The first two events of the week both had to do with Susan Pirie, who was visiting from Vancouver. She's just moved there from England (most recently Oxford) to take up a chair in Math Ed at UBC, which accounts for a certain amount of "We, I mean they..." in her conversations. Her research field, however, needs no such reorientation, because her focus is on how people at all levels learn mathematics, and any discrepancies in how the English and the Canadians do that--not to mention how we and our students do so--are at a pretty trivial level.

In the colloquium on Tuesday, Susan addressed the issue of language and how much more there is to this than meets the eye (ear?) Her underlying point was that any word that has genuine, usable, meaningful content for us has at some level an image--a metaphor--attached. It could be an image supplied by the person from whom the word was learned, or it could be one we have concocted ourselves, but it is there--it has to be. The snag is that that image can get in the way when the meaning needs to be extended. For instance, the image of solving an equation by removing objects from both sides of a balance scale is a powerful one, and can (if all goes very well) give a nice, solid foundation to a student's understanding of the expression "solve the equation." On the other hand, if it holds exclusive sway as the available metaphor, then students are going to get a little anxious when they finish shifting around the kilogram blocks in the picture and find that the cat on the first balance beam weighs the same as the cow on a later one, and may (in fact, should) panic when they carry out the validated block-shifting procedure and discover that the duck and two kilogram bricks together on one side appear to balance an empty tray on the other. Does this mean that they shouldn't be given the balance metaphor in the first place? No. It means that when a student jams at some seemingly trivial point, the teacher should try to tune into what, in the student's working image of the situation, might be blocking the progress so as to be able to help the student step beyond the original metaphor. Also that many stages later, if something really disconcerts and confuses the student, sending her back to the thumb-in- the-mouth-and-security-blanket-over-the-shoulder stage, it is to that original image, however wildly inappropriate it may by that time have become, that she will probably return.

Susan illustrated that point with a lovely videotape of a pair of twelfth graders on a lead-in to derivatives getting totally hung up when the 0/0 situation reared its ugly head and stampeding their way right back to their respective (non-matching) images of zero (and when you have a whole LOT of nothin's together do you finally have somethin' or don't you?) Their plight eventually led Jim Morrow to suggest that an alternative title for the talk might have been "Much Ado about Nothing!"

So what's the moral of all this? Susan didn't hit us over the head with one, but it seems to me that it boils down to LISTEN! More than that, rather than automatically converting a student's statement into proper mathematics, listen to the choice of words itself and contemplate its significance. Thinking about that I have come to wonder whether the reason that consistently I will have a few students from the desperation fringe of my remedial classes converting 2x+6 into x+3 is that "factor out the 2" tends to slither into "take out the 2", and taking out is something one does to the garbage!

On Wednesday evening, Susan dealt with the understanding of understanding from another angle. For a number of years she and a colleague (Tom Kieren of the University of Alberta) have been developing a theory in which they characterize the growth of knowledge as a progress out from the center of a series of nested circles (onion skins, if you're feeling three dimensional.) The first layer out from the central "primitive knowing" (what you come into the current situation with) is "image-making", which can be anything from taking three of the eight beans OUT of the cup and seeing that five remain to producing entirely invisible three dimensional mental graphs by hand-sculpting the air in front of the blackboard (I know of at least one person who would then duck under in order to look at the other side!) At the outermost level comes the total structured understanding of the concept which can permit wild leaps of the "What if?" variety. About halfway out is formalizing, which is a very familiar landmark for one and all. The process of learning does not, however, consist of ticking off layers one at a time and progressing steadily outward. At any level a glitch may occur, and when it does, the learner folds back a layer or two or three. For the eight-year-old this could mean digging out the cup and the beans. For the research mathematician it's more like the line I once heard from one of my husband's Texan colleagues : "I'm fixin' to get skeptical about that thurem." Whatever the level of the learner, the effect is the same--he folds back to an inner layer or two. But it is not a regression, because he arrives at the layer with a larger and presumably better understanding than he had the last time through--hence the terms "folds back" to a "thicker understanding."

Susan demonstrated a major element of their research method by leading us through analyses of dialogues between students in the process of learning various topics. She also completely wowed the assembled multitudes by letting us take a crack at one of the subjects on which we were about to see videotaped groups of students working. Taxicab geometry, it's called: what, for instance, is the set of points equidistant from some fixed point if the fixed point is on a street and the distance is being measured on the odometer of a taxi? Ramifications of that proved so entrancing that it was truly remarkable that Susan got us (well, most of us) to put down our pencils and watch the videotape.

What to do with all this? Well, it is a theory, not an algorithm. What it can give us at this stage is a handle on thinking about our students' thinking. It makes it a good deal less horrifying when we see a "good" student suddenly working at a level that we were convinced the student was past. It provides a way to consider what kind of question the student needs from us--one that will contribute to outward momentum, or one that will encourage folding back. It also provides an image which could be useful in the constant struggle to keep oneself from, as Susan put it, "picking the student up out of one level and simply plopping them down several layers further out." As a cautionary illustration she told us the tale she heard from Kath Hart, of the student whose teacher jumped the gun on formalizing the notion of area. Students had been determining areas of rectangles by covering them with bricks and counting the bricks. When they had gotten to the stage of putting just one row and one column and doing the reckoning from there, she decided to tell them about multiplying the number in the width by the number in the length. Then she gave them a practice sheet with a column of areas to be calculated as "sums" (=computations) and a column beside it for checking the results of the computation by counting bricks. One young man swiftly and confidently filled both columns. The researcher pointed out that the numbers in the two columns appeared unrelated, to which the student cheerfully agreed. He could see no problem with his results, "'cos bricks is bricks and sums is sums."

And on that cheery note, we de-camped up the stairs of the Faculty Club to proceed to the second part of the evening's Pew Festive Forum--the eating and socializing part. Both highly enjoyable pursuits, but in this case I would say especially the latter, because we had representatives from Seattle University, Seattle Central Community College and Norway, yet even, and from on campus we had math and applied math, and we had faculty, graduate student and administration. We sat at tables of six, and no table had fewer than two categories represented--most had more. I call that super mixing!

But that didn't end the events of the week. Thursday we had one of the best attended and liveliest Brown Bags of the year. The topic at hand was working in groups. First off Brian Hopkins described his group quizzes--a tactic inspired by his PFF visits to Carl Swenson's class at SU. Carl characteristically randomizes his groups by dealing out a deck of cards (except that he removes a suit so as to have groups of three.) He also uses the suits to determine some of the tasks (diamonds are recorders today, and clubs should be in charge of calculations,...) Brian has picked up all of that, though I gather he has not yet mastered Carl's tactic of occasionally dealing from the bottom of the deck when there is some combination of students he wishes to produce or avoid! Response from Brian's students has been very positive. Steve Monk supplied us with a list produced by his students laying out the positive and negative features of group work. We spent a bit of time on those, especially the negative, because it needs to be acknowledged that the gains produced by working in groups are not cost-free. Michael Keynes, who has taught sections for lecture courses both by performing homework problems at the board and by getting the students to work on them together felt that in that context it is a pretty unambiguous gain. I chimed in with my conviction that in working with students who are going to be elementary teachers it is absolutely essential to provide that experience, partly because they need to be aware that it is a valid method of teaching, and even more because they are likely to be the portion of the student population most in need of the gains that definitely do accrue from groups: increased ability to communicate and articulate mathematical ideas, and increased confidence. Steve had previously made similar comments about the Algebra course for future high school teachers.

Conversation became more general at that point, and all sorts of interesting remarks and queries went by in a connected--well, fairly connected--sequence that began in the group work arena and wound up rather elsewhere. Toward the end, David Pengelley listed for us a number of courses he teaches at the University of New Mexico, with mini- descriptions of what he does in several of them. It didn't bounce us back into the group work issue, though--it bounced instead into the highly thorny area of assessment. As that thicket loomed up on either side of us, I looked at the clock and discovered we had run overtime, so we secured a promise from David to lead us onward, marked the subject "to be continued", and disbanded.

You shall hear more on the topic--though in an asynchronous newsletter covering Brown Bags which have an asynchronous sequence of subjects heaven alone knows when!