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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: Mathman
Date: Nov 4 2006
On Nov  4 2006, markovchaney wrote:
> David, you claim not to want to argue

...and I won't, at least not simply for the sake of argument.  I was saying that
the method works, but is very limited, and so fro mthat not as useful as other
methods.  As shown it was used as an alternative **when all else had failed**.
Good enough.  But let's not establish it as a method to used on a regular basis
in this case.  Young students, seeing this stuff for the first time are
impressionable.  If they do it as an initial method, and it works, they will not
be as willing to then move onto other methods that will serve them better later
on.

There is always the claim that I also admit to myself of "Whatever works".
Neither will I argue, for similar reasons, that one teacher's preference is
necessarily better or worse than another's, for that will depend always upon the
circumstances of the classroom at the time.  But I will push my own preference
...and be ready to back it up with my own knowledge/skill base.  However, in all
things there is always room for comparison.  Some methods do, from my own
experience, bring about a better understanding than others in the long run, and
so I must not be reluctant to say so when the situation allows.

In this particular instance I did allow that the method does bring about a
solution.  I am still not convinced that it will extrapolate into a skill that
is useful in more complex situations ...in later study... as are other methods.
You mention the FOIL rule, and your own reluctance.  I agree 100%.  I do not
deny that it gives results.  However, I am always of the opinion that serious
study involves a little pain of effort.  The FOIL rule does work for
multiplication of two linear factors.  It does not apply so readily to
multiplication of factors with three terms or more [multimomials].  However,
understanding the distributive law enables understanding and ability to resolve
larger problems as well as binomial multiplication.

Another example:  In trigonometry students may often be taught the rule
"SOHCAHTOA", to enable them to memorise the trig functions.  I never, ever
employed that, but always taught the students to seek the function from within
each problem ...to look for it in the problem, not in the word.  Although more
difficult in the beginning, it payed dividends later, and in fact from then on.
It is a method that worked so that they often as not retaught other students in
later years, as I was told by them.  That is, it worked.

I repeat:  I do not argue for the sake of argument, but do reserve the right to
present a better alternative if I truly believe that it is in fact a better
alternative.  Young students can count on their fingers.  It works, but should
not be supported as a serious method of doing arithmetic.

>You clearly have a lot of contempt for psychologists and university
>educators.

Not entirely.  But do note all of the serious changes that have gone on in
education over a period approaching 50 years that have not been for the better,
and which were abandonned after the experiments were over.  I speak there
noteably of the "New Math" and "Free Schools", "Destreaming", and so on.
Mathematics which is brought to the attention of the young has been taught
throughout millenia.  As I tried to say, there is nothing new under the sun,
except that it became vogue to teach statistics in grade 3, and it was not any
serious teacher of mathematics who recommended that.  Students learning one
method of doing something will tend to stick to it.  Let's make it the right
method; the one that gives perhaps not immediate results, but long-term
results.  Let us stop experimenting with the young.

David.

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