The answer is, considerable agreement and an enthusiastic round of applause. At least, that was what happened on October 2, when Oxford University mathematician Sir Roger Penrose, Canadian author and Science News writer (and former science teacher) Ivars Peterson, emeritus UC Santa Cruz mathematician and chaos pioneer Ralph Abraham, and yours truly, all converged on the campus of Oregon State University for a one-day "Math Summit".
Organized by Terry Bristol, President and CEO of Oregon's Institute for Science, Engineering, and Public Policy, at the request of Oregon Superintendent of Public Instruction Norma Paulus, the Math Summit brought together four hundred K-16 mathematics teachers and mathematics education professors.
The day began with twenty-minute presentations by each of we four guests. "Outline what you think is important in a mathematics education to prepare students for life in the next millennium," was our single instruction.
Given the remarkable degree of agreement between our four responses, an observer might have been forgiven for thinking we had conferred beforehand. But that was not the case. Conversation over dinner when we all met up the night before had been largely the usual mathematicians' gossip, ranging from who was doing what, where, and with whom (the mathematical variety, I hasten to add) to black holes (an old Penrose co-discovery) and Penrose's recent discovery of an aperiodic tiling of the plane with regular hexagons. (The hexagons have oriented edges and there are restrictions on how they fit together.)
Emphasize understanding, not algorithmic skill, all four of us cried in agreement.
Show students how mathematics all fits together and relates to other aspects of life, we insisted.
Present the big picture, not just the details, we argued.
Make sure it is fun, said Roger and Ivars.
Move away from the current emphasis on a uniform curriculum, on "standards", and on frequent testing.
Train teachers well, and then trust them to do their job.
Education is about people, we kept saying, not about curriculum or facts. Put the emphasis on math teachers, not the curriculum.
Remember that mathematics takes place in a historical and a social context. Our teaching may be more effective if we do not try to isolate mathematics from those two contexts.
People remember their teachers (or some of them -- the very good or the very bad), and when they remember what they were taught it is usually by association with a particular teacher.
Education won't happen if you try to impose a single approach. People are individuals, and what might work brilliantly for one person might not fly at all with another.
In the end, Ivars Peterson summed it up like this: The question we had been asked to address was "What will tomorrow's mathematics education look like?" A far better question would be "What will tomorrow's mathematics teacher look like?"
And so it went. Four individuals, all singing the same tune unrehearsed. But would it work? Can it happen?
Having set the agenda with our opening presentation, the job facing Sir Roger, Ivars, Ralph, and myself was to spend the remainder of the day interacting with the 400 attendees, in different sized groups as the day progressed, as everybody discussed what we had said (and sometimes what we had not said), and asked that crucial "How?" question.
The goals that most of the teachers seemed to share with the four of us was clearly tempered by a strong sense of reality all round. No one I spoke to was not acutely aware of the power of the parent lobby where education is concerned. In a democratic society, if the majority of parents want hours of drill, standardized tests, and school league tables, then by golly they'll get them. Demonstrable accountability is the buzz word, trust in the individual teacher so often seems to be non-existent.
So, enough of what three ivory-towered academics and a science writer had to say. What did the K-12 teachers themselves think? Sitting in on the various discussions, I jotted down verbatim some of their comments, and I'll reproduce some of them here. I could provide introductions and links. But on balance, I think the naked quotations speak louder than any rhetoric I could add. After all, we are all mathematicians, and mathematics is the science of patterns. You look for the pattern in the following thoughts.
"Students feel disconnected and incompetent in the math class."
"Divergent thinking is better than convergent thinking."
"Too often we teach the way we were taught."
"The really big interest killer is timed tests."
"Math doesn't make sense to 90% of kids."
"Kids need to understand what they are doing, not just doing it."
"We break math down too finely, and we lose the big picture."
"Kids need a system to make sense of their world. When we move too quickly, they revert to their previous system, and this leads to problems in understanding."
"Facilitator is a better word than teacher or professor."
"Parents don't like a non textbook approach."
"Problems with calculus come more from a lack of algebra understanding than a lack of algebra skills."
"We teach width but not depth."
I'll stop there. You surely get the picture. With so much agreement, you'd think we should be able to make sweeping changes. But, as I observed earlier, none of us present, neither we four presenters nor the 400 attendees, were under any illusions that in the world of education, changing the system is anything other than, at best, a painfully slow process. On the other hand, as one elementary school teacher said:
"It's easy to change the world of a five or six year old."
Maybe there are answers, and maybe we can find one of them by starting with that teacher's observation.
Or, take the insight of one middle school teacher who said that when you have a student who has failed (say) Algebra 2 twice, and you suspect that he or she is going to continue to fail if you continue with the drill and testing approach, then you have the enormous freedom to try a different approach. For those children -- and there are plenty of them -- there is nothing to lose and a great deal to gain by trying something different. Maybe it is with the present "failures" that we can prove the effectiveness of a different path.
I don't have answers. But I met a lot of highly committed, resourceful, good people at the Oregon Math Summit. If they are given a little freedom, I would think there is a fair chance they will generate plenty of ideas, and among them, maybe some answers will start to emerge. The question is, in the land of the free, will they be allowed that little bit of freedom?
- Keith Devlin