>
> But "multiplying out" ->is<- a practical way of checking factoring?
> How so?
Because it is usually faster and easier than the technique being
tested and it catches mistakes that occur anywhere in the process, not
just one step.
>
> > For example a common mistake kids make is accidently or
> >mistakenly cancelling part of a numerator or denominator as in a^2/(a
> >+1) --> a/1 --> a. There is no reversable step that they can use to
> >test.
>
> But this arises because they are not cancelling identical FACTORS in
> fractions, or identical SUMMANDS in sums. That is indeed a categorical
> error.
>
Yes, and the kids are being taught those principles along with others
at the same time. They will make this mistake, not because they have
not been taught the right way to do it, but because the learning
process is uneven and imperfect and generally involves mistake. They
need a way that they can identify mistakes and then hunt them down.
When the have to hunt down what they did wrong, it generally helps
them understand and remember what they did wrong.
> >Or sometimes the reverse step is just way too hard for the kid
> >such as in =A0(2x^2 + x + 2) (x^2-2x+3) =3D3D 2*x^4-3*x^3+6*x^2-x+6.
>
> What is the difficulty here that they find so hard, exactly?
Well, when they face this problem they have not yet been taught the
general method for dividing polynomials. The general method for
dividing polynomials is a multi-step, multifactorial problem that
stretches the limits most kids working memory, procedural memory,
attention span, frustration tolerance and ability to keep things
organized. As an adult who enjoys math it does not seem like much of a
challange to me, but it really is to kids. You can just see their eyes
glaze over as they are faced with a problem harder than they have ever
tackled. A child's mind is not just limited by lack of knowlege but
also lack of capacity to hold complex and abstract conceptual
constructs, If you want me to go into the neuroanatomy involved I can,
but that is beyond the scope of this discussion.
They
> presumably did not go to this equality directly, but rather went
> halfway through by first multiplying out and then adding terms with
> identical powers of x. The step to check is only the last one.
Unless we are misunderstanding eachother, the step to check by
factoring is longer and more complex than the steps involved in the
multiplication.
>
> >Like I
> >said, going over the probelm again is not goos since people are prone
> >to make the same mistake twice.
>
> And if they don't know how to add or multiply correctly, which is
> usually the case with mistakes like the above, how is plugging in -3
> going to save them?
>
Well, adding and multiplying numbers most kids find easier than adding
and multiplying algebraic expressions. Even if they are having
problems with adding and multiplying numbers, if they make a mistake
with the minus 3, they will still have a clue that they did somehing
wrong and they must check thier steps carefully. Sometimes they can
infer what they did wrong by how thier answers differ.
> Look: the major problem I have with this is that you are presenting it
> on the same level playing field as categorically different
> "checks". "Plug in an answer to check if it solves the equation" is
> simply a different category of test than "test the identity by
> plugging in -3" (or any other number, or any finite number of inputs,
> for that matter). As I mentioned elsewhere, this 'check' is
> more akin to casting out 9s in sums and products than it is to 'plug
> back in to check solution'. By presenting it in the same playing
> field, you are in effect implicitly telling them they are
> equivalent. This is a real problem. This is more, as was said, a
> "sanity check" then a "test of the simplification", much like casting
> out nines, or checking the order of magnitude of an answer to see if
> it is in the right ballpark.
Sanity check is the correct term. I was not familiar with it until you
mentioned it here. I would prefer a unversally correct check if it was
relatively quick and easy and applied a different set of procedures so
that kids would not tend to make the same mistake twice.
>
> To further complicate things, encouraging them to always use the same
> INTEGER for testing these kinds of expressions is also liable to
> reinforce mistaken impressions, such as that "number" means "integer".
>
I have tried to impresson them that this is not a totally trustworthy
check, nor is it the only number they can use. Nevertheless, you make
a good point. At other peoples suggestions I may switch to suggesting
that they try several different easier numbers.
> >If you can come up with a better
> >alternative i would be grateful.
>
> Ah, but every "alternative" is just not practical. Apparently....
Ok, I think you are being a bit petulant here. Just because I had
problems with some of your suggestions does not mean I am not open to
others. Yes, this was an ad hominum attack ;-)
>
> First, I would not present this in the same breath as "plug in" or
> "multiply out" for the reasons I've mentioned many times.
I usually don't but you are right, i should emphasize that this check
is not equivalent to those.
Present it
> as a parallel to casting out 9s, perhaps, especially if they are
> familiar with the latter and realize the problems with it.
No they are not familiar with it but thanks for reminding me of that
technique. I was looking for some good applications of modular
arithmatic that kids might appreciate.
Second,
> rather than suggest a particular integer all the time, I would suggest
> trying 'pseudo-random' numbers; sometimes -3, sometimes 2, sometimes
> -1, sometimes 0, sometimes 1/2, sometimes -1/3, etc (and push
> fractions, at least simple egyptian ones, from time to time).
>
Maybe, I'll have to see how much ambiguity they can handle.
<snip>
=2E This would
> make it (i) more useful, since it can be used in situations other than
> "simplify", and (iii) by repeating the mantra, you are driving home
> what is behind it, rather than making it a pseudo-magical property of
> the number -3.
>
Yes, i don't want them to think of -3 as magical.
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