Guy Brousseau, 1997, Kluwer Academic PublishersTHEORY OF DIDACTICALSITUATIONS IN MATHEMATICS
IntroductionSetting the scene with an example: therace to 20Brousseau G. (1978) Etude locale des processus d‟acquisition ensituations scolaires. Etudes sur l’enseignement élémentaire, Cahier18, 7-21. Bordeaux: IREM de Bordeaux (TSD pp.3-18)
Setting the scene, the race to 20 A paradigmatic situation to illustrate the general classification of didactical situations. The characters: the students, the teacher and the milieu The scene setting: the race to 20 Two players, rule: add 1 or 2 to what the number the previous player said, winner the one who reaches 20 first. Starting number: 1 or 2Hidden agenda: discovery and proof of the winning series2, 5, 8, 11, 14, 17 and the implicit theorem you master the game if youkeep on 2 
Setting the scence, the race to 20 1. explanation of the rules 2. one against one To get familiar with the rule, discovery of « 17 is winning » is contingent 3. group against Groups compete publically via a spokesperson, agreeing on a group strategy is the best approach. Advantage of saying 17 appears 4. game of discovery Children formulate propositions, collective verification, acceptation (+1), rejection (+3). Saying 17 or 14 appears as theorms
Race to 20, lessons learned Strategies are used implicitly before being Sixty (60) formulated so as to respond to the needs on an experiments, eac ongoing action (SO) h phase object of Formulation takes place after conviction and a clinical before proof in order to respond to the needs of study, study over communicating an action (SO) a period of 3 years, math, psy, Established theorems are not immediately ing. “stored” as such (SO) Losing stimulate commitment to explain or search for conditions for success (CO) Lessons come from a clinical Proofs get its value when it has been tested as a observation means of convincing or obligation of being (CO), statistical convinced (AS) observation Explanations must be necessary either (SO), axiom (AS) technically or sociologically (TS) or theorem (TS) of the TSD
Race to 20, structure of phase 1 teacher “The aim of action teacher this sequence is rule of the game still theMessage information communication situation of an instruction but student (1) information it has slipped The structure of the into an action action communication between the (2) student phase” (p.7) teacher and students, eliciting the place (3) teacher of the situation rules (1) Instruction, stating the rules content of the message communication with the child (2) Semantic of the rules by situation the first gaming (3) Semantic of the rules by student commenting on them
Race to 20, structure of phase 1The teacher transforms the “rule of teacher sender rules ofthe game” into a message appropriate linguistic code the gameto what the learner can afford – here sound codingit is oral communication. message chanelThe oral message is a source of noise message as a sourceinformation for the learner of information for the studentThe meaning given by the child doesnot necessarily coincide with the linguisticmeaning the teacher intend to convey codeThe role of playing the game at the student receiver learningsame time as providing the message decodingis to leverage the convergence of message asmeaning. meaning
Race to 20, structure of phase 1 “We call an influence of the teacher sender linguistic rules of situation on the pupil “feedback”. code the game The child receives this influence coding message sound chanel as a positive or negative sanction relative to her action, which noise message as a source allows her to adjust this action, to of information for the student accept or reject a hypothesis, to choose the best solution from linguistic among several (the one which code improves the satisfaction learning student receiver obtained during the action).” decoding (p.7) message as meaning“This feedback must be closely associated with the learning which the teacher istrying to make happen” (p.8)
Race to 20, structure of phase 2 feedback From a cK¢ perspective a conception is an action invariant property situation student of this schema for a information family of problemsThe phase 2 of the sequence realizes a “situation of action” where the learnerforms her strategies and construct a model of the situation by experimentingan noticing successes and failures. These strategies and model are mostlyimplicit.“Within a situation of action, everything that acts on the student or that sheacts on is called the “milieu”. It can be that it includes the teacher or anotherstudent. This is a general pattern. Nearly all teaching situation are particularcases of it.” (p.9) “The succession of interactions between the student and themilieu constitutes what we call a “dialectic of action” ” (ibid.)
Race to 20, structure of phase 3 feedbackThe student mustcommunicate about thestrategy to use for the student situation student at her deskat the chalkboard, theconstruction of a commonlanguage is needed. feedback actionDuring this situation there aretwo types of feedback: student at - immediate (discussion) chalkboard - delayed (round played) Related process: dialectic of formulationThis second phase is called asituation of formulation: it Establishing a language /code thathas the existence of a everyone could understand, which couldcommon language specific to take into account the objects andthe situation as a constraint of relevant relationships of the situation soviability (in cK¢ terms). as to permitting useful reasoning and actions.
Race to 20, structure of phase 4 R1At stake is “the passagefrom natural thought to [a] proposingthe use of logical rounds played studentthought” needed toestablish the scientific [b]validity of a statement statements on theoryor a result. R1 formulated strategies es agThis needs the ss meconstruction, rejectionor use of differentmethods of proof: R2rhetorical, pragmatic, se studentmantic or syntactic. theory opposing R2 “If one wishes to avoid having sophistries, rhetoric and authority take the place of consistency, logic and the efficacy of proof, one must not let the discussion lose touch with the situation which reflects the students‟ discourse and gives it meaning. Motivation must make this double confrontation (R1 and R2) necessary.” (p.16)
Race to 20, structure of phase 4 R1The situation ofvalidation “motivates [a] proposingthe students to discuss a rounds played studentsituation and favour theformulation of their [b]implicit validation”. statements on theory“[It] must lead them to R1 formulated strategies esevolve, to revise their ag ssopinions, to replace metheir false theory with atrue one.This evolution R2as a dialectic character studentas well.” (p.17) theory opposing R2 Dialectic of validation << Dialectic of formulation << Dialectic of action Their outcome is compulsory but not The situation of validation permits the the corresponding situations organization of proofs
Chapter 1Foundations and methods of didactiqueBrousseau G. (1986) Fondations et méthodes des la didactique desmathématiques. Recherches en didactique des mathématiques 7 (2)33-115 (TSD pp.21-75)
Object of study of didactiqueCaveat: “didactic” in English has a negative connotation. Its meanings are : 1.Intended to instruct. 2. Morally instructive. 3. Inclined to teach or moralizeexcessively. Although the first meaning is close to the French, the third sensedominated in the 80s at point which suggested to spell the word in the French wayso as to minimize the effect.Didactique is the science of the specific conditions of thedissemination of a given knowledge domain. It deals withthe project of an institution, here called the “teachingagent” aiming at modifying the knowledge of an otherone, here called the “learning agent” whereas the latter isnot able to do it on its own or does not feel the need for it.A didactical project is the social project to make a subjectappropriating an established knowledge. Teachingincludes all the actions needed to achieve this didacticalproject. Adapted from Brousseau, Glossaire de didactique, 25/02/2003)
Didactical postulates The knowledge at stake in a didactical project cannot be taught as it appears in the institutions of reference (academic, professional, cultural) : (P1) Meaning is not given by a text/discourse, but emerges from the activity which is required by this knowledge Institutionalized knowledge is depersonalized, decontextualized and detemporalized. (P2) A specification of the knowledge stake of a didactical project requires a process of transposition in order to fulfill the teaching/training and learning constraints (p.21)
Didactical postulates “A faithful reproduction of “The teacher must imagine the scientific activity by the and present students student would require that situations within which she they can live and within reproduces, formulates, pro which the knowledge will ves, and constructs appear as the optimal and models, languages, concept discoverable solution of the s and theories ; that she problem posed” (p.22) exchanges them with other The teacher‟s work is to people ; that she recognizes some extent the opposite of those which conform to the the knowledge producer culture ; that she borrows Recontextualization those which are useful to Repersonalization her; and so on” (p.22)(P3) “Each item of knowledge must originate from theadaptation to a specific situation” (p.23)
Didactical phenomenaThe didactical phenomena witness the complexity of didacticalprocesses, there elicitation frames the objectives of research indidactique.1. Topaze effect (p.25) obtaining a behavior at the cost of the meaning of the knowledge at stake2. Jourdain effect (p.25) acknowledgement of a piece of knowledge based on a surface characteristics of behaviors3. Metacognitive shift (p.26) the teaching method or means becomes the object of teaching4. Improper use of analogy (p.27) pointing similarities to facilitate which are not relevant in themselves5. Ageing of teaching situations (p.27) feeling of the need to change lessons organizations, discourse, behaviours -- teacher does repeat a text (see also the Actor paradox)6. Dienes effect (p.35) freeing the teacher from his or her responsibility towards learning
The core didactical structure“Between the moment the student accepts the problem as if it wereher own and the moment when she produces her answer, theteacher refrains from interfering and suggesting the knowledgethat she wants to see appear” (p.30) adidactical situation actual teaching situation devolution institutionalization didactical situation
The core didactical structure “Each item of knowledge can be characterized by a (or some) adidactical situation(s) which preserve(s) meaning ; we shall call this a fundamental situation” (p.30) fundamental situation Knowledge analysis restriction deformation adidactical situation actual teaching situation devolution institutionalization didactical situation“[The teacher] is involved in a game with the system of interaction of the student with theproblem she gives her […] This game, or broader situation, is the didactical situation” (p.31)
The didactical contract“The teacher must therefore arrange not the communication of knowledge, but thedevolution of a good problem” (p.31)“The didactical contract is the rule of the game and the strategy of thedidactical situation” (p.31)“[There is a] system of obligations which resembles a contract” (p.31)but “[which] is not exactly a contract it cannot be made completely explicit […] The teacher must however accept responsibility […] similarly, the student must accept responsibility clauses concerning the breaking and the stake of the contract cannot be written in advance” (p.32)“Knowledge will be exactly the thing that will solve the crisis causedby such breakdowns […] it cannot be defined in advance” (p.32)
“Game” the key modeling tool “Modeling a teaching situation consists of producing a game specific to the target knowledge among different subsystems: the educational system, the student system, the milieu, etc.” (p.47)“To consider the teacher as a player faced with a system, itself built up from a pair ofsystems: the student and, let us say for the moment, a „milieu‟ that lacks any didacticalintentions with regards to the student” (p.40) “In the student‟s game with the milieu, knowledge is the means of understanding the ground rules and strategies and, later, the means of elaborating winning strategies and obtaining the results being sought” (p.40)The game must allow a representation of all situations […] solong as they manage to make the students learn one form of thetarget knowledge” (p.48)
“Game” the key modeling toolG1: situations in which “decisions and actions […] are determined only by pleasure [either derived] from accomplishing them, [or derived] from their effect”G2: “organization of this activity within a system of rules defining a success and a failure, a gain or a loss”G3: “whatever is used for playing, the instruments of the game”G4: “the way in which one plays”G5: “the set of possible positions from among which the player can choose in a given state of the [G2-game]” (pp.48-49)
The game of adaptation: issues “Is knowing this property the only way of shifting from agiven strategy to another one? “why should the student look for a way of replacing thisstrategy with that one? “what cognitive motivation leads to the production ofsuch-and-such a formulation of a property or to such-and-such a mathematical proof? “Is the given reason for producing this knowledgebetter, more correct, more accessible or more effectivethan any other reason?” (pp. 47-48)
“Game” the key modeling toolGame1: situations Stake, function of reference determined by / information associated to predicted state milieu player pleasure action, decision game (meaning 3 and 5)Game2: organization game (meaning 4) of the activity within a system of rules players rules; constraints ofGame3: instruments strategies, knowledge game (meaning 2) the milieu of the gameGame4: the way in which one plays formal rulesGame5: the set of game possible positions (meaning 1)
Paradox raised by the TSD1. Paradox of the devolution of situations (p.41) result from the tensions between the necessary student autonomy and the teacher responsibility to teach which is known from both. The teacher must refrain from teaching even if the student asks for it.2. Paradox of the adaptation of situation (p.42) the knowledge appropriated by adaptation may be… 1. Maladjusted to correctness 2. Maladjusted to a later adaptation3. Paradox of learning by adaptation (p.44-45) 1. Negation of knowledge: knowledge deems to be trivial 2. Destruction of the cause of knowledge: lost of motivation4. Paradox of the actor (p.46) “[the knowledge] whose text already exists is no longer a direct production of the teacher, it is a cultural object, quoted and re-quoted”
“Game” the key modeling tool Stake, function of reference information milieu(A) formalisation of the game player predicted state game action, decision (meaning 3 and 5)1. X set of distinct “positions”, J set of players game (meaning 4)2. rules of the game [Γ : X → P(X)] players rules; constraints of3. initial state I and final states F strategies, knowledge game (meaning 2) the milieu4. turn taking [θ : JxX→ J] formal rules5. gain, stake, preference [F A X f: A → R] game (meaning 1)Round : a finite sequence of states (from I to F).Strategy : any mapping X→X that determines choices from permissible statesTactic : strategy defined on a subset A of XPlayer s state of knowing : mapping of X →Γ(X) such that [ x C(x) Γ(X)]Determining knowledge reduces the player‟s choice to a single stateAcquisition of knowledge is a modification of the state of knowing
“Game” the key modeling tool (A) formalisation of the game Model for action: every strategy 1. X set of distinct “positions”, J set of players or calculation procedure giving rise to a strategy or a tactic 2. rules of the game [Γ : X → P(X)] Winning strategy : round with 3. initial state I and final states F positive payoff. It comes with… - a cost 4. turn taking [θ : JxX→ J] - a gain 5. gain, stake, preference [F A X f: A → R] A non-systematically-winning strategy can be better in terms Round : a finite sequence of states… of the risk of loss that it Strategy : any mapping X→X that… entails, the gains that it allows Tactic : strategy defined on a subset A of X… one to hope for, etc. Player s state of knowing : mapping of X … Determining knowledge reduces … Game theory allows the study of Acquisition of knowledge is a modification of… the dilemmas that arise.An acknowledged reference today is : Fudenberg D., Levine D. K. (1998) The theory of learning in games. The MIT Press. Thelimitation Brousseau makes in his choice of a game type is the same in that classical book.
From the model to a method The study of the adequacy of a situation for a particular piece of knowledge K has the aim… To show that the optimum strategy can be brought about by K and not by another one To state hypotheses about the variables of the situation and their influence on strategies and changes of strategies The meaning of a decision made by the student can be modeled with: a) the set of choices the student considers and rejects as a consequence of the choice made; b) the set of possible strategies considered and excluded, and in particular the sequence of choices or replacement strategies the student considers; c) the very conditions of the game that appear to be determining the choice made, and in particular the space of situations brought about by the values of the pertinent variables which give the decision a character of optimality, validity, or relevance.
Adidactical situations Two distinct types of games: a) the student‟s games with the adidactical milieu (games S student specific to each piece of E knowledge) teacher T S b) the teacher‟s games as organizers of these student‟s game. These games concern at least : M milieu the teacher, the student, the student‟s immediate environment “The milieu is the system the cultural milieu opposing the taught system They include the game of devolution or, rather, the previously-taught and of institutionalization. system”. (p.57)“As the students progress gradually continues, this cultural and didactical representation of the milieuwill be assumed to approach “reality”, and the subjects relationships with this milieu will have to becomefree of didactical intentions.” (p.57)
Interaction – knowing - situation The relationships between a student and the milieu can be classified into at least three major categories (p.61)  Action → actions and decisions that act directly  Formulation → exchange of info coded into a language  Validation → exchange of judgment They correspond to different forms of knowledge  the forms of knowledge which allow the explicit “control” of the subjects interactions in relation to the validity of her statements. It is composed of… a description or model expressed in a certain “language” a judgment about the adequacy of this description  the formulation of the descriptions and models  the models for action governing decisions“The fact that different types of interaction with the milieu and different formsof knowledge are justified a priori and independently allows us to discuss theparticularities of the milieu which are necessary for them.” (p.65)
Interaction – knowing – situation By pragmatic questions like “Why would the student do or say this rather than that?” “What must happen if she does it or doesn‟t do it?” “What meaning would the answer have if she had been given it?” it is possible to elicit the conditions which the typologies impose on the milieu.Design and engineering  “Does the milieu include an opponent (or a proponent ) with whom the subject must be confronted in order to attain the fixed goal in an exchange of opinions?”  “Does the milieu include a receiver of messages that the student must send in order to attain the target goal?”  Does the milieu include a feedback function adapted to the need for adjustment of the interaction to the targeted knowledge? “The answer to these two questions determines the layout of the milieu and the rules of the games, which are totally different.” (p.65)
Fundamental patterns (1) actionCheck list for a game based situation of action [The race to 20 phase 2] of Can the situation be perceived as devoid of didactical the sequence realizes a intentions? “situation of action” where Must students effectively chose a state from among several ones? Do they know which states they can select from? the learner forms her Can students lose? Do they know that they can? Do they strategies and construct a know the final states in advance (including the winning model of the situation by ones)? experimenting an noticing Do they know the rules without knowing a winning strategy? Can they be taught the rules without being given a solution? successes and failures. Is the target knowledge necessary? These strategies and model Can students start again? Does the game “gratify” are mostly implicit. anticipation? Have students any chance of finding out the sought strategy for themselves if they borrow it (from other students)? student Are feedback to the students choices relevant to the S E construction of the knowledge? teacher T S Is the control of decisions possible? Is a reflective attitude useful necessary for progress in the M milieu solution?
Fundamental patterns (1) action Stake, function of referenceCheck list for a game based situation of action information Can the situation be perceived as devoid of didactical predicted state milieu intentions? player game action, decision (meaning 3 and 5) Must students effectively chose a state from among several game (meaning 4) ones? Do they know which states they can select from? Can students lose? Do they know that they can? Do they know the final states in advance (including the winning players rules; constraints of strategies, ones)? knowledge game the milieu (meaning 2) Do they know the rules without knowing a winning strategy? Can they be taught the rules without being given a solution? formal rules Is the target knowledge necessary? game Can students start again? Does the game “gratify” (meaning 1) anticipation? Have students any chance of finding out the sought strategy for themselves if they borrow it (from other students)? student Are feedback to the students choices relevant to the S E construction of the knowledge? teacher T S Is the control of decisions possible? Is a reflective attitude useful necessary for progress in the M milieu solution?
Fundamental patterns (2) formulation stake about the milieuA milieu for communicationinclude a receiver/sender and informationa receiver/sender/executor player A milieu for action 1. insufficient means of action: sender and receiver actions A must describe to B the action which she had to carry milieu for communication out and often a part of the milieu as well so that the message is intelligible, messages information actions 2. insufficient information for A but sufficient means of action: B must describe the milieu As repertoire stake and A must decode the of transmission description and direct the repertoire of observation messages player B 3. means of action and Bs repertoire receiver, sender, information insufficient for A. executor “The messages exchanged are under the control of linguistic, formal or graphical codes and therefore make them function” (p.68)
Fundamental patterns (2) formulation stake about the milieuA milieu for communication player A information milieu for actioninclude a receiver/sender and sender and receiver actions milieu for communicationa receiver/sender/executor messages information actions 1. insufficient means of action: As repertoire stake of A must describe to B the transmission repertoire of messages player B action which she had to carry Bs repertoire receiver, sender, executor out and often a part of the milieu as well so that the “It is necessary to emphasize the importance: message is intelligible, • of the quality of the game with the 2. insufficient information for A but sufficient means of action: milieu in order to ensure and to maintain B must describe the milieu the relevance and the richness of students‟ and A must decode the description and direct the discourse; observation • of the frequency of use that it creates in 3. means of action and communications; information insufficient for A. • of the possibility of analyzing the messages produced.” (p.68) “The messages exchanged are under the control of linguistic, formal or graphical codes and therefore make them function” (p.68)
Fundamental patterns (2) validation As stake informationOnly valid knowing can be player A proposer, opposer actions action milieu messagesrecognized within the teachingsituation, it makes situation of statements proofs Bs stakevalidation an ultimate objective refutations information actionsof the didactical process. stake statements, theories constraints allowed by A of debate player B statements, theories opposer, proposer, allowed by B executorProponent and opponent musthave a symmetric position The situation of validation “it should not be possible for “motivates the students to discuss a one of the players to obtain situation and favors the formulation the agreement of the other by “illegitimate” means such of their implicit validation”. “[It] as must lead them to evolve, to revise authority, seduction, force, e their opinions, to replace their false tc. theory with a true one. This Knowledge should be the evolution as a dialectic character as only legitimate reference for well.” (p.17) decision making
Fundamental patterns (2) validation As stakeOnly valid knowing can berecognized within the teaching informationsituation, it makes situation of player A actionvalidation an ultimate objective proposer, opposer milieu messagesof the didactical process. actionsProponent and opponent must statementshave a symmetric position proofs Bs stake refutations information actions “it should not be possible for one of the players to obtain stake the agreement of the other statements, theories constraints by “illegitimate” means such allowed by A of debate as authority, seduction, force, e player B tc.” (p.70) statements, theories allowed by B opposer, proposer, executor Knowledge should be the only legitimate reference for decision making