```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 totoss in this incident from a recent ARML team practice we had. Nowthese are *strong* students -- they all love math, they are incrediblysuccessful at it, they have a very solid understanding of theprocesses 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 setof ten problems. Cooperative work is allowed. For the last few years,calculators have been permitted. In this practice, we are using an oldARML competition written before calculators were permitted -- but I amletting them use calculators anyway, since they will be able to do soin the actual competition.one of the problems in the set was:Find all numbers between 90 and 100 (inclusive) that cannot be written in the form                   a + b + abwhere a and b are positive integers.If you have spent some time playing around with multiplyingand factoring polynomials, you may recognize that a+b+ablooks a lot like (a+1)*(b+1)... in fact             a+b+ab = (a+1)*(b+1) - 1And this means that a number can be written in the desired form unlessit is one less than a prime. It's easy to confirm that the only primesbetween 91 and 101 are 97 and 101, so only 96 and 100 can't be writtenin the desired form.There are a number of other ways to do this problem, some of themminor variants of the above, others less efficient and even anexhaustive search can be done in a few minutes if one sees a fewtricks to narrow the range of values for the pair (a,b).In our practice, our first team, which included eight USAMOparticipants, elected to solve the problem by writing a quick programfor their calculator to enumerate the cases (without even seeing thereduction mentioned above -- they simply had it run through all a andb from 1 to 100) and print out all the numbers between 90 and 100 thatthey got. This would work, too, of course -- but they made a mistakesomewhere in the details of the program (they were in a hurry) and soblew the question.I'm not sure what to think about this. On the one hand, it's a prettynarrow and somewhat artificial problem, and the time constraints andother problems in the set make the circumstances still moreartificial.  Then, too, I have no objection in principle to using aprogrammable 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, ifyou see a tractable way to a solution, go for it.On the other hand, I would have hoped that they would not haveconsidered the problem done when the machine spat out its answer --that at least one of them would have followed up and tried to confirmtheir answer directly, algebraically. It also bothers me a bit that theydidn'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 andseveral are sharper than I am now.  But they don't, in general, lookfor algebraic patterns first, and they may not have as complete acollection of algebraic patterns in their heads to draw upon as I feelthey should.Littlewood wrote somewhere (in his "Miscellany"?) that he consideredevery integer to be his personal friend. By which I think he meantthat he felt intimately familiar with the properties of numbers as aresult of long hours spent playing, computing, and working problemswith, through, over, and around them. It may well be possible toacquire such familiarity even while using calculators and computersextensively -- and indeed, to capture altogether new properties anddetails -- but I think there is substantial risk of missing some corenotions, too.[As an aside, note that the form of the problem ensures that it wouldn't beenough just to plug a + b + ab into MAPLE or some symbolic computationsoftware and try to factor it --- without "completing the product" theexpression doesn't factor. This sort of limitation to symboliccomputation software really does crop up all the time in real work --you need not only a grasp of the fundamentals, but experience born ofcranking through simpler cases to guide the software. Maybe, in a fewyears, more intelligent symbolic computation software will beavailable... maybe not.] I'd like to see students who can do it all  -- who don't shy away fromcalculators OR algebra and who can use each to reinforce the other.Ted Alperalper@epgy.stanford.edu
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