Some ideas on teaching math

This is mostly a sketch but I will try to improve it over time. For a introduction to this blog read this post.

Mathematics is about understanding real things. Formulas, methods, and pure logic are just some of the tools mathematics uses for that purpose. One of the purposes of this blog is to stress the importance of learning how to use your intuition, to learn how to think by yourself and understand what those formulas, methods, and pure logic actually do and how they do it.

What could make a better class? Just some general ideas:

Conway's, Arnold's, Moore's, Polya's comments on teaching.(last two are to wikipedia I will try to find better ones later).

1) Simple, intuitive, and natural

Simple ideas are more important than complicated ones. Motivate the ideas. Use natural logic, that is, sometimes reason as if you never saw that topic before. Communicate with the students!! They will tell you the natural logic, try to use those more natural ideas to construct the theory. "Natural" ideas can be wrong but they are always instructive for the teacher. Look for easier ways to explain, maybe organizing the topics differently.

2) Be honest, don't hide anything

Sometimes it is not possible to explain the whole subject with a natural logic. Anyways, managing to be honest an giving most of our ideas in an organized fashion it's a huge achievement. Sometimes I fell the urge to hide some idea to make a cleaner presentation, for me, that is cheating. If you can't find a better way, just be honest, and tell how you thought about it at the first time you saw it, maybe that is the best presentation after all.

3) Makes the student think and participate in creating math.

It is not so impossible, present the theory showing simple examples and problems, give some orientation and let them explore the possibilities too. It is hard to make it work but it gives much more satisfaction to the student and for the teacher.

4) Make math thinking more down to earth.

I am not talking about applications, for example:

In the trigonometric circle notice (make a drawing) that small angles have big cosines and small sines, as the angle grows initially the cosine doesn't change much.... but latter it changes faster...Why? What is happening? ...It is because of the changing slope of the circle! For instance the cosine is 1/2 only at 60 ^o (after 45 ^o =90 ^o /2, because of the slow beginning) and at that point the sine is big (0.85) or sqrt(3)/2. Why? (leave that to you) Make a imaginary movie of it. There are many things to observe there. Question the students, make them think. That exercise gives a sense of proportions and makes math more real.


My teaching experience is mainly from tutoring individual students, I find it amazing to discover how people think math. Finding more natural thinking is not an easy job. We have to 'unlearn' what we know, understand it in different ways, and to make more connections.

What do you think makes a good class? Give your opinions and ideas!