Three Principles of Learning and Teaching Mathematics

Summary

I have touched upon certain aspects of each one of the LACSG recommendations. I have suggested that the focus on proofs should not begin in the first course in linear algebra, but should be emphasized throughout the mathematics curricula in all grade levels.

To compensate for the insufficient time allocated to linear algebra in the current undergraduate mathematics curriculum, I have suggested the incorporation of basic linear algebra ideas in high-school mathematics programs. This suggestion, if implemented in the spirit of the three pedagogical principles I have just discussed, would make the first course in linear algebra a natural continuation of what students have learned in high-school. Accordingly, it could build on rich concept images of linear algebra already possessed by students.

While the inclusion of geometry can aid students in building a strong understanding of linear algebra, current results suggest that the incorporation of geometry in linear algebra in the college level must be sequenced in such a way that students understand the context of investigation.

In line with the LACSG recommendations to utilize technology in linear algebra courses, I have suggested incorporating MATLAB (or any other similar software package) in the teaching of linear algebra. In particular, because MATLAB’s basic data element is a matrix, programming in MATLAB can help students make n-tuples and matrices concrete, where by implementing the Concreteness Principle, as I have discussed earlier. In addition, this would prepare students to a matrix-oriented course as was suggested in the LACSG Recommendation.

Finally, I have formulated three pedagogical principles for designing and implementing mathematics curricula: the Concreteness Principle, the Necessity Principle, and the Generalizibility Principle.