Adding new Representations of
Mathematical Objects to Aplusix
Denis Bouhineau, Hamid Chaachoua, JeanFrancois Nicaud & Christophe Viudez
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What’s next ?
 The ReMath project
 Natural representation of algebraic expressions in Aplusix
 Tree & Natural representation of algebraic expressions in Aplusix
 Motivations
 Questions raised
 Answers
 Experiments
 Graphical representation of algebraic expressions in Aplusix
 Conclusion
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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
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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
 Representation of the reasoning processes with a tree
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Natural representation of algebraic expressions in Aplusix
Two fundamental feedbacks
 Semantic equivalence between successive steps
 Syntax of the final expression
 Users (students)
 Gain autonomy
 Learn algebra
 Feel happy
 Available
 free for research, http://aplusix.imag.fr/Dir-Vers-Rech
 or see publishers : Chartwell&Yorke (uk), Les éditions
Archimède (fr), MediaDirect (it)
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Tree & Natural representation of algebraic expressions in Aplusix
 Motivations (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
 Motivations (pragmatic)
 ReMath
 Didactician’s ask
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Tree & Natural representation of algebraic expressions in Aplusix
 Questions about the kind of trees:
 internal trees used by Aplusix ?
 special algebraic trees ?
 abstract trees ?
 Questions about the link between tree representations and
natural representations :
 just a way to display object / edit ?
 ill-formed ?
 Mathematical questions :
 ‘-’ operator ?
 ‘(‘ and ‘)’ ?
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Tree & Natural representation of algebraic expressions in Aplusix
 Answers
 authentic objects of our microworld
 abstract trees
 4 modes ( 4 representations)
 natural
 mixed
 free mode
 controlled mode
 Other answers
 -, (): depends on mode
 Demos of the current prototype
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Tree & Natural representation of algebraic expressions 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
 Prototypes
 First in Dec 2006 (for demo)
 Current (June 2007) for colleagues and tests
 Experiments
 planed in France and Italy in late 2007
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Graphical representations of algebraic expressions 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 ?
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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 !)
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