In my opinion, there are a lot of problems with how we teach math. The first is that we only teach the decimal system. Really, a lot of math is just easier to do in oct or hex. In fact, all math is really easy when you change the system to another base. What's 16^4? 0x10000 (0x does not mean 0*, it's short hand for hex).
The second is that we don't start with 0 when teaching kids to count. We really should. So, 0,1,2,3,4,5,6,7,8,9. After that, you add another number to the front: 10, 11, 12... Same things with Oct: 0,1,2,3,4,5,6,7,10,11,12,13... Same thing with hex: 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F,10,12,13... Understanding this from Kindergarten would make math so much easier later on.
We also don't teach logic. This is a big one for me. Logic is the basis for rational thought, but it's also the first step in any mathematical problem. Logic is the difference between going about a problem the hard way and the easy way. Take the above problem. 16x16x16x16 is a pretty big number. I have it memorized from years of doing programming. Computers calculate in binary (well, they do everything in binary...), with data clustered into 8-bit increments called Bytes. Half of a byte is a nibble, and thus this is 4 bits. 2^4 is 16 values (0-15). Every bit doubles the number of values, so 8 bits is 256 (16x16). 16 bits is thus 256x256, 65536. Looking at the problem in binary, that's (2^4)^4, or 2^16. I can write that in a large number of ways. I can write it as 1000000000000000 in binary, 200000 base 8, 10000 base 16. However, the simplest is hex. 16 = 0x10. 10^4 is 10000. Done. I could spend a lot of time calculating in dec, or I could convert to hex and be done. This of course is a simplistic and math-centric use of logic.
Perhaps my biggest problem is that we don't teach why. It took till calculus to learn where 4/3*pi*r^3 comes from. Until then, we're asked to just memorize formulas. I never did memorize formulas well. If I understood something, I was good, but wrote memorization of formulas didn't work for me. I failed some math tests in 3rd grade because they were memorizing multiplication tables. I could do multiplication, but not as quickly as the other students could write the answers to pre-memorized tables. I kept running out of time.
the problem reversed when we started division. Because I truly understood multiplication (didn't just memorize tables), I immediately understood division. Then comes fractions. Most students dread fractions at first. Why? Because they don't understand them. 10/4 is 10 divided by 4. If you know division, you know fractions, it's just a different way of writing the problem. In fact, I prefer fractions to division. It'd hate to write out in decimal 13/9, but I can simplify that to 1 4/9 in a heart been and keep going...
Over-all, there's a belief that children can't understand complicated material. We start teaching simple material and make is more complicated as we go. The problem is, it's like making a foundation out of packed dirt because that's all you need for a hut, and then building a house on it. Really, we should spend more time teaching the basics: how numbers work, and then teaching the rest would take much less time.
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