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Re: Algebraic Manipulation
Posted:
Jan 15, 1999 2:14 PM
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Tony-
This is an eloquent plea for symbolic manipulation - one of the best I've seen.
It highlights the need for us to try to distinguish between NSM (necessary
symbolic manipulation) and MSM (mindless symbolic manipulation). The
courses of the past did not make this distiction and left students (and
some teachers) thinking that all symbolic manipulation was NSM.
The kinds of symbolic manipulation you decribe in your post fall into the
NSM range. But lots of the symbolic manipulation we do in math classes
(such as integrate Sec[x]^3) are well into the MSM range.
A part of NSM very poorly represented today is complex numbers and the
complex exponential function, an important part of physics and engineering
whose use actually simplifies a lot of former MSM. (Would you rather
integrate E^(3 x) Cos[4 x] by parts or would you rather take the real part
of the integral of E^((3 + 4 i)x)?
Another overriding point is that this country spends millions (probably
billions) per year in failing to teach symbolic manipulation to our
students. If we can't hack it the old way, then let's try tto get around
this by demphasizing MSM in favor of the smaller but more important NSM.
-Jerry
At 1:07 PM -0500 1/15/99, tony asdourian wrote:
>Hello. I have taught AP/IB Mathematics and AP B/C Physics for 10
>years now, and I have been reading this and the discussion at AP-Calc
>with great interest.
>
>Although I do agree that the new technology has been of great benefit
>in certain ways, those ways mostly involve developing a superior
>intuition and graphical sense of a problem (as per the so-called rule
>of four). Certainly, like Lin, I have learned many new things using
>the graphing calculator, and am very glad to have it as an
>indispensable tool.
>
>However, I have not found that a student's pure analytical ability has
>demonstrably improved with the technology. To be more specific, their
>ability to "manipulate algebra" has not improved just becuase they
>have a deeper understanding of the problem from another valid
>perspective. Now, I understand that "manipulating algebra" is perhaps
>the most denigrated skill in this discussion group, but I remain
>extremely unconvinced that this is not an important skill.
>
>Sure, elaborate factoring techniques or row-reducing matrices or
>interpolation of log tables is destined to fade away as a waste of
>time and not particularly helpful to student understanding, but as a
>Physics major in college, I spent an enormous amount of time dealing
>with Algebraic manipulation, not for its own sake, but so that a
>problem could be solved IN GENERAL, with an answer in terms of
>variables like m, v, and q (mass, velocity and charge), which is alot
>more meaningful than a specific numeric answer. Further, having a
>TI-89-like object do the calculation for me would have prevented
>developing a feel for the algebra of the equations. Working through
>such details made me much more proficient in solving future problems.
>
>Lin asks if perhaps the burden should be put on "symbol manipulators"
>to justify the validity of their approach, but surely this is ignoring
>the vast successes of over 150 years of Theoretical Research in Math,
>Physics, and Engineering, a great deal of which culminates in papers
>in journals that use almost EXCLUSIVELY "symbol manipulation". Can we
>as high school teachers be so cavalier about what the end-product is
>at the university?
>
>Advocates of "Algebra Reform" say that students need not do the
>intermediate steps to appreciate the significance of the final answer,
>but to be honest this rings somewhat false to me. Appreciating the
>subtlety of a piece of music or painting is not the same as knowing
>how to compose or paint(as I can certainly attest to). Showing
>students that the calculation "can be done by some algorithm, which is
>a little tedious so we won't make you learn it" implies that all
>Algebraic manipulation is trivial and that it never leads to a deeper
>understanding of what you are studying. This is patently untrue, in
>my opinion. Algebra can be a grind, but it can also provide insight.
>
>I think I should address one other point often made here. Perhaps
>requiring much, much less Algebraic manipulation is appropriate for
>the average Calculus student of today who goes on to Medicine or
>Business. But serious students of Mathematics, Physics, and
>Engineering are being done no favors when we dramatically reduce
>Algebra proficiency, as we are reducing their ability to explore the
>mathemtical/scientific world rigorously on their own. Galileo said
>that the language of the universe is Mathematics, and as a Physics
>major and Math/Physics teacher, I hate to see the students most
>capable of learning that language encouraged not to do so.
>
>Put one last way-- are there really scores of Math/Physics teachers at
>University who are supportive of their incoming students having much
>reduced Algebra skills? Surely they would be supportive of the
>increased intuitive and graphical understanding of problems Calculus
>Reform advocates, but would they want this at the expense of a solid
>background in Algebra?
>
>(This posting is similar, with some additional material, to one at
>AP-Calc, but I am intersted in reading responses from both forums,
>since the issues involved are obviously similar.)
>
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------------------------------------------------------------------
Jerry Uhl juhl@ncsa.uiuc.edu
Professor of Mathematics 1409 West Green Street
University of Illinois Urbana,Illinois 61801
Calculus&Mathematica, DiffEq&Mathematica, Matrices, Geometry&Mathematica
http://www-cm.math.uiuc.edu and http://netmath.math.uiuc.edu
All that is human must retrograde if it does not advance.
-----Edward Gibbon
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