Where is mathematics going and what is it doing in that hand basket?

I was browsing through my book shelves and came across an old arithmetic book. Samuel G. Kimball of Thompson, Connecticut had written his name several times on the fly page, as is the wont of young men, in 1836. He could well have fought in the Civil War.

The book has a little over 200 pages and is about 5'' by 7''. The author has given some thought as to the order topics are presented and in the preface explains his choice.

A topic is presented, say, how to change the currency of New England or Virginia into Federal Currency. Some examples are given which are followed by about a dozen problems.

It seems to me that the teaching of mathematics hasn't changed much since 1836. The books have gotten too heavy to lift, too many topics to cover, too many examples to go over, and too many problems for a student to work before the instructor beserks to the next section, but other than these differences, mathematics books of today are pretty much the same.

The main difference between then and now is an extreme application of the 'more is better' priciple. I think that the general teaching of mathematics hasn't changed except for the worse.

I hear of conferences held by educationists, I read the papers given in these conferences but I don't see any of these ideas reflected in the classroom.

I don't see the question: Why do we teach mathematics? addressed in any conferences.

There are students who will learn and understand mathematics no matter how bad the teacher or the book. There are students who will never understand mathematics no matter how good the teacher or the book. There are a multitude of reasons for this: hormones, parents and sports, for example.

I could go into detail on the problems students face but I will first give my ideas on why mathematics should be taught and on how it should be taught.

First, "How".

It seems to me that almost all mathematics, certainly elementary mathematics, deals with the same concepts.

There are a collection of objects, for example, counting numbers, or fractions, or real numbers, or vectors. A collection of objects is given a name, for example, the integers, or the rational numbers, or vector space. Some of these collections are called groups, or rings or fields.

At this point we have a set of objects. Now a way is given to combine the objects in a collection. Take for example, the integers. Being a greedy species we want to get a number that measures our existing herd of cows and the cows we rustled last night. We add integers. We want to know how many cows we have left after the coyotes killed some last night. We subtract integers. If I rustle 15 cows every night this week, I want an easier way to represent my ill gotten gains than 15+15+15+15+15+15+15 so I invent fast addition, also called multiplication and write 7x15.


We then study the methods of combination. Is 4+7=7+4 ? Is 4x7=7x4 ? Regardless of what the collection of objects is, we look for general properties of the combination methods for that collection.

Finally, there are functions. We want to associate a cow to a price it will bring at the slaughter house. We want to associate the time it is to where we are. We want to associate the current through a resistor with the voltage across the resistor.

It is my contention that if a collection of objects, the ways of combining the objects and the functions defined on these objects are studied deeply and understood, then the students knows how to approach any collection of objects with its own rules of combination and functions.

Instead of touching on a lot of subjects, I suggest going deeply into one subject. I think that if a student knew one collection of objects well, he would be prepared to learn any collection of objects with their rules of combination and functions.

Indeed, they would see that functions are just another collection of objects and that we can combine them. We can add functions, compose them, multiply them-these are just ways to combine objects called functions.

Instead of having the students memorize a bunch of stuff, we should be teaching students how to learn mathematics.

I have talked to graduate students in physics or engineering that didn't realize that a linear differential equation is essentially a problem in functions and that it is essentially no different than solving ax=b.

Mathematics studies a set, the rules of combination for the elements of the set and functions defined on the set. End of story.