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Topic: The Deprecation of Algebra
Replies: 36   Last Post: May 9, 2010 5:10 AM

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 Dave L. Renfro Posts: 4,792 Registered: 12/3/04
Re: The Deprecation of Algebra
Posted: May 3, 2010 5:56 PM

Jonathan Groves wrote (in part):

http://mathforum.org/kb/message.jspa?messageID=7056235

> I would also wonder how to grade such questions, and students
> might be puzzled as well how to answer such questions. Does
> this alternative sound better? I would be sure that students
> know what I mean by "algebraic methods" of solving quadratic
> equations.
>
> Discuss each of the methods of solving quadratic equations
> and disadvantages of each of these methods. Give an example
> of solving an equation by each of these methods. Can you see
> any advantages of the algebraic methods over the other methods?
>
> I think this version of the question still asks students to
> think but requires the student to give meaningful answers
> and requires students to know the mathematics to answer
> the question.

It's certainly better, but it also runs the risk (mine as well)
I learned in trying things like this (having students explain
stuff as part of their assignment) is that often the best math
students are the least interested in doing it, seeing it as
an unnecessary obstacle to simply learning methods to solve
problems. This especially applies to those talented enough
and interested enough to pursue math competitions such as
the AHSME and AIME (the math olympiad qualifying tests).
Although a very small minority, this is not a minority I
wished to alienate. It's not really a problem if done rarely
(a fairly short essay answer for every assignment, or a
more lengthy one like the above every few weeks), but if students
have to write explanations for just about every problem they
have to solve (like I've seen in some school texts), it becomes
excessively tedious for them and the quality of work you'll
get from them on these kind of assignments goes way down.

A possibly better approach might be a hybrid version that
I've sometimes tried, where I outlined how to solve
something (usually something they haven't seen before
or something a bit harder than I'd expect them to be able
to come up with on their own) and ask them to "fill in the
blanks" in some way. It occurs to me that something I once
tried as an assignment for a high school "math topics" class
(that I once taught to two very strong students, where "very
strong" means roughly U.S. math olympiad math qualifier level)
might work well. See the handout titled "obtusity.pdf" that
I posted at the following math-teach post:

http://mathforum.org/kb/message.jspa?messageID=6767154

Namely, the teacher (or textbook author) writes up a short and
equations and leave "extended blanks" in places where students
are to come up with examples to illustrate a point, or to
further explain a point that is made but not made very well.

Incidentally, writing an explanation is just one way of
having a student "work the material" in non-traditional ways.
In Fall 1996 I began teaching at a state supported math/science
boarding high school and was teaching an honors algebra 2 class
that was mainly filled with incoming Juniors gifted in other
areas than math/science, such as art, drama, humanities, etc.
During the first week I was covering polynomial multiplication
and gave two explanations, one that I called the "left brain
explanation" and another that I called the "right brain
explanation". The left brain explanation was the use of
the distributive law twice, as in (a + b)(x + y + z) is
(a + b)x + (a + b)y + (a + b)z = ax + bx + ay + ...
The right brain explanation was by using a rectangle
diagram, where one side was divided into a, b lengths
and the adjacent side was divided into x, y, z lengths.
I then mentioned how something like (a + b)^3 could be
exhibited geometrically by dividing the three dimensions
of a 3-dim cube into a, b lengths and adding up
the volumes of the 8 sub-rectangular solids these generate,
and drew a rough picture of it on the board. This seemed
to generate some interest with a few of the "artsy students"
so, on the spur of the moment, I assigned as extra credit
(up to 5 points extra on the first major test, I think)
the following: Draw a neat and carefully labeled diagram
of what I outlined, making use of different colors and/or
shading to better illustrate the various parts, maybe
drawing different views of the 3-dim cube (side view,
back view, etc.). I said that only very well done drawings
would get full extra credit points. And holy smokes, I
sure scored a bull's-eye with that assignment! I got some
absolutely excellent drawings from several of the "artsy
students", showing they understood perfectly how (a + b)^3
decomposes the way it does rather than as a^3 + b^3. Several
took other math classes from me, and one was one of my
January "Special Projects" students (a week of no classes
where faculty run "special projects" for students to sign
up for) when my topic was math in science fiction, and
he wound up making a poster of a hypercube showing the
16 parts that arise in the geometric explanation of the
expansion of (a + b)^4 (which was motivated by one of the
stories we read, Robert A. Heinlein's "And He Built a Crooked
House"). Unfortunately, I don't have any tips on replicating
things like this, even for myself. Sometimes things just
work out well, but usually (for me, at least) trying to do
things that you hope will generate high interest in certain
students is like trying too hard to get someone you like to
be interested in you (to continue the analogy I began in my
previous post today) -- it almost never works, and in fact
it usually backfires, resulting in the situation being worse
than if you didn't even try.

Dave L. Renfro