Thursday morning at 9:30 in my office, Judith Arms and I learned

from Deborah Hughes-Hallett about a project using case studies to help

graduate students learn to teach which fits beautifully with what Judith

does in our TA-training program and potentially also with what I will be

doing with our PFF project. Thursday evening at 9:30 at the airport, I

learned from Deborah about some games for deepening understanding of

arithmetic which sound superb for at least two courses I plan to teach and

for our CML project (engaging enough so that third graders play them and

research mathematicians argue the strategies for them.) For a day of high

intensity learning about learning, I'm not sure Thursday could be beaten.

All this leads to an obvious danger of turning the newsletter into

an information overload, but I shall restrain myself, aided in the

restraint by the fact that my memory hit saturation point from time to


The last tidbit of the morning (after we had kicked the case study

issue around considerably) was Deb reeling off, for the sake of someone

who is about to go there, the list of projects in and around the

University of Arizona which might be of interest to a visiting

mathematician with an interest in educational issues. No, tidbit is too

diminutive a word--the list goes on and on. What an incredibly live place

Arizona is!

Then came the Brown Bag, with a wonderful collection of people

from in and out of the department and on and off campus. Deb wanted to

provoke a discussion about international trends in the teaching of

mathematics, so she led off with her own background: schooling in England

(at a boys' boarding school, she said gleefully, but didn't elucidate),

graduate study at Harvard, two years of teaching in Turkey, some teaching

in China, a lot of work with mathematicians in South Africa as they dealt

with the aftermath of Apartheid, co-organizer of an international

education conference on the island of Samos--you get the picture. What she

wanted to explore was an idea arising from conversations with a Kirghiz

couple and experiences in South Africa. In both places the curriculum in

mathematics is undergoing major revision motivated in part by the fact

that what is now taught has too strong a flavor of Moscow in one case and

of Apartheid in the other. In effect, both feel that many topics are in

the curriculum simply to help mathematics serve as a weeder to keep the

elite class small and distinct. But given the freedom to change, what do

you change to? Meanwhile, here in the US, one of the major tenets of the

current math reform is "mathematics for all"--the idea that our focus

should be more on seeing to it that everyone has a chance for some good

and real mathematics than on accelerating the elite who are heading for

college and beyond. How do these things relate, and what else has anyone

seen? Those were the questions Deb rolled out to us. And since we had

representatives of India, Spain, New Zealand, Canada and the Philippines,

and a whole bunch of other thoughtful people as well, they got inspected

from many perspectives. One tidbit that stuck with me is that in England

the weeding job used to be shared with classical Latin, but now the Latin

requirement has gone, leaving math to bear the burden solo. Another tidbit

(from a slight excursion out from the basic questions) is a follow-up to

one of last year's Brown Bags: we watched a videotape of a geometry lesson

in a Japanese eighth grade and we all drooled: the students were tackling

a challenging problem, diving in on their own, then conferring with each

other, and eventually watching intently and commenting as a classmate

presented a (slightly flawed) solution. Just lovely. Then we gritted our

teeth and watched a few minutes of an American class in which students

clomped their way through a long drill sheet on supplementary and

complementary angles, while the teacher coaxed them along ("Look out,

you're going to have to subtract from something different this time.") The

follow-up is that American high school students watching the film speak

very ill of the Japanese teacher. He's obviously a bad teacher, because he

made his class do a problem he hadn't shown them how to do. OW!

Which leads us neatly into the next event of the day: the Calculus

Education seminar. Deborah's choice of topic was based on the observation

that somewhere between elementary school and the university there comes

about a shift of perspective from teaching PEOPLE mathematics to teaching

people MATHEMATICS. But those still are people out there in those rows of

bolted down desks, and the more insight we have into their expectations

and responses, the better we can teach them. Not that we necessarily want

to fulfill each and every expectation. There was, for instance, the

student filled with righteous indignation at her having asked how far a

particle had traveled along a curve, thereby cheating him of the points

that were rightfully his because she should have said "arc length".

Spurred on by that, Deb did a do-it-yourself survey including the

statement "A well written problem makes it clear what method to use to

solve it." On a scale from 0 (don't agree) to 5 (totally agree) the

calculus classes she surveyed gave it a 4.1, and the pre-calculus a 4.6.

That doesn't mean that we should convert and start feeding them problems

that deprive them of the opportunity to figure out what to do (Heaven

forfend!) But it might have some impact on what we say to the students as

we present them with those problems. And another expectation we would just

as soon not fulfill: very few students, in Deb's observation, find it

possible to imagine that doing mathematics can be fun or interesting.

That's one to ponder.

And then again there was the bright-eyed student who finally

figured out what he was supposed to have been doing on his test. "Oh, you

want us to do this the way we do English or Social Studies","???", "You

want me to THINK!" Well, at least he did figure it out.

On the other hand, a different type of thought to keep around is

the one generated by Deb's most recent overhearing of a "You know what

else they can't do" type sentence--the kind of sentence with which the

walls of the math lounge are saturated. Only she didn't hear it from

faculty members this time. It was two students discussing their parents'

computer skills, or lack thereof. They may not have a bunch of the skills

we wish they had, but they have others, and we need to be aware of that.

Further anecdotes and insights were numerous, but I think I will

hold it to the insight she stated most firmly and with the most resounding

conviction: the best way to get a student to engage with the mathematics

of your class, and to be an active learner in it, is to be interested in

the student. Stand by to do some listening--maybe even listening on

subjects that are not inherently interesting to you. The pay-off is


One might think that after leading both of those seminars, Deborah

would be prepared to call it a day. Ha! She spent the afternoon in sundry

conferences and conversations, then joined nine of us for dinner at the

Sun Ya restaurant. There she regaled us with tales of her activities

around Arizona's remarkable department (like for instance taking a hefty

array of folks from outside the department to lunch, because that way they

are the most comfortable talking about what it is that they need

mathematically from their students.) Eventually I whisked her off to the

airport, where she boarded a red-eye flight home, complete with exams she

was planning to grade and return to her class in mid-morning. Talk about

dynamos (dynami?) (or maybe dynamae?)!!