# theory of didactical situations

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• 1. Guy Brousseau, 1997, Kluwer Academic PublishersTHEORY OF DIDACTICALSITUATIONS IN MATHEMATICS

2. IntroductionSetting the scene with an example: therace to 20Brousseau G. (1978) Etude locale des processus dacquisition ensituations scolaires. Etudes sur lenseignement lmentaire, Cahier18, 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 andthe 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 [3] 4. Setting the scence, the race to 20 1. explanation of therules 2. one against one To get familiar with therule, discovery of 17 iswinning is contingent 3. group against Groups compete publically via aspokesperson, agreeing on agroup strategy is the best approach.Advantage of saying 17 appears 4. game of discovery Children formulatepropositions, collectiveverification, acceptation(+1), rejection (+3). Saying 17 or14 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 anexperiments, eacongoing action (SO) h phase object of Formulation takes place after conviction anda clinicalbefore proof in order to respond to the needs ofstudy, study overcommunicating an action (SO)a period of 3years, math, psy, Established theorems are not immediately ing.stored as such (SO)Losing stimulate commitment to explain orsearch for conditions for success (CO)Lessons comefrom a clinical Proofs get its value when it has been tested as a observationmeans of convincing or obligation of being(CO), statisticalconvinced (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 1teacher The aim of action teacherthis sequence isrule ofthe gamestill theMessageinformationcommunicationsituationof aninstruction butstudent(1)informationit 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 messagecommunicationwith the child (2) Semantic of the rules bysituation the first gaming (3) Semantic of the rules by student commenting on them 7. Race to 20, structure of phase 1The teacher transforms the rule ofteachersender rules ofthe game into a message appropriatelinguisticcode the gameto what the learner can afford here sound codingit is oral communication.messagechanelThe oral message is a source ofnoisemessage as a sourceinformation for the learner of information forthe studentThe meaning given by the child doesnot necessarily coincide with thelinguisticmeaning the teacher intend to conveycodeThe role of playing the game at the student receiver learningsame time as providing the message decodingis to leverage the convergence of message asmeaning.meaning 8. Race to 20, structure of phase 1 We call an influence of the teachersender linguistic rules of situation on the pupil feedback.code the game The child receives this influencecodingmessage sound chanel as a positive or negative sanction relative to her action, whichnoise message as a source allows her to adjust this action, toof information for the student accept or reject a hypothesis, to choose the best solution from linguistic among several (the one whichcode improves the satisfactionlearning student receiver obtained during the action).decoding (p.7) message as meaningThis feedback must be closely associated with the learning which the teacher istrying to make happen (p.8) 9. Race to 20, structure of phase 2feedback From a cK perspective a conception is anaction invariant propertysituation studentof this schema for ainformationfamily 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.) 10. 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 totake 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. 11. Race to 20, structure of phase 4R1At stake is the passagefrom natural thought to [a] proposingthe use of logicalrounds playedstudentthought needed toestablish the scientific[b]validity of a statementstatements ontheoryor a result. R1formulated strategies es agThis needs thess meconstruction, rejectionor use of differentmethods of proof: R2rhetorical, pragmatic, se studentmantic or syntactic. theoryopposing 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) 12. Race to 20, structure of phase 4 R1The situation ofvalidation motivates[a] proposingthe students to discuss arounds playedstudentsituation and favour theformulation of their [b]implicit validation. statements ontheory[It] must lead them to R1formulated strategiesesevolve, to revise theirag ssopinions, to replacemetheir false theory with atrue one.This evolutionR2as a dialectic character studentas well. (p.17) theory opposingR2Dialectic of validation