Date: Jun 21, 1995 6:51 PM
Author: Ted Alper
Subject: Math & Calculators: perhaps an irrelevant story


On the subject of calculator use and mathematics education, I want to
toss in this incident from a recent ARML team practice we had. Now
these are *strong* students -- they all love math, they are incredibly
successful at it, they have a very solid understanding of the
processes underlying computation and yet.... I *still* think they are
too quick to use their calculators.

Context: we're practicing the "team" round of the ARML competition.
In this round, the fifteen-member team has twenty minutes to do a set
of ten problems. Cooperative work is allowed. For the last few years,
calculators have been permitted. In this practice, we are using an old
ARML competition written before calculators were permitted -- but I am
letting them use calculators anyway, since they will be able to do so
in the actual competition.

one of the problems in the set was:
Find all numbers between 90 and 100 (inclusive) that can
not be written in the form
a + b + ab
where a and b are positive integers.

If you have spent some time playing around with multiplying
and factoring polynomials, you may recognize that a+b+ab
looks a lot like (a+1)*(b+1)... in fact
a+b+ab = (a+1)*(b+1) - 1
And this means that a number can be written in the desired form unless
it is one less than a prime. It's easy to confirm that the only primes
between 91 and 101 are 97 and 101, so only 96 and 100 can't be written
in the desired form.

There are a number of other ways to do this problem, some of them
minor variants of the above, others less efficient and even an
exhaustive search can be done in a few minutes if one sees a few
tricks to narrow the range of values for the pair (a,b).

In our practice, our first team, which included eight USAMO
participants, elected to solve the problem by writing a quick program
for their calculator to enumerate the cases (without even seeing the
reduction mentioned above -- they simply had it run through all a and
b from 1 to 100) and print out all the numbers between 90 and 100 that
they got. This would work, too, of course -- but they made a mistake
somewhere in the details of the program (they were in a hurry) and so
blew the question.

I'm not sure what to think about this. On the one hand, it's a pretty
narrow and somewhat artificial problem, and the time constraints and
other problems in the set make the circumstances still more
artificial. Then, too, I have no objection in principle to using a
programmable calculator either to check one's work or as a shortcut.
Time is short, once can't rely on waiting for an insight to hit, if
you see a tractable way to a solution, go for it.

On the other hand, I would have hoped that they would not have
considered the problem done when the machine spat out its answer --
that at least one of them would have followed up and tried to confirm
their answer directly, algebraically. It also bothers me a bit that they
didn't see the algebraic pattern instantly.

I'm not trying to imply that I'm smarter than these students are --
quite the contrary, they are all sharper than I was at their age and
several are sharper than I am now. But they don't, in general, look
for algebraic patterns first, and they may not have as complete a
collection of algebraic patterns in their heads to draw upon as I feel
they should.

Littlewood wrote somewhere (in his "Miscellany"?) that he considered
every integer to be his personal friend. By which I think he meant
that he felt intimately familiar with the properties of numbers as a
result of long hours spent playing, computing, and working problems
with, through, over, and around them. It may well be possible to
acquire such familiarity even while using calculators and computers
extensively -- and indeed, to capture altogether new properties and
details -- but I think there is substantial risk of missing some core
notions, too.

[As an aside, note that the form of the problem ensures that it wouldn't be
enough just to plug a + b + ab into MAPLE or some symbolic computation
software and try to factor it --- without "completing the product" the
expression doesn't factor. This sort of limitation to symbolic
computation software really does crop up all the time in real work --
you need not only a grasp of the fundamentals, but experience born of
cranking through simpler cases to guide the software. Maybe, in a few
years, more intelligent symbolic computation software will be
available... maybe not.]


I'd like to see students who can do it all -- who don't shy away from
calculators OR algebra and who can use each to reinforce the other.



Ted Alper
alper@epgy.stanford.edu