Abstract
Linear algebra is a core subject for mathematics students and is required for many STEM majors. Research reveals that many students struggle grasping the more theoretical aspects of linear algebra which are unavoidable features of the course. Working with vectors and understanding new concepts through definitions, theorems, and proofs all indicate that a sudden shift has occurred, and despite carrying the name “algebra,” in many respects linear algebra is significantly more complex than school algebra. In this chapter we will employ the Framework of Advanced Mathematical Thinking (FAMT) to describe the type of thinking that is required for linear algebra students to succeed at college level.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
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.
Briton, S., & Henderson, J. (2009). Linear algebra revisited: An attempt to understand students’ conceptual difficulties. International Journal of Mathematical Education in Science and Technology, 40(7), 963–974.
Carlson, D. (1997). Teaching linear algebra: Must the fog always roll in? In D. Carlson, C. R. Johnson, D. C. Lay, A. D. Porter, A. Watkins, & W. Watkins (Eds.), Resources for teaching linear algebra (MAA notes, Vol. 42, pp. 39–51). Washington: Mathematical Association of America.
Day, J. M. (1997). Teaching linear algebra new ways. In D. Carlson, C. R. Johnson, D. C. Lay, A. D. Porter, A. Watkins, & W. Watkins (Eds.), Resources for teaching linear algebra (MAA notes, Vol. 42, pp. 71–82). Washington: Mathematical Association of America.
Dorier, J. L. (1990). Continuous analysis of one year of science students’ work, in linear algebra, in first year of French University. Proceedings of the 14th Annual Conference for the Psychology of Mathematics Education, Oaxtepex, Mexico, II, pp. 35–42.
Dorier, J. L. (2000). On the teaching of linear algebra. Dordrecht: Kluwer Academic.
Dorier, J. L., Robert, A., Robinet, J., & Rogalski, M. (1994). The teaching of linear algebra in first year of French Science University: Epistemological difficulties, use of the “Meta Lever” long-term organisation. Proceedings of the 18th International Conference for the Psychology of Mathematics Education, Lisbon, Portugal, IV, pp. 137–144.
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.
Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61, 103–131.
Hannah, J., Stewart, S., & Thomas, M. O. J. (2013). Conflicting goals and decision making: The deliberations of a new lecturer. In A. M. Lindmeier & A. Heinze (Eds.), Proceedings of the 37th Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 425–432). Kiel, Germany: PME.
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 S. Oesterle, P. Liljedahl, C. Nicol, & D. Allan (Eds.), Proceedings of the Joint Meeting of PME 38 and PME-NA 36 (Vol. 3, pp. 241–248). Vancouver, Canada: PME.
Hannah, J., Stewart, S., & Thomas, M. (2015). Linear algebra in the three worlds of mathematical thinking: The effect of permuting worlds on students’ performance. In Proceedings of the 18th Annual Conference on Research in Undergraduate Mathematics Education (pp. 581–586). Pittsburgh, PA.
Hannah, J., Stewart, S., & Thomas, M. (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. doi:10.1093/teamat/hrw001.
Harel, G., & Sowder, L. (2005). Advanced mathematical-thinking at any age: Its nature and its development. Mathematical Thinking and Learning, 7, 27–50.
Keith, S. (2001). Visualizing linear algebra using Maple. Prentice Hall.
Meel, D. E. (2005). Concept maps: A tool for assessing understanding? In G. M. Lioyd, M. Wilson, J. L. M. Wilkins, & S. L. Behm (Eds.), Proceedings of the 27th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. Retrieved June 27, 2008, from http://www.allacademic.com/meta/p24713index.html
Sierpinska, A., Nnadozie, A., & Okta, A. (2002). A study of relationships between theoretical thinking and high achievement in linear algebra. Manuscript: Concordia University.
Stewart, S. (2008). Understanding linear algebra concepts through the embodied symbolic and Formal worlds of mathematical thinking. Doctoral thesis, University of Auckland. Retrieved from http://hdl.handle.net/2292/2912
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.
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.
Tall, D. O. (2008). The transition to formal thinking in mathematics. Mathematics Education Research Journal, 20, 5–24.
Tall, D. O. (2004). Building theories: The three worlds of mathematics. For the Learning of Mathematics, 24(1), 29–32.
Tall, D. O. (2010). Perceptions operations and proof in undergraduate mathematics. Community for Undergraduate Learning in the Mathematical Sciences (CULMS) Newsletter, 2, 21–28.
Tall, D. O. (2013). How humans learn to think mathematically: Exploring the three worlds of mathematics. Cambridge: Cambridge University Press.
Thomas, M. O. J., & Stewart, S. (2011). Eigenvalues and eigenvectors: Embodied, symbolic and formal thinking. Mathematics Education Research Journal, 23, 275–296.
Thurston, W. (1994). On proof and progress in mathematics. Bulletin (New Series) of the American Mathematical Society, 30(2), 161–177.
Uhlig, F. (2002). The role of proof in comprehending and teaching elementary linear algebra. Educational Studies in Mathematics, 50, 335–346.
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, 78, 1–19. doi: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.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Stewart, S. (2017). School Algebra to Linear Algebra: Advancing Through the Worlds of Mathematical Thinking. In: Stewart, S. (eds) And the Rest is Just Algebra. Springer, Cham. https://doi.org/10.1007/978-3-319-45053-7_12
Download citation
DOI: https://doi.org/10.1007/978-3-319-45053-7_12
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-45052-0
Online ISBN: 978-3-319-45053-7
eBook Packages: EducationEducation (R0)