Technique Versus Thematic Approaches to Mathematics

Prepared by:

Joseph Malkevitch
Department of Mathematics
York College (CUNY)
Jamaica, New York 11451

email:

malkevitch@york.cuny.edu

web page:

http://york.cuny.edu/~malk


To do mathematics one needs to learn how to use mathematical tools. In light of this it is not surprising that much attention has been given to what tools lay people need of a mathematical kind. With its usual concern for slick organization, the mathematics community has developed curriculum which is highly structured and efficient for the purpose of pursing mathematics. This has lead to a curriculum which is largely concerned with mathematical techniques and the conceptual framework in which these techniques fit. This curriculum served the mathematics community quite well until relatively recently. However, a combination of circumstances ranging from the development of Computer Science as an alternative to majoring in Mathematics for students with mathematical talent to the changing standards for admission to colleges and universities (e.g. admission of large numbers of students who would not have been able to attend universities by the standards of admission in the 1960's) has created difficulties for the traditional approach. One alternative to the traditional approach is to emphasize the themes that mathematics concerns itself with. Techniques of different kinds can be put to use to obtain insights and results in these various thematic areas. Areas of technique and thematic areas are spelled out in outline form below.


Techniques:

0. Arithmetic
1. Geometry
2. Algebra
3. Trigonometry
4. Calculus (Single Variable and Multivariate)
5. Differential Equations
6. Linear (Matrix) Algebra
7. Modern Algebra
8. Probability and Statistics
9. Real Variables
10. Complex Variables
11. Graph Theory
12. Coding Theory
13. Knot Theory
14. Partial Differential Equations

(Many more!)

Themes:

1. Optimization
2. Growth and Change
3. Information
4. Fairness and Equity
5. Risk
6. Shape and Space
7. Pattern and Symmetry
8. Order and Disorder
9. Reconstruction (from partial information)
10. Conflict and Cooperation
11. Unintuitive behavior

One advantage of the themes approach is that it helps students at all grade levels, in particular K-12, to know "when to call a mathematician." When a pipe bursts in your home you need to call a plumber, not a carpenter, pharmacist or mathematician to get it fixed. It is hard to describe exactly what mathematicians are "experts" at but the theme approach is more likely to provide an informed answer. Furthermore, one can use and appreciate mathematics even if one is not a "mathematician."

Mathematicians should aim to make more people ambassadors for mathematics. You don't need to be "good" at mathematics to realize how important it is to your well being and life.