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Re: [Phys-l] crutches versus shoes

On 04/09/2011 08:19 AM, Bernard Cleyet wrote:
Finally, I have found in the past that various memory devices in math
(e.g. FOIL, cross multiplication, etc.) have limited function and
students tend to use them incorrectly much of the time.

I would like to emphasize (again)
-- we are talking about shades of gray
-- the devil is in the details
-- the angels are also in the details
-- at some superficial level, crutches and shoes and bicycles are all
in the same category, namely devices to help us get around ... but at
the non-superficial level they are not the same, not by a wide margin.

The same goes for memory devices, formalisms, algorithms, et cetera.

FOIL is an interesting example, illustrating the non-extreme case, far
from either extreme.

Suppose we want to multiply some binomial (a + b) by another binomial
(x + y). There will be four terms in the product. A /tableau/ is a
good "device" for making sure we have got all the terms:

a b
+-------------
x | az bx
|
y | ay by
|

More generally, if we want to multiply something with N terms by something
with M terms, there will be MN terms in the product. Now the tableau is
even more of an advantage:

a b c
+--------------------
x | ax bx cx
|
y | ay by cy
|
z | az bz cz
|


In the two-by-two case (a + b)(x + y) FOIL is a mnemonic for the product terms:

a b
+-------------
x | az bx
|
y | ay by
|

can be described as:

a b
+---------------
x | first inner
|
y | outer last
|


There are two possibilities:
*) If FOIL is taught properly, there is an upside with minimal downside.
The 2x2 case is a special case, but it is a quite common special case,
and it can't hurt to give names to the four terms that arise in this case.
*) If FOIL is taught improperly -- "do it this way because I told you to" --
then it is a special case disconnected from the general case. It's just
one more thing to remember.

I reckon FOIL is taught improperly more often than properly, and I don't
know how to solve that problem, but that's a much longer conversation that
I'm not going to pursue right now.

The point for today is that formalism is a tool. Although any tool can be
abused, tools are still -- in general -- good things. The ability to make
and use tools is one of the things that makes us human. Formalism can
sometimes be applied by rote, in the absence of understanding, but that
does not mean formalism is the same as rote. Formalism, when properly
used, promotes understanding and provides a basis for deeper understanding.

The word "algorithm" also applies here. It means almost the same thing
as formalism. It is a recipe for doing things systematically. Writing
the MxN product as a tableau is an example of a very simple algorithm.
Most students I know would benefit from being more systematic, not less.

I mention this because sometimes in connection with so-called research
in education, the word "algorithm" is used as a curse word, more-or-less
synonymous with plug-and-chug. This is an abuse of the word. This is
offensive to the people who build algorithms for a living. Some people
who abuse this word have, when challenged, exhibited an astonishing lack
of understanding of the algorithms they complain about.

If you mean rote, say "rote".
If you mean plug-and-chug, say "plug-and-chug".
If you mean lack of understanding, say "lack of understanding".

As a general rule: Do not confuse the presence of one thing with the
absence of another. If a formalism is being used without understanding,
do not confuse the presence of one thing (formalism) with the absence
of another (understanding). Deal with the lack of understanding. Do
not blame the formalism.

Methods, devices, formalisms, and algorithms are by-and-large good things,
although some of them are better than others, depending on details of the
situation ... and any tool can be abused.