1. Guy Brousseau, 1997, Kluwer Academic Publishers
THEORY OF DIDACTICAL
SITUATIONS IN MATHEMATICS

2. Introduction
Setting the scene with an example: the
race to 20
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)

3. 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]

4. 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

5. 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

6. Race to 20, structure of phase 1
teacher “The aim of
action teacher
this sequence is
rule of
the game
still the
Message
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

7. Race to 20, structure of phase 1
The teacher transforms the “rule of teacher sender
rules of
the game” into a message appropriate linguistic
code the game
to what the learner can afford – here sound
coding
it is oral communication. message chanel
The oral message is a source of noise
message as a source
information for the learner of information for
the student
The meaning given by the child does
not necessarily coincide with the
linguistic
meaning the teacher intend to convey code
The role of playing the game at the student receiver
learning
same time as providing the message
decoding
is to leverage the convergence of message as
meaning. meaning

8. 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 is
trying to make happen” (p.8)

9. 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 problems
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.)

10. Race to 20, structure of phase 3
feedback
The student must
communicate about the
strategy to use for the student situation student at her desk
at the chalkboard, the
construction of a common
language is needed. feedback action
During this situation there are
two types of feedback: student at
- immediate (discussion) chalkboard
- delayed (round played)
Related process: dialectic of formulation
This second phase is called a
situation of formulation: it Establishing a language /code that
has the existence of a everyone could understand, which could
common language specific to take into account the objects and
the situation as a constraint of relevant relationships of the situation so
viability (in cK¢ terms). as to permitting useful reasoning and
actions.

11. Race to 20, structure of phase 4
R1
At stake is “the passage
from natural thought to [a] proposing
the use of logical rounds played student
thought” needed to
establish the scientific [b]
validity of a statement statements on theory
or a result. R1 formulated strategies
es
ag
This needs the
ss
me
construction, rejection
or use of different
methods of proof: R2
rhetorical, pragmatic, se student
mantic or syntactic. theory opposing
R'2
“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)

12. Race to 20, structure of phase 4
R1
The situation of
validation “motivates [a] proposing
the students to discuss a rounds played student
situation and favour the
formulation of their [b]
implicit validation”. statements on theory
“[It] must lead them to R1 formulated strategies
es
evolve, to revise their
ag
ss
opinions, to replace
me
their false theory with a
true one.This evolution R2
as a dialectic character student
as well.” (p.17) theory opposing
R'2
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

13. Chapter 1
Foundations and methods of didactique
Brousseau G. (1986) Fondations et méthodes des la didactique des
mathématiques. Recherches en didactique des mathématiques 7 (2)
33-115 (TSD pp.21-75)

14. 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)

15. 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)

16. 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 the
adaptation to a specific situation” (p.23)

17. 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 stake
2. Jourdain effect (p.25) acknowledgement of a piece of knowledge
based on a surface characteristics of behaviors
3. Metacognitive shift (p.26) the teaching method or means
becomes the object of teaching
4. Improper use of analogy (p.27) pointing similarities to facilitate
which are not relevant in themselves
5. 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

18. 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)
adidactical situation
actual teaching situation
devolution institutionalization
didactical situation

19. 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 the
problem she gives her […] This game, or broader situation, is the didactical situation” (p.31)

20. 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)

21. “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)

22. “Game” the key modeling tool
G1: 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)

23. 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)

24. “Game” the key modeling tool
Game1: 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 player's rules; constraints of
Game3: instruments strategies,
knowledge
game
(meaning 2)
the milieu
of the game
Game4: the way in
which one plays formal rules
Game5: the set of game
possible positions (meaning 1)

25. 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 correctness
2. Maladjusted to a later adaptation
3. 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 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”

26. “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)]
player's rules; constraints of
3. initial state I and final states F strategies,
knowledge
game
(meaning 2)
the milieu
4. turn taking [θ : JxX→ J]
formal rules
5. 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 states
Tactic : strategy defined on a subset A of X
Player '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 state
Acquisition of knowledge is a modification of the state of knowing

27. “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. The
limitation Brousseau makes in his choice of a game type is the same in that classical book.

28. 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.

29. 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 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)

30. 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)

31. 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
 [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)

32. Fundamental patterns (1) action
Check 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?

33. Fundamental patterns (1) action
Stake, function of reference
Check 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 player's 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?

34. Fundamental patterns (2) formulation
stake about the milieu
A milieu for communication
include a receiver/sender and information
a 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 A's repertoire stake
and A must decode the of
transmission
description and direct the repertoire of
observation messages
player B
3. means of action and B's 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)

35. Fundamental patterns (2) formulation
stake about the milieu
A milieu for communication player A
information
milieu for action
include a receiver/sender and
sender and receiver
actions
milieu for communication
a receiver/sender/executor messages information actions
1. insufficient means of action: A's repertoire stake
of
A must describe to B the
transmission
repertoire of
messages
player B
action which she had to carry B's 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)

36. Fundamental patterns (2) validation
A's stake
information
Only valid knowing can be player A
proposer, opposer
actions
action
milieu messages
recognized within the teaching
situation, it makes situation of statements
proofs
B's stake
validation an ultimate objective
refutations information actions
of the didactical process.
stake
statements, theories constraints
allowed by A of
debate
player B
statements, theories opposer, proposer,
allowed by B executor
Proponent and opponent must
have 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

37. Fundamental patterns (2) validation
A's stake
Only valid knowing can be
recognized within the teaching
information
situation, it makes situation of player A action
validation an ultimate objective proposer, opposer
milieu messages
of the didactical process. actions
Proponent and opponent must statements
have a symmetric position proofs
B's 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