Some analysis on a first course in linear algebra

Abstract

The aim of this paper is to analyze some topics about linear algebra course, along with researches and opinions which are original and useful. The considered topics are the content, textbooks, students’ learning profiles, teaching methods, using computer programs, and connections with other mathematics courses .According to the main results of the analyses, it is an oversimplification to think that there is a unique right way to teach this course. Although many mathematicians could expect that the first linear algebra as if were the same everywhere, the reality is different from this idea. The recent editions of linear algebra textbooks are usually good materials for what is being taught at the introductory level. It seems that only expressing and showing of teacher may not significantly improve students’ learning of an abstract course. In recent years, linear algebra researchers have formulated some efficient teaching methods in order to facilitate meaningful learning. Software provides helpful visualization in two or three dimensional vector spaces. By the creating interactive environment of the computer programs, students can explore with matrices, linear transformations and numerical representations. And finally, there is an obvious connection between linear algebra, calculus, differential equations, and statistics

Keywords

• learning and teaching linear algebra, textbook, computer program

References

1. Anton, H. (1973). Elementary Linear Algebra, New York: Wiley. Bogomonly, M. (1999, December 21). Racing Students’ Understanding: Linear Algebra. Retrieved from ftp://192.43.228.178/pub/EMIS/proceedings/PME31/2/65.pdf
2. Carlson, D. (1993). Teaching linear algebra: must the fog always roll in? College Mathematics Journal, 12(1), 29-40.
3. Carlson, D., Charles R. J., David C. L., & Porter, A.D. (1993).The Linear Algebra Curriculum Study Group Recommendations for the First Course in Linear Algebra. The College Mathematics Journal, 24(1), 41-46.
4. Cowen, C.C. (1997, July 21). On the centrality of linear algebra in the Curriculum. Retrieved from www.maa.org/features/cowen.html
5. Day, J.M., & Kalman, D. (1999, July 15). Teaching Linear Algebra: What are the Questions? Retrieved from http://pcmi.ias.edu/1998/1998-questions2.pdf
6. Davis, R. B., & Vinner, S. (1986). The notion of limit: Some seemingly unavoidable misconception stages. Journal of Mathematical Behavior, 5, 281-303.
7. Day, J.M. (1997). Teaching Linear Algebra: New Ways. In Carlson D., Johnson, C, Lay, D., Porter, D.,
8. Watkins, A, & Watkins, W. (Eds.), Resources for Teaching Linear Algebra, (pp.107-126). Washington: MAA Society. Dorier, J.-L., Robert, A., Robinet, J., & Rogalski, M. (2000).The obstacle of formalism in linear algebra. In

9. Dorier, J.-L. (Ed.), On the teaching of linear algebra, (pp. 85-124). Dordrecht, the Netherlands: Kluwer Academic Publishers.
10. Dubinsky, E. (1997). Some thoughts on a first course in linear algebra on the college level. In Carlson D., Johnson, C, Lay, D., Porter, D., Watkins, A, & Watkins, W. (Eds., Resources for Teaching Linear Algebra, (pp.107-126). Washington: MAA Society.
11. Halmos, P.R (1942). Finite Dimensional Vector Spaces. Princeton: Van Nostrand.
12. Harel, G. (1997). The Linear Algebra Curriculum Study Group Recommendations: Moving Beyond Concept Definition. In Carlson D., Johnson, C, Lay, D., Porter, D., Watkins, A, & Watkins, W. (Eds., Resources for Teaching Linear Algebra, (pp.107-126). Washington: MAA Society.
13. Harel, G. (1998). Two Dual Assertions: The First on Learning and The Second on Teaching (or Vice Versa). American Mathematical Monthly, 105(6), 497-507.
14. Harel, G., & Sowder, G. (2003, June 22). Students' Proof Schemes: Results from Exploratory Studies. Retrieved from http://class.pedf.cuni.cz/katedra/yerme/clanky_expert/Harel/Proof.pdf
15. Herrero, M.P. (2000). Strategies and computer projects for teaching linear algebra. International Journal of Mathematics Education and Science Technology, 31(2), 181-186.
16. Howard, A. (1997). Elementary Linear Algebra with Applications. New York: John-Wiley & Sons.
17. Lay, D.C. (1994). Linear Algebra and its Applications. Reading: Addison-Wesley.
18. Katz, V.J. (1995). Historical Ideas in Teaching Linear Algebra. In F. Swetz, J. Fauvel, O. Bekken, B. Johansson, & Victor Katz, (Eds.),Learn from the Masters. Washington: MAA Society.
19. Kolman, B., & Hill, D. (1999).Elementary Linear Algebra with Applications. New York: Kluwer Academic Publishers.
20. Lawrence, E.S., Insel, A.J., & Friedberg, S.H. (2008). Elementary Linear Algebra. USA: Pearson Education.
21. Mathwright, (2008). Effective mathematical software. Retrieved from: http://www.mathwright.com
22. Schneider, H, & Barker, G.P. (1968). Matrices and Linear Algebra, New York: Dover Publications.
23. Strang, G. (2005). Introduction to Linear Algebra. USA: Wellesley-Cambridge Press.
24. Poole, D. (2007). Linear Algebra: A Modern Introduction. London: Brooks Coole.
25. Richard, P.C. (1997). Linear Algebra: Ideas and Applications: Amsterdam: Willey-Interscience.
26. Tucker, A. (1993). The Growing Importance of Linear Algebra in Undergraduate Mathematics. The College Mathematics Journal, 1, 3-9.
27. Waerden, B.L. (1936). Algebra. Berlin: Springer Verlag.