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International Journal of Computers for Mathematical Learning (2005) 10: 143–189 DOI 10.1007/s10758-005-4850-7 Springer 2005 J.B. LAGRANGE

CURRICULUM, CLASSROOM PRACTICES, AND TOOL DESIGN IN THE LEARNING OF FUNCTIONS THROUGH TECHNOLOGY-AIDED EXPERIMENTAL APPROACHES

ABSTRACT. The paper starts from classroom situations about the study of a functional relationship with help of technological tools as a ‘transposition’ of experimental approaches from research mathematical practices. It considers the limitation of this transposition in existing curricula and practices based on the use of non-symbolic software like dynamic geometry and spreadsheets. The paper focuses then on the potentialities of classroom use of computer algebra packages that could help to go beyond this shortcoming. It looks at a contradiction: while symbolic calculation is a basic tool for mathematicians, curricula and teachers are very cautious regarding their use by students. The rest of the paper considers the design and experiment of a computer environment Casyopeé as means to contribute to an evolution of curricula and classroom practices to achieve the transposition in the domain of algebraic activities linked to functions. KEY WORDS: classroom practices, computer algebra, computer tool design, curriculum, dynamic geometry, experimental approaches, functions, instrumentation, praxeologies, spreadsheet, transposition

1. INTRODUCTION At the present time, mathematics curricula – especially at secondary level in France – recommend the use of computer tools and this use is not uncommon in classrooms. It is then possible to go beyond a ‘laboratory approach’ of the educational use of computers and to address as a whole the issues of curriculum, classroom practices, and relevance of software applications offered for students. Research about technology in education indeed broke off with considering a mechanical influence of new tools on curricula and classroom practices. To be successful, technological innovation has to offer new opportunities for current curricular concern and to be compatible with viable classroom practices (Ruthven and Hennessy, 2002). At a more general level, Human Computer Interaction (HCI) researchers stressed for 10 years that ‘Laboratory-based usability studies are (only) part of the solution’, and are best preceded by ‘‘careful field

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studies’’ to address question like: ‘how technology can fit into users’ actual social and material environments; the problems users have that technology can remedy; the applications that will promote creativity and enlightenment. . .’ (Nardi, 1996). The approach of this paper will be first to consider existing combinations of curriculum, practices and tools, and then, analysing possible limitations, to look into potential new combinations. These new combinations include new curricula and practices but also new tools. That is why a concern of this paper will be about technological tool’s design. Taking into account the limitations of existing software in view of curriculum and practices, and of their possible evolution, this paper will consider a project that a team including the author is currently developing: the Casyopeé environment. ‘Traditional’ software design in mathematics educational research often put a strong emphasis on the analysis of mathematical content, and take teaching practices and curriculum into account only as a second dimension. I choose another approach because of the evidence that ‘good epistemology’ is not a guarantee in itself of consistent practices, like for instance Lins (2003) showed about teachers’ conception and use of dynamic geometry. There is a general belief in research that classroom use of technology has to be seen as a systemic interaction of a number of dimensions, while individual researchers might disagree about the weight of each dimension. In this paper I try to put some weight on dimensions sometimes overlooked like curriculum and practices, considering mathematical content when analysing these. The relationship between curriculum, classroom practices and software applications is a wide topic and is best approached by choosing a mathematical domain and a field in this domain. Designing Casyopeé, we choose functions as a field of student practices to learn algebra at upper secondary level using this environment. In this field a class of situations – the study of a functional relationship with help of technological tools – offers a good example of a combination of curricular guidelines, existing and potential classroom practices and the use of varied software tools. These situations will be the topic of this paper. They will be presented and analysed in details further in the paper. Here I mention general features in order to specify questions. A first characteristic is that they put at stake models of a relationship and favour experimental approaches. Software used in these approaches can be numerical/graphic tools – like spreadsheets or graphers – but dynamic geometry recently appeared as means to study and model relationships. These tools are challenged

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by symbolic software generally named Computer Algebra Systems (CAS). This variety of tools will be part of a discussion on technological design. Figure 1, parts 1 and 2 is a summary of this general frame and topic. Sections 2 and 3 will specify the questions addressed in the paper and theoretical frameworks. They are outlined in parts 3 and 4 of the same figure. Sections 4–6 will analyse the above-presented situations and Sections 7 and 8 will discuss the possible contribution of existing tools showing difficulties likely to happen. The last section

Figure 1.

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will present design and experiment currently carried out by the Casyopeé team. This organisation aims to show how the team’s work on Casyopeé is connected to a reflection about limitations of existing of existing curriculum and practices, rather than to provide evidence of Casyopeé’s capacity to go beyond these limitations. Like many projects developers, the team started with general ideas of what a classroom tool should be. As the project progressed, we had to explain our motivations and to take crucial options. We strongly felt the need for a reflection towards a more comprehensive and detailed understanding of the contribution of experimental approaches using technology to students’ conceptualisation. In consequence, this paper is essentially philosophical, presenting our reflection as a central object and offering fragments of the Casyopeé project as means to show how this reflection can be actualised by way of a tangible software development. It leaves room for further studies based on this reflection, providing a more detailed presentation of Casyopeé and more systematic experiment and observation of uses. 2. QUESTIONS Because situations of study of a functional relationship involve experimental approaches, a general issue will be about the role of these approaches in conceptualisation. This issue is not new. In the conclusion of a paper about the didactic role of experimentation, Joshua and Joshua (1988, p. 24) observed that Experimentation is a basic choice in physics teaching and it is more and more often mentioned in mathematics teaching. But, passing from research to classroom, experimentation is much changed. In classroom, the teacher has to present the observational basis (the ‘experiment’) to the students without connection to a theorisation. He has to make the theorisation appear as built from the experiment.1

Joshua and Joshua’s reflection starts from the relationship between research practices, and mathematics or physics teaching. In research, experimentation2 and theorisation are interweaved processes. Theoretical concerns and frameworks help to design useful experiment. Experimentation provides for new conjectures inducing new theoretical developments. Constraints of teaching/learning make this interweaving more difficult, and the result is often a ‘poor induction:’3 some experimentation is done as an entry to a new subject and then a theory is


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presented to the students as a model of the experimental situation. In this schema, no real problem is offered to students and thus they have no occasion to really build some theoretical knowledge. A general question is how far technology can change this situation, giving teachers and students new means to steer clear of poorly inductive practices. Because we saw above that there is a problematic relationship between mathematical research practices and teaching/ learning, especially when dealing with experimentation, a specific question will be how far teaching and learning can be inspired by new practices in mathematical research. The introduction pointed out that curricula and practices could not be separated from tool design. Yerushalmy (1999, p. 169) reminds us that technological tool designers clearly worked towards providing means for a better approach of experimentation (or exploration 4) as a support for students’ conceptualisation when they initiated ‘this revolutionary view of software that meant to introduce new ways of building and reflecting on knowledge (that) was the essence of micro worlds. . .’ She nevertheless stresses that, beyond the stimulating prospects that such new design offers, its insertion into a curriculum remains problematic: ‘For (teachers), educational software that offers new functionality for delivering innovative didactic ideas appears to demand a substantial change in the curriculum and is therefore too big a leap to attempt’. The work presented in this paper is motivated by similar concern: reflecting on curriculum and practices’ evolution while designing a tool for experimental approach and algebraic modelling. More precisely, I will look how technological design can provide new student experimental approaches and avoid abrupt changes incompatible with curriculum and classroom’s constraints. The nature of the technology will be also matter for discussion. As software for doing mathematics can roughly be separated between numerical/graphic applications – based on approximate arithmetic, and computer algebra – performing exact symbolic calculation, I will analyse what specific support to classroom practices can be expected from each and how they fit into curriculum and practices. 3. THEORETICAL FRAMEWORKS This paper deals with students’ experimental activity related to functions, induced by combinations of curriculum, practices and

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tools, and theoretical frameworks will help to analyse this activity. First its nature and its relationship with the development of mathematical knowledge have to be analysed from a didactical and epistemological point of view. Then, looking at its evolution, especially towards ‘experimental approaches’, another framework is needed to bring some understanding of the changes in curricula and practices. Finally, introducing technological tools implies two more frameworks, one accounting for the links between students’ knowledge about the tool and his (her) mathematical knowledge in this activity and the other considering tools design and its influence. This set of four linked issues – didactical and epistemological analysis, changes in curricula and practices, tool and mathematics relationship, design – calls for a ‘multidimensional’ approach, consistent with Lagrange et al.’s (2003, p. 239) claim that many research studies or reports of innovation about technology in mathematics education fail to be relevant when they consider only one framework. Theoretical frameworks that will be used in this paper are chosen to provide specific insight into each issue. The first one will be an epistemological and didactical approach of functions in algebra teaching/learning drawn from Kieran (2001). The second and third will be the ‘praxeological’ and ‘instrumental’ approaches. As a fourth approach, Yerushalmy’s (2001, pp. 183–185) conclusion will provide a framework to consider software design. The order of presentation will set these frameworks on an axis oriented between the two poles mathematical knowledge and tool. The second one, the praxeological approach is central in this paper and will be presented in detail because it is sometimes controversial and its use in research studies involving technology is not common. The paper will only provide an outline of the three other approaches because they are more consensual, and papers and books exist for a detailed presentation. Kieran’s framework will help to look at the relevance of classroom activities that the curriculum and the tool induce, especially with regards to the type of algebraic activity involved. Kieran sees ‘functions’ as one of the two main approach that are used in a majority of algebra classes to provide meaning to algebraic activity, the other being ‘generalized arithmetic’. These two approaches each provide a unique transversal thread to three categories of algebraic activity: generational, transformational, and global/meta-level. The generational activities of algebra involve the forming of the expressions and equations that are the objects of algebra (. . .)


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The transformational (rule-based) activities includes, for instance, collecting like terms, factoring, expanding, substituting, solving equations, simplifying expressions, working with equivalent expressions and equations, and so on. A great deal of this type of activity is concerned with changing the form of an expression or equation in order to maintain equivalence. The global/meta-level mathematical activities include problem solving, modelling, noting structure, studying change, justifying, proving, and predicting – activities that could be engaged in without requiring any algebra at all. However (. . .) from the point of view of the curriculum, the global, meta-level activities cannot be separated from the other activities (. . .) otherwise the algebraic purpose is lost.

The praxeological approach (Chevallard, 1985, 1994, 1999) aims to give account of the conditions in which mathematical objects exist and live in institutions or more precisely how they are ‘known and understood’ as entities arising from practices. The word ‘institution’ has to be understood in a very broad sense as any social or cultural practice takes place within an institution. In this paper, we will consider scientific research institutions devoted to producing knowledge and didactic institutions devoted to apprenticeship. Chevallard (1985) introduced the notion of transposition to consider ‘genesis, filiations, gaps, and reorganizations’ interrelating these two types of institutions. I will use this notion to discuss the relationship between mathematical research practices and teaching/learning when dealing with technology aided experimentation and modelling. Researchers often thought of Chevallard’s idea of didactical transposition mainly through the impoverishing effects of the transition from official science to classroom taught knowledge. For instance Mason and Spencer (1999) noted ‘‘what is taught is what can be tested, so expert awareness is transformed into instruction in behaviour (transposition didactique) (. . .) Consequently all the forces are towards inert knowledge as trained behaviour’’. Freudenthal (1986, p. 326) had a similar opinion when he said that Chevallard’s description of the knowledge resulting of the transposition is ‘distressing’ (affligeante). This view of the transposition is not irrelevant and I mentioned above the risk of impoverishment of experimental practices into ‘poor induction’. But, in my understanding, the notion of didactical transposition can bring more. Another view was more explained by Chevallard (1994). It does not consider a general transposition but rather processes of didactical transposition. These processes reflect constraints of the organisation of knowledge, different in research and in the classroom but also the need for a legitimacy of taught knowledge. As Chevallard (1994) says:

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Why does curriculum evolve at given historical periods, absorbing streams of knowledge from the outside? It is because taught knowledge claims to derive from ‘secular’5 knowledge and distance between them has to be short enough. Taught knowledge has to show enough similarity with knowledge able to act in some sort as an epistemological bail under the control of Society.

In this view, the transposition has to be understood in its sociological dimension. It has not a mechanical effect. All depends of social processes involved in the creation of new school knowledge. The appearance of new knowledge in teaching is always the outcome of social expectations, more ore less supported by social groups.

Other notions of the praxeological approach, especially useful to clarify the influence of technology on teaching/learning are the three components of practices in a institution: a type of task in which the object is embedded; the techniques used to solve this type of task and the ‘theory’ which is first the discourse used in order to explain and justify the techniques and then provides a structural basis for this discourse.6 In a more recent text Chevallard (1999, p.231) explains that praxeologies (i.e. the above components of practices) are the matter of the transposition, and this helps to see that transposition cannot be only seen as a deteriorating process. Institutional transposition processes do not necessarily produce deteriorated versions (. . .) of transposed praxeological organisations. In contrary, (. . .) it happens frequently that the work of transposition is an opportunity for improving a praxeology by simplifying, specifying some elements, etc. In all cases, transposition enriches the world of socially available praxeologies.

The central component – techniques – is often misunderstood and Artigue (ibid. p. 248) explains: It has to be given a wider meaning than is usual in educational discourse. A technique is a manner of solving a task and, as soon as one goes beyond the body of routine tasks for a given institution, each technique is a complex assembly of reasoning and routine work. (. . .) techniques are most often perceived and evaluated in terms of pragmatic value. . . (efficiency, cost, field of validity). But they have also an epistemic value, as they contribute to the (theoretical) understanding of the objects they involve.

In my view, mathematics education cannot make sense of potentialities of computer applications for new didactical situations without considering their effect at the three levels of tasks, techniques and


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theorisations. The choice of an ‘approach by the techniques’ helped to perceive limitations in discourses about potentialities of computer tools for students’ direct access to ‘conceptual knowledge’ (Lagrange, 2000). These discourses indeed neglected the necessity of an elaboration of techniques and of a theoretical reflection on these techniques by which mathematical objects and relationships are known. The ‘instrumental’ approach (Guin and Trouche, 1998; Trouche, 2000) is now widely considered by researchers in the field of educational use of technology especially in mathematics. It derives from conceptualisation about technologies in social life distinguishing a technological artefact and the instrument that a human being is able to build from this artefact. While the artefact refers to the objective tool, the instrument refers to a mental construction of the tool by the user. The instrument is not given with the artefact, it is built in a complex instrumental genesis and it shapes the mathematical activity and thinking. The instrumental dimension is important for mathematical conceptualisation: because mathematics teaching usually works in an environment without rich technology, introducing technology changes radically this environment. Teaching should consider students’ transformation of technological tools into mathematical instruments and the associated processes of instrumentation – integrating the instrument in the user’s processes of thought and instrumentalisation – discovering functionalities of the tool, inventing practical uses (Artigue, 1998). In this paper, the instrumental approach will help to analyse mathematicians’ and students’ experimental practices. If technology is not ‘only a tool’, experimenting with technology will be different because knowledge acquired through an instrumented experiment will involve knowledge about the instrument. Regarding software design, the introduction mentioned how HCI researchers are concerned by the limitations of tool design only based on laboratory studies. In the questions, I exposed also Yerushalmy’s view that similar difficulties exist with software for mathematical exploration by students. In her conclusion (ibid, pp. 183–185) she notes that ‘‘there is encouraging evidence about the impact of various specific software capabilities on exploration (and also) discouraging evidence about work with educational software that does not always act as the idea generator it was designed to be’’. What is at stake is the visibility of educational intentions in software design. Designers should ‘‘realize and articulate their perhaps unconscious decisions

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and turn them into conscious design considerations’’ and help ‘‘teachers focus on the finer properties and messages of the tools they use in the classroom’’. Yerushalmy highlights three issues in which software design should make educational intentions visible. Tool design could first give the learner a control over exploration by helping him (her) to develop methods. It could also support the organisation of the curriculum by being consistent, using ‘‘the same language of objects and actions that form the grid along which the curriculum is mapped’’. Finally, she regrets the ‘confusion’ about the role of solution tools like four-operation calculators and CAS: because results provided by these tools are often ‘shortcuts’ to what students are supposed to learn, they are not easy to connect to explicit learning goals. Thus, to make exploration visible, ‘‘software should deliver a clear message about the (educational) role of technology’’. These three issues – help for students’ construction of methods, support to curriculum organisation and clear message about learning goals – will be used as guidelines to investigate tool design in this paper. Note that Yerushalmy’s concern for means to help students to develop methods is consistent with the importance of techniques in the praxeological approach while control can be seen as an instrumental requirement. Globally the four frameworks are consistent and complementary means to approach the reality of students’ activity in computer-aided experimental approaches to functions.7 4. CONJECTURING AND PROVING: A TYPE OF TASK IN THE FRENCH CURRICULUM This section will consider more precisely the type of situations announced in the introduction, taking an example of functional relationship between geometrical objects and measures whose study is prescribed by the French curriculum. I will discuss classroom implementations and look at associated practices and conceptualisations. 4.1. Modelling a Functional Relationship This is an extract from the French official curriculum for upper secondary level (10th grade, ‘Seconde’).8


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It is possible to study for instance geometrical situations, the independent variable being a length and the dependant variable an area. The problem is then often to look for a maximum, a minimum or simply a value.

Working on the situation that this extract recommends, students can learn about functions by approaching a formalisation of a functional relationship. They can explore the dependency by calculating values of the dependant variable at given values of the independent variable and then explicit the dependency by expressing the relationship in specific registers (in the sense of Duval, 1996): graphic – a curve representing the values of the area relatively to a position – , numerical – a table with values of the independent variable in a column and the corresponding values of the area in another\and algebraic – a formula involving a literal corresponding to the independent variable. These expressions of the relationship can then be used for the task of finding and proving extrema. This is an example of a problem on which situations found in textbooks and professional websites are often based. The text of the problem is written in a geometrical language and it is possible to study the relationship in geometrical setting (in the sense of Douady, 1986) like in the first solution or in algebraic setting like in the second solution. C P

ABC is right-angled in A. AMNP is a rectangle, M, N and P being respectively on [BC], [BA] and [AC].

M I

The goal is to find a maximum area for AMNP.

Q A

N

B

A geometrical solution: let I be the middle point of [BC]. Suppose that M is on [IC]. Let Q be the intersection point of [AI] et [MN]. IAC is isosceles thus also the trapeze AQMC. This trapeze and the rectangle AMNP have then equal areas. The area of the trapeze is maximum when the area of MIQ is null, which appends when M equals I. Supposing now that M is on [IB], the reasoning is the same by considering the intersection point of [AI] and [MP]. An algebraic solution. Let x, b and c be the lengths AN, AB and AC. Let f be a function defined on [0 ; b] such as f(x) equals the area of MNAP. An algebraic expression of f(x) is c x (b-x)/b. Algebraic techniques can be used to show that f(x) is maximum for x= b/2.

4.2. The Curriculum: Learning Algebra with Technology in a ‘QuasiExperimental’ Approach Let us see now more precisely how this problem can be implemented in accordance with the curriculum. In France the ‘Seconde’ is the first year

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of the ‘lyceé’ which corresponds to the upper secondary level. During this year students really encounter the notion of ‘function’ as an algebraic framework, whereas in previous years ‘generalised arithmetic’ and ‘equation solving’ were privileged. The curriculum specifies the learning objectives about this notion and its expression in three registers. (Students should be able) to identify the independent variable and its set of values for a numerical function defined by a curve, a table of data or a formula, to establish the value of the function for a given value of the variable in each register, to describe the behaviour of a function given by a curve, using a relevant vocabulary or a sketch.

The notion has to be draught from activities in varied mathematical and non-mathematical fields. Learning situations will come for instance from geometry, physics, actual life or historical problems. Students will have to reflect on language expressions like a depends on b in the common language and in mathematics.

Students have also to learn how to use the functional formalism. The notations f(x), already introduced before, and f will be systematically used. It is important to introduce progressively the use of these notations: passing from the number f(x) to the function f as a mathematical object is difficult and time consuming. The time necessary for a personal maturation can be longer than a year.

The curriculum also prescribes an approach consistent with the above problem. By solving problems, modelling situations and progressively learning to prove, students can begin to understand the nature of mathematical activity: identifying a problem, experimenting on examples, conjecturing a property, building an argumentation, writing out a solution, verifying the properties and evaluating their relevance regarding the problem.

According to the curriculum, technology is naturally able to support this approach. Computer tools can help a quasi-experimental approach to the fields of numbers and of geometrical objects. It favours students’ more active attitude and commitment to the task. Possibilities for observing and manipulating are much wider. The opportunity of doing a great number of computations and to study as many cases as wanted helps to observe and verify properties.

The curriculum recommends the use of spreadsheets and dynamic geometry. In contrast, CAS are mentioned to warn teachers against


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misuses of calculators by students rather than to promote a wise use. Thus the curriculum promotes ‘non-algebraic’ tools, a choice that I will analyse later. Dynamic geometry helps (. . .) to link the viewpoints (numeric, functional, graphical, geometrical) and contribute to unify students’ knowledge. Many uses of a spreadsheet are interesting: introducing the notion of function, linking expression and function, function and graphic representation. CAS use is not now included in the curriculum. However, because symbolic calculators exist, students might use CAS and then teachers have to consider this fact.

Assuming that technological tools – especially dynamic geometry and spreadsheet – will easily support experimental approaches to concepts, the curriculum brings back into fashion a traditional view of students mathematical activity transposed from professional mathematicians’ practices. Next section will discuss this assumption by questioning the transposition. 5. THE TRANSPOSITION The notion of transposition was presented above. It considers mathematics in research and in school as a set of knowledge and practices in relation between two institutions, the first one aiming at the production of knowledge and the other at its study. The transposition considered here is not just about mathematical contents – mathematical objects and their relations, but about the process of conceptualisation that links practices and contents. It is consistent with Chevallard’s view that not just pure knowledge is transposed, but also the associated practices and conceptualisation processes, as praxeologies involve both. Processes of transposition particularly occur when finalities of a subject to teach are under public discussion. In some sense it is the case now for mathematics in the context of a ‘technological world’. Since the time of ‘new maths’, it is known that it is not easy to reduce the distance between new contents driving mathematical research and taught knowledge and then, contents are not so much discussed than approaches to knowledge. That is why discourses about mathematics teaching and curricula put emphasize on new approaches possibly close to professional mathematicians’ practices rather than on new contents to teach.

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This aspect of the transposition is generally overlooked. A reason is that little is really known of the mathematicians’ practices with regard to conceptualisation. In a very relevant but isolated study, Burton (1999) did an interview-based survey of 70 research mathematicians, focusing on their epistemologies. She does not directly address the issue of experimental practices but she rather stresses the variety of ways mathematicians take to discover truth that she opposes to uniform common classroom practices. In her conclusion (p. 141), she notes. Out of the interviews with the research mathematicians, I have a clear image of how impossible it is to speak about mathematics as if it is one thing, mathematical practices as if they are uniform(. . .). Would it not be enlightening for more learners, (. . .) to discover that they, too, can practise mathematics in the many different ways that I have described ..?

In my understanding, Burton proposes that mathematics education consider seriously a transposition of the diversity of mathematicians’ ways of thinking and practices into classroom. In some sense it is consistent with the curriculum’s emphasize on experimental approaches as an alternative to uniform ‘learn and apply’ practices. As a researcher, my goal is to understand the conditions and limits of this transposition. It is not easy because practices in mathematical research are diverse and not well known or studied. As we shall see below, ‘experimental mathematics’ is a domain of activity relatively uniform and theorized comparatively to other domains and thus it is a good start. 5.1. Experimenting, Conjecturing and Proving Mathematical research conceived algorithms and software applications for its own needs, especially to make experimental approaches more effective. Discourses promoting the use of these applications for teaching and learning mathematics emphasize the necessity of transposing experimental approaches. We saw that above when the curriculum wished that students ‘begin to understand the nature of mathematical activity’ and emphasised the interest of ICT for a ‘quasiexperimental approach’. De Guzman (2002) developed this view, stressing the specificity of the mathematician’s work and postulating a homology between this work and the learner’s activity. The core of the ordinary work of the research mathematician consists essentially in experimenting, conjecturing and proving. The main task of the one who tries to


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learn what mathematics is and how to become in any degree a mathematician should be to look at the mathematician’s way to act when he or she is at work and to try to imitate and follow the same steps.

De Guzman also stressed the support of technology to the mathematician’s work and – by homology – to the learner’s. Experimentation in mathematics has become, through the presence of the recent technological instruments something very different, much easier, more interesting and more fruitful than it never before has been. One can experiment numerically, algebraically, visually... with such an ease as it was never dreamed of a few years ago.

5.2. Experimental Mathematics in Research and Education In my view, a transposition postulating a direct homology between mathematicians’ practices and students’ practices would be too naı¨ ve. A real transposition should discuss the similarities and dissimilarities in technology supported processes of conceptualisation appearing in research and in classrooms. Let us thus question de Guzman’s point.9 He quoted Euler.10 (. . .) as a matter of fact, as I shall show with very valid reasons, the numbers properties today known to us have been discovered in great part by observation.

My first comment is that we should make a distinction between the contribution of observation to discovery and its contribution to conceptualisation. Like Euler in number theory a mathematician may have to carry out some patient experimental evaluation of probabilities of success of a conjecture. This obligation does not transfer to a learner. Concepts and properties that (s)he has to learn have already been discovered by mathematicians. If experimental approaches have a sense in education, it is not for their empirical ‘utility’ in discovering conjectures but as a contribution to conceptualisation. Traditionally, professional mathematicians do not emphasize the contribution of their experimental observation to the development of concepts and properties. In this tradition, the means by which properties have been conjectured is omitted and mathematicians present their theories deductively. They build concepts and prove properties often by interweaving experimentation and, after, they

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justify their theory by its consistency and the problems it helps to solve, rather than by the circumstances of its conception. In education, a tradition exists that refers to professional mathematicians’ practices and stresses the role of experimentation in conceptualisation. This tradition is visible in curricula and, in France, a conspicuous example is the introduction of the ‘Travaux pratiques (TP)’ in the 1950s. Assude (2002, p.4) made a study of the role of these TP in curricular changes. She stressed that their purpose was to develop new approaches to mathematics not by changing curricular contents, but by offering students a new practice, as close as possible to real mathematical practices. The TP have a didactical role: the goal is to make students want to follow scientific courses not by way of discourses, but by ‘showing them’ what a scientific activity is. The new curriculum does not propose to change the mathematics taught but to change the working methods.

Assude’s analysis is that, while successive curricula confirmed and extended the TP, actual classroom situations tended not to really link practical activities to conceptualisation. Teachers most often confused TP with existing practices related to doing exercises under the supervision of the teacher or empirically approaching concepts in the poor induction that Joshua and Joshua (ibid.) mentioned as a possible consequence of constraints of teaching/learning. Thus ‘experimenting like mathematicians’ was a stimulating idea but not so easy to implement in classrooms. To have a closer look to the transposition, let us consider the three components that characterize practices in the praxeological approach. Analysing similarities and dissimilarities in these three components will help to go beyond a homology of practices in research and education. 1. Tasks. Observing and conjecturing are natural tasks for a researcher but they do not make spontaneously sense for a learner. Gelis and Assude (2002) showed for instance that primary students need time to understand a task of observation of invariant geometrical properties using dynamic geometry. 2. Techniques. In research, methods exist because of their efficiency. In contrast, educational institutions do not privilege efficiency and remain attached to traditional techniques as a part of inherited culture. Time is needed to conceive new technology aided


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experimental techniques adapted to classroom tasks and to students’ possibilities that could challenge the ‘old’ techniques. 3. Theorisations. In research, even when experimenting, the mathematical objects and their properties are manipulated with the traditional deductive activity in mind. ‘Experimental mathematicians’ know that their conjectures have to comply with certain principles to fit into a deductive construction. From the beginning they use the theoretical apparatus of the domain and look for the most general expression of the conjectures. In contrast students can understand their conjectures as local results about ordinary – not mathematical – objects or objects different from usual mathematics. This reflection is consistent with Joshua and Joshua’s (ibid.) concern about classroom poor inductive approaches: if experimental activity has little theoretic component, a theory will have to be presented without connection with this activity.

5.3. The Contribution of Technology 5.3.1. What Reference for the Transposition? I stressed above that the transposition has not a mechanical influence, acting through multiple competing or converging social processes. Two streams are very active and actually dominate these processes, especially in France. I will begin this section by discussing their possible influence. The first stream is represented in France by authors like CREM (2002) who claim that nowadays ‘mathematics is far from being just a matter for mathematicians’ and that ‘contemporary mathematics can be described as a pumping, distilling and irrigating process involving physicists, computer scientists, engineers, biologists and economists together with mathematicians in the strict sense of the word’. Because the influence of computers is central in this evolution, considering contemporary not strictly research mathematics would be a means to make sense of a possible evolution of teaching and learning. CREM (ibid) names ‘mathematical sciences’ this new trend.11 ‘Mathematical sciences’ has not to be understood as a new vocabulary for ‘applied mathematics’ but rather as an evolution transcending ‘pure’ and ‘applied’ mathematics. This evolution already had some influence for instance when the curriculum recommends that problems posed in classroom evoke situations related to physics, actual life or historical problems.

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However, regarding practices, this influence is now limited because the process of transposition works on the long term and time will pass before new non-strictly research practices will become an official body of reference in the transposition process. At present, research mathematics keeps a dominant position and, looking for changes that can be a reference for curricular evolution, the best choice is to consider how technology affects the practices of research mathematicians. In my opinion a major effect of computer use in research, likely to be transposed into education, is that more researchers recognize the impact of new experimental practices on mathematical thinking. Borwein and et al. (1996, p. 16) for instance stressed this qualitative effect. In Euler’s days, producing experimental conjectures was the researcher’s private non-rewarding work. Nowadays the use of computers tends to promote methods for experimenting. Borwein et al. (2003) provide a number of examples of application of a method to discover and prove exact values of series. This method consists first in obtaining a numerical approximation for the series. Then the method uses this value and the ‘Inverse Symbolic Calculator’ (ISC) to conjecture an exact value of the series. The authors describe the ISC (available at the URL http://www.cecm.sfu.ca/ projects/ISC) as a ‘a constant recognition facility’, i.e. an application that, given a decimal number tries as to find an expression involving mathematical constants (p, e. . .) whose approximate value is the same for the given precision. When the method succeeds they consider that they have an interesting conjecture. They usually reach ‘practical certainty’ by using more accurate approximation. Then they look for a formal proof. 5.3.2. A Praxeological Analysis Like Euler’s, a first reason of success of Borwein’s experimentation is that it occurs in a domain well structured by a theory. In this domain, sums of series are likely to express in terms of constants and the combinatory power of the computer makes numerical approximations really productive whereas deductive speculation would give no insight into a possible value. This aspect of experimental mathematics cannot be transposed in teaching/learning when theorisation is at stake because students explore a domain that they do not master. In this situation, powerful tools can produce much data but little insight. Yerushalmy (ibid. p. 80) reminds us of Lakatos’ warning: ‘If you believe that the longer the table the more conjectures it will suggest, you


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may waste your time compiling unnecessary data’. Thus on one hand I will not necessarily follow de Guzman when he says that ‘experimentation has become much more fruitful’ also for learners. On the other hand, thanks to the power of computer tools, methods appear at a much more open level than in Euler’s days. In these days, experimentation was the individual task of a researcher, when now everybody interested can for instance access and use the ISC on the Internet. Borwein et al. (ibid.) say in their introduction: Today computer experiments are an integral part of doing mathematics. This allows for a more systematic approach to conducting and replicating experiments.

Not all mathematicians will agree that computers changed fundamentally the way they work, but there is no denying that widely accessible computer tools give an official statute to methods systematizing experimental practices and that this systematisation complements proof as means for conceptualising. I see in these systematised computer aided methods a matter for transposition: methods for experimenting could help to identify tasks, develop techniques and connect exploration to theorisation, keeping clear of poor induction. 5.3.3. An Instrumental Analysis Experimental mathematics has another characteristics, which I think also interesting for teaching/learning. There are instances where the way the tool operates can give insight into a proof. In this case, researchers’ knowledge about the tools helps them to get this insight. In Appendix B, I analyse an example from Borwein and al. (ibid.), the Knuth series problem. This example is of course far from subjects taught at secondary level, but it will help to see how a transposition of experimental approaches can be productive from an instrumental point of view. Applying their method for a given series, the authors were surprised by the quick calculation of an approximation of this slow converging series by the CAS they were using. They ‘looked under the hood’ and found that this CAS was programmed to formally transform the series in order to gain speed. It happened that this formal transformation was a key to proof. An instrumental interpretation is that experimental mathematics is something really new and productive not just because of the power of the tool, but also because of the researchers’ instrumentation of the tool, interweaving mathematical knowledge about series’ convergence and knowledge about the tool. Transposing to the classroom, one could conjecture

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that learner’s instrumentation of a tool (s)he uses for experimenting will contribute to proof and thus to conceptualisation. Of course, this contribution is by no means automatic. Again the three components – curriculum, classroom situations and tool design – have an essential role to play. As a conclusion of these two analyses, I will retain that, to make sense of a transposition of new experimental approaches, mathematics education should consider practices based on systematised methods and instrumentation of tools. It should also be aware that at the three levels of praxeologies and especially at the techniques-theory articulation, dissimilarities exist as possible obstacles to a transposition. In the next section, I will return to the problem considered at the beginning of this paper to see how, in a given situation, teaching could overcome these obstacles. 6. AN ‘OFFICIAL’ IMPLEMENTATION We saw above that a problem about a functional dependency between the position of a point and an area fits well into experimental technology aided approach to functions. We saw also that the curriculum privileges this problem and approach, as well as the use of dynamic geometry and spreadsheets. A consequence of this privileging is that many teaching materials and textbooks propose classroom implementations of this problem very close one to another. In this section I will analyse the situation proposed by the student worksheet of Appendix A.12 In this worksheet, numerical values are given for the triangle’s lengths (4 and 5 cm). The position of the variable point is given as independent algebraic literal. Students have to carry out an exploration of the functional relationship and then gradually formalize this relationship to tackle the problem of the maximum area. For exploration and for numerical and graphic formalisation (questions 2–4) dynamic geometry software is used. Students are introduced to specific software actions like dragging the variable point. In these questions, the vocabulary induces experimentation and conjecture and little algebraic formalisation is used. In contrast, the last question is totally algebraic and involves no exploration. The goal of this section is to analyse classroom situations based on this student worksheet in order to open a discussion on the nature of students’ activity and its contribution to learning. To pass from a


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student worksheet to classroom situations, I will rely on ‘representative’ statements from teachers – a notion I will specify below, rather than on specific experimentation. This choice is consistent with the idea of considering situations anchored into actual classroom practices and with the type of analysis required for the discussion. 6.1. How Teachers see the Situation Induced by this Implementation? I rely on a text accompanying the student worksheet and on the comments by Meyrier, a teacher of the Casyopeé team who experimented this situation.13 As explained above, my goal is not to provide research conclusions regarding this implementation but to explicit how teachers see the resulting situation. The teachers’ comments are ‘representative’ in the sense that the text was published on an official web site as a outcome of the work of a group of teacher evaluated by a regional authority and that Meyrier’s address was delivered in a European conference with the participation of many researchers and teachers working in the field. The text emphasises the student’s activity. It also stresses that the situation should not be isolated. The goal is to make students create the objects in a dynamic situation. Another similar situation should be proposed in order to make students use their new knowledge. Two classroom sessions seem a minimum for long-lasting acquisition of this knowledge.

Meyrier specifies student software use and his view of its contribution to the experimental work’s devolution. Using dynamic geometry, a student can build, see and manipulate objects. This activity helps him (her) to understand the problem, to represent the mathematical objects involved in the solution (distance and area, independent and dependent variables, graphic representation. . .) on the basis of the geometric situation. This activity also legitimates a mathematical work: for instance dragging the point illustrates the idea of maximum. It makes the student want to conjecture.

He proposes to look further into the problem using a spreadsheet. Entering formulas in cells for the dependant and independent variable helps to make sense of literal expressions. This is a new exploration of the problem, abstracting from the geometrical context towards an entirely numeric approach.

He emphasises the distance between computer uses and usual mathematical activity. He stresses that usual mathematical activity is

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the actual reference for conceptualisations and that distance is often underestimated. The situation relies on the hypothesis that, even when the ‘computer objects’ – dynamic points, cells of a spreadsheet… are not similar to usual taught objects and even when this dissimilarity is an obstacle to students, the computer aided manipulation helps the student to conceptualise the usual objects.

He finally insists on the difficult role of the teacher. The teacher has to organize classroom activities to insure that students make a link between these objects. This is the actual difficulty of integration.

6.2. A Discussion Three issues – experimental activity, algebraic activity and techniques are considered here to discuss the nature of students’ activity in this situation and its contribution to learning. 6.2.1. An Experimental Activity? The situation relies on software applications’ use – dynamic geometry and spreadsheet – recommended by the official curriculum. Comments by the teachers strongly suggest that these applications will make the students actively explore with the goal of expressing a conjecture. Given suitable conditions (preparation of the students in the use of software, students’ previous knowledge. . .), dynamic geometry will encourage an active approach of the problem and the situation will help to focus on functional properties (dependence, growth and decay, graphs. . .). Beyond the search for a conjecture, the situation’s purpose is to help students to make sense of the problem and notions. It is a condition for a real experimental activity, connecting exploration to conceptualisation. As Meyrier stresses, students’ software use potentially helps conceptualisation but there is a distance between the objects manipulated and conceptualised within software and the notions that the student should learn about. That is to say that even when students acquire some ‘enactive’ knowledge about functional properties, this knowledge is far from the algebraic conception of functions of their everyday mathematics. 6.2.2. An Algebraic Activity? Let us discuss the situation regarding its contribution to Kieran’s three categories of algebraic activity outlined in the introduction. In this


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situation, dynamic geometry and spreadsheet contribute to the generational activities. The dynamic character of these applications helps to understand the dependency between position and area. At the beginning of the situation, using dynamic geometry, it is a pure geometrical dependency. The geometrical context of the problem and the use of dynamic geometry might orient students towards a geometrical study of the maximum, similar to the proof above. The teaching goals however privilege the numerical dependency and its algebraic expression in the approach of functions. By displaying numerical values of the dependant variable, the dynamic geometry application helps to pass from a geometric to a numerical relationship. ‘Tracing’ these values helps to express this relationship into the graphic register. Provided that students could master the specific formalism, using a spreadsheet could help to see the relevance of a formula to make the relationship works without a figure. In the worksheet of Appendix A, global/meta activities are limited by the fixed numerical values of the lengths of the triangle’s sides. The position of the optimal rectangle at the middle of the sides does not appear as a general property. Exploration, conjecture and – possibly – conceptualisation occurs in the limited context of a fixed size triangle. A ‘generalized problem’ with a triangle whose sides could vary, would induce classroom activities without this limitation and emphasize the fact that optimum area is for a position of the independent point in the middle of the side, a more interesting outcome as compared to mere numerical value. For instance, with dynamic geometry, dragging a vertex could change one side of the triangle. ‘Tracing’ with different values of the side, the student would obtain different parabolas, each with a maximum position at the middle. Using a spreadsheet, the independent variable and the sides of the triangle could be entered as relative and absolute references in the formula. These applications actually provide means to express a family of functions and thus could contribute to global/meta activities. The drawback is that these means are far from means used in everyday mathematics: an independent variable and parameters. Meyrier’s comment about distance between ‘technological’ and ‘ordinary notions’ is then really appropriate. Students have to carry on transformational activities only in the last question – proving the maximum. In the proposed situation (s)he has to do it with paper and pencil and there is no technology aided experimentation or problem solving. In dynamic geometry and in spreadsheets, formulas exist only to be evaluated numerically and these applications cannot contribute to a work on expressions.

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6.2.3. Techniques for Experimenting? Students have to develop new specific ‘instrumented gestures’ – i.e. activation of specific software commands as means to reach goals related to a mathematical task – in order to solve the varied sub-tasks proposed in the worksheet of Appendix A: determination of the independent variable’s domain, computation of values, approach of dependent variable’s maximum value and of corresponding value of the independent variable. . . These gestures are potentially the elementary components of experimental techniques. As the accompanying text insists on need of a repetition of the activity to solve similar problems, it is an indication that teachers who created and implemented the situation are aware that students should be helped to organise their computer aided experimental activity and that this organisation implies the development of specific techniques. Certainly it is here that classroom use of technology has similarities with Borwein et al. (ibid.) experimental mathematics. Teachers expect that techniques for exploring properties of the function will help students to understand associated concepts: set of definition, values, graph, extremum. . . Instrumentation is also involved as efficiency of techniques depends on students’ appropriation of the tools’ functionalities: dragging points, tracing, managing approximation. . . Meyrier’s comment stresses on possible difficulties in linking these instrumented techniques with the ordinary mathematics. A student has to understand the relationship between these techniques and paper/pencil techniques that (s)he uses in other situations. Even when the global organisation of these techniques is similar, elementary gestures can be very dissimilar. For instance a paper/pencil technique to study a function begins by determining the domain of the independent variable, which is done by an algebraic work on the definition. With dynamic geometry the same task exists, but the corresponding gesture – dragging a point on a line – is very different. Other gestures, like determining a maximum by graphic exploration have no correspondent in paper/pencil techniques. They can be then really difficult to integrate into ‘ordinary’ mathematical knowledge.

6.3. Limits of the Implementation I will conclude this section by observing that the actual implementation of the problem chosen as an example in this paper illustrates concerns mentioned in the study of the transposition of mathematicians’


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practices. Teachers seem to be aware of the need for developing techniques of exploration. The scope of these techniques is however limited to the generational category of algebraic activity, possibly creating an obstacle to a full conceptualisation of the notion. The discussion highlights difficulties resulting from a distance between objects in the experimental activity and ‘ordinary’ objects. To students, instrumented gestures and techniques evoke objects existing mainly in the ‘enactive’ interaction with the computer whereas mathematicians directly make sense of computer processing in terms of mathematical notions. In actual teaching/learning, students learn about functions also with paper/pencil and thus a contribution of instrumented experimental activity to conceptualisation supposes that they link the objects in the computer and the ‘ordinary’ mathematical notions. The teacher did not find it so easy with this implementation. In the next section I will consider the design of tools that could allow new implementations.

7. LEARNING ALGEBRA WITH COMPUTER ALGEBRA SOFTWARE? 7.1. The Potential Contribution of CAS The above analysis gives a mixed opinion of the contribution of an implementation of the curriculum using non-symbolic software. Students can develop an experimental activity and build associated techniques, but this activity is mostly of the generational type and involves objects that exist mainly in the interaction with the computer. It can help a first understanding of the notion of function, but it covers very partially the objectives of the curriculum, particularly transformational understanding: Students should be able to recognize the form of an expression (sum, product, square, difference of two squares), to recognize various forms of an expression and to choose the most relevant form for a given work.

The curriculum does not recommend developing transformational skills but rather a reflection on forms. Mastering algebra implies the development of abilities for anticipating the effects of transformations. For this purpose, students should reflect on the varied forms of an expression and on the questions that each form can help to answer.

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As CAS potentially contributes to help students concentrate on the sense of transformations rather than on transformational skills, it seems odd that the curriculum, like we saw above, is reluctant towards classroom use of these symbolic tools. I will discuss CAS’ potential before coming back to the reasons of the curriculum’s reluctance. In paper/pencil transformational activities, algebraic manipulations are necessary in order to get a given form, possibly hiding the interest of the outcome as compared to the initial form. Waits et al. (1997, p. 5) stress wisely the potential contribution of symbolic tools to overcome this difficulty. What is really much more important (…) is understanding why the factored form of (a) cubic tells us more about the behaviour (graph) of the function than the standard form. We must recognize that easy to use computer symbolic algebra tools like Expand and Factor offer a great advantage to both the student and the teacher.

Computer algebra was created to help mathematicians to go beyond mere numerical exploration. In education, these tools could be really useful to overcome students’ manipulative difficulties not only in the transformational activities but also at global/meta level. As compared with dynamic geometry or spreadsheet expression of a family of functions in the ‘generalized’ triangle problem, CAS offers expressive means much closer to ordinary mathematical notation. We experimented interesting ‘generalized problems’ in the TI-92 experiment (Artigue, 2004). Students’ use of CAS in an experimental algebraic activity would also help to better articulate the varied algebraic activities. Yerushalmy (1997) studies a classroom exploration of the asymptotic behaviour of functions with help of a graphing tool. They had to link the perceptive evidence of an asymptotic line on the graph – a generational activity – and the partial fractional expansion of the function – a transformational activity. But the graphing tool was of no help to get the expansion and students had to use paper/pencil polynomial division. Few students could do it alone and, as it was a long process, they lost view of the goal. If students had used CAS, they could have freely explored algebraic transformations focusing on the goal to corroborate their graphical observation. 7.2. The Difficult Classroom Implementation of CAS In spite of the above potentialities, there is no wide classroom use of CAS. Didactical research considered this technology and introduced


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the instrumental framework to make sense of classroom CAS experimentation. The instrumental concern is not limited to CAS, but students’ genesis of CAS seemed to be particularly intricate. Actually CAS’ design does not take the user’s knowledge and the task into account. Using CAS, a student is confronted with multiple capabilities offering new potentialities for action, but also to difficulties to take advantage of these. (S)he needs much time and specific situations to sort out these capabilities, integrating them as instrumented gestures into techniques and link these to his(er) mathematical knowledge (Lagrange, 1999). Monaghan’s (2004, p. 339) comparison of classroom software use shows that this characteristic of CAS also creates obstacles for the teacher. A teacher who primarily used Derive with his class found using a computergraphing package (Omnigraph) much more akin to his ‘normal’ lessons. This had several dimensions. To use Derive he had to devote several lessons to attend to simple mastery of the commands and syntax but this was a five-minute task with Omnigraph. Perhaps more important than this each differentially affected the way he interpreted the mathematics his students were doing. He viewed Omnigraph as a tool to do straightforward tasks that fitted in with his ‘normal’ lesson ideas. He viewed Derive, however, as a ‘monster’ that could do virtually everything and this forced him to completely re-interpret his lesson plans. This was a disquieting experience for him.

This is consistent with Yerusalmy’s (ibid. p. 172) observation that ‘‘the design (of CAS) serves the agenda of the tool designers – reaching a result in the smoothest possible way’’ and this design contradicts with an educational agenda that should give students control over exploration and provide for support to the organisation of the curriculum. Coming back to the French curriculum, I can see its lack of enthusiasm towards CAS as recognition of difficulties in CAS’ classroom implementation. 8. THE CASYOPEÉ PROJECT 8.1. Purpose and Organisation of the Environment In the Casyopeé project our14 purpose is both to offer symbolic calculation for classroom mathematical activities and to support a ’sustainable’ curricular evolution. The four theoretical frames presented above are involved. One goal is to provide students a

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technological aid in the three categories of algebraic activity, generational but also meta-level and transformational. We aim also easier instrumentation and better curricular adaptation by creating symbolic capabilities easy to connect with usual secondary mathematical practice, and to encourage methods or techniques as a way to conceptualisation. As a start, because CAS non-specificity was found to create obstacles to classroom integration, we choose a domain to implement these ideas. Our choice, consistent with this paper’s topic was the study of real functions, including parametrical functions – or families of functions. Students can use Casyopeé as technological help to solve a range of problems typical at upper secondary level including signs and variations of functions, problems of optimisation by algebraic methods or derivation, intersections of graphs, finding functions with respect to constraints, proving zeros by way of the intermediate values theorem… As compared with the ordinary paper/pencil environment, no manipulative difficulties prevent students to tackle all sorts of functions and families of functions and thus, even if the type of problems remains the same, problems themselves can be more ambitious and students explore more freely. Examples are presented in Lagrange (2002) and Le Feuvre et al. (2004). Casyopeé’s organisation is designed to help students to keep clear of erratic behaviour by concentrating on relevant objects for a problem (s)he wants to solve, to make sense of exploration and to develop methods. As a difference with standard CAS, which operate mainly on symbols, each object has a clear statute with regard to the curriculum: real number, function, parameter. . . Functions are defined on IR or on reunion of intervals. While standard CAS’ window is just a memory of commands and feedback, Casyopeé’s interface displays the objects relevant for a problem – real numbers, functions with their definition and standard forms (factored, developed. . .), equations, parameters – and their properties. These objects are dynamically updated like in a spreadsheet when some change is done. We wanted to facilitate graphic and numerical exploration like with a grapher, while encouraging transition to global/meta activities by generalisation using symbolic computation. We developed a special feature-instantiating and de-instantiating parameters. Together with the dynamic organisation of objects, it is an aid to students’ exploration and problem solving involving


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families of functions. Dynamic instantiation – named ‘pilotage’ – helps to study numeric cases and deinstantiation – ‘de´pilotage’ – corresponds to generalization. Developing meaningful techniques and transformational understanding implies practices of exploration in several registers (graphic, numerical, algebraic…), interweaved with algebraic reasoning and writing. Thus we wanted that students could go beyond merely reading properties on a standard graph or table. Of course this implies special situations with no direct numerical or graphic evidence but also a support for strategic approaches to properties. That is why, in addition to providing graphs and numeric table, Casyopeé helps the student to use algebraic reasoning and to keep track of properties conjectured or proved. Theorems in secondary algebra and calculus relevant for properties of sign, variations and zeros of functions are implemented as elementary steps of proof – ‘justifications’ – that students can operate as consistently as possible with the pragmatic at this level. Steps of proof are based on premises found from a form of a function or already proved or conjectured. To complete this presentation of Casyopeé it is necessary to specify its ‘knowledge’ of algebra and how this knowledge can help the student. Casyopeé uses the algebraic knowledge of a CAS kernel (Mupad). The kernel is not directly visible to students. Its more obvious aid is of course to algebraic transformation and calculations, but it is also a support to the whole students algebraic activity. Here are some examples: – Knowing whether two numbers can be compared and how. For instance, the interval [a;b], has no sense without a condition on the parameters a and b. – Checking the definition of a function. For instance, the existence of the function over [a;¥, ½x; ! 1=x depends on the sign of a. – Checking whether the same function has not already be defined with an equivalent expression and same set of definition. – Solving equations. . . The Mupad kernel is also much used by Casyopeé in the proof process. This is an example of a ‘justification’ that a student can obtain with Casyopeé’s help.

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This ‘justification’ is based on the recognition of a class of functions (fonctions de reference). Here the class of homographic functions has been chosen. It is known in the curriculum that, depending on their definition, functions in this class have not more than one zero (ze´ro) and one discontinuity (valeur interdite15). It is also known that their sign does not change in intervals limited by these values (Elle est de signe constant. . .). Thus a technique to prove the signs is to compute a value (valeur) somewhere in each interval and to conclude. In this ‘justification’ Casyopeé can check or give help. – Checking premises or giving hints on a possible justification. Thanks to the kernel, Casyopeé is able to ‘know’ that a function is homographic, whatever the algebraic form given to evaluation. – Checking conclusions or helping the student to find them. In the above example, Casyopeé ‘knows’ that )a and a are respectively, the only zero and the only discontinuity of the function and can calculate the sign of a value somewhere on the three intervals they determine. Casyopeé is still in development. A first working version has been experimented in classrooms and is ready for delivery (Appendix C outlines features of Casyopeé). More information about the genesis


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of the Casyopeé project and about a similar project whose scope is limits can be found in Lenne et al. (2003). 8.2. The Maximum Rectangle Area Problem using Casyopeé In this section, I study the actualisation of the above principles by drawing from classroom experiment. It was one of the many iterations16 between design and classroom that were necessary to correct and further develop the environment. The purpose of reporting this experiment is more to demonstrate the design method than to provide actual evidence of Casyopeé’s potential. The experimentation was done at the end of the year of Seconde with an early version of Casyopeé. Students had worked before on problems similar to the above-discussed situation. The goal was to offer them a more general problem – with a parameter – and a more self-directed activity – no detailed guiding. The maximum rectangle area problem was the first stage of three problems. In this stage, student where guided by a table (Appendix D). We choose to have only one side of the triangle as a parameter. We found that it was a sufficient generalisation while limiting the complexity of expressions. The general problem was presented at the beginning together with the capabilities of the environment. The worksheet (Appendix D) asked students to: 1. enter the function x ! 5xða xÞ=a (Figure 2), 2. instantiate and animate the parameter a (Figure 3), 3. do a graphic and numerical exploration for several values of a in order to conjecture the optimal value of f. 4. define for each of these values the function x ! fðaÞ fðxÞ; fðaÞ being the optimum value (Figure 4), 5. do an algebraic study of these functions in order to find their signs (Figure 4). and conjecture a general property regarding optimality, 6. deinstanciate a and build a general proof using a justification (Figure 5). Some gestures that students were expected to do are specified in Figures 2–5. At step 4, all calculations being done by Casyopeé, students had to obtain a table whose first and last lines are similar to the following:

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Valeur de a

Valeurs possibles pour x

Abscisse a du maximum

Expression de´veloppeé de f(a) – f(x)

Expression factoriseé de f(a) – f(x)

Signe de f(a) – f(x) sur l’intervalle d’e´tude

8 ...

[0; 4]

2

5x2 8

5/8 (x)4)2

Positif

23 2

½0; 23 2

23/4

10x2 23

5/184 (4x)23)2

Positif

5x þ 10 5x þ 115=8

We expected that, from this table, students could understand the connection between a value of x for f(x) maximum and the factorization of f(a) ) f(x) as a square (times a positive constant), connect this also to a graph -a parabola tangent to the x-axis, and think of a proof (step 5).

Figure 2. Definition of a function. Students had to enter a set of definition and an algebraic formula –involving the variable x and the parameter a - in a dialog box (Entreé Fonction). The definition appears in the central window and feedback is given in the right window.


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Figure 3. Instantiation. An interval and a step had to be chosen for the parameter in a dialog box (Exploration). It had to be declared ‘pilote´’ (animated) so students could change its value using the slider (top right). The graph and some algebraic results were dynamically updated. The table above the graph displays exact values of the function.

At step 6, the final proof of the general result, could be written out like the following. Let us define g : x ! f a2 fðxÞ onthe interval [0; a]. The factorised form of g is x ! 45a ð2x aÞ2 : It is the product of a positive number and a square. Thus it is positive. We proved that for all x in the interval [0;a] f(a/2) ‡ f(x) and thus f(a/2) is the maximum value of f(x). The maximum rectangle’s side lengths are 5/2 and a/2.17 8.3. Observation and Evolution of Casyopeé The observations reported and discussed here are about students’ understanding of Casyopeé instrumented gestures. This is consistent

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Figure 4. Algebraic transformations. Students had to define a new function (Nouvelle fonction) g as the difference between the value of f at the conjectured maximum and the current value of f At step 4, they had to compare the expansion (de´veloppement) and the factorisation for several values of a, and decide of a sign.

with the goal of the experiment – to help making the environment a relevant tool for students and teachers – and with the focus of this paper on tool’s design. Defining a function and its interval of definition, instantiating and de-instantiating a parameter, using commands for the values of functions and for algebraic transformations were relatively easily recognized by students as corresponding to usual mathematical gestures. Generalising by means of deinstantiation was not a difficulty and clearly emphasized the general property and proof. Exploring efficiently the graphic window was more difficult. For instance, students confused the maximum of the function with the maximum value of the variable though they did a similar exploration before when using other software or calculators. Some instrumented gestures, as I mentioned before, have no correspondence in the usual mathematics. In spite of previous practice with other software, students did not appropriate these gestures enough to be able to do an easy transfer into a new environment. Proving with Casyopeé was supposed to encourage transformational activities and focus on methods. In this experiment students tended to perceive the corresponding gestures as more constraining than ordinary paper/pencil proof. In the version of Casyopeé devel-


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Figure 5. Proof. At steps 6, students had to do a proof of the sign of g. Using Casyopeé, this proof in done in three steps demonstrated above. 1. Choosing a ‘justification’ in a menu. 2. Entering evidence for the premises in a dialog box (Signe…Conditions d’application). Casyopee then issues some information about a conclusion. 3. Entering conclusions in another dialog box (not seen on the picture above). Shortcuts are possible in the recent version.

oped at this time, students had to look into a menu for the relevant item and fill two successive dialog boxes whereas reason for the maximum was clear on a factorised form of a function (Figure 5). We nevertheless thought that these gestures are important in order that the symbolic resolution helps conceptualisation. Further developing the environment, we created bypasses making Casyopeé do a part of the proof when requested by the student and allowed by the teacher. We also made Casyopeé issue detailed messages about the proof, as close as possible to usual formulations, that students might insert more or less directly when writing out the proof. Thus proof is now less constraining and Casyopeé offers students a valuable help in the writing of the proof, generally a hard task for them. In the maximum area rectangle problem, the geometrical framework is important for the generational activity, as a first step in modelling and for the interpretation. In the present development of

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Casyopeé, the activity is entirely on an algebraic definition of the function. Thus, using dynamic geometry together with the environment would be valuable. To avoid the complexity of using two applications and to provide a transition from geometrical generation to algebraic modelling, we are thinking on the possibility of diversifying the way users can introduce a function: as an algebraic formula but also from geometrical definitions – lines and other geometrical objects, best fit to a set of points (functions of regression) – , or from a relationship between variable geometrical objects.

9. CONCLUSION This paper tried to consider together curriculum, classroom practices and software design by particularising to situations involving experimental situations for the learning of functions. More precisely, it was looking for the help that a reflection on curriculum and practices can bring to the design of a new environment dedicated to these situations. The praxeological approach helped to see that one cannot think of a naı¨ ve transposition based on a homology between mathematical researchers’ and students’ practices. An important obstacle is at techniques-theory articulation, which are interweaved in research practices but most often separated by a wide gap for students. Technology cannot change this situation simply by offering easier exploration. However, systematised experimental methods and instrumentation of the tools could be some bridge other the gap. The French curriculum is much influenced by the naı¨ ve transposition but innovative or reflexive teachers seem to feel the need for student experimental techniques and instrumentation as a basis for conceptualisation by bridging experimentation and theorisation. This view of a possible gap between experimental practices and theorisation contrasts with recent studies like Sinclair et al. (2004, p. 258) that conclude on a positive influence of experimentation on learner’s understanding of concepts. It is however not inconsistent as Sinclair et al.’s study is about tertiary students and number theory. These students certainly have difficulties to understand essential aspects of relationships between numbers, but they know about numbers since a long time and experimentation makes directly sense, whereas algebraic objects like functions are very new to secondary


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students considered in this paper. Sinclair et al. (ibid., p. 250) also raise an important issue: all students do not take the same advantage of experimentation when conceptualising. This issue is not addressed in this paper, but it is important to mention that the contribution of experimentation, often considered as straightforward in naı¨ ve approaches, certainly needs to be carefully discussed. The French curriculum stresses the transformational and meta/ level categories of the algebraic activity while being very cautious regarding the use of CAS. It is an illustration of a ‘cul-de-sac’: CAS could provide students necessary support for problem solving in these categories but existing software seems really difficult to integrate. The outcome is that experimentation is based on the generational activities and is then clearly separated from theorisation, which relies much on the two other categories. The way is then paved to poor inductive classroom practices: giving students ‘a feeling’ of a functional relationship by way of experimental practices and then introducing an algebraic model of this relationship without real exploration or problem solving. Literature about educational use of CAS stresses on the possible contribution of these applications to transformational and global/ meta activities, especially in experimental approaches of problem solving. This contribution is however mainly potential because of difficult classroom integration of actually available standard CAS. I analysed this situation using the instrumental approach and tool design issues raised by Yerushalmy (1999). Standard CAS were created for mathematicians without concern for teaching/learning specificities like students’ knowledge or educational goal. Using standard CAS does not directly help students to develop methods and it is really difficult to think of a curricular evolution providing consistency with these tools’ particularities. I can see French curriculum’s reluctance to classroom use of CAS as a consequence of this. The stake is then to articulate curricular evolution and Computer Agebra based tool design, in view of ‘sustainable’ classroom practices. In the French curriculum context, providing students for means to experiment in the transformational and global/meta dimensions could help to develop a better articulation between exploration and conceptualisation. This is an ambition of the Casyopeé project. Though this project is far from complete, observation of classroom use of Casyopeé shows the advantage of embedding a Computer Algebra kernel while designing computer objects and instrumented gestures consistent with

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the curriculum’s objects and actions. Working to make algebraic proof more accessible and to articulate all categories of algebraic activities are stimulating prospects for the continuation of the project. As a final conclusion I come back to a key issue: the contribution of experimental approaches using technology to students’ conceptualisation. This contribution can certainly exist when students have a clear theoretical basis. In other cases, like teaching/learning of algebra with beginners, tool use techniques and instrumentation, considered in view of their epistemic role, could open a successful way. The design of Casyopeé is a means to try this hypothesis. This is not a short path: the experiment I described in the paper was one of many iterations that were necessary to make the environment a relevant tool for teachers and students. It seems however that there is no quick way to deal with the complexity of educational use of technology. The project will go on and I expect that wider classroom experiment will bring more evidence of Casyopeé’s potential to support a real experimental activity.

ACKNOWLEDGMENTS The Casyopeé project is supported by the IREM (Institut de Recherche sur l’Enseignement des Mathe´matiques) of Rennes and the INRP (Institut National de Recherche Pe´dagogique). The author thanks B. Le Feuvre and X. Meyrier who are working on this project, M. Artigue and C. Kieran for their critical comments on a first version of this paper and the anonymous referees for their constructive critics and propositions.

NOTES 1

All extracts from texts in French are my translation. Experimentation can be understood restrictively as the action of carrying out an experiment to test an hypothesis. Here, it has the wider meaning of gathering data in order to make ideas appear. We shall see below that Yerushalmy (1999) uses the word exploration with a similar meaning. She explains what exploration is, especially in the field of equivalent equations and how students should do exploration «that do not focus on ‘looking at a numerical result’, but on mathematical ideas » (p. 175) and refers to Lakatos for possible pitfalls. Thus, Yerushalmy’s exploration of equivalent equations is close to experimentation about functions considered in this paper. I choose the word ‘experimentation’ because of the adjective ‘experimental’ used in ‘quasi-experimental

2


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approaches’ by the curriculum and in ‘experimental mathematics’ by a number of authors. The adjective ‘exploratory’ would have been misleading. 3 The words ‘inductive’ and ‘induction’ have to be explained here. In English, one speaks of ‘mathematical induction’ to mean ‘inductive proof’, whereas in French, ‘preuve par recurrence’ is used, judiciously emphasizing the deductive nature of this proof. I will not consider this acceptation. In Science, induction is a movement of the thought – or a method of argument- from data -or facts- to ideas -or laws- and is opposed to deduction which goes from ideas to ideas. Induction in this sense is fundamental in an experimental process of conceptualisation even in Mathematics. When authors speak of ‘poor induction’ as a classroom practice, they refer to this acceptation and mean that this process has been impoverished because of teaching/learning constraints: students can gather data or facts, but not really build ideas. 4 See note 2. 5 ‘Laı¨ que’ in the original text. It means: independent of religion. Here Chevallard uses the metaphor of the church (inside) and the laic society (outside). 6 Here ‘theory’ associates two components (‘technology’ and ‘theory’) that the praxeological approach distinguishes (see Artigue 2002, p. 248). 7 This set of frameworks is intended to bring understanding of students’ activity and thus the reader might wonder why I do not consider ‘activity theory’ (Nardi ibid.) as a unifying framework like researchers in HCI do. Actually, I prefer to stick to ‘mathematics education’ frameworks as complementary means to approach a reality, leaving to more theoretically oriented papers the task of confronting them and looking for an unification. 8 Extracts of the French curriculum are my translation. The curriculum can be found at http://www.cndp.fr/secondaire/mathematiques/ 9 Professor de Guzman who sadly died last year was a mathematician with strong interest in Mathematics Education. His contribution was always original and stimulating, and it is with profound respect that I take the liberty of elaborating from his address to the 2002 Vienna International Symposium on Integrating Technology. 10 L. Euler, Specimen de usu observationum in mathesi pura. Opera Omnia, ser.1, vol.2, p.459, cite´ par de Guzman (op. cit.). 11 See Lagrange (2004) for a wider discussion about « mathematical sciences » and their implication for teaching and learning. 12 The original text comes from an official web site of a ministry of education regional authority http://www.ac-orleans-tours.fr/. It was proposed for the end of the year of « Troisie`me » (ninth grade). I changed it slightly introducing the functional symbolism and the algebraic proof related to the ‘‘Seconde’’ level (10th grade). In a real teaching situation, the problem would actually be split up into parts proposed at different stages of the Troisie`me and the Seconde. 13 Meyrier X., unspublished talk at the plenary panel, ITEM conference June 2003. 14 A group of teachers of the IREM of Rennes and the author (Le Feuvre et al., 2004). 15 Continuity does not exist in the curriculum at this stage. Hence ‘valeur interdite’ (forbidden value) instead of ‘discontinuite´’ (discontinuity). 16 Iterative design was a team’s basic pragmatic choice, because we felt that very little could be really anticipated about the impact of this software on classroom practices and we wanted to try small implementations and study the effect on teachers and students. We are aware that the idea of iterative design has been conceptualized by HCI researchers like Schaffer (2001). This author develops a series of methods to systematize this type of design that could be a real improvement although perhaps not in the reach of a small team.

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17

This is actually a small implementation of a wider algebraic method for finding polynomial and rational function turning points with the help of symbolic computation, as exposed by Cherkas (2003).

REFERENCES Artigue, M. (2004). The integration of symbolic calculators into secondary education: some lessons from didactic engineering. Chapitre 9 In D. Guin, K. Ruthven and L. Trouche (Eds), The Didactical Challenge of Symbolic Calculators: Turning a Computational Device into a Mathematical Instrument. Dordrecht, The Netherlands: Kluwer Academic Publishers. Artigue, M. (2002). Learning 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 7(3): 254–274. Artigue, M. (1998). Teacher training as a key issue for the integration of computer technologies. In Tinsley and Johson (Eds), Information and Communications Technologies in School Mathematics. pp. 121–129. IFIP. Chapman and Hall. Assude, T. (2002). Analyse didactique de la vie et la disparition du dispositif des travaux pratiques au colle`ge. Actes de la rencontre sur les nouveaux dispositifs organiseé par la Commission Didactique Inter-Irem (IREM de Dijon). Borwein, J., Bailey, D. and Girgensohn, R. (2003). Experimentation in Mathematics: Computational Paths to Discovery. Boston, MA: AK Peters. Borwein J., Borwein P., Girgensohn R. and Parnes, S. (1996). Making sense of Experimental Mathematics. The mathematical Intelligencer (Vol. 18 no 4. pp. 12—17). New York: Springler. Burton, L. (1999). The practices of mathematicians: What do they tell us about coming to know mathematics? Educational Studies in Mathematics 37: 121–143. Cherkas, B. (2003). Finding polynomial and rational function turning points in pre calculus. International Journal of Computers for Mathematical Learning. 8(2), pp. 215–234. Chevallard, Y. (1985). La transposition didactique, Editions La Penseé Sauvage, Grenoble. Chevallard, Y. (1994). Les processus de transposition didactique et leur theórisation, in Arsac, Martinand and Tiberghien (Eds), La Transposition didactique a` l’e´preuve des faits, Editions La Penseé Sauvage, Grenoble. Chevallard, Y. (1999). L’analyse des pratiques enseignantes en theórie anthropologique du didactique. Recherches en Didactique des Mathe´matiques 19: 221–266. CREM, (2002). Commission de re´flexion sur l’enseignement des mathe´matiques. In J.P. Kahane, (Ed). L’Enseignement des sciences mathe´matiques. Paris: Odile Jacob. Douady, R. (1986). Jeux de cadres et dialectique outil/objet. Recherches en Didactique des Mathe´matiques 7(2): 5–31. Duval, R. (1996). Quel cognitif retenir en didactique?. Recherches en Didactique des Mathe´matiques 16(3), 349–382. Freudenthal, H. (1986). Book Review: Yves Chevallard: La Transposition Didactique du Savoir Savant au Savoir Enseigne´. Grenoble: La Pensee´ Sauvage. In: Educational Studies in Mathematics 17: 323–327. Ge´lis, J-M, and Assude, T. (2002). Indicateurs et modes d’inte´gration du logiciel Cabri en CM2. Sciences et Techniques Educatives 9(3.4): 457–490.


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Guin, D. and Trouche, L. (1998). The complex process of converting tools into mathematical instruments: The case of calculators. International Journal of Computers for Mathematical Learning 3(3): 195–227. de Guzman, M. (2002). Experiment, conjecture and proof in geometry with DERIVE. In J. Bohm (Ed), Proceedings of Vienna International Symposium Integrating Technology in Mathematics Education (CD Rom), BK teachware. Joshua, S. and Joshua, M.A. (1988). Les fonctions didactiques de l’expe´rimental dans l’enseignement scientifique. Recherches en Didactique des Mathe´matiques 9(1) pp. 5–27. Kieran, C. (2001). Contribution to the opening plenary panel. 12th ICMI Study Conference. The University of Melbourne. Lagrange, J.B. (2004). Transposing computer tools from ‘‘mathematical sciences’’ into teaching. Some possible obstacles. Chapter 3 In D. Guin, K. Ruthven and L. Trouche (Eds), The Didactical Challenge of Symbolic Calculators: Turning a Computational Device into a Mathematical Instrument. Dordrecht, The Netherlands: Kluwer Acadamic Publisher. Lagrange, J.B., Artigue M., Laborde C., and Trouche L. (2003) Technology and math education: a multidimensional overview of recent research and innovation. In J. Bishop, Clements, Keitel Kilpatrick and Leung (Eds), Second International Handbook of Mathematics Education (pp. 237–270). Dordrecht, The Netherlands: Kluwer Academic Publisher. Lagrange, J.-B. (2002). Casyopeé, a computer algebra environment for secondary students., Vienne Juillet 2002. In J. Bohm (Ed), Proceedings of Vienna International Symposium Integrating Technology in Mathematics Education (CD Rom), BK teachware. Lagrange, J.-B. (2000). L’inte´gration d’instruments informatiques dans l’enseignement : Une approche par les techniques. Educational Studies in Mathematics 43: 1–30. Le Feuvre, B., Heilbronner, L., Lagrange, J.B. and Meyrier, X. (2004). Casyopeé, un environnement logiciel pour l’analyse en lyceé. A In Lagrange J.B. et al. (Eds), On line proceedings of the ITEM conference, Reims, June 2003. (www.reims.iufm.fr /Recherche). Lenne, D., Lagrange, J.B., Py, D. and Gelis, J.M. (2003). The design of software learning environments using symbolic computation: Two case studies in pre-calculus. International Journal of Computer Algebra in Mathematics Education 9(4): 285–303 Lins, B. (2003). Actual meanings, possible uses: Secondary mathematics teachers and cabri-geóme`tre. In Mariotti et al. (Eds) Proceedings of European Research In Mathematics Education. Italy: Universty of Pisa. Mason, J. and Spencer, M. (1999). Beyond mere knowledge of mathematics: The importance of knowing-to act in the moment. Educational Studies in Mathematics 38(1–3): 135–161. Monaghan, J. (2004). Teachers’ activities in technology-based mathematics lessons International Journal of Computers for Mathematical Learning 9: 327–357. Nardi, B. A. (1996). Introduction to Context and Consciousness: Activity Theory and Human- Computer Interactions. Cambridge MA: MIT press. Ruthven, K. and Hennessy, S. (2002). A practitioner model of the use of computer-based tools and resources to support mathematics teaching and learning. Educational studies in mathematics 49: 47–88, 2002. Schaffer, S. (2001). Learning system design considerations in creating an online learning environment. Performance Improvement 40(9): 36–42.

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Trouche, L., (2000). La parabole du gaucher et de la casserole ˆ bec verseur: Etude des processus d’apprentissage dans un environnement de calculatrices complexes. Educational Studies in Mathematics 41: 239–264. Waits, B., Demana, F. and Kutzler, B. (1997). Guest editorial. International Journal of Computer Algebra in Mathematics Education 4(1): 4–6. Yerushalmy, M. (1997). Reaching the unreachable: Technology and the semantics of asymptotes. International Journal of Computers for Mathematical Learning 2: pp. 1–25. Yerushalmy, M. (1999). Making exploration visible: On software design and school algebra curriculum. International Journal of Computers for Mathematical Learning 4: 169–189.

APPENDIX A. A STANDARD STUDENT WORKSHEET L’objectif est de de´terminer les dimensions du rectangle ayant la plus grande aire. Eńonce´ ABC est un triangle rectangle en A tel que AB ¼ 4 et AC ¼ 5. Le point N appartient au segment [AB] et AN ¼ x. ANMP est un rectangle avec M sur [BC] et P sur [AV]. 1. Faire la figure pour x ¼ 3 (unite´ : le cm) et calculer l’aire du triangle AEF. 1. Observer l’e´volution de l’aire a(x) du rectangle quand le point N parcourt le segment [AB] . Pour cela, cliquer sur la figure Geóplan puis de´placer le point N au clavier (touche ou !). (a) Quelles sont les valeurs possibles de x ? (b) Que peut-on dire de l’aire a(x) du rectangle quand le point N est en A ? en B ? (c) Quelle semble eˆtre la valeur maximale de l’aire ? Pour quelle valeur de x ? 2. Comple´ter le tableau suivant avec les valeurs de l’aire indiqueés par le logiciel. (a) L’aire a du rectangle est-elle proportionnelle a` x ? (b) Repre´senter sur un graphique les points de coordonneés (x ; a ) correspondant aux donneés releveés dans le tableau. X a(x)

0

1

1,5

2

2,5

3

3,5

4


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4. Pour ve´rifier : Le logiciel peut repre´senter, dans le repe`re, le point O de coordonneés (x ; a) et en laisser une trace quand on de´place le point N. Pour cela, double-cliquer sur le repe`re puis activer la commande ‘‘Afficher’’ ‘‘Mode trace (bascule) ’’. Ensuite, cliquer sur la figure et de´placer le point N au clavier pour faire varier x. (a) Quelle repre´sentation graphique obtient-on a` l’ećran ? La tracer sur votre graphique. (b) Quelle est l’abscisse du point O quand l’aire a semble maximale ? Quelle est alors la valeur de a ? Faire apparaıˆ tre votre re´ponse sur le graphique. (Tracer les pointille´s correspondants). 5. Relation entre x et a. (a) Sachant que (AC) // (MN), justifier l’e´galite´ MN/AC ¼ NB/AB En de´duire la valeur de MN en fonction de x. En de´duire une formule pour a(x). (b) Aire maximale (a) A l’aide des formules preće´dentes, calculer pour x = 2 la valeur de l’aire a du rectangle ( retrouve-t-on la valeur donneé par le logiciel ?) et la valeur de AN. Quelle est alors la position des points M et P sur les segments [BC] et [AC] ? (b) Montrer que a(2)-a(x) = 5/4 (x)2)2. En de´duire que a(x) est maximum pour x = 2.

APPENDIX B. THE KNUTH’S SERIES PROBLEM (FROM BORWEIN ET AL. 2003, SECTION 1.5) This is a nice example on how technology assisted exploration may go beyond the production of a conjecture. The authors’ goal was to investigate a particular series 1 k X k 1 pffiffiffiffiffiffiffiffi : S¼ k!ek 2pk k¼1 The first step was an occurrence of a method that became a sort of routine because the authors used it for a number of other series

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before. This method consists first in obtaining a numerical approximation for S. In this case, they used Maple, but at this stage the use of this CAS was not significant. Any numerical software could have been used, provided that precision was sufficient. Then the method uses this value and the ISC to conjecture an exact value of the series. They describe the ISC (available at the URL http:// www.cecm.sfu.ca/projects/ISC) as a ‘a constant recognition facility’, that is an application that, given a decimal number tries as to find an expression involving mathematical constants (p, e. . .) whose approximate value is the same for the given precision. The ISC yielded the result 2 1 1 : S pffiffiffiffiffiffi f 3 2p 2 As they say ‘within a few minutes we ‘knew’ the answer’: mathematically speaking, it was a conjecture, but by further approximation they got a ‘quasi-certainty’. This first step is a pure empirical approach, and, would a proof have been obtained by ‘ordinary means’, there would be not much to say about the impact of technology. The proof began by a mathematical clue, based on the fact that f(1/2) was in the answer. By a classical reasoning (‘ordinary means’), they knew that 1 X k¼1

1 Pð1=2; k 1Þ pffiffiffiffiffiffiffiffi pffiffiffi ðk 1Þ! 2 2pk

!

1 1 ¼ f ; 2p 2

P being the Pochammer function. The authors’ reflection went on, based on a CAS ‘clue’: It was provided by the surprising speed with which Maple was able to calculate a high-precision value of this slowly convergent infinite sum. . . Peering under the covers, we found that Maple was using the Lambert W function, which is the functional inverse of w(z) ¼ z ez.

P1 kk xk1 1 ¼ pffiffiffiffiffiffiffiffi and ¼ k¼1 22x k!ek W0 ðx=eÞ W0 ðx=eÞ 1 2 ; they had only to prove that lim pffiffiffiffiffiffiffiffi ¼ 3; e e 22x Knowing that

P1

k¼1

Pð1=2;k1Þxk1 pffiffi ðk1Þ! 2

x!1

which they did by classical means.


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For me, this example shows perfectly what new dimension the CAS use can give to experimentation. Obtaining easily a conjecture is not the most important. The real change is when CAS operation gives a clue into the mathematical properties of the sum. Maple is able to recognize the relationship between a sum and a function. Because this function is ‘built in’ the CAS, approximations of the sum will be calculated more efficiently by using this function and Maple ‘decides’ to do it rather than an ordinary approximation. Of course, it is ‘transparent’ to the user and an indifferent user of Maple would not have got the clue. The clue is based both on a mathematical observation and on some knowledge of CAS operation: an ordinary approximation would be slow and CAS involve algorithms to quicken approximations by way of built-in functions. This interwoven mathematical and machine related knowledge is what researchers in technology in Math Education call instrumentation. This example shows what instrumentation can give such a powerful clue. APPENDIX C. FEATURES OF CASYOPEÉ The environment Casyopeé for the study of real functions is still in development but a first working version has been experimented in classrooms and is ready for delivery with a complete help file (in French). Download is available at http://www.irem.univ-rennes1.fr/recherches/groupes/ groupe_aide_logiciel/casyopee/casyop.htm 1. The objects are a set of real numbers, functions of the real variable x defined on intervals determined by the real numbers, with their definition and standard forms (factored, developed. . .) and equations. They are created (‘creér menu’) and manipulated (‘calculer’ menu) by the user with the help of computer symbolic computation. The definition may include parameters (a, b, c, d). 2. The environment displays the state of the objects (left window), optionally a set of ‘symbolic values’ as a table of values of functions for real numbers (central window) and a ‘history’ of the actions of the users (creations, manipulations…) and feedback of Casyopeé, that can be annotated by the user (right window). 3. Graphs and a numeric table are also available for exploring. 4. Parameters can be instantiated and animated (‘explorer’ menu). During the animation objects and graphs are dynamically

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updated. This is help for the exploration of problems involving families of function. 5. The environment helps the student to build proofs of sign, variations and zeros of functions. Twelve different elementary steps of proof (‘justifier’ menu) are proposed. A student can be helped by ‘suggestions’. Elementary proofs can be performed on ‘conjectures’ let to be proved later. If a proof fails, the environment displays information for understanding the failure. If it succeeds, information is also given in the ‘notepad’ as help for writing out a proof. 6. The environment uses the algebraic transformational knowledge (structure of expressions, manipulation of expressions, properties of numbers and literals, equation solving. . .) of a CAS kernel (Mupad), which it is linked to.

APPENDIX D. A STUDENT WORKSHEET WITH CASYOPEÉ Recherche d’ extremum Soit une voile triangulaire ABC dont les dimensions sont AC ¼ 5 et AB ¼ a (a reél positif). On construit un encart publicitaire rectangulaire AMNP ou` les point M, N, P se trouvent respectivement sur les coˆte´s [AB], [BC] et [AC]. On pose AM ¼ x. Pour a fixe´, on cherche a` de´terminer les dimensions de l’encart AMNP ayant la plus grande aire possible ? Partie A De´montre que l’aire de AMNP est e´gale a`: 5 xða xÞ. a Partie B : exploration en utilisant Casyopeé Creé la fonction f qui a` x associe l’aire de AMNP. En « pilotant le parame`tre a » tu vas e´tudier f pour le valeurs dans la premie`re colonne du tableau. Pour chaque valeur, cherche avec la trace sur le graphe de f l’abscisse a du maximum, de´finis une fonction x > fðaÞ fðxÞ, calcule avec le logiciel sa forme de´veloppeé et sa forme factoriseé. Conclus sur son signe.


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Interpre`te le signe de f(a) – f(x). D’apre`s le tableau exprime l’abscisse du maximum en fonction de a. Valeur de a

Valeurs possibles

pour x

Abscisse a du maximum

Expression de´veloppeé de fðÞ fðxÞ

Signe de fðÞ fðxÞ sur l’intervalle d’e´tude

8 9 10 11 23/4

Partie C : Vers la de´monstration d’un re´sultat geńe´ral Utilise Casyopeé pour prouver le signe de fða=2Þ fðxÞ. Conclus sur l’abscisse du maximum et donne les dimensions du rectangle d’aire maximale. E-mail: [email protected]