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I found the Gower's weblog interesting. Curious that his example was the same one I was asking about. He makes the distinction between teaching how to understand math and teaching mechanical manipulations (my cheap tricks). I can share at least one student's reaction to this distinction. When I was taking high school algebra we started coming to some things that didn't make immediate intuitive sense to me. The first one was that a minus times a minus equaled a plus. In one way I could understand this, yet it seemed to me that if the minus and the plus areas on a number line were symmetrical extensions from 0, then the way it worked should be symmetrical on each side of 0. (If this makes sense.) I asked about this and was told that I needed to simply accept it and do the problems. As other things came up that also weren't intuitive to me, things that I was just to accept, I turned off on the whole process and stopped making an effort to learn algebra. A very similar thing happened with Physics. It had to do with the speed of light being the same as measured from earth regardless of the speed at which different sources of the light were traveling in relation to the earth. This made no sense whatsoever to me. It was impossible, and I told the physics teacher it was. He said that it simply was true. I argued with him. It wasn't true of anything else, I pointed out. I was eventually told I just had to accept it. And at that point I turned off on physics. Much later I learned about the special theory of relativity -- and then about the general theory and then quantum physics, etc. This was fascinating stuff, which I could more or less grasp intuitively, but not mathematically. I regretted my earlier turning off on math and some 50 years after those fateful high-school courses, I took a pre-calculus and then a calculus course. They have hardly made me a math wizard. But learning this this math does promise to give me the tools for a better understanding of some fascinating stuff.

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