It was with some trepidation that I proposed another Brown Bag on calculus reform. All Seattleites love recycling, but... As it turned out, though, this was one of the best discussions we have had on the subject, and at least to me had remarkably little flavor of re-hash.

The most helpful conversation-starter was not so much the article I supplied (though all early arrivals dutifully read it) as the correspondence Chris Hillman had launched a few days ago. He took exception to an ad saying "Don't toil over tedious calculus homework assignments when you can solve problems instantly with CALCULUS WIZ. Calculus Wiz practically does your homework for you". Since Calculus Wiz is produced by Wolfram, the conversation started there, but by now he is hearing from four or five Calculus Wizzers. Their position is clear enough, and extreme enough, to anchor our explorations of ours. It is: the technology is here and getting steadily more accessible. There is no technique of integration the student can do that this program cannot do vastly more swiftly. Therefore the teaching of these techniques is uniformly obsolete and should be jettisoned altogether.

"No, no!", we cried, "Not that!!" Well, but why not? One reason raised is that some of them have generalizations that are needed in some 300 level courses. Several people commented that the students have mostly forgotten them by then, but Boris Solomyak put in firmly that if we do our course designing in terms of what the weaker students forget rather than what the stronger students remember we are doing them all a disservice. Point well taken. Judith Arms brought up the point from an earlier Brown Bag that what has been learned and forgotten is much more easily re-learned. But then which techniques is it that they really do need? John Roth put in a highly articulate argument that every one of the techniques be done without, and it doesn't take anything as glitzy as the Calculus Wiz to replace them. That then clears time and space in the curriculum to do some of the rest of the topics in something much closer to the depth they deserve. Donna Calhoun pointed out the parallel between this line of reasoning and the criticism now being levelled at the entire American K-12 system for having a curriculum that is "a mile wide and an inch deep." It was one of a number of occasions on which I observed that the Standards- based Reform discussions in K-12 settings and the Calculus Reform discussions in university settings are overlapping more and more.

Another aspect that arose in sundry forms is that students need to have the experience of having to dig in and toil over learning something mathematical and to deal with frustration in doing so, or alternatively that students ought to have the experience of seeing a concept develop from the ground up. Both unambiguously true, but if you step back a little from them, what is happening is that the integrals are being used as a vehicle for the teaching of a quite different aspect of mathematics. Perhaps one needs to consider teaching that aspect through some other aspect. Though that image opens the door just a crack on an absolutely overwhelming image of a totally revamped curriculum instead of simply revamped calculus.

For me, one argument stood out as ringing the truest, though it may be because of the number of different camps I have a foot in (don't think too hard about that image.) Jack Lee described the process by which children learn about fractions: they start with fraction pieces and play with them--shove them around, combine them, stack them to see how many of which are equal to some other one. Only after this process has made the fractions a solid, dependable part of their intellectual scheme of things do they go on to learn the rules. Once they get to there, they can go on and use the rules to deal with fractions like 3/17, which no self-respecting fraction block set would consider including. After which they lose all need for the blocks themselves because their concept of fractions is built on a solid foundation. But the "short cut" of bypassing the blocks produces kids who are manipulating a black box into which you throw fractions and from which you receive other fractions.

Techniques of integration are our fraction blocks, and if the parallel holds good, we omit them at our peril.

A few screens of prose can't do anything like justice to an hour of pretty intense discussion. In particular, at least a dozen more people made notable contributions which my sketchy powers of recall can't quite slot in. All I can hope is that this will convey a little of the content and of the range of ideas--and of how very enjoyable the whole conversation was.