Adding new Representations of Mathematical Objects to Aplusix
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Adding new Representations of Mathematical Objects to Aplusix. Denis Bouhineau, Hamid Chaachoua, Jean-Francois Nicaud & Christophe Viudez. 1. ICTMT’2007. What’s next ?. The ReMath project Natural representation of algebraic expression in Aplusix

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Adding new Representations of Mathematical Objects to Aplusix

Denis Bouhineau, Hamid Chaachoua, Jean-Francois Nicaud & Christophe Viudez

1

ICTMT’2007


What’s next ? Aplusix

  • The ReMath project

  • Natural representation of algebraic expression in Aplusix

  • Tree & Natural representation of algebraic expression in Aplusix

    • Motivations

    • Questions raised

    • Answers

    • Experiments

  • Graphical representations of algebraic expression in Aplusix

  • Conclusion

2

ICTMT’2007


The ReMath project Aplusix

  • The ReMath project (IST4-26751 European project, Dec 2005)

    • Representing Mathematics with Digital Media

    • ITD-CNR (Genova), NKUA – ETL, Talent S.A (Athens), UNISI (Sienna), METAH (Grenoble), Didirem (Paris), LKL-UOL (London)

  • Objectives

    • Enrich state-of-the-art dynamic digital artefacts for doing mathematics with new representations of mathematical objects

    • Work on scenarios for the use of these artefacts

    • Carry out empirical research involving cross-experiments in realistic educational contexts

3

ICTMT’2007


Natural representation of algebraic expressions in Aplusix Aplusix

  • Aplusix

    • A microworld and an exerciser for doing algebra

    • Students freely write algebraic expressions

  • Algebraic expression

    • Natural representation of algebraic expressions

    • Natural editing of algebraic expressions

4

ICTMT’2007


Natural representation of algebraic expressions in Aplusix Aplusix

  • Representation of the reasoning processes with a tree

  • Two fundamental feedbacks

    • Semantic equivalence between successive steps

    • Syntax of the final expression

  • Users (students)

    • Gain autonomy

    • Learn algebra

  • Available

    • for research, http://aplusix.imag.fr/Dir-Vers-Rech

    • or see publishers : Chartwell&Yorke (uk), Les éditions Archimède (fr), MediaDirect (it)

5

ICTMT’2007


Tree & Natural representation of algebraic expression in Aplusix

  • Motivation (ideal)

    • epistemological : trees are natural representations of algebraic expressions

    • didactical :

      • introduction of trees = change of register

      • mapping between natural & tree object  understand the syntactical structure of algebraic expression

    • computer science : trees are fundamental objects

  • Motivation (pragmatic)

    • ReMath

    • Didactician colleague’s ask

6

ICTMT’2007


Tree & Natural representation of algebraic expression in Aplusix

  • Questions about the kind of tree :

    • internal trees used by Aplusix ?

    • special algebraic trees ?

    • abstract trees ?

  • Questions about the link between tree representation and natural representation :

    • just a way to display object / edit ?

    • ill-formed ?

  • Mathematical questions :

    • ‘-’ operator ?

    • ‘(‘ and ‘)’ ?

7

ICTMT’2007


Tree & Natural representation of algebraic expression in Aplusix

  • Answers

    • authentic objects of our microworld

    • abstract trees

    • 4 modes (representation)

      • natural

      • mixed

      • free mode

      • controlled mode

    • Other answers (mode)

  • First prototype : Dec 2006

8

ICTMT’2007


Tree & Natural representation of algebraic expression in Aplusix

  • New sorts of exercise

    • build the tree representation of an expression given in the natural representation

    • build the natural representation of an expression given in the tree representation

  • Experiments

    • planed in France and italie in late 2007

9

ICTMT’2007


Graphical representations of algebraic expression in Aplusix Aplusix

  • Objective : only display

  • Motivations

    • asked by teachers

    • present in curriculum

    • combining symbolic and graphical representations

  • Questions raised

    • How to represent the solution of equations ?

    • How to represent identical objects ?

10

ICTMT’2007


Conclusion Aplusix

  • Adding new Representations of Mathematical Objects

    • Decide whether the representation will be an object or just a new way of displaying object

    • Think about experiments and use cases (à la UML)

    • Work with colleagues from other laboratories and different cultures

    • (plan time enough for debugging !)

11

ICTMT’2007


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