>Well...this is, of course, a possibility...but it all goes back to students >taking some responsibility for their own education. I feel that you probably >will not find a bunch of teachers out there that have taught students how to >divided 2 by 8 in an incorrect manner. I would be shocked to find 1 in 100 >teachers who completely blew this lesson with the kids. The high school >students who will answer that 2/8=4 will tell you themselves that they were >not taught incorrectly...rather....THEY made a silly error due to >thoughtlessness at the time. > >Harv Becker >
I certainly must agree with you about students taking more responsibility for their own education. As has been said in numerous places, "Mathematics is not a spectator sport."
My implication was that if there are a lot of students who answer that 2/8 is 4, then maybe much of the blame is on the teachers of those students. I think teachers may be doing a tremendous job in showing how to perform the algorithm. "Divide, multiply, subtract, bring down, repeat." And I think a lot of students can perform those steps. But one of the problems is that students are given so many division exercises, page after page, and most of them are pure number problems, with no context or application whatsoever. And, if you look carefully at a majority of math books, these exercises typically begin with a barrage of examples in which the dividend is greater than the divisor. Needless to say, I think, students begin to get the impression that, in division, you always divide the big number by the little number. So, "2 divided by 8" must really mean "8 divided by 2" right? So the answer must be 4.
I think this results from placing too much emphasis on the steps of the algorithm, and too little emphasis on translating back and forth between real life and the algorithm. Thus, a student may know how to share $2 among eight people, because there is a context against which he can compare his answer, but may answer 4 when asked to divide 2 by 8.
If you give a page of division exercises involving the long division algorithm, and then check to make sure the steps of the algorithm were performed correctly, but spend no time finding out if the student really knows how those problems relate to real life, then there is a good chance, in my opinion, that that student will not really understand what division is about. (Is that a run-on sentence?)