From arithmetical thought to algebraic thought: The role of the "variable"

Educational Studies in Mathematics

Volumn 71, Issue 1, 2009, Pages 19-41

  Malisani, Elsa ^a   Spagnolo, Filippo ^a  

^a UNIVERSITY OF PALERMO   (Italy)

Author keywords

Arithmetic algebra transition; Concept of variable; Translation from algebraic language into natural language; Unknown; Variable in functional relation

Indexed keywords

EID: 64949141652     PISSN: 00131954     EISSN: 15730816     Source Type: Journal    
DOI: 10.1007/s10649-008-9157-x     Document Type: Article

References (49)

1. Arzarello, F., Bazzini, L., & Chiappini, G. (1994). L'Algebra come strumento di pensiero. Analisi teorica e considerazioni didattiche. Progetto Strategico del C.N.R. Pavia: Dipartimento di Matematica: Università di Pavia, Quaderno n. 6.
2. Bardini, C., Radford, L., & Sabena, C. (2005). Struggling with variables, parameters, and indeterminate objects or how to go insane in mathematics. Proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp.129-136). Melbourne, Australia.
3. Brousseau, G. (1997). Theory of didactical situations in mathematics. N. Balacheff, M. Cooper, R. Sutherland & V. Warfield, Eds. and Trans.). Dordrecht: Kluwer.
4. Carraher, D., Schliemann, A., & Brizuela, B. (2001). Can young students operate on unknowns? Proceedings of the 25th Conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 130-140). Utrecht, The Netherlands.
5. Chevallard, Y. (1989). Arithmetique, algebre, modelisation. Aix, France: IREM.
6. Collins English Dictionary. (1991). Bologna, Italy: Zanichelli.
7. Eisenberg, T., & Dreyfus, T. (1991). On the reluctance to visualize in mathematics. In W. Zimmermann, & S. Cunningham (Eds.), Visualization in teaching and learning mathematics (pp. 25-37). Washington, DC: Mathematical Association of America.
8. Freudenthal, H. (1983). Didactical phenomenology of mathematical structures. Dordrecht: Reidel.
9. Furinghetti, F., & Somaglia, A. (1994). Functions in algebraic and graphical environments. Proceedings of 46th Commission Internationale pour l'Étude et l'Amélioration de l'Enseignement des Mathématiques. Toulouse, France.
10. R. Gras 1997 Metodologia di analisi di indagine Quaderni di Ricerca in Didattica (G.R.I.M.) 7 99 109
11. R. Gras 2000 Les fondements de l'analyse statistique implicative Quaderni di Ricerca in Didattica (G.R.I.M.) 9 189 209
12. Gras, R., et al. (1996). L'implication statistique (Nouvelle méthode exploratoire de données). Grenoble, France: La Pensée Sauvage.
13. Gras, R., Couturier, R., Guillet, F., & Spagnolo, F. (2005). Extraction de règles en incertain par la méthode statistique implicative. Comptes rendus des 12èmes Rencontres de la Société Francophone de Classification, (pp. 148-151). Montréal, Canada: UQAM.
14. Gras, R., Diday, E., Kuntz, P., & Couturier, R. (2001). Variables sur intervalles et variables-intervalles en analyse implicative. Actes du 8ème Congrès de la Société Francophone de Classification de Pointe à Pitre (pp. 166-173).
15. G. Harel L. Sowder 2005 Advanced mathematical-thinking at any age: its nature and its development Mathematical Thinking and Learning 7 1 27 50
16. G. Harel D. Tall 1991 The general, the abstract, and the generic in advanced mathematics For the Learning of Mathematics 11 38 42
17. E. Harper 1987 Ghosts of Diophantus Educational Studies in Mathematics 18 1 75 90
18. N. Herscovics L. Linchevski 1994 A cognitive gap between arithmetic and algebra Educational Studies in Mathematics 27 1 59 78
19. C. Hoyles R. Noss R. Adamson 2002 Rethinking the microworld idea Journal of Educational Computing Research 27 1-2 29 53
20. Janvier, C., Charbonneau, L., & Cotret, S. (1989). Obstacles épistémologiques à la notion de variable: Perspectives historiques. Construction des savoirs, obstacles et conflits, (pp. 64-75). Montréal, Canada: CIRADE-Université de Québec.
21. Küchemann, D. (1980). The understanding of generalised arithmetic by secondary school children. (Unpublished doctoral dissertation, Chelsea College, University of London).
22. Kücherman, D. (1981). Algebra. In K. M. Hart (Ed.), Children's understanding of mathematics: 11-16 (pp. 102-119). London: John Murray.
23. Lagrange, J. P., & Chiappini, G. (2007). Integrating the learning of algebra with technology at the European level: two examples in the remath project. Proceedings of the Fifth Congress of the European Society for Research in Mathematics Education (pp. 903-912). Larnaca, Cyprus.
24. L. Linchevski 1995 Algebra with numbers and arithmetic with letters: A definition of pre-algebra The Journal of Mathematical Behavior 14 113 120
25. E. Malisani 1996 Storia del pensiero algebrico fino al cinquecento. Costruzione del simbolismo e risoluzione di equazioni Quaderni di Ricerca in Didattica (G.R.I.M.) 6 26 77
26. E. Malisani 1999 Los obstáculos epistemológicos en el desarrollo del pensamiento algebraico. Visión histórica Revista IRICE 13 105 132
27. Malisani, E. (2006). The concept of variable in the passage from the arithmetic language to the algebraic language in different semiotic contexts (Doctoral Dissertation, Comenius University of Bratislava, 2006). Quaderni di Ricerca in Didattica (G.R.I.M.), 16(1). Available from: http://math.unipa.it/ ~grim/ quad-16-suppl-1.htm ).
28. Malisani, E., Scimone, A., & Spagnolo, F. (2008). Modelling by statistic in research of mathematics education. In F. Spagnolo, R. Gras, E. Suzuki, & F. Guillet (Eds.), Statistical implicative analysis: theory and applications (pp. 247-276). Berlin: Springer.
29. Malisani, E., & Spagnolo, F. (2005). Difficulty and obstacles with the concept of variable. Proceedings of 57th Commission Internationale pour l'Étude et l'Amélioration de l'Enseignement des Mathématiques (pp. 226-231). Piazza Armerina, Italy.
30. Matz, M. (1982). Towards a process model for high school algebra errors. In D. Sleeman, & J. S. Brown (Eds.), Intelligent tutoring systems (pp. 25-50). London: Academic Press.
31. Nesselman, G. H. F. (1842). Die algebra der Griechen (pp. 302-306). Berlin.
32. M. Panizza P. Sadovsky C. Sessa 1999 La ecuación lineal con dos variables: entre la unicidad y el infinito Ensenañza de las Ciencias 17 3 453 461
33. Radford, L. (1996). The roles of geometry and arithmetic in the development of algebra: historical remarks form a didactic perspective. In N. Bednarz, C. Kieran, & L. Lee (Eds.), Approaches to algebra. Perspectives for research and teaching (pp. 39-53). Dordrecht, The Netherlands: Kluwer.
34. Radford, L. (2002). On heroes and the collapse of narratives: a contribution to the study of symbolic thinking. Proceedings of the 26th Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 81-88). Norwich, UK.
35. L. Radford 2003 Gestures, speech and the sprouting of signs Mathematical Thinking and Learning 5 1 37 70
36. Radford, L. (2006). Algebraic thinking and the generalization of patterns: a semiotic perspective. Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 2-21). Prague, Czech Republic.
37. P. Sadovsky C. Sessa 2005 The adidactic interaction with the procedures of peers in the transition from arithmetic to algebra: a milieu for the emergence of new questions Educational Studies in Mathematics 59 85 112
38. A. Sfard 1991 On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin Educational Studies in Mathematics 22 1 1 36
39. Sfard, A. (1992). The development of algebra. Confronting historical and psychological perspectives. Algebra Working Group of Seventh International Congress on Mathematical Education. Quebec, Canada.
40. A. Sfard L. Linchevski 1994 The gains and the pitfalls of reification-the case of algebra Educational Studies in Mathematics 26 191 228
41. Sierpinska, A. (1994). Understanding in mathematics. London: Falmer Press.
42. Spagnolo, F. (1998). Insegnare le Matematiche nella scuola secondaria. Florence: La Nuova Italia.
43. F. Spagnolo 2006 La modélisation dans la recherche en didactiques des mathématiques: Les obstacles épistémologiques Recherches en Didactiques des Mathématiques 26 3 337 379
44. Spagnolo, F., Gras, R., Suzuki, E., & Guillet, F. (Eds.). (2008). Statistical implicative analysis: Theory and applications. Berlin: Springer.
45. M. Trigueros A. Reyes S. Ursini R. Quintero 1996 Diseño di un cuestionario de diágnostico acerca del manejo del concepto de variable en el álgebra Ensenañza de las Ciencias 14 3 351 363
46. Usiskin, Z. (1988). Conceptions of school algebra and uses of variables. In A. F. Coxford, & A. P. Shulte (Eds.), The ideas of algebra, K-12 (pp. 8-19). Reston, VA: NCTM.
47. S. Vinner 1989 Avoidance of visual considerations in calculus students FOCUS on Learning Problems in Mathematics 11 2 149 156
48. Wagner, S. (1981). An analytical framework for mathematical variables. Proceedings of the 5th Conference of the International Group for the Psychology of Mathematics Education (pp. 165-170). Grenoble, France.
49. S. Wagner 1983 What are these things called variables Mathematics Teacher 76 7 474 479