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.