Theory of didactical situations

THEORY OF DIDACTICAL SITUATIONS IN MATHEMATICS

Guy Brousseau, 1997, Kluwer Academic Publishers

Brousseau G. (1978) Etude locale des processus d’acquisition en situations scolaires. Etudes sur l’enseignement élémentaire, Cahier 18, 7-21. Bordeaux: IREM de Bordeaux (TSD pp.3-18)

IntroductionSetting the scene with an example: the race to 20

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 2

Hidden agenda: discovery and proof of the winning series 2, 5, 8, 11, 14, 17 and the implicit theorem you master the game if you keep on 2 [3]

Setting the scence, the race to 20 1. explanation of

the rules 2. one against one

3. group against group

4. game of discovery

To get familiar with the rule, discovery of « 17 is winning » is contingent

Groups compete publically via a spokesperson, agreeing on a strategy is the best approach. Advantage of saying 17 appears

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 formulated so as to respond to the needs on an ongoing action (SO)

Formulation takes place after conviction and before proof in order to respond to the needs of communicating an action (SO)

Established theorems are not immediately “stored” as such (SO)

Losing stimulate commitment to explain or search for conditions for success (CO)

Proofs get its value when it has been tested as a means of convincing or obligation of being convinced (AS)

Explanations must be necessary either technically or sociologically (TS)

Sixty (60) experiments, each phase object of a clinical study, study over a period of 3 years, math, psy, ing.

Lessons come from a clinical observation (CO), statistical observation (SO), axiom (AS) or theorem (TS) of the TSD

Race to 20, structure of phase 1

Message

teacher

student

rule of the game

teacher

student

situation

information

action

information

action

teacher

student

situation

rules

content of the message

communication with the child

(1)

(2)

(3)

The structure of the communication between the teacher and students, eliciting the place of the situation(1)Instruction, stating

the rules(2)Semantic of the rules

by the first gaming(3)Semantic of the rules

by commenting on them

“The aim of this sequence is still the communication of an instruction but it has slipped into an action phase” (p.7)


The teacher transforms the “rule of the game” into a message appropriate to what the learner can afford – here it is oral communication.

The oral message is a source of information for the learner

The meaning given by the child does not necessarily coincide with the meaning the teacher intend to convey

The role of playing the game at the same time as providing the message is to leverage the convergence of meaning.

linguistic code

rules of the game

teacher

student

sender

receiver

sound chanelmessage

noisemessage as a source of information for the student

decodingmessage as meaning

linguistic code

learning

coding


“We call an influence of the situation on the pupil “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).” (p.7)

linguistic code

rules of the game

teacher

student

sender

receiver

sound chanelmessage

noisemessage as a source of information for the student

decodingmessage as meaning

linguistic code

learning

coding

“This feedback must be closely associated with the learning which the teacher is trying to make happen” (p.8)


The phase 2 of the sequence realizes a “situation of action” where the learner forms her strategies and construct a model of the situation by experimenting an noticing successes and failures. These strategies and model are mostly implicit.

“Within a situation of action, everything that acts on the student or that she acts on is called the “milieu”. It can be that it includes the teacher or another student. This is a general pattern. Nearly all teaching situation are particular cases of it.” (p.9) “The succession of interactions between the student and the milieu constitutes what we call a “dialectic of action” ” (ibid.)

situation studentaction

feedback

information

From a cK¢ perspective a conception is an invariant property of this schema for a family of problems

Race to 20, structure of phase 3The student must

communicate about the strategy to use for the student at the chalkboard, the construction of a common language is needed.During this situation there are two types of feedback:

- immediate (discussion)- delayed (round played)

This second phase is called a situation of formulation: it has the existence of a common language specific to the situation as a constraint of viability (in cK¢ terms).

situation student at her desk

action

feedback

student at chalkboard

feedback

Related process: dialectic of formulation

Establishing a language /code that everyone could understand, which could take into account the objects and relevant relationships of the situation so as to permitting useful reasoning and actions.

Race to 20, structure of phase 4At stake is “the passage

from natural thought to the use of logical thought” needed to establish the scientific validity of a statement or a result.This needs the construction, rejection or use of different methods of proof: rhetorical, pragmatic, semantic or syntactic.

rounds played

student

student

R1

statements on formulated strategies

mes

sage

s

proposing

opposing

theory

theory

R1

R'2

R2

[a]

[b]

“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 4The situation of

validation “motivates the students to discuss a situation and favour the formulation of their implicit validation”. “[It] must lead them to evolve, to revise their opinions, to replace their false theory with a true one.This evolution as a dialectic character as well.” (p.17)

rounds played

student

student

R1

statements on formulated strategies

mes

sage

s

proposing

opposing

theory

theory

R1

R'2

R2

[a]

[b]

Dialectic of validation << Dialectic of formulation << Dialectic of action Their outcome is

compulsory but not the corresponding situations

The situation of validation permits the organization of proofs

Brousseau G. (1986) Fondations et méthodes des la didactique des mathématiques. Recherches en didactique des mathématiques 7 (2) 33-115 (TSD pp.21-75)

Chapter 1Foundations and methods of didactique

Object of study of didactique

Caveat: “didactic” in English has a negative connotation. Its meanings are : 1. Intended to instruct. 2. Morally instructive. 3. Inclined to teach or moralize excessively. Although the first meaning is close to the French, the third sense dominated in the 80s at point which suggested to spell the word in the French way so as to minimize the effect.

Didactique is the science of the specific conditions of the dissemination of a given knowledge domain. It deals with the project of an institution, here called the “teaching agent” aiming at modifying the knowledge of an other one, here called the “learning agent” whereas the latter is not 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 subject appropriating an established knowledge. Teaching includes all the actions needed to achieve this didactical project.

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 scientific activity by the student would require that she reproduces, formulates, proves, and constructs models, languages, concepts and theories ; that she exchanges them with other people ; that she recognizes those which conform to the culture ; that she borrows those which are useful to her; and so on” (p.22)

“The teacher must imagine and present students situations within which they can live and within which the knowledge will appear as the optimal and discoverable solution of the problem posed” (p.22)

The teacher’s work is to some extent the opposite of the knowledge producer Recontextualization Repersonalization

(P3) “Each item of knowledge must originate from the adaptation to a specific situation” (p.23)

Didactical phenomena

The didactical phenomena witness the complexity of didactical processes, there elicitation frames the objectives of research in didactique. 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 were her own and the moment when she produces her answer, the teacher refrains from interfering and suggesting the knowledge that she wants to see appear” (p.30)

devolution institutionalization

adidactical situation

didactical situation

actual teaching 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)

devolution institutionalization

adidactical situation

didactical situation

actual teaching situation

fundamental situation

restrictiondeformation

Knowledge analysis

“[The teacher] is involved in a game with the system of interaction of the student with the problem 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 the devolution of a good problem” (p.31)

“The didactical contract is the rule of the game and the strategy of the didactical 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 caused by 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 of systems: the student and, let us say for the moment, a ‘milieu’ that lacks any didactical intentions 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 […] so long as they manage to make the students learn one form of the target 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 a given strategy to another one?

“why should the student look for a way of replacing this strategy with that one?

“what cognitive motivation leads to the production of such-and-such a formulation of a property or to such-and-such a mathematical proof?

“Is the given reason for producing this knowledge better, more correct, more accessible or more effective than any other reason?”

(pp. 47-48)

“Game” the key modeling tool

Game1: situations determined by / associated to pleasure

Game2: organization of the activity within a system of rules

Game3: instruments of the game

Game4: the way in which one plays

Game5: the set of possible positions

player

milieu

game (meaning 3 and 5)

game (meaning 2)

Stake, function of reference

informationpredicted state

action, decision

game (meaning 4)

constraints of the milieu

player's rules; strategies, knowledge

formal rules

game (meaning 1)

Paradox raised by the TSD

1. 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 correctness2. Maladjusted to a later adaptation

3. Paradox of learning by adaptation (p.44-45) 1. Negation of knowledge: knowledge deems to be trivial2. Destruction of the cause of knowledge: lost of motivation

4. 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”


(A) formalisation of the game

1. X set of distinct “positions”, J set of players

2. rules of the game [Γ : X → P(X)]

3. initial state I and final states F

4. turn taking [θ : JxX→ J]

5. gain, stake, preference [F AX f: A → R]

player

milieu


game (meaning 2)



action, decision

game (meaning 4)



formal rules

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


(A) formalisation of the game

1. X set of distinct “positions”, J set of players

2. rules of the game [Γ : X → P(X)]

3. initial state I and final states F

4. turn taking [θ : JxX→ J]

5. gain, stake, preference [F AX f: A → R]

Round : a finite sequence of states…Strategy : any mapping X→X that…Tactic : strategy defined on a subset A of X…Player 's state of knowing : mapping of X …Determining knowledge reduces …Acquisition of knowledge is a modification of…

Model for action: every strategy or calculation procedure giving rise to a strategy or a tacticWinning strategy : round with positive payoff. It comes with…

- a cost- a gain

A non-systematically-winning strategy can be better in terms of the risk of loss that it entails, the gains that it allows one to hope for, etc.

Game theory allows the study 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. The limitation 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 situationsTwo distinct types of games: a) the student’s games with

the adidactical milieu (games specific to each piece of knowledge)

b) the teacher’s games as organizers of these student’s game. These games concern at least : the teacher, the student, the student’s immediate

environment the cultural milieu

They include the game of devolution and of institutionalization.

S

E

M

student

milieu

teacher

S

T

“The milieu is the system opposing the taught system or, rather, the previously-taught system”. (p.57)

“As the student's progress gradually continues, this cultural and didactical representation of the milieu will be assumed to approach “reality”, and the subject's relationships with this milieu will have to become free 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)

[1] Action → actions and decisions that act directly [2] Formulation → exchange of info coded into a language [3] Validation → exchange of judgment

They correspond to different forms of knowledge [3] the forms of knowledge which allow the explicit “control” of the

subject's 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

[2] the formulation of the descriptions and models [1] the models for action governing decisions

“The fact that different types of interaction with the milieu and different forms of knowledge are justified a priori and independently allows us to discuss the particularities of the milieu which are necessary for them.” (p.65)

Interaction – knowing – situationBy 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.

[3] “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?”

[2] “Does the milieu include a receiver of messages that the student must send in order to attain the target goal?”

[1] 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)

Desi

gn

an

d e

ng

ineeri

ng

Fundamental patterns (1) action

Check list for a game based situation of action

Can the situation be perceived as devoid of didactical intentions?

Must students effectively chose a state from among several 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 ones)?

Do they know the rules without knowing a winning strategy? Can they be taught the rules without being given a solution?

Is the target knowledge necessary?

Can students start again? Does the game “gratify” anticipation?

Have students any chance of finding out the sought strategy for themselves if they borrow it (from other students)?

Are feedback to the students choices relevant to the construction of the knowledge?

Is the control of decisions possible?

Is a reflective attitude useful necessary for progress in the solution?

S

E

M

student

milieu

teacher

S

T

[The race to 20 phase 2] of the sequence

realizes a “situation of action” where the learner forms her

strategies and construct a model of

the situation by experimenting an

noticing successes and failures. These

strategies and model are mostly implicit.

Fundamental patterns (1) action

Check list for a game based situation of action

Can the situation be perceived as devoid of didactical intentions?

Must students effectively chose a state from among several 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 ones)?

Do they know the rules without knowing a winning strategy? Can they be taught the rules without being given a solution?

Is the target knowledge necessary?

Can students start again? Does the game “gratify” anticipation?

Have students any chance of finding out the sought strategy for themselves if they borrow it (from other students)?

Are feedback to the students choices relevant to the construction of the knowledge?

Is the control of decisions possible?

Is a reflective attitude useful necessary for progress in the solution?

player

milieu


game (meaning 2)



action, decision

game (meaning 4)



formal rules

game (meaning 1)

S

E

M

student

milieu

teacher

S

T

Fundamental patterns (2) formulation

A milieu for communication include a receiver/sender and a receiver/sender/executor

1. insufficient means of action: A must describe to B the action which she had to carry out and often a part of the milieu as well so that the message is intelligible,

2. insufficient information for A but sufficient means of action: B must describe the milieu and A must decode the description and direct the observation

3.means of action and information insufficient for A.

stake about the milieu

player A

sender and receiver

player B

receiver, sender, executor

milieu for action

messages

A's repertoire

B's repertoire

repertoire of messages

information

actions

actionsinformation

stake of transmission

mili

eu f

or c

omm

unic

atio

n

“The messages exchanged are under the control of linguistic, formal or graphical codes and therefore make them function” (p.68)

Fundamental patterns (2) formulation

A milieu for communication include a receiver/sender and a receiver/sender/executor

1. insufficient means of action: A must describe to B the action which she had to carry out and often a part of the milieu as well so that the message is intelligible,

2. insufficient information for A but sufficient means of action: B must describe the milieu and A must decode the description and direct the observation

3.means of action and information insufficient for A.

stake about the milieu

player A

sender and receiver

player B

receiver, sender, executor

milieu for action

messages

A's repertoire

B's repertoire

repertoire of messages

information

actions

actionsinformation

stake of transmission

mili

eu f

or c

omm

unic

atio

n

“The messages exchanged are under the control of linguistic, formal or graphical codes and therefore make them function” (p.68)

“It is necessary to emphasize the importance:• of the quality of the game with the

milieu in order to ensure and to maintain the relevance and the richness of students’ discourse;

• of the frequency of use that it creates in communications;

• of the possibility of analyzing the messages produced.” (p.68)

Fundamental patterns (2) validation A's stake

player A

proposer, opposer

player B

opposer, proposer, executor

statements proofs refutations

statements, theories allowed by A

information

actions

actionsinformation

stake constraints of debate

B's

sta

ke

statements, theories allowed by B

action milieu messages

The situation of validation “motivates the students to discuss a situation and favors the formulation of their implicit validation”. “[It] must lead them to evolve, to revise their opinions, to replace their false theory with a true one. This evolution as a dialectic character as well.” (p.17)

Only valid knowing can be recognized within the teaching situation, it makes situation of validation an ultimate objective of the didactical process.

Proponent and opponent must have a symmetric position “it should not be possible for

one of the players to obtain the agreement of the other by “illegitimate” means such as authority, seduction, force, etc.

Knowledge should be the only legitimate reference for decision making

Fundamental patterns (2) validation

Only valid knowing can be recognized within the teaching situation, it makes situation of validation an ultimate objective of the didactical process.

Proponent and opponent must have a symmetric position “it should not be possible for

one of the players to obtain the agreement of the other by “illegitimate” means such as authority, seduction, force, etc.” (p.70)

Knowledge should be the only legitimate reference for decision making

A's stake

player A

proposer, opposer

player B

opposer, proposer, executor

statements proofs refutations

statements, theories allowed by A

information

actions

actionsinformation

stake constraints of debate

B's

sta

ke

statements, theories allowed by B

action milieu messages

Documents

Theory of didactical situations