> So much of the curriculum is filled with activities that are > simply mathematically beyond the student's maturity that > "helping with math" becomes nothing more than helping them > follow instructions. For example, a recent activity relating > to mental math was to perform subtraction of two numbers > by rounding the second number to an "easier" number, > performing the subtraction and then adding or subtracting > the difference from rounding. If we had 153 minus 47 then > we could do this as 153 minus 50, which is 103, and then > add 3 to get 106, the final answer. This is certainly a > valid technique, as is completing the square, but if the > kids are not yet mature enough to understand its significance > they will not own this after all is said and done. > > Instead of a steady progression of getting better and better > with numbers, the curriculum jumps from one technique to another. > Techniques which I am sure pleased the adult authors but which > I am also sure do not have the intended effect on third graders. > And it isn't just about "technique", even borrowing and carrying > is a technique, but if every week is a new technique then the > result isn't the understanding of addition or subtraction it > is how to follow instructions. Math is not about following > instructions. In fact, nothing academic is about following > instructions.
This is an excellent account of why I think many neat ideas fail in education. In this case we have certain arithmetic coping strategies, such as to add 9 to an integer you can add 10 to the integer (which is easy) and then take away 1 (which is easy). Better students tend to discover many of these on their own, so we think maybe if we step in and show other students these strategies then they too will become better. Then mix in a little of: educator grows bored of the same thing over and over again, comes across something new, and expects (perhaps at an unconscious level) students to also be just as excited at this "neat thing" (overlooking the fact that the "same old thing over and over again" is also new and interesting to students who haven't been teaching the same thing for 20 some years). Then shake up with a little of: lots of instructions and explanations so that those who don't get it will be able to carry out the "neat thing", never mind that they never knew what the "old thing" was, let alone grew tired of it. Finally, we toss into the mix a lot of: money for textbook authors to write about the "neat thing", pilot programs to showcase (and yes, I meant to use this word) the "neat thing" in action (by top teachers trained in the best ways of presenting the "neat thing"), money for marketing the "neat thing" (which of course isn't going to be something freely available in the internet like a highly original research paper in topology, but instead must be sold in order to pay the salaries of the others involved in promoting the "neat thing"), and so on.
The things you talked about sound like my experience in trying to teach what I call "coping strategies", which I discussed a little in this post:
I don't think I mentioned what usually happened, at least in the post above (although I do recall writing about it, so perhaps this was in another post), but my experience was that almost always the students who could carry out these coping strategies were already good in math and usually discovered several on their own while the other students (with a few very inspirational counterexamples) found these coping strategies made things more difficult for them by being yet another thing they were supposed to learn how to do.