Adding new Representations of Mathematical Objects to Aplusix

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

ICTMT’2007 1

What’s next ?

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

ICTMT’2007 2

The ReMath project

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

ICTMT’2007 3

Natural representation of algebraic expressions in 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

ICTMT’2007 4

Natural representation of algebraic expressions in 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)

ICTMT’2007 5

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

ICTMT’2007 6

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 ‘)’ ?

ICTMT’2007 7

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

ICTMT’2007 8

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

ICTMT’2007 9

Graphical representations of algebraic expression in Aplusix

Objective : only displayMotivations

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 ?

ICTMT’2007 10

Conclusion

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 !)

ICTMT’2007 11