The notion of obstacle 1. Epistemological obstacles 1. Manifestation of obstacles in didactique of mathematics 1. Origin of various didactical obstacles 1. Consequences for the organization of problem-situations 1. Problems in the construction of the concept of decimals 1. History of decimals 1. History of the teaching of decimals 1. Obstacles to didactique of a construction of decimals 1.

Epistemological obstacles—didactical plan 1. Comments after a debate 79 79 79 81 2. Epistemological obstacles and didactique of mathematics 2. Why is didactique of mathematics interested in epistemological obstacles? Do epistemological obstacles exist in mathematics? Search for an epistemological obstacle: historical approach 2. The case of numbers 2. Methods and questions 2. Fractions in ancient Egypt 2. Identification of pieces of knowledge 2. What are the advantages of using unit fractions? Does the system of unit fractions constitute an obstacle? Search for an obstacle from school situations: A current unexpected obstacle, the natural numbers.

Obstacles and didactical engineering 2. Local problems: lessons. How can an identified obstacle be dealt with? Which obstacles can be avoided and which accepted? Didactical handling of obstacles 2. Obstacles and fundamental didactics 2. Problems internal to the class 2. Problems external to the class Chapter 3 prelude Chapter 3. Problems with teaching decimal numbers 1. Introduction 2.

The teaching of decimals in the s in France 2. Description of a curriculum 2. Introductory lesson 2. Metric system. Problems 2. Operations with decimal numbers 2. Decimal fractions 2. Justifications and proofs 2. Analysis of characteristic choices of this curriculum and of their consequences 2.

Dominant conception of the school decimal in 2. Consequences for the multiplication of decimals 2. The two representations of decimals 2. The order of decimal numbers 2. Approximation 2. Influence of pedagogical ideas on this conception 2. Evaluation of the results 2. Classical methods 2. Optimization 2. Other methods 2. Separation of this learning and what causes it 2.

Algorithms 3. The teaching of decimals in the s 3. Description of a curriculum 3. Introductory lesson 3. Other bases. Decomposition 3. Operations 3. Order 3. Problems 3. Approximation 3. Analysis of this curriculum 3. The decimal point 3. Identification and evaporation 3.

Product 3. Conclusion 3. The choices 3. Properties of the operations 3. Operators 3. Fractions 3. Pedagogical ideas of the reform 3. The reform targets content 3. Teaching structures 3.

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## Theory of Didactical Situations in Mathematics: Didactique des Mathématiques, 1970-1990 / Edition 1

The psychodynamic process and educational practice 3. Influence of the psychodynamic process on the teaching of decimals, critiques and comments 3. Conceptions and situations ix Chapters 3 and 4 interlude Chapter 4. Didactical problems with decimals 1. General design of a process for teaching decimals 1. Conclusions from the mathematical study 1. Axioms and implicit didactical choices 1.

Transformations of mathematical discourse 1. Metamathematics and heuristics 1. Extensions and restrictions 1. Mathematical motivations 1. Conclusion of the epistemological study 1. Different conceptions of decimals 1. Dialectical relationships between D and Q 1. Types of realized objects 1. Different meanings of the product of two rationals 1. Need for the experimental epistemological study 1. Cultural obstacles 1. Conclusions of the didactical study 1. Principles 1. The objectives of teaching decimals 1.

Consequences: types of situations 1. New objectives 1. Outline of the process 1. Notice to the reader 1. Phase I: From rational measures to decimal measures 2. Analysis of the process and its implementation 2. The pantograph 2. Introduction to pantographs: the realization of Phase 2. Examples of different didactical situations based on this schema of a situation 2.

Place of this situation in the process 2. Composition of mappings two sessions 2. About research on didactique 2. Summary of the remainder of the process 2 sessions 2. Limits of the process of reprise 2. The puzzle 2. The problem -situation 2. Summary of the rest of the process 2.

Affective and social foundations of mathematical proof 2. Decimal approach to rational numbers five sessions 2. Location of a rational number within a natural -number interval 2. Rational -number intervals 2. Remainder of the process 2. Experimentation with the process 2. Methodological observations 2. The experimental situation 2. School results 2. Reproducibility—obsolescence 2.

Brief commentary 3. Analysis of a situation: The thickness of a sheet of a paper 3. Description of the didactical situation Session 1, Phase 1. Preparation of the materials and the setting 3. First phase: search for a code about minutes 3. Second phase: communication game 10 to 15 minutes 3. Third phase: result of the games and the codes 20 to 25 minutes [confrontation] 3. Results 3. Comparison of thicknesses and equivalent pairs Activity 1, Session 2 3.

Preparation of materials and scene 3. Second phase: Completion of table; search for missing values 20—25 minutes 3. Third phase: Communication game 15 minutes 3. Summary of the rest of the sequence Session 3 3. Analysis of the situation—the game 3. The problem-situation 3. The didactical situation 3. The maintenance of conditions of opening and their relationship with the meaning of the knowledge 3. Analysis of didactical variables. Choice of game 3. The type of situation 3. The choice of thicknesses: implicit model 3.

From implicit model to explanation 4. Questions about didactique of decimals 4. The objects of didactical discourse 4. Some didactical concepts 4. The components of meaning 4. The didactical properties of a problem-situation 4. Situations, knowledge, behaviour 4. Return to certain characteristics of the process 4. Inadequacies of the process 4. Return to decimal-measurement 4. Remarks about the number of elements that allow the generation of a set 4.

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Partitioning and proportioning 4. Questions about methodology of research on didactique on decimals 4. Models of errors 4. Levels of complexity 4. Dependencies and implications xi 22 1 Chapters 3 and 4 postlude—Didactique and teaching problems Chapter 5 prelude Chapter 5. The didactical contract: the teacher, the student and the milieu 1. Contextualization and decontextualization of knowledge 2. The problem of meaning of intentional knowledge 2.

Teaching and learning 2. Engineering devolution: subtraction 3. The search for the unknown term of a sum 3. First stage: devolution of the riddle 3. Second stage: anticipation of the solution 3. Third stage: the statement and the proof 3. Fourth stage: devolution and institutionalization of an adidactical learning situation 3. Fifth stage: anticipation of the proof 4. Institutionalization 4. Knowing 4.

## Theory of didactical situations in mathematics : didactique des mathématiques (1970-1990)

Meaning 4. Epistemology 4. Memory, time 5. Conclusions Chapter 6 prelude Chapter 6. Didactique: What use is it to a teacher? Objects of didactique 2. Usefulness of didactique 2. Techniques for the teacher 2.

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Knowledge about teaching 2. Conclusions 3. Difficulties with disseminating didactique 3. How one research finding reached the teaching profession 3. What lesson can we draw from this adventure? Didactique and innovation Appendix. With very few exceptions, what has been available until now have been interpretations of the works of Brousseau rather than the works themselves. It was in response to this need that two of us, in the euphoria of an unforgettable Mexican evening at the time of the PME conference, decided to undertake the task of translating into English most of the works of Guy Brousseau.

The ceuvre is immense, and once past the initial moments of enthusiasm, with the accompanying ambition to produce the entire of it, we recognized the need to choose both the texts and a method of proceeding. As far as the texts go, we chose to take the period from to , in the course of which it seemed to us that Brousseau had forged the essentials of the Theory of Didactical Situations. But even there the collection is huge. So, after an initial translation of most of the publications of the period, we carved out a selection, retaining the texts which gave the best presentation of the principles and key concepts of the Theory.

At the heart of the book we put two works which demonstrate in detail the articulation between theoretical work and experimental research which is the source of the richness of the Theory of Didactical Situations. The texts we chose, which came from a variety of sources, occasionally overlapped. In the interests of creating a book rather than a collection of papers, we have permitted ourselves to recompose some of them to avoid redundancies and fuse some previously distinct texts. It was, however, out of the question to rewrite absolutely everything, and the attentive reader will observe that a few redundencies remain.

Perhaps, though, they will prove to be an aid to comprehension. We have composed preludes and interludes to situate the chosen texts and clarify the construction of the book. And finally, footnotes here and there fill out references from the original texts or elucidate for the reader certain points which seem to us particularly specific to the French educational context of the research presented.

The most important of these choices are pointed out in the course of the text. But the English language itself has variations depending on the country in which it is spoken and the culture in which it has developed. Finally, this work could not have been accomplished without the collaboration of Nadine Brousseau, who accompanied us in our research and found the answers to many questions, and Guy Brousseau, who produced for this translation a number of amendments which a connoisseur may spot.

As the son of a soldier, he had an early education marked by frequent changes from one school to another. He was fascinated by both mathematics and physics. He knew already what interested him and what he wanted to study: the way in which children learn mathematics. He explained this to his mathematics professor, Mr. He reports with relish that he may have been only a fifteen-year-old in short pants, but he had great powers of discernment! It was during this period that his reflections on the acquisition of mathematics and the teaching of mathematics really started.

During the next two years from to , he taught, observed children, prepared sheets of lessons, analyses and problems and continued to learn mathematics. This activity was interrupted in October , when he was called up for military service. He was stationed briefly in Paris, where he was able to take some courses at the Sorbonne from Mr. In Algeria, in , he created worksheets which he was anxious to try out in his class on his return.

From January, to October, , he experimented and created new texts, all the while carrying out the tasks of all sorts which were expected of teachers at that period. He wrote in a series of grey notebooks his cahiers gris , sometimes all through the night: lesson plans, problems, reflections. The grey notebooks became a kind of talisman for him. In October, , he was encouraged by an article in Sciences et Avenir, in which he learned that in Belgium G.

She replied, asking him to send her copies of his texts to look at. He had, at that time, about three hundred lesson-sheets. There he had minor disagreements with G. Choquet, who was leaving the presidency, but met G. Papy and W. He continued his writing, producing a textbook for fourth and fifth grades. The primary-school inspector whom he consulted before conducting a larger experiment in the class had himself to obtain authority from the General Inspector, Mr. Degeorge, who was very reluctant and replied that this teacher G. Brousseau ought to start by studying mathematics which he had, in fact, started and which he continued to do.

During this year, Brousseau also became interested in the mathematical problems posed by the management of agricultural business, which he worked on with the farmers in the village. At issue were problems of optimisation applications of operational calculation or of representations and treatment of numerous variables which must be considered in order to modify the practices of polyculture. He became aware at that point of the importance and the difficulty of the diffusion of mathematical knowledge in the population at large. Brousseau was accepted and granted leave for the following school year —63 to attend this course in Bordeaux.

At the same time, he applied to the university to resume officially his mathematical studies interrupted in The director, Mr. Research on these — technical conditions theories, methodology, fields of experience , — sociological conditions administrative organisation, contacts among researchers, subjects, domains of reference, fundamental concepts , and — pedagogical conditions acceptable forms of teaching, ethical aspects led him to deepen his knowledge in various domains, and to seek the guidance of specialists whom he encountered at the University of Bordeaux or thanks to the activities of the CRDP: cinematic linguistics and semiology with Christian Metz, later the Science of Education with J.

Wittwer, etc. In February, , at a colloquium at Amiens, Brousseau presented, with J. Becker and J. The project was studied in a systemic perspective. In order to permit it to function, its relations with all the organisations concerned were examined. It was necessary to propose one or more initial theoretical models to be demolished, but consistent and large enough , research methods, an initial group of researchers The most delicate point was imagining a relationship between the researchers and their object of study—teaching—which would neither compromise the academic research nor be detrimental to the teaching.

The COREM would provide the milieu in contact with which a composite team would be able to carry out the first whorls of the spiraling development of this academic research. Guy Brousseau, a licensed teacher of mathematics, was recruited by the University as an Assistant in Mathematics to participate in the realisation of the announced project. The first article on methodology on quasi-implications was published in the bulletin of psychology of the University of Paris in The first example of a predictive mathematical model relative to a modification of teaching, experimentally verified the teaching of the calculation of multiplication and division , was communicated in at the Sixth International Congress on the Science of Education.

Research was carried out there —74 on the teaching of the natural numbers, the operations on the natural numbers and fundamental structures. Other topics were the teaching of probability and statistics, in collaboration with P. Hennequin, statistical studies under the direction of H. Rouanet —74 , the teaching of rationals and decimals —80 , In a doctoral program in didactique was created at Paris, Strasbourg and Bordeaux.

Brousseau was in charge of theses and several principal courses. NOTES 1. IPES was a kind of institute allowing students access, through a competition, to a state position while they prepared themselves to become secondary school teachers. The training in IPES was essentially in the discipline, in this case mathematics. First year of schooling. This page intentionally left blank.

Such a warning seemed, however, likely to prove counterproductive. Actually, we do not think that it should come as a terrible surprize to the reader for a scientific text to require some effort to read. Mathematicians especially must be aware that the reading of mathematics is in itself work.

Nonetheless, this particular difficulty seemed worthy of attention. As Perrin-Glorian emphasizes , p. The presentation speaks to the teacher as well as to the researcher or the mathematics educator.

It is at the core of the whole theory. The roles of the teacher and the related didactical problems are identified and described in a way which prepares the reader for the powerful concepts of Didactical contract and of Institutionalization which were coined later by Brousseau. The Editors This page intentionally left blank. From this study, we shall derive a general classification of didactical situations. The game The game is played by pairs of players.

Description of the phases of the game This first section is a nutshell description of the phases of the game, all of which will be discussed in more detail later. We include it in order to make the reading easier. Phase 1: Explanation of the rules The teacher explains the rules of the game and starts playing a round at the chalkboard against one of the children, then relinquishes her place to a second child.

They write the numbers chosen on a sheet of paper on opposite sides of a line. This phase should consist of about four rounds and take no more than ten minutes. Remark: During this phase, the children apply the rules. Certain children, without being conscious of it, realise that saying numbers at random is not the best strategy; they test the constraints of the game at the level of action and immediate decisions, and provide themselves with a sequence of examples.

Phase 3: Playing Group-against-group six to eight rounds, 15 to 20 minutes The children are divided into two groups. In each one, the teacher nominates one child as team representative for each round, naming her at random. Any child might be called upon to defend her group at the chalkboard, in front of the whole assembly; if she wins, her group will receive one point.

The children very quickly realize the necessity of planning together and discussing within each group so as to share strategies among themsleves. Phase 4 : Game of discovery 20 to 25 minutes The teacher then asks the children to put forward propositions. These are the discoveries that they made which allowed them to win. The teacher writes these discoveries on the chalkboard as they are presented by each group in turn; they are then verified by the other group and either accepted or rejected.

If they are accepted, they remain on the chalkboard. Figure 1 Each child who puts forward a proposition must prove to an opponent that it is either true or false, either by playing or by an intellectual demonstration. In order to make the game more interesting, the following rule can be adopted: — each proposition accepted by the class is worth one point; — each proposition proved false is worth three points to the group that does so.

If she did not know how to play, clearly she might lose. These proofs were always carried out by further play starting with 5, for example. After an hour, the children had discovered that, to win, they must say 2, 5, 8, 11, 14 or Remarks This situation was reproduced under observation sixty times. In addition, each of its phases was the object of experimentation and clinical study1. The observations were carried out over a period of three years by a group from the IREM of Bordeaux consisting of a Teaching Assistant in Mathematics, seven academic psychologists and a middle school teacher specializing in computer science.

As a result, we are able to introduce our remarks as properties of the situation. Strategies and discoveries are used implicitly before being formulated so as to respond to the needs of an ongoing action. Formulation takes place after conviction and before proof2 in order to respond to the needs of communicating an action.

Several formulations precede the proof and are supported at the same time by effectiveness and rationality. Established theorems do not immediately serve to support each other; their articulation is discovered only at the end. The same proof is discovered several times, even by the same child. It is the children who lose a round who most want to explain their failure or the conditions for success. The teacher must send the questions back to the groups.

Explanation must be necessary, technically and sociologically; if the result is obvious or—as it was here in the beginning—generally accepted, only a recipe is obtained. The teacher talked about a game; she communicated a message which contained the rules of the game so that the students could internalize and apply them. This message did not contain any new words; it is assumed that the children understand the terms and their organization that is to say, the phases. Figure 2 The statement of the whole set of rules might have been too long for some children.

So the communication of the message is accompanied by an action of the child by making her play. The teacher simulates the situation which the child will meet during the course of a normal game. In this case, the situation for the child is the sequence of numbers in the game. Example: The child plays 1; the teacher, 3; the child, 5; the teacher, 6. For the child, the situation at this moment consists of the sequence of moves illustrated below: 3 5 6 Figure 3 This is a source of information for her and she acts on the situation by deciding to add a number.

She talks about these rules by matching them with the circumstances of the situation. Figure 5 Earlier, when the instructions were given, the teacher stated the rules and the child had to imagine the corresponding game-situations. Now, the rules are communicated to her with the corresponding and alternating circumstances; 2 is played and 3 is communicated.

The aim of this sequence is still the communication of an instruction but it has slipped into an action phase. Remark: The teacher wants to communicate to the child a rule of action that she has in her repertoire. She transforms this rule into a message appropriate to the medium by means of which she can communicate with the child. Here, she communicates by sound. For the child, this message is a source of information which she interprets following her own codes starting with which she reconstructs a message having meaning. The meaning given by the child does not necessarily coincide with the meaning the teacher intended to convey.

In the case of an instruction, it is hoped as well that the pupil interprets the content of the message communicated by the teacher as a rule of action. She must therefore internalize and remember it. The aim of practising a game at the same time as giving the intruction is to ensure that the rules internalized by the child are the same as those given by the teacher; action reduces the ambiguity of the message by introducing feedback. The child receives this influence as a positive or negative sanction relative to her action, which allows her to adjust this action, to accept or reject a hypothesis, to choose the best solution from among several the one which improves the satisfaction obtained during the action.

The student, for her part, has feedback which is the evaluation given by the teacher about the validity of the rules and the propositions that she has made. The children play two by two, one against the other; each student is faced with a situation: the sequence of numbers already played.

When her partner has played, she must make a decision and act on the situation by proposing a number herself after having, in this case, analyzed the situation and drawn information from it. At the end of a few moves, the penalty occurs: the round is won or lost. It can be that it includes neither the teacher nor another student. This is a very general pattern. Nearly all teaching situations are particular cases of it.

Dialectic of action While the student is playing new rounds, she develops strategies, that is to say reasons for playing one number rather than another. For instance, she will play 10rather than 9 because she thinks, incorrectly, that the game has something to with decimal counting. Perhaps she plays 13 because she considers it to be a lucky, magic number, or 17 because she has intuitively noted that she has already won after playing 17 or, on the other hand, that she found herself in bad shape after her opponent had played She may also adopt the strategy that consists of randomly choosing one of the two possible numbers.

Generally, a strategy is adopted by intuitive or rational rejection of an earlier strategy. A new strategy is therefore adopted as the result of experimentation. Example: At the start of a game, all numbers appear to the child to be of equal importance. The majority of didactical situations arise from a particular scheme of action. Observation: This implicit model does not coincide with the whole of the knowhow.

It can happen that during the learning of algorithms the child develops, unknown to the teacher, incorrect models which justify in a good or not-so-good way the know-how acquired. TS and SO. In the sixth round, they preferentially play 14 long before stating why. Second part of the game group against group In this second part of the game, the children are formed into two equal groups, and two different phases can be alternately observed: a when the group representative is at the chalkboard and playing; b when there is discussion within the group.

In the former phase, a child who is not at the chalkboard records all the information present by observing what the two representatives write down, but she can neither act play, herself nor intervene transmit information to her teammate. Whoever is playing at the chalkboard is in the didactical situation of action. In order to win, it is not enough for a student to know how to play that is to say, have an implicit model —she must indicate to her teammates which strategy she proposes; this is the only way she has of acting on the situation to come.

Each child is therefore led to anticipate, that is to say, to be conscious of the strategies which she would use first phase, 2a. Her only means of action is to formulate these strategies. She is subjected to two types of feedback: — an immediate feedback at the time of formulation from the people with whom she has the discussion, who show that they do or do not understand her suggestion phase 2b ; — a feedback from the milieu at the time of the next round played, if the formulated, applied strategy is a winning one or not. Figure 9 The formulation of a strategy is the only means that a student has of getting it to be applied by the student at the chalkboard.

This second phase b: discussion within the group achieves what we call the didactical situation of formulation. Figure 9 is a special case of the scheme of action, a case in which this action is possible only if it is given a formulation. In the present case, the formulations are very easy and do not give rise to the construction of any special language. Dialectic of formulation A dialectic of formulation would consist of progressively establishing a language that everyone could understand, which would take into account the objects and the relevant relationships of the situation in an adequate way in other words, by permitting useful reasoning and actions.

At each moment, this constructed language would be tested from the point of view of intelligibility, ease of construction and the length of the messages that it allows to be exchanged. The construction of such a language or code repertoire, vocabulary, sometimes syntax into an ordinary language or a formalized language makes possible the explanation of actions and models of action. The scheme of formalization is governed by the laws of communication: — qualititative conditions of intelligibility on repertoire and syntax , — quantitative conditions of intelligibility on utterance, noise, ambiguity, redundancy, capacity for collating and control conditions for beginning redundancy, etc.

They appear here as means of action. The students use them as means of encouraging their partners to carry out the proposed action. The means of transmitting conviction can vary widely authority, rhetoric, pragmaticism, validity, logic. In order to obtain the latter, one must organize a new type of didactical situation. Third part of the game establishment of theorems In this third part of the game, the pupils are still divided into two groups—A and B, the same groups as for the previous part.

When this conjecture has been accepted by everybody, then it will become our theorem.

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Depending what happens after that, the point will be added to the score of Group A or Group B. You may confer with each other before proposing a statement. Remark: Unlike phases 1 and 2, this one does not proceed in a standardized way. Only the rules and results stay the same. But they can only be explained to the students progressively, as the game goes along, and depending on how it goes.

Listing off the rules is fussy and hard to understand. Here we will communicate them to the reader by describing the development of a fictional class. Teacher: Group A, can you make a declaration which is true and will help you win? You may consult with each other before you make your proposal. You are the Proposers.

People have played 17 and lost. I can play 17 and lose if I want to. Teacher: Wait! When a conjecture has been written up on the board, the other team—B in this case—has to decide — whether to declare it true, in which case the other group wins the stake of one point; — whether to declare it false, in which case that group becomes in turn a proposer, but of the opposite conjecture. Then there will be two points at stake. Discussion among the children Pupil in B: And if we say we doubt it?

Teacher: You become the opposer. You put two more points in play, to be added to the stakes. The opposer can — make the proposer play 5 rounds of the Race to 20 in which the proposer is required to apply her conjecture. Points won by the proposer or opposer are marked on the score. The opposer can make the proposer go on playing until one or the other retracts her conjecture. The other one then adds to her score the points at stake.

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In this case, the one who convinces the other gets 5 points, the one who gets convinced gets 2 points. The conjecture becomes a theorem. Pupil b in B: We four do not agree with the rest of group B. We want to say we doubt the theorem. Pupil b in B: Ummm We want to request a mathematical proof! Other pupils in B: No! Teacher: OK—as an example, you are going to carry out a discussion between members of Group B with everyone listening following the same rules.

Teacher to the others in B: Figure out the proof that A could give b so you can save your points. In any case, people can always ask for proofs of theorems, even when everybody agrees to them. Doing mathematics does not consist only of receiving, learning and sending correct, relevant appropriate mathematical messages. To state a theorem is not to communicate information, it is always to confirm that what one says is true in a certain system; it is to declare oneself ready to support an opinion, to be ready to prove it. This attitude is not innate. It is developed and sustained by particular didactical situations which we shall now discuss.

## Theory of Didactical Situations in Mathematics - Guy Brousseau - Häftad () | Bokus

It requires an adherance, a personal conviction, an internalization which by definition cannot be received from others without losing its very value. We think that knowledge starts being constructed in a genesis of which Piaget has pointed out the essential features, but which also involves specific relationships with the milieu, particularly after the start of schooling. We therefore consider that for the child, making mathematics is primarily a social activity and not just an individual one. The passage from natural thought to the use of logical thought like that which regulates mathematical reasoning is accompanied by construction, rejection, the use of different methods of proof rhetorical, pragmatic, semantic or syntactic.

The consideration of a proof is a reflexive attitude. The proof must be formulated and present while being considered, and therefore most often written, and must be able to be compared with other written proofs also dealing with the same situation. In general, proof will be formulable only after having been used and tested as an implicit rule either in action or in discussion. These statements can be messages previously exchanged at the time of a dialectic of formulation, bearing on strategies and descriptions as well as on judgements.

The child must make statements about these relationships. The favourable situation will therefore obey a scheme of formulation. This means that the interlocutor must be able to provide feedback to the child who is judging; she must be able to protest, reject a reason which she judges false, prove in her turn. They must therefore both be in a priori symmetrical positions as much from the point of view of the information which they retain as that of the means of feedback.

In addition, discussion between teacher and student is highly disadvantageous, even when the teacher practises a refined maieutic in order to reduce her authority. Motivation must make this double confrontation R1 and R2 necessary. Figure 11 results from these considerations. Figure I1 Remark: This pattern is characteristic of social situations involving the construction or reorganization of a repertoire, and in particular of a repertoire of theorems, i.

Dialectic of validation The didactical scheme of validation motivates the students to discuss a situation and favours the formulation of their implicit validations, but their reasoning is often insufficient, incorrect, clumsy. They adopt false theories, accept insufficient or false proofs. The didactical situation must lead them to evolve, to revise their opinions, to replace their false theory with a true one.

This evolution has a dialectic character as well; a hypothesis must be sufficiently accepted—at least provisionally—even to show that it is false. The system of proof functions alternatively — as an implicit means; for example children tacitly accept an unformulated fact or a method of proof logical or implicit model ; — as a means of explicitly communicating a reason being advanced; — as an object of study consciously subjected to logical, semantic or pragmatic proof.

In the reality of the classroom, it is not possible to establish an order of appearance. Examples of good and bad reasons: Intellectual reasons: If I play 17, I win because my opponent can play 18 or 19 and I play 20 in both cases. Pragmatic reasons: By playing 14, I win; proof, let us do it, I win the child tests what she says by really playing a game. Children must be given the chance to discover their errors. We shall see later that this is a necessity in the construction of knowledge. We have distinguished within the didactical situation several kinds of feedback. They correspond to the different types of proof intellectual, semantic, pragmatic.

A dialectic of validation will consist of various particular dialectics of action or of formulation in order to establish a terminology, for example. They occurred in sequence, but slow down as they get further away. The scheme of proof is not self-reinforcing, and does not get reapplied very well, though reasoning by recurrence appears in the situation of proof.

Formulations do not appear until later than the corresponding implicit theorem. Informal debates among students in phase two do not produce any new theorems and d o not reinforce convictions—on the contrary, they cause the students who do have a proof to become dubious. The situation of validation permits the organization of proofs and of a mathematical proof which itself follows the progression of the game.

Institutionalization is in any case necessary to shore up the practices and their use elsewhere. The pupil implements cognitive strategies. For example, she establishes the terms starting at the end, then plays systematically up to a number six or seven less than the last theorem and proceeds from there by trial and error. For example, children played and re-played some of them quite a number of turns in the conditions of the first phase.

It was the first comprehensive presentation of the Theory of Didactical Situations. The purpose of this text was to gather the concepts Brousseau had coined in the course of more than 20 years of research, to formulate them and to organise them in a coherent theoretical framework. The first is relevance. What is at issue first is to describe certain kinds of human relationships in such a way that the concepts of didactique are made to appear in order to serve as useful means of description.

Here the issue is for all the relevant phenomena to be taken into consideration. It may be the newest because if teachers in their professional practice use relevant concepts which tend to allow the treatment of all the cases, they do not—and they are not supposed to—take responsibility for the consistency of these concepts. Mathematical knowledge and didactical transposition Established knowledge appears in many forms. It can, for example, take the form of questions and answers. For mathematics, one of the classical forms is axiomatic presentation. In addition to its more apparent scientific virtues, this presentation appears to be marvellously well adapted to the needs of teaching.

Axiomatics make it possible for us to define the objects of study in terms of previously introduced notions, thus allowing the organization of the acquisition of new items of knowledge in relation to that already acquired. Clearly, to complete the process, one must fill in with examples and problems whose solution requires the use of the knowledge in question.

But such a presentation removes all trace of the history of this knowledge, that is, of the succession of difficulties and questions which provoked the appearance of the fundamental concepts, their use in posing new problems, the intrusion of techniques and questions resulting from progress in other sectors, the rejection of points of view found to be false or clumsy, and the very many quarrels about them.

To make teaching easier, it isolates certain notions and properties, taking them away from the network of activities which provide their origin, meaning, motivation and use. It transposes them into a classroom context. Epistemologists call this didactical transposition. It is at the same time inevitable, necessary and, in a sense, regrettable. It must be kept under surveillance. The work of the mathematician Before communicating what she thinks she has discovered, a mathematician must first identify it.

A whole readjustment of similar and related knowledge, old or new, must be started. One must conceal the reasons which led her in these directions and the personal influences which guided success. One must skillfully contextualize even ordinary remarks, while avoiding trivialities. One must look, too, for the most general theory within which the results remain valid.

Thus, the producer of knowledge depersonalizes, decontextualizes and detemporalizes her results as much as possible. This work is essential if the reader is to be able to gain knowledge of these results and convince herself of their validity without having to go through the same procedures in order to discover them, and at the same time to benefit from the possibilities offered by their use.

Other readers also transform these results, reformulate them, apply them, and generalize them according to their needs. Occasionally, they destroy them by identifying them with previous knowledge, by including them within stronger results, or by simply forgetting them; or even by showing them to be false. Thus, from the time of its discovery, the organization of knowledge depends on the requirements of communication imposed on the author. The knowledge never ceases being modified for the same reasons, so much so that its meaning changes quite profoundly.

The functioning of this community is based on the relationships that exist between the personal and contextual investment and renewal of mathematical questions, and the rejection of this investment for the production of a text of knowledge that is as objective as possible. Knowing mathematics is not simply learning definitions and theorems in order to recognize when to use and apply them.

We know very well that doing mathematics properly implies that one is dealing with problems. We do mathematics only when we are dealing with problems—but we forget at times that solving a problem is only a part of the work; finding good questions is just as important as finding their solutions. A faithful reproduction of a scientific activity by the student would require that she produce, formulate, prove, and construct models, languages, concepts and theories; that she exchange them with other people; that she recognize those which conform to the culture; that she borrow those which are useful to her; and so on.

To make such an activity possible the teacher must imagine and present to the students situations within which they can live and within which the knowledge will appear as the optimal and discoverable solution to the problems posed. Each item of knowledge must originate from adaptation to a specific situation; we do not create probability theory, for example, in the same kind of context and relationship with the milieu as that in which we invent or use arithmetic or algebra. But within the story which the students are reliving the teacher must also provide the means of discovering the cultural and communicable knowledge that she wanted to teach them.

Problems with Teaching Decimal Numbers. Didactical Problems with Decimals. Didactique - What Use is it to a Teacher? Index of Names. Index of Subjects. Du kanske gillar. Spara som favorit. Skickas inom vardagar. Laddas ned direkt. This book is unique. It gathers texts which give the best presentation of the principles and key concepts of the Theory of Didactical Situations that Guy Brousseau developed in the period from to