A FOUR-DIMENSIONAL MODEL OF ALGEBRAIC SYMBOLIC SKILL Jean-Philippe DROUHARD1 GECO2 I.U.F.M. de Nice3 We present a four-dimensional model of knowledge involved in algebraic symbolic skill. The first dimension is linguistic, the second related to “meta” levels and the third related to the generation of objects. Most parts of this model are involved when solving symbolic algebraic tasks, so it provides a possible guide to understand students’ difficulties.

1.

INTRODUCTION AND BACKGROUND

For the last ten years, algebra learning studies have been widely increasing. Most promising features of this development come from domain-specific theories, based upon extensive analyses of what is taught algebra. In particular, many researchers looked at some dimensions of algebra learning processes, for instance: • time of learning processes (Mercier 1992) • “ecology” of algebraic knowledge (Assude 1991, Chevallard 1984, Tonnelle 1979) • natural/formal language in algebra (Arzarello, Bazzini & Chiappini 1992a, 1992b, Kirshner & McDonald 1992, Kirshner 1992, Norman 1987) • arithmetic/algebric relation (Arzarello 1991, Arzarello, Bazzini & Chiappini 1992a, 1992b, Chevallard 1984, 1989, 1990, Linchevski & Sfard 1991, Norman 1987, Sfard & Linchevski, 1992). • psychology of algebraic concepts learning (Vergnaud, Cortes & Favre-Artigue 1988, Vergnaud 1991) … During this period, we focused our attention on linguistic dimension. We described first syntax of elementary Algebra Symbolic Expressions (“ASEs”). Now we look at semantics of these ASEs (Drouhard 1991, 1992a, 1992b). Concerning the semantics, the major point is that we look at the meaning of expression instead of concepts. In this paper, following Frege’s distinction between sense and denotation, we focus our attention on the sense component of meaning. For this purpose, we aim to describe the way students not just manipulate ASEs, but how they may answer to the question “why”

1

Internet: [email protected]

2

Association pour le Développement du Génie COgnitif 1bis rue Charles de Foucault, 06000 NICE FRANCE

3

Institut Universitaire de Formation des Maîtres (♠ Teachers’ Training University) 89 av. George-V, 06000 NICE FRANCE

1

Algebraic Processes and the Role of Symbolism / Working Conference / London, Sept. 1993. as well. In other words, we want this model to help us to identify students’ local knowledge (Léonard & Sackur 1991) in algebra. This looks necessary, for this model to be plausible (relevant, realistic, human-like), in a Computer Aided Learning perspective. 2. 2.1

FOUR DIMENSIONS

Linguistic dimensions

We “linearize” the Fregean triangle: Sense

Denotation Expression

and we split denotation into “actual objects” (e. g. real numbers) and “virtual objects” (e. g. “unknown” real numbers). We obtain the following scale:

Actual objects Denotation Virtual objects

Sense

Expressions

2.2.

“meta” dimension

“Why?” may be synonymous of “How?”. For example, “Why x(x-2)?” may be understood as “How” (=by what means, with what transformations) “did you obtain it from x2-2x ?” Then, we were led to combine the syntax/semantic dimension with a “meta” dimension, so as to integrate transformations on ASEs. 2

Algebraic Processes and the Role of Symbolism / Working Conference / London, Sept. 1993. Moreover, in such a bi-dimensional model, transformations themselves are governed by tactics and strategies which may be considered as meta-transformation. It is related to another meaning of “Why?” which is similar to “Wherefore?” (=for what purpose?).

3

Algebraic Processes and the Role of Symbolism / Working Conference / London, Sept. 1993.

2.2. “Analysis/Synthesis” dimension However “Why?” may also mean: “What gives you the right to write x2-2x ?” This question is on legitimacy of transformations, not on their choice (as “how?”). We introduced then the relation between objects and their construction: construction of real numbers legitimates the field axioms of R, and therefore the licit transformations of Expressions. 2.4.

“frame” dimension

“Why?” = “from what?” Example: “I wrote that because I remembered Pascal Triangle...” We propose to consider: Linguistic frames (where representations of objects are linguistic expressions) among which: Natural (mathematical) Language frames: arithmetic frames Symbolic language frames: abstract arithmetic frames (ex. numerical fractions) algebra for arithmetic frames (ex. even/odd numbers) the abstract algebra frame (where are equations...) Semiotic frames (ex: geometry)

Actual objects Denotation Virtual objects Sense Litteral exp.

Numerical exp.

Representation

arithmetics

algebra for arithmetics

4

abstract algebra

Other frame

Algebraic Processes and the Role of Symbolism / Working Conference / London, Sept. 1993. 3.

THE ABSTRACT ALGEBRA FRAME

5

Algebraic Processes and the Role of Symbolism / Working Conference / London, Sept. 1993.

1. The central part of the model is the Algebraic Symbolic Expression (ASE). Example: x2 - 2x.

1 3

2

4

5

6

2. No grammar can straight generate ASE with all graphological characteristics (smaller and higher exponents, vinculums, wide spacing etc.). One needs Intermediate Linear Forms (Drouhard 1992, Kirshner 1987). Example: x^2-2x.

3. Intermediate Linear Forms are generated by a grammar (ibid.) which associate them to a tree structure:

x^2-2x Many students fail to identify structures (Norman 1987) and the rules and conventions used in algebra (Bliss & Sakonidis 1988). Computer representations may help them (Larkin 1989, Thompson 1987, Thompson & Thompson 1989).

4. Manipulations of ASE are modelized by transformations of Intermediate Linear Forms (Gélis 1993). Example: x2 - 2x. x (x - 2) x^2-2x. ∅ x(x-2)

5. Transformations are governed by meta-rules (Bundy & Welham 1979) called tactics. Example: To factorize by common factors: Identify a common factor (“cf”) Transform each term in a product where the cf appears etc.

6. Tactics themselves are ruled by meta-tactics called strategies. Example: Try first to factorize by a common factors tactic, and if that fails, try find a difference of two squares.

6

Algebraic Processes and the Role of Symbolism / Working Conference / London, Sept. 1993.

7

9 10

8

11

1 3

2

4

5

6

7. Usually, ASE are said representing (unknown) real numbers. However, the property of being “unknown” is not strictly mathematical4 as being prime or odd (one cannot define a set U of unknown numbers and a set K of “known” numbers).

8. Real numbers are generated by ordinal (or cardinal) construction and Cauchy series (or Dedekind “cuts”).

9. Transformations of ASE correspond to operations (+, -, ∞ …) and relations (<, >, =) between real numbers.

10. Operations and relations are defined by mathematical construction too.

11. Operations and relations on real numbers are ruled themselves by field (or at east “shellettes”5) properties. Example: distributivity. EXTENSION (Arzarello, Bazzini & Chiappini 1992): 7 8

9 10

11

12

3

2

1 4

5

6

12. It seems not fully relevant to consider ASE as representing unknown numbers. We prefer consider that they denote functions (we call them denotations, Frege’s “Bedeutung”). Example: the denotation of “x2-2x” is the function: δ(“x2-2x”):

{~R N∅ ~R ,(uj)j N ∅ ui2 - 2ui}

Denotations may be equally represented in the formalism of “λexpressions” (ibid.): δ(“x2-2x”) = λx . x2 - 2x

4

«unknown» is an “epimathematical” term (Lacombe)

5

structures with internal and external multiplication and exponentiation (see Drouhard 1992a)

7

Algebraic Processes and the Role of Symbolism / Working Conference / London, Sept. 1993.

7

9 10

8

11

13. a denotational component: an ASE E is defined by paradigmatic opposition with all other ASEs having same denotation.

12 13 14

15

1 3

2

4

5

INTENSION (ibid.): ASEs have also a sense (Frege’s “Sinn”) which has:

6

14. a transformational component: within a paradigm we may define a pseudo-distance d(E, E’) based on the number and the “evidence” of transformations to apply for linking E to E’ (Gélis 1993). Sfard & Linchevski (1992) showed how students fail to recognize the equivalence of “remote” expressions. This component is related to the sense, as to the question: «why “x(x-2)”?» one may answer: «because I obtained it from “x2-2x”.» (“why”=“how”). 15. a pragmatic component: The sense of E is related to the strategies which may be applied to E. For instance, you can apply trinomial formulae to “x2-2x”, but if you search for a root, “x(x-2)” is better. To the question «why “x(x-2)”?» one may answer: «because that allows me using AB = 0 so as to find the root.» (“why”=“for what”). This component is related to “connotation”.

Within this model, some shemas are commutative. For instance, if you transform an expression E into an expression t(E), the denotation δ(E) of E is equal to the denotation δ(t(E)): δ(E) = δ(t(E)) ≠ ≠ E ∅ t(E) All symbolic algebra relies on this fundamental principle! However, most shemas are not commutative. For instance, operations on real numbers are not homomorphic with transformations, neither field/shelette rules with tactics, though there is a relationship between correct transformations and field/shelette rules. Students have to gradually master the levels and the relations within the model.

8

Algebraic Processes and the Role of Symbolism / Working Conference / London, Sept. 1993.

4.

OTHER FRAMES

ASEs may receive denotations in various frames: geometry, analytic geometry, pseudoconcrete world, etc. Example: x2 - 2x may denote a parabola in the analytic geometry setting. These desnotations may also receive other non-linguistic representations (graphical ...). Spreadsheets settings (Sutherland 1992) are particularly interesting: representing a formula as a pair of columns focuses on the functional aspect of numerical denotation.

formula

Litteral exp.

abstract algebra

Spreadsheet frame (formulae)

pair of columns

Spreadsheet frame (cells)

Transformations of ASEs are linked to transformations of related denotations; computers may show dynamically this linkage (Kaput 1989). Tactics and strategies govern (in relation with connotation) transformations of related representations. For instance, to solve a geometrical problem you may need a geometric reasonning (with its specific strategies). 3.

PROVISIONAL CONCLUSION

Abstract algebraic denotation is the keystone of this model. We claim it is the exact difference between symbolic computation and abstract algebra. “Pseudoformalist” attitudes (Linchevski & Sfard 1991) (= “rigid designation”) may be interpreted as a lack of understanding of the role abstract algebraic denotation plays in algebra. 4. REFERENCES ARZARELLO F. (1991): “Procedural and relational Aspects of Algebraic Thinking”, Proceedings of PME XIV (Assisi), FURINGHETTI F., Ed., Dipartimento di Matematica dell’Università di Genova, Italy. ARZARELLO F., BAZZINI L. & CHIAPPINI GP. (1992a): “L’algebra come strumento di pensiero: stato e metodologie del la ricerca - tendenze e prospettive”, IX Seminario Nazionale di ricerca in didattica della matematica, Pisa, Italy. ARZARELLO F., BAZZINI L. & CHIAPPINI GP. (1992b): “Intensional semantics as a tool to analyse algebraic thinking”, to be published in Rendiconti del Seminario Matematico dell’Università e del Politecnico di Torino, University of Turin, Italy. 9

Algebraic Processes and the Role of Symbolism / Working Conference / London, Sept. 1993. BLISS J. & SAKONIDIS H. (1988): “Teachers’ written explanations to pupils about algebra”, Proceedings of the XIIth PME Conference, Borbás A. (Ed.), O.O.K., Veszprém, Hungary. BUNDY A. & WELHAM B. (1979): Using meta-level descriptions for selective application of multiple rewrite rules in algebraic manipulation, DAI research Paper N°121 Edinburgh (ou DAI Working paper N°55, ou Artificial Intelligence Journal 1979) CHEVALLARD Y. (1984): “Le passage de l’arithmétique à l’algèbre dans l’enseignement des mathématiques au collège”, première partie, Petit x , n° 5, IREM de Grenoble. CHEVALLARD Y. (1989): “Le passage de l’arithmétique à l’algèbre dans l’enseignement des mathématiques au collège”, deuxième partie, Petit x , n° 19, IREM de Grenoble. CHEVALLARD Y. (1990): “Le passage de l’arithmétique à l’algèbre dans l’enseignement des mathématiques au collège”, troisième partie, Petit x , n° 23,b IREM de Grenoble. DROUHARD J-Ph. (1991): “Reaction to the «Proposal for a Working Group»”, PME XV (Assisi), Working Group “Algebraic Processes and Structures” , Draft Paper. DROUHARD J-Ph. (1992a): “Shells, Shellettes and Free Shells: Towards a Model of Elementary Algebraic Knowledge Representation”, to be published in Rendiconti del Seminario Matematico dell’Università e del Politecnico di Torino, University of Turin, Italy. DROUHARD J-Ph. (1992b): Les expressions symboliques de l’algèbre élémentaire, Thèse de doctorat, Université Paris 7. GELIS J-M. (1993): “Un cadre général pour une modélisation cognitive et computationnelle de l’algèbre”, Proceedings of RFIA ‘94. (?) KAPUT J. (1989): “Linking Representations in the Symbol System of Algebra”, Research Issues in the Learning and Teaching of Algebra, S. Wagner & c. Kieran (Eds.), Lawrence Erlbaum Associates & National Council of Teachers of Mathematics, Reston (Va). KIERAN C. (1991): “A Procedural-Structural Perpective on Algebra Research”, Proceedings of PME XIV (Assisi), FURINGHETTI F., Ed., Dipartimento di Matematica dell’Università di Genova, Italy. KIRSHNER D. (1987): Linguistic analysis of symbolic elementary algebra, Unpublished Doctoral Dissertation, University of British Columbia. KIRSHNER D. (1989): “The Visual Syntax of Algebra”, Journal for Research in Mathematics Education, 20-3, 276-287. KIRSHNER D. & McDonald J. (1992): “Translating English Sentences to Algebraic Notation: methodological and epistemological considerations”,Journal for the Learning Science, in press. KIRSHNER D. (1992): “Syntactically based methods of theory testing: in the shadows of the conscious”, paper presented to the Working Group on Methodologies for Research in Mathematics Education, ICME-7, Québec, Canada. KIRSHNER D. (1993): The Structural Algebra Option: A Discussion Paper, Dept. of Education and Curriculum, Louisiana State University, Baton Rouge (La). LARKIN (1989): “Robust Performance in Algebra: the Role of the Problem Representation”, Research Issues in the Learning and Teaching of Algebra, S. Wagner & c. Kieran (Eds.), Lawrence Erlbaum Associates & National Council of Teachers of Mathematics, Reston (Va). 10

Algebraic Processes and the Role of Symbolism / Working Conference / London, Sept. 1993. LEONARD F. & SACKUR C. (1991): “Connaissances locales et triple approche, une méthodologie de recherche”, Recherches en Didactique des Mathématiques, 10, 2.3, 205-240. LINCHEVSKI L. & SFARD A. (1991): “Rules without reason as processes without objects: The case of equations and inequalities”, Proceedings of PME XIV (Assisi), FURINGHETTI F., Ed., Dipartimento di Matematica dell’Università di Genova, Italy. MERCIER A. (1992): L’élève et les contraintes temporelles de l’enseignement, un cas en calcul algébrique, Thèse de Doctorat, Université Bordeaux 1. NORMAN F. A. (1987): “A Psycholinguistic Perspective on Algebraic Language”, Proceedings of PME XI, J. Bergeron, N. Herscovics & C. Kieran (Eds.), Montréal. SFARD A. & LINCHEVSKI L. (1992): “Equations and Inequalities: Processes without Objects?”, to be published in Rendiconti del Seminario Matematico dell’Università e del Politecnico di Torino, University of Turin, Italy. SUTHERLAND R. (1992): “The Influence of Teaching on the Practice of School Algebra: What can we Learn from Work with Computers?”, to be published in Rendiconti del Seminario Matematico dell’Università e del Politecnico di Torino, University of Turin, Italy. THOMPSON P. W. & THOMPSON A. G. (1987): “Computer representation of structures in algebra”, Proceedings of PME XI, J. Bergeron, N. Herscovics & C. Kieran (Eds.), Montréal. THOMPSON P. W. (1989): “Artificial Intelligence, Advanced Technology and Learning and Teaching Algebra”, Research Issues in the Learning and Teaching of Algebra, S. Wagner & c. Kieran (Eds.), Lawrence Erlbaum Associates & National Council of Teachers of Mathematics, Reston (Va). TONNELLE J. (1979): Le monde clos de la factorisation au premier cycle, Mémoire de DEA de Didactique des Mathématiques, IREM de Bordeaux et IREM d'Aix-Marseille. VERGNAUD G. (1991): “La théorie des champs conceptuels”, Recherches en Didactique des Mathématiques , 10/2-3, 133-169. VERGNAUD G., CORTES A. & FAVRE-ARTIGUE P. (1988): « Introduction de l'algèbre auprès de débutants faibles: problèmes épistémologiques et didactiques », Didactique et acquisition des connaissances scientifiques - Actes du Colloque de Sèvres - Mai 87, La Pensée Sauvage, Grenoble.

11