learning mathematics as modelling

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N. Balacheff

CNRS

Laboratoire Leibniz

IMAG, Grenoble

 

Introduction

The main evolution of a large part of computer-science during the last decade could be well described as the development of an increasing awareness of the need of a shift of emphasis from information processing to knowledge processing.

The initial definition of information relies on a clear separation between the meaning and the form of a message. The development of computer-sciences since the beginning of the seventies demonstrates an increasing awareness of the need to take meaning into account–in particular when what is at stake is the interaction between machines and humanbeings. This evolution culminates, from the point of view of education, when it appeared clearly that the expertise of machines could not be merely transfered back to humanbeings (Clancey 1983).

But the awareness of the need to take into account relationships between meaning and form (symbols and their organisation) has not received all the attention it deserved. A dominant problématique of knowledge representation is the fidelity of the means used with respect to the knowledge represented (Wenger 1987, especially pp.84-85 & 312-314). This problématique ignores the transmutation of knowledge when passing from one media to another, from one semiotic system to another. I suggest that the issue is not that of fidelity but of being able to make explicit in which way the new representation could interfere with the intended meaning.

Knowledge cannot be merely read on the screen, it is the result of a construction in the process of the interaction with the machine. Whether this interaction addresses this aim directly or indirectly, in all cases the way the interaction is managed constraints the knowledge constructed by the learner. I will exemplify this difficulty and finally I will propose that one may have to look at the problem raised from the point of view of modelling.

the impossible fidelity

The relation between knowledge representation and the treatment of these representations is one of the basic issues of in computer-science (in the case of Algebra, for example, considering a formula as a string or a tree modifies the possible manipulation, see an example in Balacheff 1993). More recently a large number of research in computer graphics have dealt with the consequences of computational and interface constraints. The following figure is an axample of such problems, here we observe the effect of a representation of real numbers (or their approximation) and of sampling the pixels on the screen (deciding whether a pixel represents or not an element of the graph).

Beyond x=3.5, the representation of the graph of xÆsin(exp x) is "clearly false". One cannot use the same means in order to plot xÆsin x and xÆsin(exp x).

This phenomena must not be confused with the fact that properties of a given function demonstrated by a graph ploter depend on the choice of the scale on the axis (what led Hillel 1993 to coin the expression window shopping to designate the activity to be engaged by students, see also Tall 1997). What makes crucial the above example is that it is out of the control of the learner.

Reacting to what might be seen as technological limitations, one sometimes suggests other implementations so that such effects no longer exist. But such suggestions miss the fact that other implementations would give rise to other "side" effects or unintended effects. The question is not to avoid completely all these effects but to be able to say in as much detail as possible what they are or could be. The technology literature, especially related to the development of interfaces which reify abstract concepts and their direct manipulation, raises the problem of the relationship between a representation and what it represents in terms of fidelity. Indeed, some sort of fidelity is necessary. No one would expect a circle not being round. But the problem is far more complex because it is related to the complexity and the specificity of the computer as a medium.

Let us take the example of Cabri-like environments (I coined this expression to stress a distinction between such environments and environments allowing dynamic transformation of figures whithout a direct manipulation of their components). Such software allow a point, let us call it P, to be drawn on a segment without any other constraint than being one of the points of the segment (the basic relationship is here: point on an object). When one extremity of the segment is dragged on the screen, P must move too; so a decision must be taken about its behaviour. One can take this decision following what might happen with paper and pencil, choosing randomly a new point for each new position of the extremities of the segment. But in this case P will suprisingly "jump" from place to place. One might prefer P to follow a continuous trajectory like the other points. This is obtained by constraining P to always divide the segment according to the same ratio. The consequence is that, so to say, P is no longer a random point on the segment: when one extremity of the segment moves while staying on a given line, the trajectory of P is an homothetic line... Although being a perceptual invariant, this property must not be considered as a geometrical property, it is a result of an arbitrary choice in the process of computerization.

The decision taken by designers might be the object of an endless discussion, the issue here is that in any case such decisions must be taken by people. And whatever they are the essential issue, for us, in mathematics education, is to characterize their effects and possible consequences.

I have call computational transposition (Balacheff 1993) the process which leads to the specification and then the implementation of a knowledge model. Computational transposition refers to the work needed to fit the requirements of symbolic representation and of computation. Since it is not possible to find a solution which avoids a bias between representations and what they intend to represent, the way to deal with these problems could be to shift from the issue of fidelity to the issue of delineating the epistemological domain of validity of the chosen modelling or representation (Balacheff 1991, Balacheff and Sutherland 1994). In the case of geometry, this is close to the question raised by Goldenberg and Cuoco (1996) : "What is dynamic geometry?"

If students are to construct their knowledge through the interaction with such environments, then the issues I address here are crucial. The features of the software behaviour, including the unintended ones are likely to turn into specific characteristics of the meaning constructed by students.

In the first version of Geometer-Sketchpad the drawing of a segment was constituted by a set of three objects: a stick and two points. This representation is no longer used, but one may find somes ign of its existence. The next pictures shows the symetrical image of a segment through a line symetry.

 

 

 

 

 

The image of the segment has no extremities, then one may raise the question of knowing "what will be a segment for the learners in such an environment?." Let us take the case of erasing a segment in Cabri-geometry, it preserves the extremities which then have clearly not the same status as the other points of the segment, whereas erasing an extremity destroyes the "stick" but not the other extremity… Geometer Sketchpad has the same behaviour. I may continue to explore more elementary objects like points, or more complex objects like circles or conics. From an external point of view, it appears clearly that an object is characterised by both its representation as a set of pixels on the screen and by the behaviour of this set as a result of its direct manipulation.


Students constructions in these environments raise have the same characteristics. For example a right triangle will not have the same behaviour depending whether it will be constructed on its circum-circle or using a perpendicular line. The issue is then to decide whether the two different representations account for the same set of geometrical objects.

 

My claim is that the interface of these environments offers to the learner a domain of phenomenology within which they cannot simply read the mathematics, instead they have to develop a modelling relationships in which mathematics is a tool to make sense of what is experienced. To turn it in an other way, if mathematical exploration is a way to support learning as a construction of new knowings (connaissances), knowings are necessary in order to guide and organise the mathematical exploration. Modelling and learning as construction of meaning cannot be separated during the course of an interaction with a computer-based learning environment.

Computational transposition and the domain of epistemological validity are intrinsically related. An essential research issue for the coming decade is to understand the related processes, especially their intrinsic characteristics (the ones which will not be modified by technical progress), and to develop theoretical frameworks and methodologies for the identification of an epistemological domain of validity.

The source of possible misunderstandings in teaching

Can these elements of a new complexity have a serious effect on teaching? I would claim that it is so in two ways. First, these computer-based environments may favour new students’ conceptions of mathematical objects, in ways we are not prepared to face so easily. Second, they raise for the teacher new problems of diagnostic of students understandings and production due to the fact that it is necessary in order to understand software productions, to have a view of the underlying processes and knowledge structures. The following example, taken from a study of teachers activities in the context of a distance learning environment, aims at illustrating this claim.

Let us consider the following problem:

Construct a triangle ABC. Construct a point P and its symmetrical point P1 through A. Construct the symmetrical point P2 of P through B, construct the symmetrical point P3 of P through C. Move P. What can be said about the figure when P3 and P are coincident? Construct the point I, the midpoint of [PP3]. What can be said about the point I when P is moved? Explain.

From Capponi and Laborde (1995) Cabri-classe, sheet 4-10

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The figure being constructed with Cabri-geometry, the direct manipulation of the point P shows an obvious fact: the point I does not move. Since I depends directly on P and P3, two points which move when one moves P, this fact seems surprising. The challenge of the situation is to propose an explanation.

Let us examine the interaction between a teacher and a student, Fabien, about this problem (a more detailed analysis can be found in Balacheff and Soury-Lavergne 1995). Fabien, has observed the fact but he has no insight about the reason: "The point I does not move, but so what?..." Since Fabien has noticed spontaneously the parallelogram ABCI, the tutor encourages him to focus on it–the student proves that ABCI is a parallelogram. At this stage, from the point of view of geometry (and of the tutor), the reason why I stands immobile while P moves, is obvious. The tutor then provides Fabien with several hints:

" Hey! What does is mean that when one moves P, I does not move? It tells you that I is how?" [prot.113]

" you have used a lot of intermediate points but if we consider the conclusion that they don’t play any role any more, the points P, P1, P2, P3." [prot.117]

"But if it does not move when you move P. That tells you what? I, the point I, you have told me that it moved according to which points?" [prot.139]

"The others, they do not move. You see what I mean? Then how could you define the point I, finally, without using the points P, P1, P2, P3?" [prot.143.]

But Fabien still does not see the "obviousness". It is only later in the interaction that the tutor tells him the mathematical reasons of the immobility of I, provoking a genuine Ah-ha effect...

In order to explain the immobility of I, the teacher had to obtain from the student the construction of a link between a mechanical world–which is that of the interface of Cabri-geometry, and a theoretical world–which is the world of geometry. Only this link can turn the observed fact of the immobility of I into a phenomenon–i.e. the property of invariance of I. The tutor’s interventions follow a kind of maïeutic under the constraints of a didactical contract which functioned as a paradoxical injunction: the more precisely the tutor would tell to Fabien what he had to do, the more she risked provoking the disappearance of the expected learning (Brousseau 1997, p.66).

Then, sharing the screen as a common field of experimentation, the student and the teacher may share facts, but not phenomena. Simply said, it is not enough to look at the screen of the DGE to see the geometry. In a way, the teacher might have said "can’t you see it", not realizing that seeing is knowing–what sheds a different light on the issue of mathematics and visualization.

Let us listen to an other student, Sébatien [prot. 78-84]:

Sébatien: So... I have said... But is not very clear... That when, for example, we put P to the left, then P3 compensates to the right. If it goes up, then the other goes down...

Tutor: So what...

Sébatien: But I have not found a mathematical proof, hmm

Tutor: OK, so what can you say about I. That... why I is invariant? Why I does not move?

Sébatien: It is exactly what I have not found... How to prove that ... in fact P3... it negates the move of P.

This excerpt from Sébastien’s protocol gives more explicit evidence that the passage from the mechanical world to the world of geometry, even if the need for this passage has been perceived, is not obvious.

My claim is that this passage from the screen of the computer to the mathematics is a process of modelling.

The computer, as a physical machine, creates three different universes which interact heavily and which are not so easy to delineate: its internal universe (inside the box), the interface, and the external universe in which we stand. The following diagram illustrates this division. If we consider it from the point of view of the problem we study, we could describe the situation as it follows:

The invariance of I is a result of the geometrical model which has been implemented in the machine and which allows it to maintain the coherence and the constraints of a given construction while allowing direct manipulation of free objects. This invariance is "translated" dynamically for the user by the fact of the immobility of I at the interface. The meaning of this fact has to be reconstructed by the user on the basis of the knowledge he or she has, but this meaning is by no means given.

The issue of computational transposition has shown the problem of the expression of a model under computational constraints. This problem appears again at the level of users who want to express their knowledge in the context of a given software. To some extent they have to perform a local transposition of their knowledge in order to cope with the specific constraints of the system they use (i.e. software and computer).

In geometry, there is not one way to produce a geometrical construction. Let us take the case of a right triangle, its construction could implement the fact that two sides are perpendicular, or the fact that a triangle with a vertex on a circle and whose opposite side is the diameter of this circle is a right triangle. The direct manipulation of these two constructions will behave in different ways whereas the drawings will remain the same (any set of pixels obtained in one way can be obtained in an other way). This transposition of the user knowledge in the computerized media, which is a type of modelling activity, may be the source of difficulties in communication as is illustrated by the following example.

In another domain than geometry, the teaching of fractions, Ohlsson (1991) gives us a very clear example of a widely shared belief among computer-scientists or software designers. This author describes a software for the learning of fractions which provides the student with two different windows, one window displays fractions (mathematical register) and is associated to another one which displays a world of related illustrations (concrete world of partitioning strips). Then Ohlsson claims that "[this] yoking feature enabled [the student] to get mathematical feedback on her physical actions" (ibid. p.55, italics in the text), but he also reports that a student could nevertheless achieve a "wrong insight" (ibid.). So he raises the following questions (among several others): "Why did the student construct one idea rather than another? Would it have been possible to predict before the event that this exercise would predict this particular insight?" (ibid.).

These questions are crucial for the teacher’s control of learning. Ohlsson looks for an answer in the direction of learning theories, but I suggest here that one may get a better understanding of this complexity by looking at the mathematical characteristics of both the environment and the learning situation since modern learning theories emphasize, although in different maners, the centrality of the process of adaptation of the learner to his or her environment. The characteristics of this environment are then crucial in the construction of meaning. Thus to explore them from the point of view of the conceptualizations they may stimulate is as essential as exploring the possible cognitive processes of the learner. The outcome of the latter depends largely on the characteristics of the former.

Conclusion

Many questions, both fundamental and technical, are currently considered by the research community in the field of design and implementation of computer based teaching/learning environments. Some of them are now classical, such as student modelling, knowledge elicitation, automatic production of explanation, etc. Some questions seem to be missing, perhaps because of the lack of solid links between computer scientists and researchers in mathematics education in the current research scene. I would like to conclude by raising some of these questions.

The introduction of educational software, of whatever type, makes more complex the teaching/learning situation from a didactical point of view because a computer based system is first of all the materialization of a symbolic technology. This particularity plays a role in two ways:

A question I would very much like to address is of developing frameworks and tools to allow a study of computer-based learning environments beyond usual comercial comparisons. There is a need to inform teachers and math educators in an other way than simply describing the a priori specifications of the software or providing users with a user manual.

Teachers will not be fully able to insert such technologies into their daily practice, if they are not well informed on all the aspects which could determine its place and its precise role in a didactical process. I would claim that teachers must know the computer based learning environments from a didactical point of view.

A key issue concerns that of the possibility of a teacher control of the learning situation, while leaving the student enough autonomy so that a genuine learning process can develop. The richness and the complexity of the outcome of the interaction between the learner and the machine is such that the teacher will hardly be sure to be able to make sense of the student production in all cases. This issue is especially important in the case of distance tutoring because of time constraints. This calls for the development of machines able to interact with the teacher in order to facilitate her activity, in particular at the level of interpreting and debugging students’ productions.

Another key issue I would like to emphasize is the need for the development of teacher training in order to improve the teacher’s understanding of the mathematics of the computer. Computer-based learning environments raise an intrinsic difficulty compared to classical material environments due to the dynamic representation they display as well as their autonomy in performing actions. These features are likely to change the relationships between the learner and its symbolic environment, but also between the teacher and his or her working environment. I would suggest that ‘Modelling’ may be the keyword for a possible answer. If it is so, one may consider this relationship as being part of the mathematical content to be taught.

References

Balacheff N. (1991) Contribution de la didactique et de l'épistémologie aux recherches en EIAO. in Bellissant C. (ed) Actes des XIII° Journées francophones sur l'informatique (pp.9-38). Grenoble : IMAG.

Balacheff, N. (1993) La transposition informatique. Note sur un nouveau problème pour la didactique. In: Artigue, M., Gras, R., Laborde, C., Tavignot, P. (eds.) 20 ans de didactique des mathématiques en France. Grenoble: La Pensée Sauvage.

Balacheff N., Soury-Lavergne S. (1995) Analyse du rôle de l’enseignant dans une situation de préceptorat à distance: TéléCabri. In : Noirfalise R., Perrin-Glorian M.-J. (eds.) Actes de la VII° Ecole d’été de didactique des mathématiques (pp.47-56).

Balacheff, N., Sutherland, R. (1994) Epistemological domain of validity of microworlds, the case of Logo and Cabri-géomètre. In: Lewis, R., Mendelshon, P. (eds.) Lessons from learning. Proceedings of the IFIP WG3 working group. A46, 137-150. Amsterdam: North-Holland/Elsevier B.V.

Bresenham, J. E. (1988). Anomalies in incremental line rastering. In: Earnshaw, R.A. (ed.) Theoretical foundation of computer graphics and CAD. NATO ASI series F. 40, 329-358. Berlin: Springer Verlag.

Brousseau G. (1997) Theory of Didactical Situations in Mathematics. Dordrecht: Kluwer Academic Publishers.

Capponi B., Laborde C. (1995) Cabri-Classe. Argenteuil : Editions Archimède.

Clancey W.J. (1983) The epistemology of a rule based expert system : A framework for explanation. Artificial Intelligence 20, 215-251.

Cuban L. (1986) Teachers and machines. New York: Teachers College Press.

Goldenberg P. E., Al Cuoco (1996) What is dynamic geometry. In: Lehrer R., Chazan D. (eds.) Designing Learning Environments for Developing Understanding of Geometry and Space. Hillsdale, NJ: Laurence Erlbaum (to appear).

Jackiw N. (1991) The Geometer’s Sketchpad. Berkeley, CA: Key Curriculum Press.

Laborde J.M. (1985) Projet d'un Cahier Brouillon Informatique de Géométrie, rapport interne LSD (IMAG) Grenoble (repris in: D. Guin, J.-F. Nicaud and D. Py (eds.) Environnements Interactifs d’Apprentissage avec Ordinateur, Eyrolles, Paris, 1995, 41.)

Miller, J. R. (1988). The Role of Human-Computer Interaction in Intelligent Tutoring Systems. In M. C. Polson & J. J. Richardson (eds.) Foundations of Intelligent Tutoring Systems. (pp.143-189). Hillsdale, NJ: Lawrence Erlbaum Associates.

Nicaud J.-F. (1989) APLUSIX : un système expert pédagogique et un environnement d'apprentissage dans le domaine du raisonnement algébrique. Technique et Science Informatique 8 (2) 145-155.

Ohlsson S. (1991) Knowledge requirements for teaching: the case of fractions. In: Goodyeart P. (ed.) Teaching knowledge and intelligent tutoring (pp.25-59). Norwood, NJ: Ablex Publishing Corporation.

Sleeman D.H. (1982) Inferring (mal) rules from pupils' protocols. Proceedings of the European Conference on Artificial Intelligence (pp.160-164). Orsay, France.

Wenger E. (1987) Artificial Intelligence and Tutoring Systems. Los Altos, CA: Morgan Kaufmann Publishers