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Re: Learning and Mathematics: Schoenfeld, Metacognition
Posted:
Mar 26, 1996 9:09 AM


Sasha Clayton wrote:
>So often, the focus of doing math, is the "doing" part, the getting it done. >In order to get a problem done, a student (and most of us have been this >student) will plug in the formula given by the teacher, or will execute the >long division, without understanding what the formula means, or why it >should be used.
Most students don't know the why's because answering the why questions are, traditionally, not part of the mathematics curriculum. I might suggest that despite college degrees in education or even in mathematics, many teachers themselves don't know the WHY's in mathematics. Many don't even realize that it's reasonable to ask why. This is not a criticism of their knowledge. Rather, it's a reflection of their perceptions of mathematics  perceptions that were built during their years as students.
I'll offer myself as an example. After a degree in elementary education, eight years as a very dedicated mathematics teacher (including two at the community college level) and at the end of my program for an M.A. in mathematics education AND acceptance into a Ph.D. program in mathematics education, I was having a conversation with a colleague who was ABD in mathematics. I was talking about how much it bothered me that we labeled x divided by zero "undefined". It bothered me that "we" could just leave it at that. "There must be more there", I said. "It seems like a big hole to me that for every other divisor there is a quotient, but for zero, all we can say is 'undefined'." At the time, I had no idea that undefined meant anything more than 'we don't know'. I had no idea how to think about division by zero in the context of repeated subtraction. I had no idea that the term 'undefined' had something to do with infinity.
How could I get through all those years of school and teaching without making the connection? Why was it that my ABD friend couldn't set me straight??? I suggest that it's because we naively  and as children reasonalbly  accept that what we we are taught in school is all there is. We trust that our teachers would not have left out such important details if they existed.
I shudder now to acknowledge my ignorance, but on the other hand, throughout all of my learning and teaching years, division by zero was just "undefined". No one ever considered looking beyond the label. It was just something one accepted. In retrospect, I suppose I should feel good that the arbitrariness of the label bothered me so much that eventually I figured out for myself WHY division by zero was undefined. Now that I know, I'll never let any prospective elementary teacher pass through my classroom without understanding why.
It takes an active effort to go back, rethink your mathematical knowledge, and construct the foundations from which that knowledge grows. Fortunately, I have found that process extremely interesting and I've taken delight in helping prospective elementary teachers explore mathematics with the objective to understand WHY. I believe it is essential for colleges and universities  particularly the mathematics departments  to take an active role in developing coursework designed to help future teachers rethink their existing mathematics knowledge, and develop strategies that will assist them in reconstructing the logic and reasoning that illuminates the WHYs underpinning elementary mathematics content. Experience like this will help prepare new teachers to teach children so that the children know not only how to do math, but why it makes sense too.
Tracy Rusch
>So often, the focus of doing math, is the "doing" part, the getting it done. >In order to get a problem done, a student (and most of us have been this >student) will plug in the formula given by the teacher, or will execute the >long division, without understanding what the formula means, or why it >should be used. I have seen this repeatedly in tutoring. My tutee will >quickly do the problem, just as the teacher has shown him. Yet, when asked >a simple "why" question, he has no idea how to answer, for the "why" has >never been discussed. To him, math is arbitrary steps that he has to do to >numbers to get a correct mark on his paper. The use of metacognition in a >class seems to move the focus to a real understanding of mathematics, >instead of just where to apply formulas. > >Students need to be aware of what they are doing and why they are doing it. >Mathematics can easily disolve into a jumble of abstract numbers and signs. >It's so important to give meaning to mathematics. It's not good enough to >just get it done. > >Sasha Clayton
*************************************************************************** * Tracy L. Rusch email: trusch@mail.utexas.edu * * Mathematics Education Phone: 5127950389 * * Dept. of Mathematics Fax: 5124719038 * * RLM 8.100 * * University of Texas, Austin * * Austin, TX 78731 * ***************************************************************************



