Abstract
This paper reports on a case study of a 16-year-old student working on transformations of functions in a computer-based, multi-representational environment. The didactic approach to reflections, translations and stretches began with visualization exercises, and then was extended to investigate the implications of visual changes in data points, and subsequently, in algebraic symbolism. A detailed analysis of the student's work during the transition from the use of visualization and analysis of discrete points to the use of algebraic symbolism is presented. Two new semiotic forms are introduced as an alternative kind of algebraic symbolism and as a means to facilitate the transition to f(x) notation: covariational equations and the horseshoe display for transformations. The implications of this case for the redesign and modification of the software are discussed.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Borba, M. C.: 1993, Students' understanding of transformations of functions using multirepresentational software. Doctoral Dissertation, Cornell University, U.S.A.. Published in 1994-Lisbon, Portugal: Associação de Professores de Matemática.
Borba, M. C.: 1994a, ‘A model for students' understanding in a multi-representational environment’, Proceedings of the XVIII Psychology of Mathematics Education (Vol. II, pp. 104–111), Lisbon, Portugal: University of Lisbon.
Borba, M. C.: 1994b, ‘Computadores, Representações Múltiplas e a Construção de Idéias Matemáticas’, BOLEMA — Boletim de Educação Matemática 9 (Especial 3), 83–101, São Paulo, Brazil: UNESP.
Borba, M. C.: 1995a, ‘Overcoming Limits of software tools: A student's solution for a problem involving transformation of functions’, in L.Meira et al (Editors) Proceedings of the XVIII Psychology of Mathematics Education (Vol. II, pp. 248–255), Recife, Brazil: UFPE.
Borba, M. C.: 1995b, ‘What's new in teaching mathematics: computers in the classroom’, Clearing House 68(6), 333–334, D.C., U.S.A.
Borba, M. C. and Confrey, J.: 1992, ‘Transformations of functions using multirepresentational software’, Proceedings of the XVI Psychology of Mathematics Education (Vol. III, pp. 149), Durham: University of New Hampshire.
Cobb, P. and Steffe, L.: 1983, ‘The constructivist reseacher as teacher and model builder’, Journal for Research in Mathematics Education 14(2), 83–94, U.S.A.
Confrey, J.: 1991a, ‘Function Probe© [computer program]’, Santa Barbara, Intellimation Library for the Macintosh.
Confrey, J.: 1991b, ‘Learning to listen: A student's understanding of powers of ten’, in E. von Glasersfeld (ed.), Radical Constructivism In Mathematics Education, Kluwer Academic Publishers, Dordrecht, Holand, pp. 111–138.
Confrey, J.: 1992, ‘Using computers to promote students' inventions of the function concept’, in S.Malcom, L.Roberts, and K.Sheingold (eds), This Year In School Science 1991, Washington D.C: American Association for the Advancement of Science, pp. 141–174.
Confrey, J.: 1993a, ‘The role of technology in reconceptualizing functions and algebra’, in J. R.Becker and B. J.Pence (eds.), Proceedings of the XV Psychology of Mathematics Education-NA (Vol. I, pp. 47–74), San Jose, U.S.A.: Center for Mathematics and Computer Science Education at San Jose State University.
Confrey, J.: 1993b, ‘Diversity, tools, and new approaches to teaching functions’, paper presented at the China-Japan-U.S. Meeting on Mathematics Education, Shanghai: East-China Normal University.
Confrey, J.: 1994a, ‘Six approaches to transformation of functions using multirepresentational software’, Proceedings of the XVIII Psychology of Mathematics Education (Vol. I, pp. 47–74), Lisbon: Lisbon University.
Confrey, J.: 1994b, ‘Voice and Perspective: Hearing epistemological innovation in students' words’. In N. Bednarz, M. Larochelle, and J. Desautels (eds.) Revue des Sciences de l'education, Special Issue: Constructivism in Education 20(1), 115–133.
Confrey, J., Rizutti, J., Scher, D., and Piliero, S.: 1991, ‘Documentation of Function Probe’, unpublished manuscript, Cornell University, Ithaca, NY.
Dennis, D. and Confrey, J.: 1995, ‘Functions of a curve: Leibniz's original notion of functions and its meaning for the parabola’, The College Mathematics Journal, 26(3).
Eisenberg, T. and Dreyfus, T.: 1991, ‘On visualizing functions transformations’, Technical Report, Rehovot: The Weizmann Institute of Science.
Goldenberg, E. P., Harvey, W., Lewis, P., Umiker, R., West, J., and Zodhiates, P.: 1988, ‘Mathematical, technical, and pedagogical challenges in the graphical representation of functions’, Technical Report, Cambridge: Educational Technology Center.
Goldenberg, E. P. and Kliman, M.: 1990, ‘What you see is what you see’, Technical Report, Newton: Educational Technology Center.
Kirshner, D.: 1989, ‘The visual syntax of algebra’, Journal for Research in Mathematics Education 20(3), 274–287, U.S.A.
Rizzuti, J.: 1991, ‘Students' conceptualizations of functions: Effects of a pedagogical approach involving multiple representations’, Doctoral dissertation, Cornell University, Ithaca, NY.
Smith, E. and Confrey, J.: 1992, ‘Using a dynamic software tool to teach transformations of functions’, Proceedings of the fifth Annual Conference on Technology in Collegiate Mathematics (pp. 115–242), Reading: Addison-Wesley.
Wertsch, J.: 1985, Vygotsky and the Social Formation of Mind, Cambridge: Harvard Universty Press.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Borba, M.C., Confrey, J. A student's construction of transformations of functions in a multiple representational environment. Educ Stud Math 31, 319–337 (1996). https://doi.org/10.1007/BF00376325
Issue Date:
DOI: https://doi.org/10.1007/BF00376325