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Adding new Representations of Mathematical Objects to Aplusix

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

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  1. Adding new Representations of Mathematical Objects to Aplusix Denis Bouhineau, Hamid Chaachoua, Jean-Francois Nicaud & Christophe Viudez 1 ICTMT’2007

  2. 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 2 ICTMT’2007

  3. 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 3 ICTMT’2007

  4. 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 4 ICTMT’2007

  5. 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) 5 ICTMT’2007

  6. 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

  7. 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

  8. 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

  9. 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

  10. Graphical representations of algebraic expression in 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

  11. 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 !) 11 ICTMT’2007

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