Abstract
This article considers the nature and growth of mathematical understanding within the context of the workplace-training classroom. In doing this, it draws on elements of the Pirie-Kieren Theory for the Dynamical Growth of Mathematical Understanding, in particular the notions of “Image Making” and “Image Having.” These theoretical ideas are elaborated to allow the more appropriate description of how the understanding of construction industry apprentices is observed to grow in their training as they engage with tasks with significant mathematical components. Through doing this we illustrate how the Pirie-Kieren Theory can be used to describe the growth of understanding of “mathematics-for-working” (influenced by the work of Ball and Bass (2003) on “mathematics-forteaching”), and identify some of the key elements in this growth.
Résumé
Cet article se penche sur la nature et l’évolution de la compréhension des mathématiques dans le cadre de la formation pratique en classe visant l’intégration dans le marché du travail. Ainsi, nous nous inspirons de certains éléments de la théorie de Pirie-Kieren sur la croissance dynamique de la compréhension mathématique, en particulier en ce qui concerne les notions de “création d’images” et de “possession d’images”. Ces notions théoriques sont construites pour qu’on puisse formuler une description plus adéquate des fac¸ons dont la compréhension, chez les apprentis dans le domaine de la construction, évolue au cours de leur formation au fur et à mesure qu’ils s’engagent dans des tâches comprenant des composantes mathématiques significatives. Ainsi, nous sommes en mesure d’illustrer comment la théorie de Pirie-Kieren peut servir à décrire l’évolution de la compréhension des “mathématiques pour le travail” (point de vue influencé par les travaux de Ball et Bass (2003) sur les “mathématiques pour l’apprentissage”), et à déterminer certains éléments clés de cette évolution.
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References
Ball, D. L., & Bass, H. (2003). Toward a practice-based theory of mathematical knowledge for teaching. In E. Simmt & B. Davis (Eds.), Proceedings of the 2002 Annual Meeting of the Canadian Mathematics Education Study Group/Groupe Canadien d’E´tude en Didactique des Mathématiques (pp. 3–14). Edmonton, Alberta: Canadian Mathematics Education Study Group.
Berenson, S. B., Cavey, L. O., Clark, M., & Staley, K. (2001). Adapting Pirie and Kieren’s model of mathematical understanding to teacher preparation. In M. van den Heuvel-Panhuizen (Ed.), Proceedings of the annual meeting of the International Group for the Psychology of Mathematics Education (Vol 2, pp. 137–144). Utrecht, The Netherlands: Freudenthal Institute, Utrecht University.
Bessot, A. (2000a). Geometry at work—Examples from the building industry. In A. Bessot & D. Ridgway (Eds.), Education for mathematics in the workplace (pp. 143–158). Dordrecht: Kluwer Academic Publishers.
Bessot, A. (2000b). Visibility of mathematical objects present in professional practice. In A. Bessot & D. Ridgway (Eds.), Education for mathematics in the workplace (pp. 225–240). Dordrecht: Kluwer Academic Publishers.
Bessot, A., & Ridgway, J. (2000). Education for mathematics in the workplace. Dordrecht: Kluwer Academic Publishers.
Cavey, L. O., & Berenson, S. B. (2005). Learning to teach high school mathematics: Patterns of growth in understanding right triangle trigonometry during lesson plan study. Journal of Mathematical Behavior, 24, 171–190.
Davis, B. (1996). Teaching mathematics: Toward a sound alternative. New York: Garland.
Davis, B., & Simmt, E. (2006). Mathematics-for-teaching: An ongoing investigation of the mathematics that teachers (need to) know. Educational Studies in Mathematics, 61, 293–319.
Eberhard, M. (2000). Forms of mathematical knowledge relating to measurement in vocational training for the building industry. In A. Bessot & D. Ridgway (Eds.), Education for mathematics in the workplace (pp. 37–52). Dordrecht: Kluwer Academic Publishers.
Evans, J. (2000). Adults’ mathematical thinking and emotions: A study of numerate practices. London: Falmer Press.
Kieren, T. E., Pirie, S., & Gordon Calvert, L. (1999). Growing minds, growing mathematical understanding: Mathematical understanding, abstraction and interaction. In L. Burton (Ed.), Learning mathematics: From hierarchies to networks (pp. 209–231). London: Falmer Press.
Lave, J. (1988). Cognition in practice. Cambridge: Cambridge University Press.
Martin, L. C., LaCroix, L., & Fownes, L. (2005). Folding back and the growth of mathematical understanding inworkplace training. Adults Learning Mathematics, 1(1), 19–35.
Martin, L. C., & Pirie, S. E. B. (2003). Making images and noticing properties: The role of graphing software in mathematical generalisation. Mathematics Education Research Journal, 15(2), 171–186.
Noss, R., Hoyles, C., & Pozzi, S. (2000). Working knowledge: Mathematics in use. In A. Bessot & D. Ridgway (Eds.), Education for mathematics in the workplace (pp. 17–36). Dordrecht: Kluwer Academic Publishers.
Noss, R., Hoyles, C., & Pozzi, S. (2002). Abstraction in expertise: A study of nurses’ conceptions of concentration. Journal for Research in Mathematics Education, 33(3), 204–229.
Nunes, T., Schliemann, A. D., & Carraher, D. W. (1993). Street mathematics and school mathematics. Cambridge: Cambridge University Press.
Pirie, S. E. B., & Kieren, T. E. (1994). Growth in mathematical understanding: How can we characterise it and how can we represent it? Educational Studies in Mathematics, 26(2–3), 165–190.
Pozzi, S., Noss, R., & Hoyles, C. (1998). Tools in practice, mathematics in use. Educational Studies in Mathematics, 36(2), 105–122.
Roth, W.-M. (2005). Mathematical inscriptions and the reflexive elaboration of understanding: an ethnography of graphing and numeracy in a fish hatchery. Mathematical Thinking and Learning, 7(2), 75–110.
Wedege, T. (2002). “Mathematics—That’s what I can’t do”: People’s affective and social relationship with mathematics. Literacy and Numeracy Studies, 11(9), 63–78.
Williams, J. S., Wake, G. D., & Boreham, N. C. (2001). School or college mathematics and workplace practice: An activity theory perspective. In C. Morgan & K. Jones (Eds.), Research in mathematics education (Vol. 3, pp. 69–83). London: British Society for Research into Learning Mathematics.
Ball, D. L., & Bass, H. (2003). Toward a practice-based theory of mathematical knowledge for teaching. In E. Simmt & B. Davis (Eds.), Proceedings of the 2002 Annual Meeting of the Canadian Mathematics Education Study Group/Groupe Canadien d’Étude en Didactique des Mathématiques (pp. 3–14). Edmonton, Alberta: Canadian Mathematics Education Study Group.
Berenson, S. B., Cavey, L. O., Clark, M., & Staley, K. (2001). Adapting Pirie and Kieren’s model of mathematical understanding to teacher preparation. In M. van den Heuvel-Panhuizen (Ed.), Proceedings of the annual meeting of the International Group for the Psychology of Mathematics Education (Vol 2, pp. 137–144). Utrecht, The Netherlands: Freudenthal Institute, Utrecht University.
Bessot, A. (2000a). Geometry at work—Examples from the building industry. In A. Bessot & D. Ridgway (Eds.), Education for mathematics in the workplace (pp. 143–158). Dordrecht: Kluwer Academic Publishers.
Bessot, A. (2000b). Visibility of mathematical objects present in professional practice. In A. Bessot & D. Ridgway (Eds.), Education for mathematics in the workplace (pp. 225–240). Dordrecht: Kluwer Academic Publishers.
Bessot, A., & Ridgway, J. (2000). Education for mathematics in the workplace. Dordrecht: Kluwer Academic Publishers.
Cavey, L. O., & Berenson, S. B. (2005). Learning to teach high school mathematics: Patterns of growth in understanding right triangle trigonometry during lesson plan study. Journal of Mathematical Behavior, 24, 171–190.
Davis, B. (1996). Teaching mathematics: Toward a sound alternative. New York: Garland.
Davis, B., & Simmt, E. (2006). Mathematics-for-teaching: An ongoing investigation of the mathematics that teachers (need to) know. Educational Studies in Mathematics, 61, 293–319.
Eberhard, M. (2000). Forms of mathematical knowledge relating to measurement in vocational training for the building industry. In A. Bessot & D. Ridgway (Eds.), Education for mathematics in the workplace (pp. 37–52). Dordrecht: Kluwer Academic Publishers.
Evans, J. (2000). Adults’ mathematical thinking and emotions: A study of numerate practices. London: Falmer Press.
Kieren, T. E., Pirie, S., & Gordon Calvert, L. (1999). Growing minds, growing mathematical understanding: Mathematical understanding, abstraction and interaction. In L. Burton (Ed.), Learning mathematics: From hierarchies to networks (pp. 209–231). London: Falmer Press.
Lave, J. (1988). Cognition in practice. Cambridge: Cambridge University Press.
Martin, L. C., LaCroix, L., & Fownes, L. (2005). Folding back and the growth of mathematical understanding inworkplace training. Adults Learning Mathematics, 1(1), 19–35.
Martin, L. C., & Pirie, S. E. B. (2003). Making images and noticing properties: The role of graphing software in mathematical generalisation. Mathematics Education Research Journal, 15(2), 171–186.
Noss, R., Hoyles, C., & Pozzi, S. (2000). Working knowledge: Mathematics in use. In A. Bessot & D. Ridgway (Eds.), Education for mathematics in the workplace (pp. 17–36). Dordrecht: Kluwer Academic Publishers.
Noss, R., Hoyles, C., & Pozzi, S. (2002). Abstraction in expertise: A study of nurses’ conceptions of concentration. Journal for Research in Mathematics Education, 33(3), 204–229.
Nunes, T., Schliemann, A. D., & Carraher, D. W. (1993). Street mathematics and school mathematics. Cambridge: Cambridge University Press.
Pirie, S. E. B., & Kieren, T. E. (1994). Growth in mathematical understanding: How can we characterise it and how can we represent it? Educational Studies in Mathematics, 26(2–3), 165–190.
Pozzi, S., Noss, R., & Hoyles, C. (1998). Tools in practice, mathematics in use. Educational Studies in Mathematics, 36(2), 105–122.
Roth, W.-M. (2005). Mathematical inscriptions and the reflexive elaboration of understanding: an ethnography of graphing and numeracy in a fish hatchery. Mathematical Thinking and Learning, 7(2), 75–110.
Wedege, T. (2002). “Mathematics—That’s what I can’t do”: People’s affective and social relationship with mathematics. Literacy and Numeracy Studies, 11(9), 63–78.
Williams, J. S., Wake, G. D., & Boreham, N. C. (2001). School or college mathematics and workplace practice: An activity theory perspective. In C. Morgan & K. Jones (Eds.), Research in mathematics education (Vol. 3, pp. 69–83). London: British Society for Research into Learning Mathematics.
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Martin, L.C., LaCroix, L.N. Images and the Growth of Understanding of Mathematics-for-Working. Can J Sci Math Techn 8, 121–139 (2008). https://doi.org/10.1080/14926150802169263
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DOI: https://doi.org/10.1080/14926150802169263