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Moving Between the Embodied, Symbolic and Formal Worlds of Mathematical Thinking with Specific Linear Algebra Tasks

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Challenges and Strategies in Teaching Linear Algebra

Part of the book series: ICME-13 Monographs ((ICME13Mo))

Abstract

Linear algebra is made out of many languages and representations. Instructors and text books often move between these languages and modes fluently, not allowing students time to discuss and interpret their validities as they assume that students will pick up their understandings along the way. In reality, most students do not have the cognitive framework to perform the move that is available to the expert. In this chapter, employing Tall’s three-world model, we present specific linear algebra tasks that are designed to encourage students to move between the embodied, symbolic and formal worlds of mathematical thinking. Our working hypothesize is that by creating opportunities to move between the worlds we will encourage students to think in multiple modes of thinking which result in richer conceptual understanding.

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References

  • Asiala, M., Brown, A., DeVries, D., Dubinsky, E., Mathews, D., & Thomas, K. (1996). A framework for research and curriculum development in undergraduate mathematics education. Research in Collegiate Mathematics Education II, CBMS Issues in Mathematics Education, 6, 1–32.

    Google Scholar 

  • Ausubel, D. P., Novak, J. D., and Hanesian, H. (1978). Educational Psychology-A Cognitive view (second edition) (New York: Holt Rinehardt and Winston).

    Google Scholar 

  • Cook, J. P., Pitale, A., Schmidt, R., & Stewart, S. (2014). Living it up in the formal world: An abstract algebraist’s teaching journey. In T. Fukawa-Connolly, G. Karakok, K. Keene, M. Zandieh (Eds.), Proceedings of the 17th Annual Conference on Research in Undergraduate Mathematics Education (pp. 511–516). Denver, CO.

    Google Scholar 

  • Cook, J. P., Petali, A., Schmidt, R., & Stewart, S. (2013). Talking Mathematics: An Abstract Algebra Professor’s Teaching Diaries. In S. Brown, G. Karakok, K. Hah RoH, & M. Oehrtman, Proceedings of the 16th Annual Conference on Research in Undergraduate Mathematics Education (pp. 633–536). Denver, CO.

    Google Scholar 

  • Dorier, J. L., & Sierpinska, A. (2001). Research into the teaching and learning of linear algebra. In D. Holton, M. Artigue, U. Krichgraber, J. Hillel, M. Niss & A. Schoenfeld (Eds.), The Teaching and Learning of Mathematics at University Level: An ICMI Study (pp. 255–273). Dordrecht, Netherlands: Kluwer Academic Publishers.

    Google Scholar 

  • Dreyfus, T. (1991a). Advanced Mathematical thinking processes. In D. O. Tall (ed.) Advanced Mathematical Thinking, (pp. 25–41). Dordrecht: Kluwer.

    Google Scholar 

  • Dreyfus, T. (1991b). On the status of visual reasoning in mathematics and mathematics education. In F. Furinghetti (Ed.), Proceedings of the 15th PME International Conference, 1, 33–48.

    Google Scholar 

  • Dubinsky, E. & McDonald, M. (2001). APOS: A constructivist theory of learning. In D. Holton et al. (Eds.) The Teaching and Learning of Mathematics at University Level: An ICMI Study (pp. 273–280). Dordrecht: Kluwer.

    Google Scholar 

  • Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61, 103–131.

    Google Scholar 

  • Hannah, J., Stewart, S., & Thomas, M. O. J. (2016). Developing conceptual understanding and definitional clarity in linear algebra through the three worlds of mathematical thinking, Teaching Mathematics and its Applications: An International Journal of the IMA. 35 (4), 216–235.

    Google Scholar 

  • Hannah, J., Stewart, S., & Thomas, M. O. J. (2013). Conflicting goals and decision making: the deliberations of a new lecturer, In Lindmeier, A. M. Heinze, A. (Eds.). Proceedings of the 37th Conference of the International Group for the Psychology of Mathematics Education, Vol. 2, pp. 425–432. Kiel, Germany: PME.

    Google Scholar 

  • Hannah, J., Stewart, S., & Thomas, M. O. J. (2014). Teaching linear algebra in the embodied, symbolic and formal worlds of mathematical thinking: Is there a preferred order? In Oesterle, S., Liljedahl, P., Nicol, C., & Allan, D. (Eds.) Proceedings of the Joint Meeting of PME 38 and PME-NA 36, Vol. 3, pp. 241–248. Vancouver, Canada: PME.

    Google Scholar 

  • Hannah, J., Stewart, S., & Thomas, M. (2015). Linear algebra in the three worlds of mathematical thinking: The effect of permuting worlds on students’ performance, Proceedings of the 18th Annual Conference on Research in Undergraduate Mathematics Education. (pp. 581–586). Pittsburgh, Pennsylvania.

    Google Scholar 

  • Harel, G. (1989). Applying the principle of multiple embodiments in teaching linear algebra: Aspects of familiarity and mode of representation, School Science and Mathematics, 89(1).

    Google Scholar 

  • Harel, G. (1997). The linear algebra curriculum study group recommendations: Moving beyond concept definition. In D. Carlson, C. R. Johnson, D. C. Lay, A. D. Porter, A. Watkins & W. Watkins (Eds.), Resources for Teaching Linear Algebra, (Vol. 42, pp. 107–126). MAA Notes, Washington: Mathematical Association of America.

    Google Scholar 

  • Harrel, G. (1999). Students’ understanding of proofs: A historical analysis and implications for the teaching of geometry and linear algebra, Linear Algebra and Its Applications, 302–303, 601–613.

    Google Scholar 

  • Harel, G. (2000). The Linear Algebra Curriculum Study Group Recommendations: Moving Beyond Concept Definition. In J. L. Dorier (Ed.), The Teaching of Linear Algebra in Question (pp. 107–126). Dordrecht, Netherlands: Kluwer Academic Publishers.

    Google Scholar 

  • Hillel, J. (1997). Levels of description and the problem of representation in linear algebra. In J. L. Dorier (Ed.), L’enseignement de l’algebra en question (pp. 231–237). Edition Pensee Sauvage.

    Google Scholar 

  • Kolman, B. & Hill, D. (2007). Elementary linear algebra with applications, 9th edition, Pearson.

    Google Scholar 

  • Mason, J. (2002). Mathematics Teaching Practice: a guidebook for university and college lecturers. Chichester: Horwood Publishing.

    Google Scholar 

  • Presmeg, N. C. (2006). Research on visualization in learning and teaching mathematics: emergence from psychology. In A. Gutierrez & P. Boero (Eds.), Handbook of Research on the psychology of mathematics education (pp. 205–235). Rotterdam: Sense Publishers.

    Google Scholar 

  • Stewart, S., & Thomas, M.O.J. (2009). A framework for mathematical thinking: the case of linear algebra. International Journal of Mathematical Education in Science and Technology, 40(7), 951–961.

    Google Scholar 

  • Stewart, S., & Thomas, M.O.J. (2010). Student learning of basis, span and linear independence in linear algebra. International Journal of Mathematical Education in Science and Technology 41(2), 173–188.

    Google Scholar 

  • Stewart, S., Schmidt, R., Cook, J. P., & Pitale, A. (2015). Pedagogical challenges of communicating mathematics with students: Living in the formal world of mathematical thinking, Proceedings of the 18th Annual Conference on Research in Undergraduate Mathematics Education (pp. 964–969). Pittsburgh, Pennsylvania.

    Google Scholar 

  • Stewart, S., Thompson, C. & Brady, N. (2017). Navigating through the mathematical world: Uncovering a geometer’s thought processes through his handouts and teaching journals, 10th Congress of European Research in Mathematics Education (CERME 10). Retrieved August 10, 2017, from https://keynote.conference-services.net/resources/444/5118/pdf/CERME10_0581.pdf

  • Stewart, S., & Schmidt, R. (2017). Accommodation in the formal world of mathematical thinking, International Journal of Mathematics Education in Science and Technology. 48(1): 40–49. https://doi.org//10.1080/0020739X.2017.1360527

  • Tall, D. O. (2004). Building Theories: The Three Worlds of Mathematics, For the Learning of Mathematics. 24(1): 29–32.

    Google Scholar 

  • Tall, D. O. (2008). The transition to formal thinking in mathematics. Mathematics Education Research Journal, 20, 5–24.

    Google Scholar 

  • Tall, D. O. (2010). Perceptions Operations and Proof in Undergraduate Mathematics, Community for Undergraduate Learning in the Mathematical Sciences (CULMS) Newsletter, 2, 21–28.

    Google Scholar 

  • Tall, D. O. (2013). How humans learn to think mathematically: Exploring the three worlds of mathematics, Cambridge University Press.

    Google Scholar 

  • Tall, D. O., & Mejia-Ramos, J. P. (2006). The long-term cognitive development of different types of reasoning and proof, presented at the Conference on Explanation and Proof in Mathematics: Philosophical and Educational Perspectives, Essen, Germany.

    Google Scholar 

  • Tall, D. O., & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. Educational Studies in Mathematics, 12(2), 151–169.

    Google Scholar 

  • Thomas, M.O.J., & Stewart, S. (2011). Eigenvalues and eigenvectors: Embodied, symbolic and formal thinking, Mathematics Education Research Journal, 23, 275–296.

    Google Scholar 

  • Thompson, C. A., Stewart, S., & Mason, B. (2016). Physics: Bridging the embodied and symbolic worlds of mathematical thinking. Proceedings of the 19th Annual Conference on Research in Undergraduate Mathematics Education. In T. Fukawa-Connolly, N. Engelke Infante, M. Wawro & S. Brown (Eds.), (pp. 1340–1347). Pittsburgh, Pennsylvania.

    Google Scholar 

  • Vinner, S. (1988). Subordinate and superordinate accommodations, indissociability and the case of complex numbers, International Journal of mathematics Education, Science and Technology, 19, 4, pp. 593–606.

    Google Scholar 

  • Wawro, M., Sweeney, G., & Rabin, J. (2011). Subspace in linear algebra: Investigating students’ concept images and interactions with the formal definition. Educational Studies in Mathematics. https://doi.org/10.1007/s10649-011-9307-4.

  • Wawro, M., Zandieh, M., Sweeney, G., Larson, C., & Rasmussen, C. (2011). Using the emergent model heuristic to describe the evolution of student reasoning regarding span and linear independence. Paper presented at the 14th Conference on Research in Undergraduate Mathematics Education, Portland, OR.

    Google Scholar 

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Authors and Affiliations

  1. University of Oklahoma, Norman, USA

    Sepideh Stewart

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  1. Sepideh Stewart
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Correspondence to Sepideh Stewart .

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Editors and Affiliations

  1. Department of Mathematics, University of Oklahoma, Norman, Oklahoma, USA

    Sepideh Stewart

  2. College of Education, Florida State University, Tallahassee, Florida, USA

    Christine Andrews-Larson

  3. Department of Mathematics, Technion—Israel Institute of Technology, Haifa, Israel

    Avi Berman

  4. Arizona State University, Mesa, Arizona, USA

    Michelle Zandieh

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Stewart, S. (2018). Moving Between the Embodied, Symbolic and Formal Worlds of Mathematical Thinking with Specific Linear Algebra Tasks. In: Stewart, S., Andrews-Larson, C., Berman, A., Zandieh, M. (eds) Challenges and Strategies in Teaching Linear Algebra. ICME-13 Monographs. Springer, Cham. https://doi.org/10.1007/978-3-319-66811-6_3

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  • DOI: https://doi.org/10.1007/978-3-319-66811-6_3

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  • Print ISBN: 978-3-319-66810-9

  • Online ISBN: 978-3-319-66811-6

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