Mathematical Techniques and Mathematical Themes
York College (CUNY)
Jamaica, NY 11451
Mathematics is viewed by society as having sufficient importance that the teaching of mathematics begins in kindergarten and continues typically to be required through the third year of high school. The reason for this attention to mathematics over such a long stretch of primary school education is that mathematics is an important tool for daily life and individuals who excel in mathematics have great value for society. This value goes far beyond what is provided to society by the small number of individuals who become professional mathematicians. Other professionals in fields ranging from anthropology to zoology with mathematical skills are prized.
The reality of the value of mathematics to society is played out against another reality. For many, learning mathematics is a struggle, and many individuals reach adulthood neither being able to use the mathematics they have been exposed to nor knowledgeable about the way that mathematics is affecting their daily lives. This mistrust or lack of understanding of mathematics is transmitted to children of these individuals and limits their effectiveness as workers in many situations. Many reasons have been offered for this problem ranging from evolutionary arguments (human evolution probably did not adapt humans to maximize mathematical skills), to claims that the teaching of mathematics is not as good as it might be (probably true), to a host of other possibilities.
The purpose of this note is to suggest a way that the current organization for the delivery of mathematical ideas and skills might be made more effective by changing the point of view about how to deliver the mathematics we teach as part of the primary school education process.
The teaching of mathematics is probably most fluid in the very beginning grades, K-2, although even here, school districts in the United States list "outcomes" that are being sought. Thus, very early children learn to count and begin to come to grips with the issue of the difference between a number and the name of a number. They also begin to learn about shapes, and encouraged to give examples of shapes such as circles, rectangles, squares, and spheres in the their lives. Challenges for teachers occur very early in the mathematics education that society tries to deliver to all citizens. Thus, having learned about some kinds of cookies being circles and some fruit (oranges and grapefruit) being spheres the sophisticated teacher might worry about what a child thinks makes a circle a circle and a sphere a sphere. What I am driving at is that A mathematician defines a circles as a plane figure whose points are equidistant from a point and a sphere as a 3-dimensional shape consisting of points equidistant from a point. So what to do if the teacher holds up a pear and asks what shape it has and a child answers "Its a sphere?" or holds up a plum pudding and the child answers its circular. Teachers also have to worry about the commonly used phrase "circular shaped."
Cookies are really two-dimensional not flat; the shadow of a cookie is often a circle but sometimes an ellipse. Its nice to have teachers sophisticated enough to use language discussions as a learning experience. If a teacher holds up an orange golf-ball and asks a 1st grader what color it is, and the student responds "red" the teacher will probably not say "good Mary." He will so, this golf ball's color has some resemblance to red but its the color we call orange, and the teacher might make a mental note to have Mary tested to see if her color vision is accurate. Children seem to pick up ideas about color a lot faster than they pick up ideas about shape. What are the shapes in Figure 1 below? Very young people will recognize there is an oak leaf on the left and a maple leaf on the right even though no two leaves are probably exactly alike!
But even professional mathematicians would be hard pressed to describe over the phone with words what shape an oak or maple leaf has to an intelligent alien, who happened to speak good English but had no trees of any kind on its planet. The words we use for shape: straight, curved, pointed, etc. are quite crude. Using advanced mathematics one could code the curvature of the leaf, but though two oak leaves may be very similar, not two are quite identical, or are they? How would one tell?
When describing a mathematics curriculum for a particular grade, it is nearly always the case that what the students are learning is described in terms of techniques they will be taught and expected to master. For a long time to get across what was taught in high school it was common to say: in the 9th grade elementary algebra, in the 10th grade geometry, and in the 11 grade intermediate algebra and trigonometry. For many American this summative approach brings forth distasteful memories of manipulating symbols whose meaning was not understood in the 9th grade, being asked to prove geometrical facts which seemed self evident in many cases using a complex sequence of derived theorems and axioms in a way that defied easy understanding and for the results the teacher considered important meant committing a proof to memory without insight or understanding.
If one assumes that end users of mathematics education will be professionals as well as "laymen" then this system is a failure. Although it may be that the system has worked adequately to locate and nurture those with mathematical talent, the price seems to be very high. An alarming number of adults admit to be being phobic about mathematics. If these individuals are successful in society they often become conservatives about changing mathematics curriculum because they, though they may not like mathematics themselves, know the high regard that mathematical skill and talent is held by society. They often force their children to go down the same road to negative attitudes that they traveled.
If the students one is teaching mathematics master solving mechanical problems involving mathematics but in daily life and most jobs never have occasion to use the materials that they learn the process undercuts genuine appreciation of the importance of mathematics. If they never master solving the problems in the first place, and they also not few situations where these skills are ever employed the situation is no better.
Although I am not recommending that no attention be given to mathematics techniques, I am urging that students see mathematics at a much earlier stage than is now the case, as supporting a goal structure that goes beyond technique. This goal structure has to do with the fact that mathematics can be applied in so many settings.
Suppose one is teaching urban children. One of the constant background factors for urban residents is the extent to which important services are reliably delivered to the citizens. Examples of such services are collecting of trash and garbage, delivery of mail, repair of potholes, street sweeping, snow removal and other similar services. It is not difficult for students to understand that when these services are done "efficiently" not only are they better off but also that the cost of providing these services is lower. For example, if a garbage truck which has already collected garbage along a certain stretch of streets, must retraverse this stretch to get to a section of town where the garbage has yet to be collected, then the cost is higher than if time is never wasted by retraversing sections where the garbage is already gone. When the word "efficiently" appears in the setting of a problem, very often what is involved from a mathematical perspective is an optimization problem. Optimization refers to finding a solution which is from some perspective the best. If one sets up a criterion by which to judge how successful different solutions to a problem are, this means finding the smallest (minimum) or largest (maximum) value with respect to this criterion. In most problems there are various choices for criteria for judging the success of ones approaches. In the garbage collection problem our goal might be to collect the garbage for the least cost. A different goal might be to minimize the overtime of the workers involved. The reason why these two situations may differ is that it may be cheaper to pay fewer full time workers lots of overtime than to have more full time workers but pay them little overtime. This is true because each full time worker incurs other costs such as health benefits, which may be more costly to provide than to pay existing workers the extra overtime. Minimization problems are natural for situations which involve cost, time, and effort while maximization problems are natural for situations that involve profit, income, or satisfaction.
If students starting in early grades were exposed to a broader vision of mathematics, not as a collection of meaningless techniques but as a collection of tools for obtaining insight into a wide variety of phenomena in many fields, perhaps fewer students would be phobic about mathematics and mathematics would have more fans.
Here is an over simplified accounting of various mathematical techniques and mathematical themes:
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
2. Growth and Change
4. Fairness and Equity
6. Shape and Space
7. Pattern and Symmetry
8. Order and Disorder
9. Reconstruction (from partial information)
10. Conflict and Cooperation
11. Unintuitive behavior
Malkevitch, J., Discrete Mathematics and Public Perceptions of Mathematics, in "Discrete Mathematics in the Schools," (eds.), J. Rosenstein, D. Franzblau, and F. Roberts, DIMACS Series, Vol. 36, American Mathematical Society, Providence, 1997.