of Mathematics Education, Melbourne, Australia: University of Melbourne, 4, 257-264. Thomas, M. O. J., Hong, Y. Y., Bosley, J. and delos Santos, A. (2007) Calculator use in the mathematics classroom: A longitudinal study, in Wei-Chi Yang, T., de Alwis & Jenchung Chuan (eds). Proceedings of the 12"' Asian Technology Conference in Mathematics, Taiwan: ATCM, 37-47. Available from: http://atcm.mathandtech.org/EP2007/EP2007.htm. Thomas, M. O. J., Monaghan, J. and Pierce, R. (2004) Computer algebra systems and algebra: Curriculum, assessment, teaching, and learning, in Stacey, K., Chick, H. and Kendal, M. (eds). The teaching and learning of algebra: The 12th ICMI study, Norwood, MA: Kluwer Academic Publishers, 155-186. Thomas, M. O. J. and Pahner, J. (In Press) Teaching with digital technology: Obstacles and opportunities, in ClarkWilson, A. Sinclair, N. and Robutti, O. (eds) The mathematics teacher in the digital era, Dordrecht: Springer. Thomas, M. O. J., Tyrrell, J. and Bullock, J. (1996) Using Computers in the mathematics classroom: The role of the teacher. Mathematics Education Research Journal, 8(1), 3857. © 2013 Research Information Ltd. All rights reserved.

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Revised: 1" November 2012

Graphic calculators (GC) have been widely used in teaching for around 20 years, and yet many teachers still appear to be either unconvinced, or unaware, of how to release the potential of such technology to assist students to learn mathematics. This research describes a study addressing aspects ofthe teacher's role in integrating technology into their pedagogy. It considers issues related to the didactical contract, pedagogical technology knowledge (PTK), and procedural and conceptual knowledge in GC integration in teaching by two groups of secondary teachers. The results describe a number of factors that identify teacher progress toward pedagogical integration of digital technology and from these some tentative inferences for teacher professional development may be drawn.

1

BACKGROUND

Learning and teaching mathematics with technology is a complex process requiring a teacher to marshal a number of key competencies. In this paper we outline what we believe seme of these to be and illustrate how teachers can make progress in gaining these by considering cases of specific tîachers. It seems clear that the key factor in whether, £ind if so how, technology is used in the learning of mathematics is the teacher (or lecturer). In turn, there are many factors that influence a teacher's attitude to, and practice of, using a graphic calculator (GC), as with any technology. These include the affordances and constraints ofthe environment they work in (see e.g., Greeno, 1994) and their attitude to, and beliefs about, the mathematics and the technology, as well as their confidence and ability in using it to teach mathematics. Teacher affective variables, or orientations (a broad term that includes attitudes, dispositions, beliefs, values, tastes and preferences - see Schoenfeld, 2011, p. 29), their perceptions of the nature of mathematical knowledg(; and how it should be learned, their mathematical knowledge for teaching (Ball, Hill and Bass, 2005; Hill and Ball, 2004), which includes pedagogical content knowledge (Shulman, 1986), all influence their teaching. When we add the potentially subversive element of technology to this mix (Thomas, Tyrrell and Bullock, 1996) then the situation becomes even more complex. 1.1

Pedagogical Technology Knowledge

by a teacher of what we have previously called pedagogical technology knowledge (PTK) (Thomas and Hong, 2005; Hong and Thomas, 2006; Thomas and Chinnappan, 2008). A teacher's PTK applied to mathematics includes the principles, conventions and techniques required to teach mathematics through the technology. PTK is an emerging framework that can be used by researchers to generate explanations about teachers' use of technology to scaffold mathematical learning; it comprises partly the teacher's perspective on the technology and partly their familiarity with it (see Figure 1). They need to appreciate that technological tools, such as GCs, may be employed in teaching in qualitatively different ways, such as for: property investigation; computation; transformation; data collection and analysis; visualising; and checking (Doerr and Zangor, 2000) and that students may form a variety of different relationships with them (Goos, Galbraith, Renshaw and Geiger, 2000) depending on teacher direction. This directional emphasis in the use of technology, which has been called teacher privileging (Kendall and Stacey, 1999, 2001), has been shown to shape student preferences, so that they often follow the teacher's lead. We argue that teachers who have built up a sound basis of PTK are more likely to feel comfortable in accessing tools of technology in designing mathematical learning experiences. This privileging can have a positive impact on students' uptake of technology in exploring mathematics. Another key aspect of PTK surrounds the transforming of a technological tool into one or more instruments (Rabardel, 1995). This engages the teacher in actions and decisions to produce the personal schemes required to adapt the tool to a particular mathematical task, considering what it can do and how. PTK grows as teachers progress through the stages of instrumentalisation and instrumentation of the tool, gaining a personal appreciation of its role in learning mathematics, and importantly, of ways in which students may be assisted through various teaching approaches to emulate their instrumental genesis of the technological tool. This concerns more than just the mechanics of getting the syntax and semantics of the input/output, developing algebraic insight and expectation (Pierce and Stacey, 2004a), and coping with the difficulties of navigating between screens and between menu operations. It means seeing opportunities for epistemic mediation by the technology, between the user and the mathematics.

Due to this complexity of teaching mathematics with technology we hypothesise that it requires the development

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Pedagogical Knowledge

Mathematical Content Knowledge

Mathematical Knowledge for Teaching Personal orientations

Technology instrumental genesis

Pedagogical Technology Knowledge Figure 1 An outline ofthe construction of Pedagogical Technology Knowledge (PTK). In this regard. Artigue (2002) describes pragmatic value as a "productive potential (efficiency, cost, field of validity)" (p. 248) and epistemic value as a contribution "to the understanding ofthe objects they involve" (p. 248). For example, technology may have pragmatic value in providing a means (an instrument with associated techniques) for solving quadratic equations, but it can also have epistemic value if it is used to promote mathematical thinking about the kind of objects that quadratic functions and equations are and the nature of the transformations involved in the solution process. Discussing instrumentation and conceptualisation, Guin and Trouehe (1999) stress the need for them to occur concurrently in the classroom, and hence teachers need to focus technological activity on specific conceptions. In work somewhat parallel with the general concept of PTK, Pierce and Stacey (2004b) outline four key aspects of knowledge and skills for teaching effectively with Computer Algebra System (CAS) technology: mathematical, machine, technical and personal, and maintain that the technical and personal need the most attention by new teachers. In order to assist development ofthe above range of skills, they have proposed a framework for teaching to promote effective use of CAS by monitoring progress. In summary, PTK involves a teacher understanding the principles and techniques required to incorporate technology into mathematics classrooms so that mathematical learning through the technology, in the form of epistemie rather than simply pragmatic knowledge (Artigue, 2002), is the outcome. The factors that combine to produce PTK involve instrumental genesis, strong mathematical knowledge for teaching, and positive teacher orientations and goals (Schoenfeld, 2011), especially beliefs about the value of technology and the nature of learning mathematical knowledge, along with affective aspects, such as confidence (see Figure 1 ). 1.2

Affordances Situations

and

Constraints

in

Didactical

A useful framework for examining technology use in the classroom is the Theory of Didactical Situations (TDS) of © 2013 Research Information Ltd. All rights reserved.

Brousseau (1997). Here, Brousseau describes three situations, action, communication and validation that have as their endpoint student acquisition of specific knowledge. These didactical situations arise from the students' relationship with a didactical milieu established by the teacher. The teacher's role is to orchestrate the elements of this milieu so that the specified knowledge is attained. In doing so she takes into account the dynamic didactical contract between the students and the teacher, which is the mutual recognition of the roles that the student and teacher play in the acquisition of the knowledge. This recognises, through the tacit acceptance of each party, that in any didactical situation there are reciprocal obligations in the relationship. Not least among these is the expectation by students that they will gain knowledge, and by the teacher that the students will want to learn. Of course this social contract is a dynamic entity, changing and adapting to new circumstances that arise in the classroom situation. The provision by a teacher of GCs in the milieu of a didactical situation can alter the dynamics of how students might gain the expected knowledge. Even when they have a high level of PTK teachers may still faee difficulties in implementing technology use in didactical situations. One useful way of analysing these has been presented by Gibson ( 1977), who has initiated a theory that presents relationships between agents, in our case teachers, and the environment they seek to interact with. A key feature of this interaction is the presence of affordances (attributes of the environment that contribute to the potential for an interaction to occur) and constraints (characteristics of the affordance that provide structure and guidance for the interaction - see Kennewell, 2001). In this mix, according to Greeno (1994, p. 338), "An affordance relates attributes of something in the environment to an interactive activity by an agent who has some ability". In short, affordances speak about the potential for action, while constraints impose the structure for that action. Thus, an example of an affordance, given by Gibson is the provision of mailboxes for posting letters; its location and the size of its opening would be constraints. In a technology classroom setting, the presence of technology is an affordance, with student or teacher www.technologyinm atheducation.com

Teacher Integration of Technology into Mathematics Learning instrumentation, time available to use the technology and the content of curriculum as examples of consfraints. Similarly, in terms of the present research, a graphic calculator is an affordance and the size and resolution of its screen display a consfraint. Other researchers have used different words to describe the absence of an affordance with regard to technology use in mathematics learning. For example, Forgasz (2006) talks about encouraging and inhibiting factors, vk^hile Thomas (2006) and Thomas, Hong, Bosley and delos Santos (2007) use the terminology of obstacles to technology use. In this paper we delineate between a consfraint, which implies the presence of an affordance, and an obstacle, something that inhibits or prevents the presence of an affordance-producing entity in the classroom situation. A classroom didactical situation with GCs in the milieu requires changes in the didactical confract. Factors inñuencirig the didactical contract include the teacher's affective variables (beliefs and attitudes), their perceptions of the nature; of mathematical knowledge and how it should be learned, as well as their mathematical and pedagogical content knowledge (Shulman, 1986). One may assume that if a teacher possesses a limited knowledge of a concept and its related subconcepts (Chinnappan and Thomas, 2003) then they will find it more difficult to provide the kind of environment and experiences that will assist students in the construction of rich conceptual thinking. Instead they may regress to a process-oriented approach (Thomas, 1994), presenting; students with a toolbox selection of procedures that may be applied to each problem that arises. While such procedures and skills are important, mathematical thinking is clearly much wider than this, and requires procedural and conceptual interactions with the various representational forms of mathematics (Thomas and Hong, 2001). However, teaching is not mediated simply by the mathematical understanding ofthe teacher (Cooney, 1999), but it is also influenced by the teacher's mathematical knowledge for teaching (Hill and Ball, 2004; Ball, Hill and Bass, 2005). This refers to understanding the mathematical content, the constructs and ideas involved in a particular topic, and how these relate to the principles and techniques required to teach and learn it, including appropriate structuring of content and relevant classroom discourse and activities (Simon, 1995; Cooney, 1999; Chinnappan and Thomas, 2003).

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multiple representational approach to teaching. In this sense they act as a physical medium for the purpose of reexpression and translation (re-presentation), and for mathematical communication (Noss and Hoyles, 1996; Yackel, 2000). However, when teachers use graphic calculators they may sfress different teaching goals, since these are anchored in their personal orientations, including their beliefs about the potential of the calculator to support the goals (Doerr and Zangor, 1999; Simmt, 1997; Harskamp, Suhre and van Sfreun, 1998). Such goals can include student-cenfred initiatives, interactive-inquiry teaching styles and student-cenfred views of learning, as well as contentoriented goals and a view of learning as listening. Teacher progression in their belief in the value ofthe technology in teaching mathematics leads to a key issue they have to face, namely the level of integration of the technology in learning that they will practice. This may range from using it at prescribed moments as a teacherdirected add-on, to an ever-present insfrument that is an extension of cognition; as an instrument for pragmatic mediation to one for epistemic mediation. The research described in this paper followed a group of teachers as they began, or continued, their instrumental genesis of the GC tool, thus extending their PTK. It attempted to understand teacher practice in relation to the congruency of content knowledge, mathematical knowledge for teaching, instrumentation and instrumentalisation, PTK and the didactical contract. 2

METHOD

This research comprised two case studies. In the first 22 secondary school teachers, from 22 different schools were using GCs in their teaching in Auckland, New Zealand. Of these 22 teachers who were using GCs, 17 were experienced in their use and 5 were not. The second-named researcher carried out the fieldwork and had an initial meeting with each ofthe teachers. At this initial meeting the topics to be taught, the time-lines and the technology to be used were discussed, and they were given a Likert-style attitude test (see Figure 2) with five subscales, comprising attitude to: mathematics, technology in general, personal learning, technology and GC in learning mathematics. Following this meeting a further, more detailed, discussion took place by email.

The introduction of new technology into the classroom has been shown to be capable of a subversive effect (Thomas, Tyrrell and Bullock, 1996) radically altering the didact cal confract. Thomas, Tyrrell, and Bullock (ibid. p. 49) suggest that the introduction of technology requires a new mindset on the part of teachers, a 'shift of mathematical focus ', to a broader perspective of the implications of the technology for the learning of the mathematics. As teacher and student insfrumental genesis of the technological tool proceeds teacher beliefs and attitudes are shaped, changing their teaching emphasis and the didactical confract to give increased emphasis to the insfrument. There is some evidence that if the use of technological tools, such as computers and graphic calculators, is made an integral part of a didactical contract, they can provide valuable support for a

The teachers in this first case study were also given a diary and encouraged to complete it for all their GC lessons, including their lesson aims and expected outcomes, teaching method, classroom organisation, details of GC use, and their reflection on student learning. Three of these group one teachers, whose teaching practice is featured in this paper, chosen to illustrate their comparatively positive work with the GCs, and to cover a wide range of obstacles and consfraints. These three teachers, we call A2, A6 and A8. Teacher A2 had 9 yeats' teaching experience, had made some use of technology in teaching, and taught in a middle socio-economic school (decile 5), while A8 was a relatively new teacher, with 4 years' experience, who had made only a little use of GCs in teaching, and who taught in a low socioeconomic school (decile 2). Teacher A6 was a deputy head of department (HOD) with 20 years' teaching experience.

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including a number of years implementing technology in her teaching, who worked in a high socio-economic school (decile 10). Following discussion, they chose to teach topics including families of curves, linear programming, limit, drawing graphs and derivatives. We made a total of 8 lesson observations; two in Year 13 (aged 18 years) for each of teachers A2 and A6 and three for Year 13 for teacher A8. In addition, one Year 12 (aged 17 years) lesson for teacher A6

was observed. The classroom practice of these three teachers is considered in more detail below (along with three group two teachers). The lesson videotapes were transcribed for analysis along with the data from the questionnaire, the lessons and the diaries. After these observations were completed the teachers were interviewed for about 40 minutes, and these were tape-recorded and later transcribed for analysis.

Mathematics Attitude Questionnaire Name: School: Levels of Teaching (Circle): Y7 Y8 Y9 YIO Yll Y12 Y13 Years of Teaching: Please circle the numbers on the right below corresponding to which of the following indicates your level of agreement with each statement. 5 - 1 STRONGLY AGREE (SA) with the statement 4 - 1 AGREE (A) with the statement 3 -- NEUTRAL (N) 2 - I DISAGREE (D) with the statement 1 - I STRONGLY DISAGREE (SD) with the statement

1. More interesting mathematics problems can be done when students have access to technology. 2. Students understand mathematics better if they solve problems using paper and pencil. 3.1 have lots of ideas about how I can make use of technology in mathematics. 4. Students should not be allowed to use technology during mathematics tests or examinations. 5.1 think technology is a very important tool for learning mathematics. 6. Technology can be used as a tool to solve problems students could not solve without it. 7. Technology is only a tool for doing calculations more quickly. 8. Technology can make mathematics more fun. 9. Students should use technology less often in mathematics. 10. Using technology will cause students to lose basic computational skills. 11.1 want to improve my ability to teach with technology. 12. Students rely on technology too much when solving problems. 13. Technology should only be used to check work once the problem has been worked out on paper. 14. Mathematics students need to know how to use technology. 15. Students should not be allowed to use technology until they have mastered the idea or the method. 16. Mathematics is easier if technology is used to solve problems. 17. Learning how to use technology is difficult for me. 18. Using technology makes students better problem solvers. 19.1 lack the confidence to use technology to solve mathematical problems. 20. Learning mathematics is mostly memorising a set of facts and rules. 21. When doing mathematics it is more important to know how to do a process than to understand why it works. 22. Learning mathematics means exploring problems to discover patterns and make generalisations. 23. Students would be better motivated in maths if they could use a graphic calculator. 24. Using a graphic calculator removes some learning opportunities for students. 25. Students would understand maths better if they had a graphic calculator. 26. Using a graphic calculator would make the management of data easier. 27. Students would be more confident in maths if they had a graphic calculator. 28. Since students can use a graphic calculator, they do not need to leam to draw graphs by hand. 29. I feel that computer algebra system calculators should be allowed in mathematics tests and examinations. 30. Using a graphic calculator to solve statistics makes the problems easier to understand.

S A 5

A N D S D 2 1 4

5 5 5

4 3 2 4 3 2 4 3 2

1 1 1

5 5 5 5 5 5 5 5 5

4 4 4 4 4 4 4 4 4

2 2 2 2 2 2 2 2 2

1 1 1 1 1 1 1 1 1

5 5

4 3 2 4 3 2

1 1

5 5 5 5 5 5

4 4 4 4 4 4

2 2 2 2 2 2

1 1 1 1 1 1

5

4 3 2

1

5 5 5 5 5 5

4 4 4 4 4 4

2 2 2 2 2 2

1 1 1 1 1 1

5

4 3 2

1

5

4 3 2

1

3 3 3 3 3 3 3 3 3

3 3 3 3 3 3

3 3 3 3 3 3

Figure 2 The attitude questionnaire.

© 2013 Research Information Ltd. AJl rights reserved.

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Teacher Integration of Technology into Mathematics Learning A second group of seven Auckland teachers completed a workshop covering both content and pedagogy for algebra and calculus, using the TI-83Plus GC and several downloadable memory-based Flash App[lication]s. None of these teachers had much experience of using the GC to teach mathematics, although two of the seven had previously made some use of them. Apart from one teacher trainee, all the teachers were experienced, with between 10 and 24 years teaching. Each teacher was given their own GC and a class set that they kept these for six months after the course. Once the teach(;rs were familiar with the GC they began to ask questions related to their teaching, and discussed ideas with one another. In the month following the workshop the teachers v/ere given a brief questionnaire on their perspective of the value of the TI-83P]us GC for teaching mathematics, and six teachers, four from one school and two from a second, agreed to take part in the classroom-based phase of the research. During this three-week phase we observed their teaching with the GC for Years 10-13 (age 15-18 years) recording observations and video taping their practice. Three of these t(;achers, F1, F2 and F3 are used below, along with A2, A6 and A8, to exemplify the differing rates of progress Teacher

Teacher GC Experience/Confidence

Al A2 A7 A8 AID B4 C6 É4 E5 E6

New user/Weak New user/Weak Experienced/Weak New user/Weak Experienced/Weak New user/Weak Some experience/Weak Some experience/Weak Some experience/Weak Some experience/Weak

A3 A4 A5 A6 A9 B3 B5 C3 D2 D5 El E2

Experienced/Strong Experienced/Strong Experienced/Strong Experienced/Strong Experienced/Strong Experienced/Strong Experienced/Strong New/Strong Experienced/Strong Experienced/Strong Experienced/Strong Experienced/Strong

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in constructing PTK, and its subsequent infiuence on their technology use. 3

RESULTS

Potential affordances for teachers using GCs (or other technology) arise in a number of domains. For example, there may be a supportive environment, comprising a school that supplies GCs and time for professional development, a mathematics department with a head of department who encourages GC use, and other mathematics teachers of a like mind. The level of confidence in GC use in mathematics of the 22 teachers in case study one was deduced through discussion with them and from classroom observation, and we concluded that 12 had strong, and 10 weak confidence. The teachers were from 15 different schools, and our aim was to investigate the factors influencing the different levels of confidence and to see if confidence was linked to style of teaching with the GCs. Table 1 lists some ofthe school and department factors we identified, along with the personal experience and confidence of the teachers in the study, divided into groups based on confidence.

Department HOD/Teacher Support Weak Confidence Group Weak/None Weak/None Strong/Others Weak/None Weak/None Strong/Others Weak/None Strong/Others Weak/None Moderate/Some Strong Confidence Group Strong/Others Strong/Others Strong/Others Strong/Others Weak/None Strong/Others Strong/Others Strong/Others Strong/Others Weak/None Strong/Others Strong/Others

School Use of GC/Teacher Support New/Strong New/Weak Experienced/Strong New/Weak New/Weak Experienced/Strong Experienced/Weak Experienced/Strong Experienced/Moderate Experienced/Moderate New/Strong New/Strong Experienced/Strong Experienced/Strong New/Strong Experienced/Strong Experienced/Strong New/Strong Experienced/Strong Experienced/Weak Experienced/Strong Experi enced/Strong

Table 1 Background Factors Infiuencing the 22 Teachers' Use of GCs From Table 1 we see that of the 12 teachers with strong confidence in their ability to teach with the GC, 11 were experienced users, 11 had strong school support, 10 had strong HOD support, and 11 had the support of other teachers in their department. In contrast, of the 10 teachers with weak confidence in their GC use, 8 were inexperienced users, and seven had little HOD or other teacher support in their schools. Interestingly the level of school support was split, with five being supportive and five not.

This initial analysis seems to suggest that among the key variables in producing confident users of GCs are the teacher's own experience, and the immediate affordance of support from others in the mathematics department. However, these may be neither all necessary nor sufficient, since there are exceptions to the general trend. On the one hand teacher C3, who was new to using the GC was confident, but was in a school where there was strong support from the HOD and the school, even though the school was new to using the technology, and teachers D5 and

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A9 were confident in spite of no support (D5), or no support from the department (A9). On the other hand, teacher A7 was experienced, had strong support from both department and school, and yet lacked confidence in his ability to teach with the GC; teachers A10, C6, E5 and E6 all had some (limited) experience with the GC but had little or no support from their department or school, and teacher E4 had some experience and strong support, yet they were all weak in confidence. It seems support from others in the school, and especially the mathematics department may be a primary environmental factor influencing teacher confidence. In addition to the above examination of the personal and environmental factors influencing GC use, the teachers were given an attitude questionnaire, to probe for other possible influences on teaching practice. The scale, given in Figure 1, comprised 5 subscales, on attitude to: mathematics (Q's 2, 20, 2 1 , 22); technology (Q's 7, 13, 15, 18); using technology to learn mathematics (Q's 1, 4, 5, 6, 8, 9, 10, 12,

Sfrong Confidence (JV= 12) Non-users (A'= 10) p-value

14, 16); using the GC to learn mathematics ( Q ' s 23-30); and personal learning (Q's 3,11, 17, 19). The internal consistency of the scale, measured using Cronbach's Alpha reliability coefficient, was 0.90, suggesting that it was a reliable scale. For the teachers in case study one we managed to collect data based on the scales in Figure 1 from another group often teachers in their schools who did not use GCs at all in their teaching. Comparing the 22 teachers who used the GC with the 10 who did not we found that there was no difference between these two groups on the subscales except, not surprisingly, weak evidence of a significantly more positive attitude {p' = 5. So I'll just give you an example to try yourself That should say 1 and 5. Anyway I'll give you an example today. So following the insfructions and directions on the sheet try doing this question. In summary we could say that for these two teachers insfrumental genesis was very much in progress, but since they were primarily working on their insfrumentation, as shown by their emphasis on which keys to press, this may have delayed progress on the instrumentalisation phase. They also emphasised the ability of the GCs to produce solutions rather than improve mathematical thinking. As a consequence their practice was focussed on the use of the technology rather than on the mathematics, and how understanding of it could be enhanced though the technology. Further, they both had low levels of confidence in using the technology to teach mathematics. These atfributes seem to be a feature of what we see as the early stage of progress in the acquisition of PTK. 3.3

More progress

Teacher A2, from the first case study group, worked in a medium socio-economic level school (decile 5). She had 9 years' teaching experience and had been using GCs for 3 years. We made 2 observation visits to her classroom, both involving Year 13 students (age 18 years) studying firstly frigonomefric graphs and their fransformations, and then the solution of trigonomefric equations. The school allowed GC use in examinations and encouraged students to buy their own calculator, but the financial situation at the school was not considered good enough to support technology; according to the head of the department "the budget doesn't allow for it." Hence the mathematics department did not have a class set of GCs or a viewscreen, so she had an obstacle to overcome when she wanted to demonsfrate working with a GC. When we visited teacher her class only seven of the 14 students had their own scientific or graphic calculator, and so the students shared with each other or worked without a calculator. Thus she worked under the constraint that the calculators were not all the same, and so she had to explain how to work with each model. Teacher A2 used a CASIO fx-9750G GC and an overhead projector (OHP) on which she wrote to demonsfrate and explain key points. In her interview she spoke about the visual value of the GC, how "when they have a graphics calculator, it's very useful for them to see how the graphs... what the graphs look like, and you can change numbers". She also mentioned the time-saving aspects of its use "It's much faster, quicker and easier". However, she also thinks ofthe conceptual value of the GC in helping students make connections. I think it's important for them to understand concepts in mathematics and there's got to be a balance between the skills they do and the word www.technologyinmatheducation.com

Teacher Integration of Technology into Mathematics Learning problems...Basically we want them to be able to see what the concept is, and instead of sketching it every time... so if you want them to check for '>^=3cos(4x)', they have independently seen what is happening to 'cos (4x)' and what happens when they [do] '2cos(x)', and they can put the two together and get it from the graphics calculator quicker and they can see the changes much faster. Teacher A2 did not put a lot of emphasis on student instrumentation, getting them to think about what buttons to press, etc. She says that once they have the basics students are quick to pick up what they need in each lesson: "I don't put it all up at the beginning of the lesson or they'll get confused with it. I do it as and when I feel it's necessary. But generally, switching it on, feeling the menu and how to use the cursor, most of the students know. That's why I put the instructions up and most of them caught on to it very quickly". Her emphasis on conceptual understanding was seen in aii interview comment "I think it's important for them to understand concepts in mathematics and there's got to be a balance between the skills they do and the word problems they work out." In her first lesson with the Year 13 students (age 18 years) she used the GC to allow students to investigate graphs of the form y = AsmB(x-i-C)-\-D, etc. First she concentrated on the effect of a single parameter, using >'=yísin;í:, with ^ = 2, 3 and 0.5, and asking what these numbers signified, and then moved on to graphs of the form y = cos(Bx), with ß = 2 and 3. Students were encouraged to work together, "Discuss with the person sitting next to you the effect of y = sin(.ßx).", and she also got students to come out and sketch graphs on the white board. In each case she tried to geit students to focus on the concepts of domain, range, period, amplitude and frequency, asking questions such as "V/liat does the number 2 signify?". She explained the conceptual approach, and the role of the GC this way: Instead of just sketching y = cos(2x)' and then after it doing 'v = cos(3x)', they've got to see the connection. If they see... keep changing the variables, and they see the effect ofthat, that's conceptual understanding and that's v/hat we should be getting at and the graphics calculator is really useftil for that."

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Moving on she focused on the concept of translation, asking "What sort of translation is y = sim- -I- 2? What is [sic] the domain, range and period? What does [the] graph look like?" Figure 3 shows her use of some of the key ideas on the overhead projector. In the second lesson was spent solving equations such as sinx = 0.5. To do this she got the students to draw the graphs of>' = sinx and y = 0.5 on the GC (see Figure 4(a)) and consider their intersection. Afterwards she used this same concept to get them to solve 7 - 3x = 6cosx. Since she did not want her students to be procedural users of the GC, but to think about the mathematics, she used this example as an opportunity to get them to look through what they saw on the screen. She pointed to the apparent intersection of the two graphs near the y-axis (see Figure 4(b)) and asked the students to use the GC to zoom in on that area. They could then see that the line does not actually intersect the curve (see Figure 4(c)) and that was why there was no solution given by the GC, instead "When you press [G-Solve], it always give you first intersection from left to right, then x = 3.85,

(a) y = sinx and j ; = 0.5 for sinx = 0.5

(b) y = 7 — 3x and y = 6cosx for 7 — 3x = 6cosx

(c) Zooming in Figure 3 Teacher A2 stresses the concepts of range and period for j ; = sinx + 2.

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Figure 4 Teacher A2 shows that trig equations can be solved with intersecting graphs

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Teacher A8, also in the first group, worked in a low socio-economic level school (decile 2). He was less experienced than the others, with 4 years' teaching, was new to using GCs in his teaching and had not attended any GC professional development course. He worked in the same school as teacher A2, and the school allowed GC use in examinations and encouraged students to buy them, even though the majority of the students could not afford one. However, the school was trying to support technology use so the mathematics department had purchased one class set of CASIO fx-9750G calculators and had one classroom with a computer set up for PowerPoint use. Teacher A8 commented on this constraint that "Very few of the students can afford their own graphics calculator therefore, the only chance they get to practise with them is in the lesson when we hand them out and draw them back again, so they don't get the familiarity ...they haven't had the practice in using the technology." We made 3 observations of his classroom. All three lessons involved year 13 students (age 18 years) using the class set of GCs, one studying the binomial distribution in statistics, and the others the use of power and exponential functions in statistical modelling. Once again he had to work around the obstacles ofthe lack of an overhead projector and a viewscreen to project his calculator screen, but managed this by using the affordance of a poster of the GC to show students the right key presses. However, this clearly has constraints, such as not being able to show the result of the key presses.

In his interview teacher A8 showed that he did not want students simply to use the GC in a procedural manner, but saw the value of an inter-representational approach. The graphical side of the GC was important to him since "it can also provide a visual help to understanding the overall idea.", and he talked about how "students have come to me and said, 'I now understand what you're saying', by having a little presentation of the graph." He also confirmed his desire for a non-procedural approach, saying that his "prime aim would be for them to understand the method and be able to apply it rather than to arrive at the right answer." Actually, his first lesson was rather procedural, using the computing power of the GC to calculate Pix = 6) = ^C^ 0.4* (1 - 0.4)^ and finding other probabilities, P(x > k) or P{x < k). It appeared from his interview that the purpose of this lesson was revision, since "Basically the type that they're going to get in the exam." However, for the second lesson on power and exponential functions he tried to integrate the GC into students' mathematical thinking. He gave students the ftinction y = 2e""'*^*, asking them to sketch the graph for x values from 0 to 3, by completing a table he gave them (see Figure 5(a)) and then plotting the function. Students completed the table by putting x = 0, 0.5, 1, 2, 3 into the ftinction y = 2é~°''*^' on their GC and then plotting by hand (Figure 5(b)). He used this method rather than getting the GC to draw the final graph so that the students could see how the graph was constructed.

Í 5

Figure 5(a)

Figure 5(b)

Figure 5.Teacher A8 integrates the GC into graph plotting. A similar process was then followed for graphs ofthe form y = x , for k = -2, 1.5 and -0.5. Finally a contextual problem involving the volume of water in a lake at time /, with the function V = 1275/"^^^, was considered. Here the ñmction was drawn using the GC. Teacher A8 asked the students "At what point does the amount of water in the lake drop below 200 m"? When y = 200, what is x-value? Use [Trace] key, approach to y = 200, you can find the x-value." (variables had been changed on the GC). Here the students had a choice of method. They could either use [G-solve], which gave them the x-value directly, or the [Trace] key on the GC; they preferred the former. This work led into the third lesson on using power and exponential functions to model two variable regression. In summary we might say that the instrumentation of these teachers was improved but they were still making progress on instrumentalisation of the tool, with classroom constraints a factor in this. However, rather than simply © 2013 Research Information Ltd. All rights reserved.

focussing on the instrument, such as menu operations and key presses, they were beginning to use the technology, for example through graphs, to assist students with conceptual understanding ofthe mathematics. Examples of this include getting students to think about generalising transformations of the form of the form ^ = ^ sin B(x + C) + Z) and graphs ofthe form j ; = jc* by linking algebraic, tabular and graphical representations. They were also growing in confidence in use ofthe technology to the point where they were willing to experiment with their approach. 3.4

Greater strides forward

Teacher F3 from the second group exemplifies a teacher who had made greater strides forward. She had over six years' experience of using GCs in her teaching and had been involved in a previous research study with them. In spite of her experience, in the questionnaire she admitted "Sometimes it's hard to see how to use it effectively so I www.technologyinm atheducation.com

Teacher Integration of Technology into Mathematics Learning don't use it as continuously as I should." Her motive was a rather pragmatic "We should move with the times" and she had a small reservation about the GC that "It is OK. By now expected better resolution though." Due to her relative experience she appeared confident in her use of the GC and spoke at length in her interview, describing how "In the past I have also done some exploratory graphs lessons where students get more freedom to input functions and observe the plots." Further, she explained that she was happy to loosen control of the students and let them explore the GC and help one another: "Students learn a lot by their own exploration...In past lessons I have never had a student get lost while using a graphics calculator. Sometimes friends around will assist someone." However, she acknowledged her need to progress in her types of GC use, "1 would like to see them used more frequently and beneficially in class with structured lessons." Wliile she could perceive imaginative GC usage, part of her difficulty involved the pressure of day-to-day teaching, since she admitted that "I was not relaxed enough with the term coming to an end and other aspects in this year's teaching to be inspired to use the calculator with imagination for the students." Responding to what she wanted her students to learn she replied in terms ofthe challenge ofthe depth of mathematics, "The success for me as a teacher is when they want to learn more and students show a joy either in what they are doing or in challenging themselves and their teacher with more deeper or self posed mathematical problems." She was convinced that the novel and challenging nature of :he GC could motivate students - "The calculator Appears parallel

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puts a radiant light in the class... With a graphics calculator lesson no one notices the time and no one packed up." - and that her perspective on learning mathematics influenced her PTK came out in the comment that "Today we find a lot of Maths does not need underlying understanding... I feel as teachers what we need to really be aware of is what the basics are that students must know manually... when we sit down to work with graphics calculators we need to consider carefully what still should be understood manually." In this way she addressed an important factor of the integration of technology use into mathematics, namely learning what is better done by hand and what could be done better with the technology (Thomas, Monaghan and Pierce, 2004). One of her lessons with the GC was with Year 12 students and she considered families of functions with the aim of exploring exponential and hyperbolic graphs and noting some of their features, "we're going to utilise the calculator to show that main graph and then we're going to go through families of >' = 2^". She was comfortable enough to direct them to link a second representation "Another feature of the calculator I want you to be aware of .[pause] you've got also a list of x and y values already done for you in a table." Teacher F3 had moved away from giving explicit key press instructions, instead declaring "I want you to put these functions in and graph them and see what's going on.", and "You can change the window if you want to see more detail, and if you want to see where it cuts the x-axis, you can use the "trace" function." Figure 6 shows a copy of her whiteboard working.

y=2''+l y=2'' +2 Original graph of y=2'' shifted up Cuts J'axis different places

Figure 6 Teacher F3's whiteboard working: Viewscreen projection and overwriting. She was also able to move towards an investigative mode of teaching "if you're not sure where the intercepts are, you can use the "trace" key, remember, and I want you to observe what is happening.", encouraging students to use the GC in a predictive manner, to investigate a different family. We want to do some predictions... Looking at the screen try to predict where 3 x 2' will go then press "'y=..." and see if it went where you expected it to go. You may get a shock... Can you predict where "y = 4 X 2*" will be? Now you learned from that, so can you predict where it'll lie. The gap between them gets smaller. If you're interested put in 'y = 100 x 2^". Does it go where you expect? There was also some discussion of mathematical concepts and how this could help with interpretation of the www.technology inmatheducation. com

GC graph. She linked 2 x 2' with 2'^' and then during examination of the family of equations y = 2\ y = 2^*', y = l''^- said of >'=2''^' "We expect this to shift 1 unit to the left [compared with 2']. Did it?" In this way she made a link with previous knowledge of translations of graphs parallel to the X-axis, and then reinforced this with the comment that "With this family, when you look at the graph can you see that the distance between them stays the same because it's sliding along 1 unit at a time. The whole graph shifts along 1 unit at a time." In addition, there was a discussion of the relationship between the graphs in the family of y =2' + k, and the relative sizes of 2^ and k. ... as the exponential value gets larger, because we're adding a constant term that is quite small, it lands up becoming almost negligible. So, when...all they're differing by is the constant part, you'll find that they appear to come together. Do they actually equal the International Joumal of Technology in Mathematics Education Vol 20. No 2

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Michael O J Thomas and Ye Yoon Hong same values ever? Do they ever meet at a point? No, because of the difference by a constant, but because ofthe scaling we have, they appear to merge.

The discussion on the relative size of terms in the function continued with "How significant is "+1" or "+2"? We know that 2^ is 32." and again the use of prediction was evident "I want you to predict where y = 2'' + 3" would be." Teacher A6, from the first group, worked in a high socio-economic school (decile 10), had 20 years' teaching experience and had been using GCs for three years. She had also made very good progress in her GC use. We made 3 visits to observe her classroom. The first involved year 12 students (age 17 years) studying probability simulations in statistics, and the second and third observations were of year 13 students (age 18 years) working on calculus, stationary points and trigonometry. In each lesson the teacher and all her students had access to T1-83+ calculators. While in her interview she sfressed the value of the technology for covering more ground "...the number of things that they can do in any one lesson is far greater so it's better than drawing. Usually it takes you all lesson to draw three (graphs). You can draw ten with a graphics calculator and they can really understand it for themselves...", she mentions student understanding, which was a focus for her. She also stressed the visual benefits ofthe GCs in this: ...it gives the students a visual interpretation, a handson approach; it's not all just writing, they can see things happening, particularly with the probability simulations yesterday. 1 felt they had a much better comprehension of what was actually happening so

The lesson's concepts

they got a visual picture but also every time they did it ...they could see it happening differently every time. These two ideas converge in her mind to give students the ability to generalise "I think it's the amount of visual information they can get, and the amount of examples they can get through, so they really feel they understood it because they've seen so many that they can actually accept and they can generalize for themselves and it gives a better understanding." Rather than focussing simply on getting her students to perform procedures she talked about how "they had a much better concept of what was actually happening". She does not put the emphasis in class on the instrumentation of the GC, instead "Basically if I'm organized I'll make a worksheet with the 5 keys we'll use that day or put the 5 keys we'll use on the blackboard, go over those and just spend 5 minutes on what we're going to day for the day". This approach was confirmed by the observation of her second lesson, on concavity. Her teaching was focused on using the technology to improve conceptual understanding, with an emphasis on visual representations of functions. She explained the concept of concavity on the whiteboard (see Figure 7), followed by turning point, stationary point, local maximum and minimum points. Teacher A6 also gave the definition of a point of inflection, showing that it may have a non-zero gradient. The students then worked on the function y = x'^- 2x^, to find its key features, such as concavity, being encouraged to work by-hand and on the GC (with, eg, [2"''] [Trace] to find turning points) in parallel.

The point of inflection

Figure 7 Teacher A6's conceptual emphasis in differentiation. In her third lesson she considered how the GC could help her students understand and recognise key concepts of functions y = AúnB{x+C)+D, such as amplitude, period, maximum and minimum values. This led to a question using trig functions to model temperature: A patient in the hospital had an illness in which his temperature (in degrees census) varied front it low of 37° to a high of 40.4°. The length of time between successive highs is 16 days. Determine the formula for the temperature, T, of the patient at time in days since the beginning of the illness. Assume that the function describing the temperature can be modelled with a sine function, with no phase shift.

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Teacher Integration of Technology into Mathematics Learning

a) Preliminary data

b) Beginning the curve

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c) Using the GC

Figure 8 Teacher A6 integrates the GC in modelling. In this she got the students to work by-hand on the conceptual structure of the problem, and then integrated the GC into the solution process. The students found A as \.l from (40.4 - 37)/2 (Figure 8a), D as 38.7 (37 + 1.7), and B as

of the GC and are thus more able focus on other important aspects, such as the linking of representations such as graphs, tables, and algebra, and to use other features of GCs. In turn, this better instrumentation ofthe GC produces a higher level of confidence in classroom use. Considering their PTK they — from B = 27i/period, to give y = 1.7siní — + 38.7begin to see the GC in a wider way than simply as a 8 \8 j calculator. They feel free to loosen confrol and encourage students to engage with conceptual ideas of mathematics In summary, these two teachers seem to have reached through individual and group exploration of the technology, a point where they have both strong insfrumentation and investigation of mathematical ideas, and the use of prediction insfrumentalisation of the technological tool. They are able and test methodology. For these teachers the mathematics to use the affordances of the technology (within acceptable rather than the technology has again been thrust into the consfraints) to provide an epistemic focus on the foreground, and the GC has been better integrated into the mathematics through the technology. Examples of this lessons and the didactical contract. If we think that the include the idea of testing concepts against definitions and a approach of this second group is preferable, then we must sfrong emphasis on generalisation, with the use of ask how we assist teachers to progress towards it. One investigation to form conjectures about graphs of the forms crucial answer involves the provision of pedagogicallyy — 2^^ + ic and y = fc • 2^. This focus has as its aim the focussed professional development, relevant resources and important aspect of generalisation (Mason, Graham and good lines of communication. Johnston-Wilder, 2005), with attendant modelling, of mathematii;al processes and constructs. In addition, these Although mathematics teachers often claim to be teachers had a high level of confidence in using the supportive ofthe use of technology in their teaching (Forgasz, technology to teach mathematics. 2006; Thomas, 2006) the degree and type of use in the classroom often does not correlate with this (Becker, 2000). 4 DISCUSSION Research into the uptake and implementation of technology in mathematics teaching has identified a range of factors that In summary we may describe the differences in the influence it. Goos (2005) lists some of these as: skill and progress of the teachers we have observed in terms of a previous experience in using technology; time and number of variables that delineate two different groups, with opportunities to learn (pre-service education, guidance a third proj^essing between the two. The first group may be during practicum and beginning teaching, professional identified, in terms of their insfrumental genesis (Lagrange, development); access to hardware (computers and 1999), as teachers who are still coming to grips with basic calculators), software, and computer laboratories; availability operational aspects of the technology, such as key presses of appropriate teaching materials; technical support; support and menu operations. This appears to be related to a low from colleagues and school adminisfration; curriculum and level of confidence in terms of teaching with the GC in the assessment requirements and how teachers interpret these for classroom. In terms of their PTK, this group is characterised students perceived to have different mathematical abilities; by an over-emphasis on passing on operational matters to knowledge of how to integrate technology into mathematics students, such as key presses and menu operations, to the teaching; and beliefs about mathematics and how it is learned. detriment c i a focus on the mathematical ideas. Furthermore, Forgasz (2006) agrees, with her computer survey listing the mathematics approached through the technology has an access to computers and/or computer laboratories as the most emphasis on technology, and work tends to be very processprevalent inhibiting factor (constraint), with lack of oriented; based on procedures and calculating specific professional development and technical problems, including answers to standard problems. There is little or no freedom lack of technical support next. Thomas's (2006) survey of given to students to explore with the GC, and it tends to be computer use in New Zealand secondary schools found that seen as an add-on to the lesson rather than an integral part of teachers make similar statements, citing availability of it. These features then become part of the teacher-initiated computers as the major issue, followed by a lack of software, expectations in the didactical confract. training and confidence. The first two may be described as obstacles, while the last two are consfraints. In contrast to this, the second group have advanced to the point where they are competent in basic instrumentation An analysis of possible affordances for the action of wwTv.technologyinmatheducation.com

International Joumal of Technology in Mathematics Education Vol 20, No 2

Michael O J Thomas and Ye Yoon Hong

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teaching with technology led us to consider a number of domains where the affordances arise, including the school's physical infrastructure, the school's personnel structure and relationships, and the technology itself. Hence, when considering the obstacles and constraints that might be infiuencing the teachers' use of GCs (and other technology) we considered three areas, departmental, school, and personal factors. It appears that teachers' confidence in using technology is related to the support of colleagues, including the head of department, and the school. However, we note that two teachers who did well did not have this support. Similarly, although two of the teachers had relatively weak personal instrumentation of the GC they did not allow this to prevent them from trying to introduce it in a conceptual way. In fact only teacher A6 had the profile that we would consider ideal for a successful outcome. She also was more experienced with GCs and was the only teacher to have had professional development with them, as reflected in her lessons. This research suggests that the affordances produced by supportive colleagues in a mathematics department constitute a strong infiuence on personal confidence, and that this confidence is ñirther linked to a teacher's attitude to

using technology in teaching mathematics and to personal leaming, as well as to epistemic teaching with technology. In view of this we suggest a tentative model (see Figure 9), with teacher confidence as a pivotal variable in producing conceptual teaching with technology. There has been some attempt to substantiate this (Thomas and Palmer, In Press) with positive results, but we suggest that further research to confirm or reftite this relationship would be valuable. There are some possible implications of these findings for both preand in-service professional development of teachers with regard to technology. It may be beneficial to building teacher confidence to enable them to be part of a supportive group sharing knowledge of instrumentation, practical classroom activities and ideas about the calculator use, especially in the initial stages of leaming about the technology. Hence, constructing professional development around a supportive community of practice in a manner that gives teachers the opportunity to use technology in a classroom might lead to positive changes in the perception of its value in leaming mathematics, to increased confidence in using the technology to teach mathematics, and improved classroom practice based on the epistemic value of the technology.

Attitude to: teaching maths with technology; personal learning

^ Instrumentation technology

of

1

4

Confidence in teaching with technology

i

Improved PTK

Conceptual teaching with technology

Support of school and mathematics department

Figure 9 A model of factors influencing conceptual teaching with technology. Placing confidence in a key role like this is in agreement with some research that identifies the infiuence of teacher beliefs and attitudes on teaching practice. For example, in her study, Forgasz (2006) found that teacher confidence, experience, skills or enjoyment of computers was the third highest factor encouraging computer use. While we cannot come to any definite conclusions from our small-scale studies it is interesting to note that teachers who have few school resources, are not well supported by their head of department, and who do not have strong personal GC skills can do quite well in implementing technology use. Thus, our research suggests that the teacher's personal orientations (Schoenfeld, 2011), especially their attitudes and beliefs, if strong enough, can override other negative constraints and obstacles. In particular, our results indicate that a strong belief in the value of technology in leaming mathematics coupled with a strong willingness to be open to personal leaming could be crucial factors. Further research will be necessary to test these claims.

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ACKNOWLEDGEMENT We would like to acknowledge the support of Texas Instruments and a Ministry of Education of New Zealand, Teaching, Leaming and Research Initiative (TLRl) grant for this research.

REFERENCES Artigue, M. (2002) Leaming mathematics in a CAS environment: The genesis of a reflection about instrumentation and the dialectics between technical and conceptual work. International Journal of Computers for Mathematical Learning, 1, 245-274. Ball, D. L., Hill, H. C. and Bass, H. (2005) Who knows mathematics well enough to teach third grade, and how can we decide? American Educator, Fall Issue, 14-22 and 43-46.

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Teacher Integration of Technology into Mathematics Learning Becker, H. J. (2000) Findings from the teaching, learning and computing survey: Is Larry Cuban right? paper presented at the 2000 School Technology Leadership Conference of the Council of Chief State Officers, Washington, DC. Brousseau, G. (1997) Theory of didactical situations in mathematics: Didactique des mathématiques. 1970-1990, Balacheff, N., Cooper, M., Sutherland, R. and Warfield, V. (trans, and eds), Dordrecht: Kluwer Academic Publishers. Chinnappan, M. and Thomas, M. O. J. (2003) Teachers' function schémas and their role in modelling. Mathematics Education Research Journal, 15(2). 151-170.

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Education, 35(5), 330-351. Harskamp, E. G., Suhre, C. J. M. and van Struen, A. (1998) The graphics calculator in mathematics education: An experiment in the Netherlands, Hiroshima Journal of Mathematics Education, 6, 13-31. Hong, Y. Y. and Thomas, M. O. J. (2006) Factors infiuencing teacher integration of graphic calculators in teaching. Proceedings of the 11''' Asian Technology Conference in Mathematics, Hong Kong, 234-243.

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Kendal, M. and Stacey, K. (2001) The impact of teacher privileging on learning differentiation with technology. International Journal of Computers for Mathematical Learning, 6, 143-165.

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Kennewell, S. (2001) Using affordances and constraints to evaluate the use of informal and communications technology in teaching and learning. Journal of Information Technology for Teacher Education, 10, 101-114.

Forgasz, H. (2006) Factors that encourage and inhibit computer use for secondary mathematics teaching. Journal of Computers in Mathematics and Science Teaching, 25(1), 7793. Gibson, J. I. (1977) The theory of affordances, in Shaw, R. and Bransford, J. (eds) Perceiving, acting and knowing: Towards an ecological psychology, Hillsdale, NJ: Erlbaum, 67-82. Goos, M. (2005) A sociocultural analysis ofthe development of pre-service and beginning teachers' pedagogical identities as users of technology. Journal of Mathematics Teacher Education, 8, 35-59. Goos, M., Galbraith, P., Renshaw, P. and Geiger, V. (2000) Reshaping teacher and student roles in technology-enriched classrooms. Mathematics Education Research Journal, 12(3), 303-320. Goos, M., Galbraith, P., Renshaw, P. and Geiger, V. (2000) Reshaping teacher and student roles in technology-enriched classrooms. Mathematics Education Research Journal, 12(3), 303-320.

Lagrange, J.-B. (1999) Complex calculators in the classroom: Theoretical and practical reflections on teaching pre-calculus. International Journal of Computers for Mathematical Learning, 4(1), 51-81. Mason, J., Graham, A. and Johnston-Wilder, S. (2005), Developing thinking in algebra, London, UK: Paul Chapman Publishing and The Open University. Nosss, R. and Hoyles, C. (1996) Windows on mathematical meanings: Learning culture and computers, Dordrecht: Kluwer. Pierce, R. and Stacey, K. (2004) Monitoring progress in algebra in a CAS active context: Symbol sense, algebraic insight and algebraic expectation. International Journal for Technology in Mathematics Education, 11(1), 3-11. Pierce, R. and Stacey, K. (2004b) A framework for monitoring progress and planning teaching towards the effective use of computer algebra systems. International Journal of Computers for Mathematical Learning, 9, 59-93.

Greeno, J. G.(1994) Gibson's affordances. Psychological Review, 101 (2), 336-342.

Rabardel, P. (1995) Les hommeset les technologies, approche cognitive des instruments contemporains, Paris: Armand Colin.

Guin, D. and Trouche, L. (1999) The complex process of converting tools into mathematical instruments: The case of calculators. International Journal of Computers for Mathematical Learning, 3, 195-227.

Schoenfeld, A. H. (2011) How we think. A theory of goaloriented decision making and its educational applications, Routledge: New York.

Hill, H., and Ball, D. L. (2004) Learning mathemafics for teaching: Results from California's mathematics professional development institutes. Journal for Research in Mathematics www.technologyinm atheducation.com

Shulman, L. C. (1986) Those who understand: Knowledge growth in teaching. Educational Researcher, 15, 4-41. Simmt, E. (1997) Graphing calculators in high school International Joumal of Technology in Mathematics Education Vol 20, No 2

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mathematics. Journal of Computers in Mathematics and Science Teaching, 16(2/3), 269-289. Simon, M. (1995) Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, 26(2), 114-145.

Yackel, E. (2000) Creating a mathematics classroom environment that fosters the development of mathematical argumentation, paper presented at the Ninth International Congress of Mathematical Education, Tokyo. BIOGRAPHICAL NOTES

Thomas, M. O. J. (1994) A process-oriented preference in the writing of algebraic equations, Proceedings of the 17th Mathematics Education Research Group of Australasia Conference, Lismore, Australia, 599-606. Thomas, M. O. J. (2006) Teachers using computers in the mathematics classroom: A longitudinal study. Proceedings of the 30''' Conference of the International Group for the Psychology of Mathematics Education, Prague, Czech Republic, 5, 265-272. Thomas, M. O. J. and Chinnappan, M. (2008) Teaching and learning with technology: Realising the potential, in Forgasz, H., Barkatsas, A., Bishop, A., Clarke, B., Keast, S., Seah, WT., Sullivan, P. and Willis, S. (eds). Research in Mathematics Education in Australasia 2004-2007, Sydney: Sense Publishers, 167-194. Thomas, M. O. J. and Hong, Y. Y. (2001) Representations as Conceptual Tools: Process and Structural Perspectives, Proceedings of The 25''' Conference of the International Group for the Psychology of Mathematics Education, Utrecht, The Netherlands, 257-264.

Mike Thomas is a full Professor in the Mathematics Department at The University of Auckland, New Zealand. His research interests are in using technology to improve learning, developing theories of advanced mathematical thinking, the learning and teaching of calculus and undergraduate mathematics, school and university teaching, and connections between mathematics education and cognitive neuroscience. He has given invited research seminars in a number of countries and is on the editorial boards of the international journals. Mathematics Education Research Journal and the International Journal of Mathematical Education in Science and Technology. He recently led an international survey team for the 2012 International Congress on Mathematical Education (ICME), on the mathematical difficulties inherent in the transition from school to university. Ye Yoon Hong is a lecturer in the Mathematics Department at The University of Yonsei, Korea. Her research interests are in using technology to improve learning and teaching of calculus and undergraduate mathematics, school and university teaching.

Thomas, M. O. J. and Hong, Y. Y. (2005) Teacher factors in integration of graphic calculators into mathematics learning, in Chick, . L. and Vincent, J. L. (eds). Proceedings ofthe 29th Conference of the International Group for the Psychology of Mathematics Education, Melbourne, Australia: University of Melbourne, 4, 257-264. Thomas, M. O. J., Hong, Y. Y., Bosley, J. and delos Santos, A. (2007) Calculator use in the mathematics classroom: A longitudinal study, in Wei-Chi Yang, T., de Alwis & Jenchung Chuan (eds). Proceedings of the 12"' Asian Technology Conference in Mathematics, Taiwan: ATCM, 37-47. Available from: http://atcm.mathandtech.org/EP2007/EP2007.htm. Thomas, M. O. J., Monaghan, J. and Pierce, R. (2004) Computer algebra systems and algebra: Curriculum, assessment, teaching, and learning, in Stacey, K., Chick, H. and Kendal, M. (eds). The teaching and learning of algebra: The 12th ICMI study, Norwood, MA: Kluwer Academic Publishers, 155-186. Thomas, M. O. J. and Pahner, J. (In Press) Teaching with digital technology: Obstacles and opportunities, in ClarkWilson, A. Sinclair, N. and Robutti, O. (eds) The mathematics teacher in the digital era, Dordrecht: Springer. Thomas, M. O. J., Tyrrell, J. and Bullock, J. (1996) Using Computers in the mathematics classroom: The role of the teacher. Mathematics Education Research Journal, 8(1), 3857. © 2013 Research Information Ltd. All rights reserved.

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