Current banner headlines on the local teaching/learning front definitely belong to David Clarke of the Mathematics Education department of the University of Melbourne, who was here last week. Having spent the week as his transport committee I am now so full of ideas and new perspectives that I feel obliged to move gently lest they slosh out. Perhaps if I leak some through the keyboard the danger will be less acute.

David's two main themes were "telling" as a teacher and assessment of student learning, with a few minor issues along the way like "What IS quality mathematics?" and "What is it that we really want our students to come away from our classes with?" The first theme is an outgrowth of a re-examination of the constructivist extremism to which David (with many others) used to subscribe. That position is roughly that if students are supposed to be constructing their own knowledge, then directly handing over any piece of information to them constitutes a violation of their construction rights. As J.Smith from the University of Michigan once put it, one can come out of the classroom with a feeling of "Oh, no! I just committed an act of telling!"

David presented four alternative teaching formats: 1) Socrates getting a Boy to uh-huh his way into halving the area of a square, 2) A geometry workshop with structure based entirely around problem sets worked and discussed by students, 3) A direct instruction model with the time parcelled out in exact chunks for review, skills presentation and drill, 4) A set of worked examples carefully rigged to avoid "cognitive overload"Qi.e. any complication which might interrupt the flow of mechanical imitation.

Rather than totally endorsing or rejecting any one he got us to think a bit about their various virtues and vices. Then he hit us with a problem that Prof. Fermi of the University of Chicago loves to turn people loose on: How many piano tuners are there in Chicago? After we had floundered about a bit, with great pleasure and rather varied results, he pointed out the various estimating and computational skills required if a bunch of students really get their teeth into it (and showed us a lovely solution produced by a quartet of middle-schoolers.) And it is in that kind of context that a student who asks "How can I carry out this step?" should be told how--without guilt!

From that platform, David made the following proposal (which I may have oversimplified a tad): the role of the teacher is to set up situations in which the student develops a need to know, and to be sure that the student recognizes that need him/herselfQthen to satisfy the need by whatever means seem appropriate. The means may well include simple telling.

David's other topic was assessment, which he addressed mainly at the Pew Festive Forum. Even getting started on that one may require a basic hypothesis shift: assessment tends to be viewed as simply measurementQan attempt to lower some sort of dip-stick into the student's mind and see how high the knowledge-level line is. Such a process is unpleasant for the assessor and downright beastly for the assessee. As a result, the word "assessment " tends to produce reactions ranging from horror to revulsion . (I can testify to that after watching people react to the forum announcement!)

In a sense, David's first thesis makes that image even bleaker: there is no such thing as one-way communication. Our choice of what to assess and our method of evaluation tell our students more clearly than any words what we really think is important.

The counterbalance to that dire pronouncement is that there are many, many ways to assess student learning apart from the quiz/test/exam routine. Most of them directly enrich student learning, and most of them are fun. None of them, mind you, are as easy to attach grades to as a basic crank-the-handle skill test, but then again, how many of us want our students to come out equating mathematics with handle-cranking?

On the other hand, it's a lot easier to make snide remarks about what we don't want them to think than it is to characterize what we do. Probably no two of us would be in total agreement even as regards what mathematics is, much less which aspects of it bear the most weight. (If you don't believe that, come to a Brown Bag or two!)

So where does that leave us? I don't knowQbut I do find highly attractive a model David produced and entitled "Assessment by Contextual Exhaustion". I shall reproduce his three categories along with the examples he gave for dealing with students' learning about area of a rectangleQlearning traditionally tested with the question "What is the area of a rectangle with sides of length 5 and 7?"

1) Physical representation.

In groups of four, using only the grid paper provided, construct as many rectangles as you can with an area of 12 square centimeters. Submit your results individually.

2) "Real world" context.

Fred's flat has five rooms. The total floor area is 60 square meters. Working in pairs, draw a plan of Fred's flat. Label each room, and show the dimensions (length and width) of each room. Submit your results individually.

3) Mathematical abstraction.

The area of a rectangle is 12 square centimeters. What might be its dimensions? Work individually.

Neat, no? And food for lots of thought about the extent to which we agree that those three aspects characterize math, and in particular math at the level we're dealing with, and insofar as we do agree, what the equivalent sets of tasks might be. Food, but definitely without recipes!

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3/4 c sugar

1/2 tsp salt

1 tsp cinnamon

Mix these. Then add

1 cup canned pumpkin

5 large eggs (Lightly beaten, it says--I never do)

1 can evaporated milk

1 1/2 tsp vanilla

Mix well. Turn into caramel coated 8 by 8 pan(see below.) Place in a pan of hot water. Bake at 350 degrees for 1 hour and 20 minutes (a knife poked into the middle of it should come out clean.)

To caramel coat a pan melt 1/2 cup sugar over medium low heat until it forms a golden syrup, stirring a fair percentage of the time. Pour immediately into pan, tilting around to spread (I always heat the pan first, otherwise the tilt is automatically too late. On the other hand, it is not clear to me that it makes a particle of difference anyway, because it all melts and mixes with the pumpkin juice.)

To serve, loosen the edges with a knife or spatula, then flip the flan onto a serving plate. Be absolutely sure that the diameter of the plate is larger than 8 times the square root of 2 (assuming you've obediently used an 8 inch square pan), or you will be mopping up caramel sauce for quite a while!

Two editorial notes:

- If, like me, you operate on the theory that what is good out of a can
is better straight from nature, restrain yourself here. I tried this
recipe twice with real, live ex-jack o' lanterns. It was about three
times the work for about a third the flavor.
- I have a sneaking suspicion this would be even better with rum instead
of vanilla (amazingly many things are), but I haven't actually tried it.

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