Mathematics and Education

he general purpose of teaching-learning mathematics, for those who do not require such training in their chosen fields, is to

Establish Intellectual Confidence

Thus, in any trial requiring strength of mind, one might say to oneself
"I was able to comprehend and utilize (what seemed to me at the time) the most complex and arcane principles in existence. Therefore, I can and will master the difficulties at hand."
Teachers and students who fail to understand this at the outset are in for a hard time. One should never hear the oft repeated
"Why do I have to learn this stuff?".
This is an excellent question if the teacher has not established the foregoing ... isn't it?

As for the actual teaching ...

One need look no further than "Quick Calculus" by Daniel Kleppner and Norman Ramsey. This is the finest testbook (in its format) ever penned by man. The layout-logic is as near to perfection as one can expect on this planet at this time.

Let us examine ...

I always look at textbooks for the following:

  1. Spatial - Lots of "white" space
  2. Temporal - Linear learning curve
  3. Page/section (incremental) conceptual density with obvious sectioning as with horizontal rules.
  4. Problem threading from section to section, i.e. same problem with added details.
  5. An absolute proscription against 3rd degree forced induction on the part of the student. The student does deduction and no greater than 2nd degree forced induction.
    (Third degree is the primary killer of math interest in students ... it causes "burnout".)
Kleppner and Ramsey have A, B and C in spades. There is little in the way of problem threading but there is, for the most part, no 3rd degree forced induction required. I found some in the integral calculus section and pulled hair & cursed for a few hours but that was all. A small omission - one which could be taken care of in the arena of feedback corrections to subsequent printings.


By "white" space, I mean that there must be a great deal of empty space around the text and work area

Enough to make one feel free
to focus on the text itself.

Empty space says that that which IS PRINTED is more important.

The space leaves room for notation ... the student must mark his book. The marks constitute a "diary of progress" ... something to look back on.


By linear learning curve, I mean that the student marches through the book at constant speed. He doesn't do the first hundred pages in 40 days and the second hundred in 140 days.

The student develops an "expectation" of degree of difficulty which must be met as a constant allowing him to focus on the math itself. If otherwise, the student who is having difficulty, will think "Is it me, or is the work itself getting harder?".

He must always know with certainty that "There is something wrong with me." It must never be a change in the textbook.

It should be understood here that all mathematical concepts are equal in degree of difficulty. They are all exceedingly simple. The most difficult concept in mathematics is ... 1 + 1 = 2 (and in the rawest sense, this is the only concept).

Each concept is a grain of rice. The problem occurs when one attempts to swallow 10,000 grains at once. It won't go down.


You can't read a math textbook like a newspaper or popular novel. The "conceptual density" is too great. If you can ever take the trouble ... obtain a popular book and go through it crossing out all non-essential words. You will find that there is almost nothing left. It is conceptually "vapid". Thus, you can speed read them (and newspapers too!). You can't speed read a proper textbook because there is no filler material to "skip" over.

By "incremental" conceptual density, I mean, each sectioned off area contains, in general, just one new concept to be learned. And, again in general, each page will contain just one ... maybe two ... sections.

By keeping this constant, the student makes predictable progress through the textbook.

And this is an important fact:
As a book is used, it gets dirty on the used pages (fingerprints, notes, etc.). The pages get bent, curved and stick up unlike the pages not yet visited.

The student must be able to "fan" the pages making a confirmation of his progress. Page fanning (when nobody's looking, of course) is essential to keeping the student's interest in further progress. This is his "reward" for his effort ... the right to fan the pages and think ... "This is what I have done. I will do as much again and be twice the better for it."


Problem threading is the expansion of a "basic" problem into evermore sophisticated permutations illustrating the concepts presently being learned. Thus, one might compute the area of a fenced in field. Then, in the next section, compute the area of the same field with a different polygon shape. Then, further on, with a curved fence, etc.

Such threading allows the student to focus on the new aspect by being familiar with several other non-essential factors.

The repetition of problem after unrelated problem is wasted time for the average student. It is enough to understand the concepts and be able to remember the example. Proficiency is not required unles one plans to become an engineer/scientist and even then a fully functional hand held computer is the method of choice (provided one understands the problem and the attendant math).


This is the learning killer.

It took man about 10,000 years to develop a mind sufficiently sophisticated to do abstract addition and subtraction. Yet this is "really" simple stuff ... isn't it?

What would happen if you gave a kid a bunch of blocks and said "Kid, develop some abstract method of cataloging, keeping track of, naming objects of any type ... and bring that to me by 3PM."

Sure ... no problema ... not!

One must be very careful to teach an "obvious" fact. Especially if it took Karl Freidrich Wilhelm Gauss ten years to recognize the "simplicity". That is, don't expect the student to do overnight what you can't do (but think you can) and which took LaGrange - Abel - Cauchy - Euler - Newton - Liebniz - Laplace - Hamilton -... years to do.

This is "spirit killer".

End suggestion ...

I would much like to see an "Encyclopedia of Mathematics Learning". Multiple volumes compiled and refined over decades which serve to illustrate all fundamental "most used" concepts of mathematics. One could start at the first volume and, after some months of use, calculate (roughly) "how long it will take me to do these really cool-looking, indecipherable equations in the last volume".

As I have asserted before, the necessary time is about 5,000 hours for an excellent understanding and 2,000 for a good, well rounded general knowledge.

Wonder if Kleppner & Ramsey are still 'kickin'

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