the teaching of math (babble)
When I was taught multiplication, my teacher explained it, then said "if you do this backwards, it's division, and it looks like this: ... " then she went back to multiplication. Nothing more than that, but when division came up the people who came from her class had an easier time of it because the concept was already in their heads. I think when teaching exponents, people should say "and by the way this is what it looks like when you do it sideways." I'm not suggesting you teach logs to all 4th graders, or go into depth about how to use them, but I think attaching the concept to exponents where it belongs as soon as that concept comes up would prevent a lot of chaos later. It really really does make a difference when you plant that seed. And don't get me started about radians. I'm not so much annoyed that I was taught everything in degrees instead, but I did feel a little betrayed by math teachers everywhere when at 40 years old I heard a word I had *never* heard before and was told it was preferable to degrees. I now understand the concept of radians, and I can convert from degrees to radians and back again, but I will never ever think in radians. I probably wouldn't have either way, but if I'd been introduced to both concepts together, my brain would have meshed them to some degree, and I wouldn't have sat in trig class feeling like I'd just been told "English is for babies, we really speak Russian in this country, so catch up." There seems to be a school of thought that says certain "big" concepts shouldn't be introduced until higher math because it will confuse people; I believe those concepts confuse people mostly because they're kept a big secret and are surrounded by a cloud of mystery that could easily be dispelled.
My bigger problem with the way math is taught is the tendency to teach things that are WRONG in order to keep things simple. My husband uses an example of an absolute of math that he was taught in the 2nd grade - something that was drilled into his brain hard so he'd never forget it: "you can't take a bigger number from a smaller number." Just what did these teachers expect to happen when their students encountered negatives? Not only are they going to have a harder time with the concept than necessary, but they're going to rightly wonder if *anything* they know about math is correct. Next week you might tell me addition doesn't really work the way I think. My latest algebra teacher said at least a dozen times "you can never ever take the square root of a negative number" and I muttered "yes you can" for the people near me. Which they really appreciated when we got to the chapter on imaginary numbers *in that same class*. It's bad enough to put a wrong idea into students' heads and pass them on to someone else, but to do it knowing you'll have to contradict *yourself* in a few weeks is just dumb. I'm currently hitting my head against a wrong idea that I didn't know I had, because multiple teachers told me "you can't do that" when they meant "you don't have to do that for the purposes of this class." If I'd examined things more closely earlier I would have seen it, and I can see it clearly now, but that wrong idea is stuck well enough that what to do with a certain kind of problem doesn't jump out at me the way it should. It's a simple concept and I feel totally stupid. And a little baffled to find that I've never had a problem that challenged this idea before, or maybe I did occassionally and didn't look closely enough at why I got it wrong. I hope it's not too late to reprogram my brain.
My bigger problem with the way math is taught is the tendency to teach things that are WRONG in order to keep things simple. My husband uses an example of an absolute of math that he was taught in the 2nd grade - something that was drilled into his brain hard so he'd never forget it: "you can't take a bigger number from a smaller number." Just what did these teachers expect to happen when their students encountered negatives? Not only are they going to have a harder time with the concept than necessary, but they're going to rightly wonder if *anything* they know about math is correct. Next week you might tell me addition doesn't really work the way I think. My latest algebra teacher said at least a dozen times "you can never ever take the square root of a negative number" and I muttered "yes you can" for the people near me. Which they really appreciated when we got to the chapter on imaginary numbers *in that same class*. It's bad enough to put a wrong idea into students' heads and pass them on to someone else, but to do it knowing you'll have to contradict *yourself* in a few weeks is just dumb. I'm currently hitting my head against a wrong idea that I didn't know I had, because multiple teachers told me "you can't do that" when they meant "you don't have to do that for the purposes of this class." If I'd examined things more closely earlier I would have seen it, and I can see it clearly now, but that wrong idea is stuck well enough that what to do with a certain kind of problem doesn't jump out at me the way it should. It's a simple concept and I feel totally stupid. And a little baffled to find that I've never had a problem that challenged this idea before, or maybe I did occassionally and didn't look closely enough at why I got it wrong. I hope it's not too late to reprogram my brain.