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Enhancing Aplusix with Natural and Graphical Representations of Algebraic Expressions

The ReMath project aims to enrich the Aplusix learning environment through innovative representations of mathematical objects, specifically algebraic expressions. This initiative emphasizes the incorporation of tree and natural representations to facilitate better understanding of algebra among students. Through extensive empirical research and experiments conducted in realistic educational contexts, we explore various scenarios that promote autonomy in learning algebra. This paper discusses the project's objectives, the methodological framework, and findings from studies that reflect on the effectiveness of these representations in teaching mathematics.

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Enhancing Aplusix with Natural and Graphical Representations of Algebraic Expressions

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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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