Didactical TriangleI/didactical System
Teaching
Creating best possible conditions for
learning – Planning and Interaction
Teacher
Student
Learning
Acquisition (cognition, individual)
Participation (being part of social
praxis – classroom set ups
norms for (good) math activity)
Subject
Products (skills, knowledge) - Mastering
Proces (develop algorithms and proofs,
problemsolving) - understanding
The Theory of Didactical
Situations - TDS
Theory ?
Knowledge
TDS – Guy Brousseau
TDS is in this context
presented more like a didactic
approach than a theory
TDS – Guy Brousseau
Knowledge is transformed/exchanged between 3 main groups of
actors:
Researchers
Teachers
Students
For each group there is a distinction between two forms of subject
knowledge – depending on the actor:
• Personal knowledge (Implicit, contextual, explicit)
• Shared/common/official knowledge (explicit, e.g. textbooks, articles)
Learning means two processes:
Shared knowledge
Personal knowledge
Shared knowledge
TDS – Guy Brousseau
Brousseau:
•A teacher is normally asking questions of
which he/she knows the answer – an artificial
situation
•Outside teaching-learning situations you ask
questions in order to get an answer to what
you did not know
•Brousseau wants to overcome this paradox
TDS – Guy Brousseau
Brousseau:
•Traditional teaching in Math and Science
tend are in practice based on transfer
•Student do not learn mathematics but they
learn to decode teacher’s expectations
•So students learn a certain perception of
what mathematics is or what it is about
TDS – Guy Brousseau
Learning
Intended
Adidactical
Situation
Didactical
Unintended
Non - didactical
TDS – Guy Brousseau
Adidactical
situation
Teacher
Adjusting
Didactical
variables
acts
Student
Milieu (Subject)
Didactical variabels
Gets feed back
VERY
IMPORTANT
Fundamental didactical situation: Utilizing/exploring the knowledge in question
is a winner strategy in the situation
TDS – Guy Brousseau
Characteristics of a fundamental didactical situation:
•Sufficient pre knowledge in order to understand
the challenge and be able to suggest answers
•Feed back incorporated so students’ strategies are
validated without the teachers interference
•Sufficient uncertainty (complexity) in order to
avoid obvious beforehand judgements of suggested
strategy
•Possibility for trial and error – many attempts
•The aimed knowledge must be a condition for
reaching the new strategy
TDS – Guy Brousseau
The puzzle example
TDS – Guy Brousseau
Teacher role
Student
role
Milieu
Situation
Devolution (handling over
the challenge to the
students
Start
Clarify
Receive and
understand
Established
didactical
Action ( formulate and test
strategies)
Observe
Reflect
Act and
reflect
Field of
research
adidactical
Statement/explanation
(explains to other groups,
stating hypothesis)
Organize and
ask
Formulate,
state more
exactly
Open
discussion
Adidactical
or didactical
Validation (reject/accept
hypothesis)
Listen and
evaluate
Argue, reason
and reflect
Guidet
discussion,
evaluation
Normally
didactical
Institutionalizing (define the
achieved knowledge)
Present and
explain
Listen and
reflect
Institutional
knowledge
Didactical
TDS – Guy Brousseau
Do we know the Phases?
Construct a didactical situation – state
exactly the topic and the didactical
variables.
TDS – Guy Brousseau
Example: logistic growth
RME - Freudenthal
Realistic Mathematics Education.
Basic idea: Mathematics must be considered a
special approach to solve problems related to
the world outside
RME - Freudenthal
Methods and concepts are there but the
student’s understanding is to be developed
through more informal work with problems
related to the world outside
Guided reinvention: students are supposed to
reason, develop, generalize and systematize
their informal work
RME - Freudenthal
”The learner should reinvent
mathematising rather than mathematics,
abtracting rather than abstractions,
schematising rather than schemes,
algorithmising rather than algorithms,
verbalising rather than language”
(Freudenthal)
RME - Freudenthal
Model concept – emergent models
• activity based on the given context
• referential level – a model of the given
context is developed – reflect on the context
• general level – the model becomes a model
for situations in outside world – it gets it’s
own life - reflect on the model – thinking with
• formal level – pure mathematical
investigations can be possible – the
significance of parameters…Thinking about
the model
RME - Freudenthal
Example:Taxi functions
RME - Freudenthal
”By describing their own process students can use their
reflections to develop flexible prototypes of experiences that
can be drawn on in future problem solving”
(Lesh & Zawojewski)
”A fresh view of problem solving needs to view the learning
of mathematics and problem solving as integrated, as largely
based on modelling activity, and as a construct that is itself
continually in need of development”
(Lesh & Zawojewski)