The issue for the day was calculators--but not "Are they a Good Thing?" or even "What new and interesting mathematics do they enable us to teach?" This time it was more "For better and/or for worse, calculators are now part of our students scheme of things. How should we be dealing with that fact?'
The aspects that leap to mind are, of course, the for worse ones. The recurrent image that haunted the hour was the distressingly believable one of the student groping in the backpack for the means of calculating 4 divided by 2. Sundry variations on that theme ran by, involving graphing or trigonometric functions or adding fractions. But there were counterbalances, too. Somebody commented that in the late forties there was a large flap over the fact that slide rules were wiping out everybodys capacity to do calculations. An abacus slipped in there, too (I have a mental image of it being pulled out of a brief case)--same general point, I think. Someone else told the tale of Fred Holt on the ferry to the Victoria NWCTM meeting. As he sat there, figuring out the talents of his new graphing calculator, a fellow passenger peered over his shoulder, sniffed, and commented that if they didn't keep those things out of students hands, pretty soon absolutely nobody would be able to extract square roots.
So what do we have? Well, some evidence that the graph-to-function relationship has solidified for some students--that inflection points and maxima have been demystified, along (presumably) with other concepts of that nature. A willingness to crunch numbers, which, tamed and directed, could make (for instance) Newtons Method carry some weight (though I gather the taming and directing are definitely a current need.) An ease in producing graphs and computations which IF WE ASK THE RIGHT QUESTIONS can enable them to come up with some powerful generalizations on their own. And an on-going challenge to find out for ourselves more of the potential in the tool. An example of that potential that I found very striking was John Roth's, gleaned from PFF-ing in Jan Ray's class at SCCC. His clearest memory of student use of graphing calculators was to check whether the line whose equation they had just found really was the tangent line required, by graphing the function, graphing the line and seeing whether the latter by golly touched the former at the required point. What more do you need to know about tangency?
That doesn't actually cover the question of calculator dependency, though. I'm not sure anything really does. Perhaps the best hopes offered on that had to do with future students, and they came from Jack Beal of the College of Education and Rich Edgerton, a Roosevelt High School teacher. Jack pointed out that not only has the state already largely put together a list of Essential Academic Learnings which include an emphasis on reasoning and problem-solving skills, but by the year 2000 it is intended that every graduating senior will have achieved a mastery level in those learnings. If that works at all, then we ought to be seeing the calculator assuming the role of a tool rather than a crutch. And Rich pointed that up with a tale from the high school front. He regularly teaches classes of sophomores and juniors who are slightly below the hot shot tracks. Less pressure there, so he can take the time, for instance, to have them learn slopes by really studying roofing trusses--finding what the difficulties are and figuring out how to resolve them. Lots of problem-solving and not much of algorithm-learning. Upon occasion he also teaches a senior-level course in which his kids are mixed with algorithm-enriched honors students. He maintains firmly that the lower-trackers can reason rings around the others. Monumentally anectodal evidence, but the kind that feels really good, because, as Jack had indicated, the problem-solving approach is very much the current trend, and exactly what we ought to be seeing increasing.
In fact, taking a renewed look at the situation we met to discuss, the basic worry isn't really the turning on of the calculator--it's the turning off of the mind. There's a slight shortage of panaceas for that, but not of people trying for at least a piece of a cure. Three examples came to my attention outside of the Brown Bag via conversations or e-mail. One is David Prince's tactic, applying to all cases where a student uses a calculator in the course of a solution: the solution is simply not accepted unless it includes at each stage an explanation of how the calculator was used--and why. Another is what Fred Kucsmarski is in the process of instituting in his 120 class: at the beginning of the hour the classroom is a calculator-free zone and students have five minutes in which to complete as many as they can of a sheet of absolutely basic computations at a highly elementary level. And yet another came up in a tale from Rob Smith. During a section quiz, a student came up to him to explain:"I can't compute the max. value since I need to find 192/36, and I don't have a calculator." Rob's response was, in effect, "Yes, you can!" He sat the student down and had him (her?) carry out the division and finish. A procedure which, as he remarked, is not without risks, but one which illustrates a point that seems to me worth bearing in mind: theres a huge mixture of factors in the student/calculator scene, but one is that a student may well reach for a calculator not because of having forgotten how to carry out an operation, but because of having forgotten that they actually do know how. When that is the case (a question which can be answered with a mere soupcon of mind-reading), then a Yes, you can! in the form of words or of actions, can be a really powerful gift to the student.