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 Discussion: All Topics in Algebra Topic: Solving Equations Using Backtracking

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 Subject: RE: Solving Equations Using Backtracking Author: markovchaney Date: Nov 4 2006
David, you claim not to want to argue, but you continue to make very
argumentative statements that beg to be challenged. In this case, what does the
issue of being a mathematician or not have to do with the man in the moon? This
is essentially an argument about whether a method is mathematically sound (which
this is, within limits, like most methods) and pedagogically sound (an open

So if I want to stand up for this method I need to be a mathematician, by which
I'll take you to mean a Ph.D in pure or applied mathematics working in some
capacity as a research mathematician or high-end user of mathematics.
Tragically, I'm neither. I'm merely a mathematics educator. Never even completed
my Ph.D. So probably nothing I say counts.

But at the risk of "not counting": I first encountered this approach in the
context of finding inverse functions. It was in 1992, at the University of
Michigan, in a math for secondary teachers course taught in the mathematics
department by Roland Trapp, a Columbia University knot theorist now on the
faculty of one of the campuses of the University of California (San Bernadino, I
think). The text was by Eugene Krause, now retired and then on sabbatical,
another mathematician. Rollie seemed pretty comfortable teaching this method.
Gene seemed very comfortable writing about it in a text he used for both his
elementary and secondary teacher content courses.

So now, is this method "more better"? Is it okay to teach such a method to
students who may never deal with multivariate equations? Indeed, do we decide
what methods to expose students to based strictly or primarily on how far they
can be generalized to higher order equations, more complex equations, etc.?
Granted, I loathe the "FOIL" method, such as it is, for multiplying binomials,
because it can't be used on anything BUT multiplying two binomials, but that's
more of a mnemonic than a method. No thought involved at all. Much better to
think about what one is doing (multiplying every term in each factor by every
term in the other factors) and coming up with a more generalizable method of
keeping track of what one is doing for any given case, on my view. But the issue
is hardly whether FOIL comes from mathematicians, psychologists, math educators,
or the school custodian, is it?

You clearly have a lot of contempt for psychologists and university educators.
That's hardly unique in my experience. I'm very sorry if you had bad experiences
with one or more of such folks, but one or two of us knows his behind from third
base when it comes to mathematics, mathematics learning, and mathematics
teaching. It would be rather prejudiced of you to dismiss out of hand anything
that came up on this list because of the kind of person who came up with it. If
you had an idea that had merit, I wouldn't dismiss it simply because you are
biased against me and my colleagues. :^)

On Nov  3 2006, Mathman wrote:

Nothing seems to be forthcoming.  I
> did a Google and found a technical school, Mirani High, in
> Australia.  Further Google on "backtracking" brought up this:
>
http://www.learner.org/channel/courses/learningmath/algebra/session6/part_b/backtracking.html
> The site contains a video showing a demonstration of the technique.
> I had been aware of that sort of process as one of the products of
> our university educational institution, full of PhDs of education
> and psychologists, who are not mathematicians.  The process is
> simple enough, but would apply readily only to the simplest of
> equations involving one variable, and is not in my opinion a
> technique of any extended usefulness.  It would possibly get a
> result where all else fails, but does little towards the deeper
> understanding, and so retention and usefulness gained from more
> standard techniques.  In the same sense, one can readily use
> Cramer's Rule to solve two equations, two variables, but I'd not
> recommend that in a non-academic environment except also as one to
> possibly apply where all else fails, and as a "Do This This Way"
> approach to finding a solution.

There you have it.

David.