Technique Versus Thematic Approaches to Mathematics (2017)

Prepared by:

Joseph Malkevitch
Mathematics Department
York College (CUNY)
Jamaica, NY 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

Although I believe that a themes point of view for curriculum is vastly superior for most students to a technique approach to curriculum, this is not the approach endorsed in the recent Common Core State Standards in Mathematics (CCSS-M). These Standards emphasize the importance of technique but have limited the range of topics that is treated in K-12 mathematics to a smaller body of knowledge than was true for the NCTM Standards. One nice feature of the CCSS-M is the presence of Mathematical Practice 4: Model With Mathematics. However, because of the limitation of the topics treated, it makes it harder for teachers to show students as broad a range of modeling ideas as would be possible with a broader curriculum.

Very recently many states are no longer adhering to the CCSS-M and instead have modified the CCSS-M for "local" usage.