Towards a Problematique for Research on Mathematics TeachingAuthor(s): Nicolas BalacheffSource: Journal for Research in Mathematics Education, Vol. 21, No. 4 (Jul., 1990), pp. 258-272Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/749524Accessed: 02/10/2008 21:22

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Journal for Research in Mathematics Education 1990, Vol. 21, No. 4, 258-272

TOWARDS A PROBLEMATIQUE FOR RESEARCH ON MATHEMATICS TEACHING

NICOLAS BALACHEFF, Laboratoire IRPEACS, CNRS

This article presents the main features of the theoretical framework of French research known as recherches en didactique des mathematiques. The foundation of this approach consists

mainly of the relationships between two hypotheses and two constraints, which are presented together with some specific key words. Outlines are given of Brousseau's thdorie des situations didactiques (theory of didactical situations). An example is given that presents in some detail the rationale for the construction of a didactical situation and its analysis. This article ends with some questions addressed to research on mathematics teaching.

Kilpatrick (1981) pointed out some years ago that "one of our greatest needs in research ideas on mathematical learning and thinking is for conceptual, theory- building analyses of the assumptions we are using in our research" (p. 370). It could be added that that search for theories is not sufficient, insofar as theories are of no use if they are not related to precise problems. To say that our problem is to improve mathematics teaching or even the teaching of algebra, or that one of our problems is pupils' difficulties in thinking mathematically, is too vague. First of all, theories are tools either to solve problems or to clarify them and improve their formulation. Inversely, to solve research problems very often leads to the improve- ment of theories, or at least it puts them under question; and sometimes it leads us to consider the need for new theories. This fundamental dialectic between theories and research problems is at the core of the approach to research on mathematics teaching I would like to present here.

A prioblmatique is a set of research questions related to a specific theoretical framework. It refers to the criteria we use to assert that these research questions are to be considered and to the way we formulate them. It is not sufficient that the subject matter being studied is mathematics for one to assert that such a study is research on mathematics teaching. A problem belongs to a probldmatique of re- search on mathematics teaching if it is specifically related to the mathematical meaning of pupils' behavior in the mathematics classroom. In this article I present the main features of such a prioblmatique and an example of a research done in this framework.

The material in this article was an invited address at the research presession of the 65th annual meeting of the National Council of Teachers of Mathematics, Anaheim, CA, April 1987, written when I was a member of the Equipe de Recherche en Didactique des Math6ma- tiques et de l'Informatique from Grenoble; its content has had the benefit of discussions with many of my colleagues there. I deeply appreciate discussions with Jere Confrey and her comments on the earlier version of this article. I would also like to thank Jeremy Kilpatrick for his comments and editing remarks that helped me to carry out this final version.

259

TWO BASIC HYPOTHESES, A PROBLEM, AND TWO CONSTRAINTS

Our theoretical framework is grounded on two hypotheses: The constructivist hypothesis and the epistemological hypothesis.

The constructivist hypothesis is that pupils construct their own knowledge, their own meaning. The fact that previous knowledge is questioned, the disequilibration in the Piagetian sense, results in the construction of new knowledge as a necessary response to the pupils' environment.

The epistemological hypothesis (Vergnaud, 1982) is that problems are the source of the meaning of mathematical knowledge, but also intellectual productions turn into knowledge only if they prove to be efficient and reliable in solving problems that have been identified as being important practically (they need to be solved

frequently and thus economically) or theoretically (their solution allows a new

understanding of the related conceptual domain). These two hypotheses imply that pupils' learning depends on their recognition

and re-construction of problems as being their own. It is not sufficient that the teacher proposes a problem for this problem to become that of the pupils, because

usually the responsibility for what is true in the mathematics classroom depends on the teacher. A problem is a problem for a student only if she or he takes the

responsibility for the validity of its solution. This transfer of the responsibility for truth from teacher to pupils must occur in order to allow the construction of mean-

ing. Here is our fundamental problem: What are the conditions for the devolution

of the responsibility for truth from teacher to pupils in the mathematics classroom? If this devolution process is achieved, then we can consider that pupils' intellec-

tual activity is intrinsically justified by the problem and not by what they think is

expected by the teacher. As far as learning is a personal process, its product would be private knowledge, the pupils' conceptions. But this process conflicts with two constraints specific to the teaching process, which has to guarantee the socializa- tion of pupils' conceptions for the following reasons:

1. Mathematical knowledge is a social knowledge. Pupils should make their own the knowledge that exists outside the classroom. It has a social status in society or in smaller social groups under whose control it is used. For example, the com-

munity of mathematicians or that of engineers can be taken as a social frame of reference.

2. The mathematics class exists as a community. The teacher has to obtain a certain homogeneity in the meaning of the knowledge constructed by pupils, and she or he has to ensure its coherence. Otherwise, the functioning of the class will

hardly be possible. This constraint is quite evident if one considers the language or the means of representation specific to a given piece of mathematical knowl-

edge. Because of the constructivist hypothesis, the use of authority is not desirable. Thus the homogenization can only be the result of a negotiation or of other spe- cific social interactions such as the one Brousseau (1986a, 1986b) has described to frame his thdorie des situation didactiques.

260 Towards a Probldmatique for Research on Teaching

ELEMENTS OF BROUSSEAU'S DIDACTICAL THEORY

We consider the aim of teaching to be carrying pupils from their initial concep- tions related to a given item of mathematical knowledge to resultant conceptions through what we call a didactical process. The control and the design of this di- dactical process constitute the heart of our approach.

It follows from the two hypotheses we mentioned that the fundamental means to initiate this process are mathematical problems. Mathematical problems are fundamental insofar as they constitute means to challenge the pupils' initial con- ceptions and to initiate their evolution. Also, they are fundamental because they convey the meaning of the mathematical content to be taught mainly by making explicit the epistemological obstacles that must be overcome for the construction of that meaning.

Pupils' behaviors in the context of a classroom situation cannot be understood only through an analysis of the mathematical content involved or its related psy- chological complexity. The problems offered to pupils in a didactical situation are set in a social context dominated by both explicit and implicit rules that permit it to function but also that give meaning to pupils' behaviors. For example, consider the case of a pupil proposing a solution to a given problem without evidence of any attempt to base its solution on a proof. Before one makes any diagnosis of concep- tual understanding, cognitive level, or ability level, one has to examine whether there was any necessity for the pupil to give a proof in such a situation (Balacheff, 1982, 1988b). The rules of social interaction in the mathematics classroom include such issues as the legitimacy of the problem, its connection with the current class- room activity, and the responsibilities of both the teacher and pupils with respect to what constitutes a solution or to what is true. We call this set of rules a didacti- cal contract. A rule belongs to the set, if it plays a role in the pupils' understand- ing of the related problem and thus in the constitution of the knowledge they con- struct.

Thus, the pupils' behavior and the type of controls pupils may exert on the solu- tion they produce strongly depend on the feedback given during the situation. If there is no feedback, then the pupils' cognitive activity is different from what it could be in a situation in which the falsity of the solution could have serious con- sequences. In this last type of situation, pupils will search for a proof, the level of which could depend on both the nature of the knowledge they have available and the pressure of the situation. Perhaps they will even reconsider their own knowl- edge before producing a definitive answer.

Brousseau (1981) differentiates types of situations with respect to the kind of cognitive functioning they imply. First, there are situations imposed by the social constraints I have mentioned. Brousseau calls them situations for institutional- ization. They aim at pointing out, and giving an official status to, some piece of knowledge that has been constructed during the classroom activity. In particular they concern the knowledge, symbolic representation, and so on, to be retained for further work. A new mathematical concept has to be recognized as something to

Nicolas Balacheff 261

be kept for further activities. Otherwise pupils may soon forget it. On the other hand, although many new intellectual constructions might appear during a prob- lem-solving process, not all of them will reach the status of knowledge to be re- tained. That shows the importance of this kind of situation within which the teacher gives the status of knowledge to be retained to some new intellectual construct. Also, the new knowledge has to be proved useful: It has to function in order to es- tablish its practical interest and also to stabilize the new cognitive state of the pupil. But the processes for institutionalization and further activities (e.g., systematic problems and exercises) are not neutral, in the sense that they do not maintain the initial meaning constructed; through these activities pupils' conceptions evolve (Boschet, 1983; Robert, 1982). The control of this evolution is a didactical problem.

But such a status can be given to a piece of knowledge only if it has been con- sidered as an object explicitly recognized and not just as a tool implicitly used in problem-solving activities (Douady, 1985). Brousseau distinguishes two main types of situations that allow one to elicit the formulation of pupils' intellectual productions: situations for validation and situations for formulation.

Situations for validation require pupils to offer proofs and thus to formulate the related theories and means underlying their problem-solving processes. Situations for decision (Balacheff, 1987) are situations for validation within which there is an intrinsic need for certainty but a proof is not explicitly requested.

Situations for formulation involve the construction and the acquisition of ex- plicit models and language. Situations for communication are situations for formu- lation with explicit social dimensions. The problem of formulation is not a mere problem of encoding ready-made knowledge. In a situation that specifically re- quires a formulation (i.e., whose success depends on the quality of the formula- tion), it appears from experimental studies that the process engaged is dialectical (Laborde, 1982): The failure of a formulation chosen for the purpose of a problem- solving strategy causes a reconsideration of the underlying knowledge itself, its components, and its relationships. That is quite clear in a situation for communi- cation because of its social dimension.

Pupils cannot enter directly the situations characterized above; before the knowl- edge becomes an object of discourse, it has to exist as a tool. At that initial point, Brousseau considers another type of situation: situations for action. These situ- ations favor the development of conceptions-as models for action-necessary to initiate the teaching-learning process or the search for a solution to a given problem.

A KEY ISSUE: PUPILS' ERRORS

Pupils' errors are the most obvious indication of their difficulties with mathe- matics. The problem of the meaning of these errors is one of the key issues in the field of research on mathematics teaching.

Let us take the case of decimal numbers: To the question "Does there exist any real number between 2.746 and 2.747?" Izorche (1977) found that about 40% of

262 Towards a Probldmatique for Research on Teaching

16-year-old pupils answered that "it is not possible." This type of error has also been shown at the level of primary school and at the beginning of secondary school (Perrin-Glorian, 1986). In a task in which pupils had to order decimals, Grisvard and Leonard (1981) found that the procedures used by pupils can be described by the following rules: (a) The decimal that has a bigger number to the right of the decimal point is bigger (62% of their sample); (b) the smaller number is the one that has the longer decimal part (16% of their sample). The existence of these rules has been confirmed by other reseachers (Nesher & Peled, 1986).

The problem is not only to eliminate such errors but to identify what their origin might be. The basic hypothesis of our theory is that these errors are not mere fail- ures but symptoms of specific pupils' conceptions. In the case of decimal numbers, a hypothesis is that pupils' conceptions can be related to the errors mentioned above in the following way: (a) Decimals are integers with a decimal point that share some properties with the integers; (b) decimals are pairs of integers separated by a decimal point, a conception that can also explain errors like (2.4)2 = 4.16. If we claim that such conceptions are part of the pupils' knowledge, we have to show that they allow the pupils to solve some problems correctly.

For Conception (a), we consider problems of calculation. To succeed in learn- ing how to calculate with decimals, it is efficient to consider them as integers with a decimal point. Pupils have then only to learn how to cope with the decimal point, having added or multiplied the numbers as if they were mere integers. At a deeper level we note that decimals are often introduced to pupils in a context of measure- ment, in which they appear to be integers with the decimal point as information about a chosen unit.

For Conception (b), we consider one of the algorithms at hand for comparing decimals: First you compare the integers written on the left of the decimal point, and if they are equal you then compare the integers written on the right, provided that they have the same number of digits. But some pupils forget this constraint when they compare two decimals. It could be argued that in this case pupils will not succeed in performing comparison tasks, so their errors will be apparent to them. But more often than not the pupils do not really need to pay attention to the constraint, because the exercises that are offered to them frequently have the same number of digits to the right of the decimal. Their conception is reinforced by the fact that in everyday life decimals used to code a price are in fact understood as being a pair of integers: francs and centimes in France, dollars and cents in the U.S.

These descriptions of pupils' conceptions of decimals are hypothetical descrip- tions proposed by researchers. They are validated by experimental means and by the fact that they allow us to foresee what the pupils' productions will be for a given task. It is not possible to make a direct observation of pupils' conceptions related to a given mathematical concept; one can only infer them from the obser- vation of pupils' behaviors in specific tasks, which is one of the more difficult methodological problems we have to face.

So if pupils' conceptions have all the properties of an item of knowledge, we have to recognize that it might be because they have a domain of validity. These

Nicolas Balacheff 263

conceptions have not been taught as such, but it appears that what has been taught opens the possibility for their existence. Thus, the question is to know whether it is possible to avoid a priori any possibility of pupils constructing unintended conceptions.

THE DIDACTICAL TRANSPOSITION

I have suggested that pupils' unintended conceptions can be understood as prop- erties of the content to be taught or of the way it is taught. To overcome that diffi- culty, a first idea could be to search for a new definition of that content within the framework of mathematics as a science. In such an approach the meaning of mathematics is likely to be reduced to the text of its presentation. In that context, the content to be taught might appear essentially as being more elementary than its scientific reference. Such an approach does not take into account that mathe- matics is first of all a tool to solve problems or that problems for which mathemati- cal concepts have been forged are part of their meaning, just as part of their mean- ing resides in the context of their discovery.

Mathematical concepts cannot be fully understood if we do not know the type of problem they allow one to solve. Also, we have to know that their construction is not only a deductive process but also the result of dialectical confrontations of different points of view, together with different metaphysical conceptions. Finally, we should know that this process of construction is still not finished.

More often than not, this historical context of discovery cannot be carried into the mathematics classroom. Because it is not by means of the same mathematical activity or within the same epistemological context, the meaning constructed by pupils may be qualitatively different from that of mathematics as a scholarly topic. Consequently, we can no longer consider the relationship between mathematics as a science and mathematics as a content to be taught as being the result of a mere process of elementarization.

On the other hand, mathematicians are not solely responsible for deciding what is to be taught and how. At least in France, that is the result of a social interaction within which teachers and mathematicians as well as parents, politicians, and industrialists are involved. All these interactions contribute to constituting the specific epistemology of the content to be taught. The concept of didactical transposition coined by Chevallard (1985) aims at giving a theoretical framework to the study of this process, a process by which some mathematical knowledge is transformed to become teachable. The most general constraints that call for the didactical transposition process and at the same time assign its form are as follows: compatibility constraints between the didactical system and society, ideological constraints pertaining to the different sociological groups involved, and finally, constraints specific to didactical functioning in the strict sense (Chevallard, 1982).

It should be emphasized that the didactical transposition is unavoidable, because of constraints specific to mathematics teaching. I would like to mention two of them:

1. Any content has to be embedded in a context in order to be teachable. But later

264 Towards a Problematique for Research on Teaching

it has to be taken out of that context to exist as genuine mathematical knowledge. This context, together with the way it is moved aside, becomes part of the mean- ing for the learner of the mathematical content taught.

2. Any content has to be supported by the pupils' previous knowledge. But this old knowledge can turn into an obstacle to the constitution of new conceptions, even though it is a necessary foundation. But more often than not, to overcome this obstacle is part of the construction of the meaning of the new piece of knowledge. For this reason, following Bachelard (1938), we call it an epistemological obstacle.

Before searching for hypothetical good didactical transposition-even if we suppose that an optimal one exists-we must describe its function, answering the question: How can the didactical transposition be characterized so that we can predict which meanings it might allow learners to construct?

AN EXAMPLE OF THE DESIGN OF A DIDACTICAL SEQUENCE: THE SUM OF THE ANGLES OF A TRIANGLE

I illustrate this approach to research on mathematics education with an example taken from my own field of research, which is that of problems related to the learn- ing and teaching of mathematical proof (Balacheff, 1988a, 1988b, 1988c). I pres- ent in detail the construction of a didactical process designed to allow pupils to formulate a conjecture and then to prove it; I then present one of the results obtained.

It is well known that pupils have great difficulties in learning what a mathemati- cal proof is. Very often, teachers and researchers mention the insufficient logical maturity of pupils together with their lack of awareness of the necessity for proofs. To some extent I agree with these statements, but our probldmatique leads me to go a bit beyond these remarks to address the following questions:

1. What is a mathematical proof for mathematicians as professionals, and what is it as a content to be taught? What is a mathematical proof as part of the mathe- matical activity within the classroom?

2. On what basis can pupils construct a meaning for the notion of mathematical proof?

3. What are the contexts in which mathematical proof can appear as an efficient or relevant tool for solving problems pupils have recognized as such?

I will here concentrate on the last question. Since it is usually forgotten that as children, pupils are logical enough to cope with most of the problems they encoun- ter in everyday life, this problem is often discussed as a linguistic/formal gap be- tween the logic of common sense and mathematical logic. But this probl"matique misses a key point: Mathematics, unlike everyday life, is concerned with theory. The key word in mathematics is rigor; in everyday life it is efficiency. That means that the teaching process should allow for this shift in pupils' interest from being practitioners to becoming theoreticians (Balacheff, 1987).

Thus, to raise the problem of proof in the mathematics classroom, we need to

Nicolas Balacheff 265

shift to pupils the responsibility for the truth of some mathematical statement. Such a statement has first to be recognized as a conjecture; that means that it is not mere

speculation but that pupils consider it plausible and also share a sufficient interest in knowing whether it is true or not. The formulation of the proof should be justi- fied not by an injunction of the teacher but by an intrinsic need, which could stem from a debate among pupils about the validity of the conjecture.

Our study of the characteristics of such a situation for validation was based on the construction and analysis of a didactical process in which pupils about 12 years old discover, formulate as a conjecture, and then try to prove that the sum of the measures of the three angles of a triangle is 180'.

Outlines of the Situation

Pupils' conceptions of the notion of angle are likely to lead them to assert that the larger the triangle, the larger the sum of its angles (Close, 1982). Because of this conception, the value of a proof proposed by the teacher, even after some

manipulations, are doubtful, because (a) the assertion itself might appear arbitrary insofar as results like 182' or 178' are pragmatically as good candidates as 180', and (b) the pupils will be left with an open conflict between their intuition (Fisch- bein, 1982) and the authority of the proposed proof.

Let us try to solve this didactical problem, using these initial, wrong conceptions of the pupils in order to lead them to the intended conjecture-and thus to a new

conception-and then to cope with the problem of its truth. For that purpose we

identify four main constraints:

1. It is not possible to tell the pupils beforehand that the purpose of the sequence will be to establish that the sum of the angles of a triangle is 180'. That would

destroy the problem, because the assertion would no longer be considered as a

conjecture; the student knows the teacher always tells the truth. This is a classic

example of one of the basic beliefs held in the didactical contract.

2. The validity of the measurement of a particular set of triangles as a means to establish the conjecture should be dismissed. But this decision should be taken by the pupils on their own and not imposed by the teacher; otherwise they will seek a

proof that is acceptable to the teacher. 3. The situation we design should elicit the pupils' conceptions about the rela-

tions between the size of a triangle and the value of the sum of its angles, because it is from the contradiction between this conception and the fact that the sums are around 180' that the conjecture could stem. This requires a situation for action.

4. We should provide the classroom with a situation for validation oriented toward the construction of a proof of the conjecture. That supposes a didactical contract in which the pupils have the responsibility for the truth of the conjecture. This is possible only if they have had the responsibility for forming the statement of the conjecture itself.

It is under these constraints that a sequence of didactical situations has been designed. Note that the following teaching setting is not the solution to the prob-

266 Towards a Problematique for Research on Teaching

lem I have formulated. What is important is the relationships between the theoreti- cal analysis and the construction of this didactical process.

Conditions for the Genesis of a Conjecture

It is possible to ask the pupils to measure the three angles of a triangle without giving specific reasons: The teacher asks each pupil to carry out the task and then to propose his or her result. The possible variety of the results has no specific meaning for the pupils with respect to their conceptions because almost all tri- angles are different. I do not consider this a situation for action because it does not provide the mobilization of the conceptions specifically related to the intended conjecture.

The following activity allows the pupils to discriminate, from the variation in the obtained results, between what is due to the measurement and what is explained by their conceptions. We confront the class with the computation of the sum of the angles of a unique triangle. Each pupil gets a copy of the same triangle, and we ask them to predict the sum of its angles. The predictions are recorded by the teacher before the pupils start measuring and computing. We have proposed a tri- angle large enough to activate the expected conceptions.

After this task has been completed two things are done:

1. Each pupil is confronted with his or her prediction and asked for a comment about a discrepancy between the prediction and the result obtained. This request should elicit a formulation of the possible conceptions underlying the prediction. There is not necessarily a cognitive conflict, for as far as the pupil is concerned this

discrepancy can be regarded as unique for the chosen triangle. This situation for action prepares for the coming of the conjecture.

2. The teacher represents the collected results on the chalkboard by means of a histogram and then asks for comments. That leads to the problem of the determi- nation of the exact value of the sum of the angles of a given triangle; it will appear that measurement is not a reliable means to an answer.

Towards the Birth of a Conjecture

To raise the question of the invariance of the sum of the angles, we need to have pupils measure the angles and compute the sum in more than one triangle. Because the number of triangles manipulated will not be very large, the set chosen is very important. Taking into account the pupils' conceptions, we use the shape of the

triangles as a didactical variable: Pupils are likely to focus on the size of the tri-

angle and the type of angle within the triangle. Thus we choose three triangles with

shapes, and contrast between these shapes, sufficiently unusual to challenge pu- pils when they are asked to predict the sum of the angles (Figure 1).

The pupils work in teams of three or four, each team being asked to make one

prediction for each triangle before any measurement and computation. The debate necessary to make a decision elicits the underlying conceptions and initiates the construction of arguments for or against the assertion that the sum of the angles of

Nicolas Balacheff 267

C

B

Figure 1. Triangles with contrasting shapes.

a triangle depends on its shape or on its size. Because of the social interaction, this situation has the characteristics of a situation for decision.

After the task has been completed, each team is confronted with its prediction. The teacher asks for a comment about a possible gap between the prediction and the result obtained for each triangle. The teacher represents the set of results on the chalkboard by means of a histogram and asks for comments. Issues concerning the value of the sum of the angles for each triangle are discussed.

Actually, all that activity is not sufficient to ensure that the conjecture will be formulated and recognized collectively by the class. Two possible cases should be considered:

1. The sum of 1800 seems to be evident from a comparison between the predic- tions and the results of the measurements. But some pupils may still assert that it is because of the particular choice of triangles. In that event, the teacher challenges the class to find a triangle in which the sum of the angles is quite different from 1800. The confrontation between the robustness of pupils' conceptions and the difficulty in finding a triangle in which the sum of the angles is different from 180' leads to a formulation of the conjecture together with the problem of its proof.

2. The class supports the statement "The sum of the angles of a triangle is 180'"9 as a conjecture. But because an appeal to measurement has been dismissed, the problem of constructing a proof on the ground of rational arguments can be stated.

Whatever the case, the situation now has the characteristics of a situation for validation, because the class has the responsibility to produce a proof of the con- jecture. The teacher stays aside; she or he has managed the situation but has never offered any opinion about the validity of the results produced or of the conjecture.

268 Towards a Problematique for Research on Teaching

Closure

To show that the sum of the angles of a triangle is 180', or to refute it, is now a problem for the class. It is an open-ended problem for which there is no evidence that pupils will find any solution within the time constraints of the traditional school context.

We have then to consider possible scenarios for a conclusion:

1. The pupils agree on a proof of the conjecture. Then the teacher just has to ratify it, provided that it is acceptable. If it is not acceptable, then there is a nego- tiation to either reject it, suggest a modification of it, or even begin to develop another proof.

2. The pupils do not agree on a single proof of the conjecture. Then the teacher should manage the negotiation in order to accept some proofs and reject others.

3. The pupils do not find any solution. Then the teacher has the following alter- natives: (a) to propose a solution that is consistent with the pupils' conceptions, the

strategies they unsuccessfully initiated, and the level of proof they have revealed (Balacheff, 1988b); or (b) to propose that they admit the truth of the conjecture and

delay the production of a proof.

Even if the conjecture has not been proved by the pupils themselves, the knowl-

edge constructed throughout this sequence should be quite different from what

they might have constructed after merely observing some triangles and having a

proof presented to them. Here the proposition has been developed as a conjecture by pupils on their own. It has been discussed and settled as a genuine problem. Even if the production of a proof is now delayed, a real attempt has been made to solve the problem. The proposal of the teacher has practical reasons but does not

rely on a priori principles pupils do not know. This situation for institutionaliza- tion guarantees that what has been produced during the sequence is valid and is

genuinely considered as knowledge. It implies that pupils and teacher recover their own place and responsibility within the teaching situation.

A Few Words About the Results Obtained

This didactical process has been developed in seven 7th-grade mathematics classrooms in France, two of which have been videotaped (Balacheff, 1988a). The main result I would like to present is the one that is specifically related to our theo- retical framework: the robustness of pupils' conceptions.

In all the classrooms observed, 180' appeared to be dominant right from the first

activity, but the pupils' measurements ranged from 160' to 260'. The pupils' pre- dictions on the second activity confirmed the dominance of 180', but the range was

quite large: from 160' to 770'. For the measurement of the common triangle al- most all pupils found 180'. That is possible only if the result of the measurement has been corrected towards 180'. It might be proposed that it is possible to end the didactical process at this point. The pupils seemed, from their behavior, ready to

accept as true that "the sum of the angles of a triangle is 180'." But in drawing such a conclusion one would mistake conformist behavior for genuine knowledge. The

Nicolas Balacheff 269

third activity evidenced the robustness of the pupils' initial conceptions: Despite the fact that the pupils were working collectively, half of them predicted a value rather different from 180 for at least one of the given triangles.

The debate among the pupils, first within each team and then within the class, initiated the construction of the conjecture as such, insofar as it was challenged by the claim that a triangle could be sharpened enough to have the sum of its angles very small. Only at this stage of the didactical process were the pupils' conceptions called into question; the evolution toward a correct conception could start, having been activated by the debate about the validity of the conjecture.

The constructivist hypothesis is clearly supported by this experiment, which also brings to light the implication of social interactions in the learning process. The existence of a conformist cognitive behavior is probably one source of the diffi- culties that characterize effective teaching, for this phenomenon might allow teaching to progress despite the absence of real learning.

CONCLUSION

What I have presented gives an idea of the probldmatique and its related theo- retical framework, on which are based what we in France call the recherches en didactique des mathe'matiques. As I have tried to show, the key word of this

probldmatique is meaning. Some basic questions, which have not been considered in this article, are as follows:

*What mathematical meaning of pupils' conceptions can we infer from an obser- vation of their behavior?

*What kinds of meanings can pupils construct in the context of mathematics teaching?

*What is the relation between the meaning of the content to be taught and that of the mathematical knowledge chosen as a reference?

*What determines the transformation of mathematics to constitute it as a content to be taught? As a content taught?

*Beyond definitions, how can one characterize the meaning of mathematical concepts?

This research is essentially experimental, which means that it relies on the ob- servation of experimental settings specifically designed to answer precise ques- tions. Our aim is to construct a fundamental body of knowledge about phenomena and processes related to mathematics teaching and learning. The social purpose of such an enterprise is to enable teachers themselves to design and to control the teaching-learning situation, not to reproduce ready-made processes. This knowl- edge should allow teachers to solve the practical problems they meet, to adapt their practice to their actual classroom.

But for practical reasons this experimental approach is very difficult. Because of time constraints, the observation of a sequence like the one about the sum of the angles of a triangle can be done only one to three times a year. Given what teach-

270 Towards a Problematique for Research on Teaching

ers are planning to do in their classroom, the period during which the experiment can be conducted is quite short. Furthermore, to delegate others to do the observa- tion is very difficult, since at present it is not well known what must be said in order to allow other researchers to repeat an experiment. Let me emphasize that one of the main obstacles we meet is the communication within our research community. That is strongly related to two essential open questions that concern research on mathematics teaching as a scientific domain:

1. What does a research result consist of? When we design a teaching experiment with respect to some mathematical content, the result is not the teaching setting itself but the answer to the initial research question or a new formulation of it, or the evidence of intrinsic links between pupils' behavior and some set of variables whose control conditions the teaching process, or even the principles of the teach- ing design.

2. What is a proof in our field of research?

Other types of research exist, for example, the observation of real teaching situ- ations. This research is not as well developed in France as it is in other countries. Such research must be developed because it will be of crucial importance in mak- ing an effective relationship between research and practice. The confrontation and the discussion of both types of research projects could be organized around a meta-

probldmatique about which I would like to add a few words as a conclusion. During the observation phase of an experiment, facts and events are recorded

and then reported with an accurate description. But two major questions occur with respect to observation:

1. Not all the facts are relevant to research in the didactics of mathematics. But which ones are to be retained? On the basis of which criteria? Indeed the way rele- vant facts are recognized is strongly related to the theoretical background of the research. The discussion on this point could be organized around the concept of didactical fact: Within a teaching process what facts are relevant for the purpose of a didactical analysis? From what theoretical basis can the criteria for recogniz- ing didactical facts be derived?

2. When a fact occurs at a given moment within the didactical process, it implies that others have not occurred at that moment. That seems quite clear. But it raises an important question for our research. Can we guess the set of possible didactical facts to appear under certain conditions? This a priori analysis should be a meth- odological principle for research based on observation. It leads us to discuss the necessity of the occurrence of an event. For such an analysis we need a theoretical background-some model to predict as precisely as possible, in a given situation, what will be the pupil's behavior, the teacher's behavior, the interaction between them, and so on. The meaning of an observed fact stems from both its occurrence and the nonoccurrence of other possible facts.

Finally, research has not been completed, whatever it is, since we have not ex- amined the problem of the conditions for its reproducibility. What kind of infor-

Nicolas Balacheff 271

mation do we have to communicate to enable other researchers to repeat an experi- ment, to observe the same facts? To give an accurate description of the experimen- tal setting and of the facts observed is not sufficient. We need also all the informa- tion about the theory and the related piroblmatique that has led to this hypothesis, particularly in the case of further falsifications.

REFERENCES

Bachelard, G. (1938). La formation de l'esprit scientifique. Paris: Vrin.

Balacheff, N. (1982). Preuves et demonstrations au collbge [Proofs and demonstrations in high school]. Recherches en Didactique des Math matiques, 3, 261-304.

Balacheff, N. (1987). Processus de preuve et situations de validation [Proof processes and situations for validation]. Educational Studies in Mathematics, 18, 147-176.

Balacheff, N. (1988a). Une dtude des processus de preuve en mathematiques chez des leves de college [A study of proof processes in mathematics of high school students]. Th6se d'etat, Universit6 Jo- seph Fourier, Grenoble.

Balacheff, N. (1988b). Aspects of proof in pupils' practice of school mathematics. In D. Pimm (Ed.), Mathematics, Teachers and Children (pp. 216-135) Hodder & Stoughton.

Balacheff, N. (1988c). Treatment of refutations.: Aspects of the conmplexity of a constructivist approach of mathematics learning. Unpublished manuscript.

Boschet, F. (1983). Les suites num6riques comme objet d'enseignement [Numerical sequences as a content to be taught]. Recherches en Didactique des Mathdmatiques, 4, 141--164.

Brousseau, G. (1981). Problemes de didactique des d6cimaux [Problems in teaching decimals]. Re- cherches en Didactique des Math matiques, 2, 37-128.

Brousseau, G. (1986a). Theorisation des phInomnnes d'enseignement des mathematiques [Theoriza- tion of the phenomena of mathematics teaching]. These d'6tat, Universit6 de Bordeaux.

Brousseau, G. (1986b). Basic theory and methods in the didactics of mathematics. In P. F. L. Verstap- pen (Ed.), Proceedings of the Second Conference on Systematic Co-operation Bemteen Theory and Practice in Mathematics Education. (pp. 109-161). Enschede, The Netherlands: NICD, 1988.

Chevallard, Y. (1982). Un exemple d'analyse de la transposition didactique [An example of the analy- sis of the didactical transposition]. Recherches en Didactique des Mathematiques, 3, 157-239.

Chevallard, Y. (1985). La transposition didactique [The didactical transposition]. Grenoble: La Pens6e Sauvage.

Close, G. S. (1982). Children's understanding of angle at the primarylsecondary transfer stage. Un- published master's of science thesis: Polytechnic of the South Bank, London.

Douady, R. (1985). The interplay between different settings, tool-object dialectic in the extension of mathematical ability. In L. Streefland (Ed.), Proceedings of the Ninth International Conference for the Psychology of Mathematics Education. (Vol. II, pp. 33-52) Utrecht, The Netherlands: State University of Utrecht.

Fischbein, E. (1982). Intuition and proof. For the Learning of Mathematics, 3 (2), pp. 9-18, 24.

Galbraith, P. L. (1979). Pupils Proving. Nottingham: University of Nottingham, Shell Centre for Mathematics Education.

Grisvard, C., & L6onard, F. (1981). Sur deux r6gles impicites utilis6es dans la comparaison des nombres d6cimaux [On two implicit rules used in comparing decimal numbers]. Bulletin APMEP, 340,450-460.

Izorche, M. L. (1977). Les rnels en classe de seconde [The real numbers at the tenth grade] (M6moire de DEA) Bordeaux: IREM et Universit6 de Bordeaux.

Kilpatrick, J. (1981). Research on mathematical learning and thinking in the United States. Recherches en Didactique des Mathdmatiques, 2, 363-379.

Laborde, C. (1982). Langue naturelle et ccriture symbolique [Natural language and symbolic writing]. These d'6tat, Universit6 Joseph Fourier, Grenoble.

272 Towards a Probldmatique for Research on Teaching

Nesher, P., & Peled, I. (1986). Shifts in reasoning. Educational Studies in Mathematics, 17, 67-80.

Perrin-Glorian, M. J. (1986). Repr6sentation des fractions et des nombres d6cimaux chez des dl6ves de CM2 et du college [Representation of fractions and decimal numbers by pupils in CM, and high school]. Petit X, 10, 5-29.

Robert, A. (1982). Acquisition de la notion de convergence des suites numeriques dans l'enseignement supirieur [Acquisition of the notion of the convergence of numerical sequences in post secondary school]. These d'dtat, Universit6 de Paris VII, Paris.

Vergnaud, G. (1982) Cognitive and developmental psychology and research in mathematics education: Some theoretical and methodological issues. For the Learning of Mathematics, 3 (2).

AUTHOR NICOLAS BALACHEFF, Directeur de recherche CNRS, Laboratoire IRPEACS,CNRS, BP 167

69131 Ecully Cedex, France