Problems dealing with unknown quantities and two different levels of representing unknowns

Journal for Research in Mathematics Education

Volumn 41, Issue 1, 2010, Pages 52-80

  Filloy, Eugenio ^a   Rojano, Teresa ^a   Solares, Armando ^a  

^a DEPARTAMENTO DE FÍSICA   (Mexico)

Author keywords

Algebra; Aptitudes and abilities; Clinical interviews; Cognitive development; High school, 9 12; Language and mathematics; Problem solving

Indexed keywords

EID: 76949104790     PISSN: 00218251     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (61)

1. Ainley, J. (1996). Purposeful contexts for formal notation in a spreadsheet environment. Journal of Mathematical Behavior, 15, 405-422.
2. Ball, L. (2001). Solving equations: Will a more general approach be possible with CAS? In H. Chick, K. Stacey, J. Vincent, & J. Vincent (Eds.), The future of the teaching and learning of algebra. Proceedings of the 12th ICMI study conference (Vol. 1, pp. 48-52). Melbourne, Australia: The University of Melbourne.
3. Booth, L. R. (1984). Algebra: Children's strategies and errors. Windsor, UK: NFER-Nelson.
4. Cobb, P. (2000). From representation to symbolizing: Introductory comments on semiotics and mathematical learning. In P. Cobb, E. Yackel, & K. McClain (Eds.), Symbolizing and communicating in mathematics classrooms: Perspectives on discourse, tools, and instructional design (pp. 17-36). Mahwah, NJ: Erlbaum.
5. Cortés, A. (1998). Implicit cognitive work in putting word problems into equation form. In A. Olivier & K. Newstead (Eds.), Proceedings of the 22nd conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 208-216). Stellenbosch, South Africa: University of Stellenbosch.
6. Davis, G. E., & McGowen, M. A. (2001, July). Embodied objects and the signs of mathematics. Paper presented at the 25th conference of the International Group for the Psychology of Mathematics Education, Utrecht, The Netherlands.
7. Dettori, G., Garuti, R., & Lemut, E. (2000). From arithmetic to algebraic thinking by using a spreadsheet. In R. Sutherland, T. Rojano, A. Bell, & R. Lins (Eds.), Perspectives on school algebra (pp. 191-207). Dordrecht, The Netherlands: Kluwer.
8. Drijvers, P. H. M. (2003). Learning algebra in a computer algebra environment: Design research on the understanding of the concept of parameter. Utrecht, The Netherlands: CD-ß Press.
9. Drouhard, J. P. (1992). Les écritures symboliques de l'algèbre élémentaire [The symbolic writings of elementary algebra]. Unpublished doctoral dissertation, Université Paris 7.
10. Drouhard, J. P., & Teppo, A. R. (2004). Symbols and language. In K. Stacey, H. Chick, & M. Kendal! (Eds.), The future of the teaching and learning of algebra: The 12th ICMI study (pp. 227-264). Dordrecht, The Netherlands: Kluwer.
11. Dugdale, S. (1993). Functions and graphs-perspectives on students' thinking. In T. A. Romberg, E. Fennema, & T. P. Carpenter (Eds.), Integrating research on the graphical representation of functions (pp. 101-130). Hillsdale, NJ: Erlbaum.
12. Duval, R. (1999). Representation, visual and visualization: Cognitive functions in mathematical thinking. Basic issues for learning. In F. Hitt & M. Santos (Eds.), Proceedings of the 21st annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 3-26). Columbus OH: ERIC Clearinghouse for Science, Mathematics and Environmental Education.
13. Fey, J. T., & Heid, M. K. (with Good, R. A., Sheets, C., Blume, G., & Zbiek, R. M.). (1991, 1999). Concepts in algebra: A technological approach. Dedham, MA: Janson (republished in 1999 version, Chicago: Everyday Learning Corporation).
14. Filloy, E. (1990). PME algebra research: A working perspective. In G. Booker, P. Cobb, & T. N. de Mendicuti (Eds.), Proceedings of the 14th conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. PII.1-PII.33). Morelos, México: Program Committee of the 14th PME Conference.
15. Filloy, E. (1991). Cognitive tendencies and abstraction processes in algebra learning. In F. Furinghetti (Ed.), Proceedings of the 15th conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 48-55). Assisi, Italy: Program committee of the 15th PME Conference.
16. Filloy, E., & Rojano, T. (1984a). La aparición del lenguaje aritmético-algebraico [The emergence of arithmetic-algebraic language]. L'Educazione Matemática, 5 (3), 278-306.
17. Filloy, E., & Rojano, T. (1984b). From an arithmetical to an algebraic thought (a clinical study with 12-13 year olds). In J. M. Moser (Ed.), Proceedings of the 6th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 51-56). Madison, WI: University of Wisconsin.
18. Filloy, E., & Rojano, T. (1985a). Obstructions to the acquisition of elemental algebraic concepts and teaching strategies. In L. Streefland (Ed.), Proceedings of the 9th conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 154-158). Utrecht, The Netherlands: The State University of Utrecht.
19. Filloy, E., & Rojano, T. (1985b). Operating the unknown and models of teaching (A clinical study with 12-13 year olds with a high proficiency in pre-algebra). In S. K. Damarin & M. Shelton (Eds.), Proceedings of the 7th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 75-79). Columbus, OH: Ohio State University.
20. Filloy, E., & Rojano, T. (1989). Solving equations: The transition from arithmetic to algebra. For the Learning of Mathematics, 9 (2), 19-25.
21. Filloy, E., Rojano T., & Puig, L. (2008). Educational algebra: A theoretical and empirical approach. New York: Springer.
22. Filloy, E., Rojano, T., & Solares, A. (2003). Two meanings of the 'equal' sign and senses of comparison and substitution methods. In N. A. Pateman, B. Dougherty, & J. Zilliox (Eds.), Proceedings of the 27th conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 223-229). Honolulu, HI: University of Hawai'i.
23. Filloy, E., Rojano, T., & Solares, A. (2004). Arithmetic/algebraic problem-solving and the representation of two unknown quantities. In M. J. Hemes & A. B. Fuglestand (Eds.), Proceedings of the 28th conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 391-398). Bergen, Norway: Bergen University College.
24. Gallardo, A. (2002). The extension of the natural-number domain to the integers in the transition from arithmetic to algebra. Educational Studies in Mathematics, 49, 171-192.
25. Gallardo, A. (2008). Historical epistemological analysis in mathematical education: Negative numbers and the nothingness. In O. Figueras, J. L. Cortina, S. Alatorre, T. Rojano, & A. Sepúlveda (Eds.), Proceedings of the joint meeting of PME 32 and PME-NA XXX (Vol. 1, pp. 17-29). México: CINVESTAV-UMSNH.
26. Heid, M. K. (1996). A technology-intensive functional approach to the emergence of algebraic thinking. In N. Bednarz, C. Kieran, & L. Lee (Eds.), Approaches to algebra: Perspectives for research and teaching (pp. 239-255). Dordrecht, The Netherlands: Kluwer.
27. Herscovics, R., & Linchevski, L. (1991). Crossing the didactic cut in algebra: Grouping like terms in an equation. In R. G. Underhill (Ed.), Proceedings of the 13th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 196-202). Blacksburg, VA: Division of Curriculum & Instruction, VPI & SU.
28. Hoopes, J. (Ed.). (1991). Peirce on signs. Writings on semiotics by Charles Sanders Peirce. Chapel Hill: The University of North Carolina Press.
29. Kaput, J. J. (1992). Technology and mathematics education. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 515-556). New York: Macmillan.
30. Kieran, C. (1981). Concepts associated with the equality symbol. Educational Studies in Mathematics, 12, 317-326.
31. Kieran, C. (1996). The changing face of school algebra. In C. Alsina, J. M. Alvarez, B. Hodgson, C. Laborde, & A. Pérez (Eds.), Eighth International Congress on Mathematical Education: Selected lectures (pp. 271-290). Seville: S. A. E. M. Thales.
32. Kieran, C. (2006). Research on the learning and teaching of algebra. In A. Gutiérrez & P. Boero (Eds.), Handbook of research on the psychology of mathematics education: Past, present and future (pp. 11-49). Rotterdam, The Netherlands: Sense Publishers.
33. Kieran, C., Boileau, A., & Garançon, M. (1996). Introducing algebra by means of a technology-supported, functional approach. In N. Bednarz, C. Kieran, & L. Lee (Eds.), Approaches to algebra: Perspectives for research and teaching (pp. 257-293). Dordrecht, The Netherlands: Kluwer.
34. Kieran, C., & Filloy, E. (1989). El aprendizaje del álgebra escolar desde una perspectiva psicológica [The learning of scholastic algebra from a psychologial perspective] (L. Puig, Trans.). Enseñanza de las Ciencias, 7, 229-240.
35. Kieran, C., & Saldanha, L. (2005). Computer algebra systems (CAS) as a tool for coaxing the emergence of reasoning about equivalence of algebraic expressions. In H. L. Chick & J. L. Vincent (Eds.), Proceedings of the 29th conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 193-200). Melbourne, Australia: University of Melbourne.
36. Kirshner, D. (1987). Linguistic analysis of symbolic elementary algebra. Unpublished doctoral dissertation, University of British Columbia.
37. Kirshner, D. (1989). The visual syntax of algebra. Journal for Research in Mathematics Education, 20, 274-287.
38. Kuchemann, D. (1981). Algebra. In K. Hart (Ed.), Children's understanding of mathematics: 11-16 (pp. 102-119). London: John Murray.
39. Laborde, C. (1990). Language and mathematics. In J. Kilpatrick & P. Nesher (Eds.), Mathematics and cognition (pp. 4-19). Cambridge: Cambridge University Press.
40. Linchevski, L., & Herscovics, N. (1996). Crossing the cognitive gap between arithmetic and algebra: Operating on the unknown in the context of equations. Educational Studies in Mathematics, 30, 39-65.
41. Matz, M. (1980). Towards a computational theory of algebraic competence. Journal of Mathematical Behavior, 3 (1), 93-166.
42. Mokros, J. R., & Tinker, R. F. (1987). The impact of microcomputer-based labs on children's ability to interpret graphs. Journal of Research in Science Teaching, 24, 369-383.
43. Nemirovsky, R., Kaput, J. J., & Roschelle, J. (1998). Enlarging mathematical activity from modelling phenomena to generating phenomena. In A. Oliver & K. Newstead (Eds.), Proceedings of the 22nd conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 287-294). Stelienbosch, South Africa: Program Committee of the 22nd PME Conference.
44. Peirce, C. S. (1931-1958). Collected papers of Charles Sanders Peirce (C. Hartshorne & P. Weiss (Vols. 1-6) & A. Burks (Vols. 7-8), Eds). Cambridge, MA: The Belknap Press of Harvard University Press.
45. Peirce, C. S. (1982). Writings of Charles S. Peirce: A chronological edition (Vol. 3, C. J. W. Kloesel, Ed.). Bloomington, IN: Indiana University Press.
46. Presmeg, N. (2005). Metaphor and metonymy in processes of semiosis in mathematics education. In M. H. G. Hoffmann, J. Lenhard, & F. Seeger (Eds.), Activity and signs-Grounding mathematics education (pp. 105-115). New York: Springer.
47. Puig, L. (1994a). El De Numeris Datis de Jordanus Nemorarius como sistema matemático de signos [The De Numeris Datis from Jordanus Nemorarius considered as a mathematical sign system]. Mathesis, 10, 47-92.
48. Puig, L. (1994b). Semiótica y matemáticas [Semiotics and mathematics] (Vol. 51). Valencia: Episteme, col. Eutopias.
49. Puig, L. (2003). Signos, textos y sistemas matemáticos de signos [Signs, texts and mathematical sign systems]. In E. Filloy (Ed.), Matemática educativa: Aspectos de la investigación actual [Mathematics education: Aspects of contemporary research] (pp. 174-186). México: Fondo de Cultura Económica and Cinvestav.
50. Puig, L., & Rojano, T. (2004). The history of algebra in mathematics education. In K. Stacey, H. Chick, & M. Kendall (Eds.), The future of the teaching and learning of algebra: The 12th ICMI study (pp. 189-223). New York: Kluwer.
51. Radford, L. (2000). Signs and meanings in students' emergent algebraic thinking: A semiotic analysis. Educational Studies in Mathematics, 42, 237-268.
52. Rojano, T. (1994). The case of pre-symbolic algebra and the operation of the unknown. In J. P. da Ponte & J. Filipe (Eds.), Proceedings of the 18th conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 125-128). Lisbon, Portugal: Program Committee of the 18th PME Conference.
53. Rojano, T. (1996). The role of problems and problem solving in the development of algebra. In N. Bednarz, C. Kieran, & L. Lee (Eds.), Approaches to algebra: Perspectives for research and teaching (pp. 55-62). Dordrecht, The Netherlands: Kluwer.
54. Rojano, T. (2004). Local theoretical models in algebra learning: A meeting point in mathematics education. In D. E. McDougall & J. A. Ross (Eds.), Proceedings of the 26th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 37-53). Toronto, Canada: Ontario Institute for Studies in Education of the University of Toronto.
55. Rojano, T., & Sutherland, R. (2001). Arithmetic world-algebraic world. In H. Chick, K. Stacey, J. Vincent, & J. Vincent (Eds.), Proceedings of the 12th ICMI study conference: The future of the teaching and learning of algebra (Vol. 2, pp. 515-522). Melbourne, Australia: The University of Melbourne.
56. Sfard, A., & Linchevski, L. (1994). The gains and pitfalls of reification-the case of algebra. Educational Studies in Mathematics, 26, 191-228.
57. Stacey, K., & MacGregor, M. (1997). Multiple referents and shifting meanings of unknowns in students' use of algebra. In E. Pehkonen (Ed.), Proceedings of the 21st conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 190-197). Lahti, Finland: University of Helsinki.
58. Talens, J., & Company, J. M. (1984). The textual space: On the notion of text. The Journal of the Midwest Modern Language Association, 17 (2), 24-36.
59. Vlassis, J. (2002). The balance model: Hindrance or support for the solving of linear equations with one unknown. Educational Studies in Mathematics, 49, 341-359.
60. Yerushalmy, M., & Chazan, D. (2002). Flux in school algebra: Curricular change, graphing, technology, and research on student learning and teacher knowledge. In L. D. English (Ed.), Handbook of international research in mathematics education (pp. 725-755). Mahwah, NJ: Erlbaum.
61. Yerushalmy, M., & Shternberg, B. (2001). Charting a visual course to the concept of function. In A. A. Cuoco & F. R. Curcio (Eds.), The roles of representation in school mathematics: 2001 Yearbook of the National Council of Teachers of Mathematics (pp. 251-268). Reston, VA: NCTM.