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Topic: Re: [math-learn] - new question for discussion
Replies: 1   Last Post: Jan 30, 2001 1:20 PM

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Posts: 346
Registered: 12/4/04
Re: [math-learn] - new question for discussion
Posted: Jan 30, 2001 7:07 AM
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I think the real power of any tool we use is in its demonstration of the math
our students think they know how to do, but do wrong. For example, as a
seventh grade teacher, I introduce my 11 to 13-year-olds to the order of
operations by giving them a problem in which they must divide before they
multiply, as well as in which they will divide and multiply before they
subtract, and with addition last. I ask them first to calculate the answer
by hand, and then to calculate it on their calculator. Immediately, students
remember PMDAS. However, they still do X before / and + before -. Of
course, afterwards, most of the calcs come up with the right answer, and we
then discuss the difference between what they were taught in previous years
(the difference is not in what you were taught, but rather in the problems
you were given--or not given--to solve) and the difference in calcs (some are
"cheaper" and are not programmed with the correct order of operations. If
you are using a calc that does not do the correct order of operations, then
you must know the correct order of operations, or you will get the wrong
answer, even when you use your calc).

In general, my handling of students and calcs is slightly different from your
perspective of "I have always believed that children should be able to use
calculators as they wish..." I allow my students to use their calculators
(in fact, our school provides a tub of a dozen calculators for those who do
not have one). However, there are times, when my purpose is to check up on a
basic skill, when I will not allow the use. I also stress the difference
between "showing process" which is required, and "doing scratch work" which
does not need to be shown.

In addition, on the order of "...but that they certainly need to be
discerning about their use. This they need to be taught," I find that calc
use is very high at the beginning of the year, and then as we concentrate on
calc short-cuts, and problem-solving, and a good problem-solver's time-line;
and as they see me using the short-cuts to come up w/ answers before they do,
using the calcs, they begin to realize that math is so much more than
arithmetic, and the calcs begin to fall into a less prominent role.

I have not yet learned to use a graphing calculator (with which our 8th grade
classes do a unit), or an "algebraic calculator" (of which I have heard, but
with which I am not at all familiar). However, as we go through the year, we
introduce things like using the memory to hold a constant for repetitive
calculations, pressing the "equals key" over and over to find a power, and so
on. I find this a most powerful tool for demonstrating why we must learn the
"rules"...b/c the entire rest of the world knows them and uses them, and if
we don't we will be the ones that are out of sync.

One final comment re: elementary textbooks. When my daughter was in 3rd
grade, I ended up having a run-in w/ her teacher b/c she had used the correct
order of operations on a problem on a quiz, but it was marked wrong and she
lost credit. Knowing I was a math teacher, she had come to me to complain
about the unfairness of being marked down for something she had done right.
What I eventually found out is that most elementary math series, at that
time, did what they called "chain problems" to help the student practice
retaining one number in their heads while operating on it w/ another. We
finally reached a compromise in which my daughter, at least, would never be
given "chain problems" b/c I did not want her unlearning what she already
knew correctly. I gave the teacher a step pattern to use for that skill,
instead. I have, since that time, wondered if chain calculations are still
in the elementary textbooks? At that time, I researched a half dozen elem
math books, and found the concept, which was taught prior to the order of

The problem, as I see it, is that such "pre-existing" pseudo-skills cause
problems for those of us further up the chain, when we try to teach what was
wrong about it. The kids come to us with some faulty knowledge, and we are
stuck not just teaching, but unteaching first. I do hope that situation has
changed by now.

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