I recently bought myself a Math book that I'm delighted with (Mathematics 1001 by Elwes). I'm finally understanding many math concepts I had not understood in my education and I've reached an few interesting theories on how math should be taught.
There should be three things that a math class should teach at the same time:
2. Visual (Graph-Based)
3. Procedural (Equation-Based)
All four are part of a whole and some students will respond better on one than the other. These are four avenues to teach the same thing by the way not three different things.
Memorization should begin really early. There is no need to explain how multiplication works to have the kids start to memorize the multiplication table. The reason is not that you *need* to memorize it but that by doing so or encouraging kids to do so, they'll be faster at doing calculations. Here are the things I think should be memorized in math:
a. all the pairs of numbers that add up to 10.
b. the first 32 square roots.
c. the multiplication tables up to 12 for 3, 4, 7, 8 and 12, because these are the trickiest ones.
d. the prime numbers with 1 or 2 digits.
e. the Sine, Cosine triangle calculations.
Of those I only memorized the first and the last, but he key here isn't to force the kids to all memorize them but to encourage them to.
Visual (Graph-based) math is my favorite and I have a natural proclivity for it. I did extremely well in geometry, but graph based math allowed me to understand calculous. To this day when I do a limit, I visualize the graph in my mind. It is my theory that some people have a more visual mind that process math better graphically than procedurally (equation-based).
Procedural (Equation-based) seems to be the physics way to teach math. Everything gets boiled down to an equation. Some people need to visualize the relationship between numbers while others find the visual clutter, well, cluttering. Some crave a purely mathematical description of the relationship, because it creates the relationship instantly. It's like sight reading code. I can describe a pattern with words and leave you confused but the code, is unambiguous. And some people really enjoy equations that way. There is a beauty in the sparseness of an equation that is undeniable.
Historical perspective has helped me understand some of the more basic concepts of math that seem just weird. For example what are rational numbers? Or really who cares about them? Really? We live in a world of real numbers so the previous historical divisions of numbers don't necessarily map well to the reality of today. A historical perspective can help. Once I understood the strange almost spiritual fascination the Greeks had for integers, suddenly rational numbers made more sense. Considering most numbers we encounter today are real numbers and not integers it can be confusing to go from rare numbers (integers) to rarer (rational) to not-really used in real life (irrational) to the most common (real) numbers in number theory.