Romance, Precision and Generalization

Romance, Precision and Generalization

Peter Taylor
1994 3M Teaching Fellow

or most introductory courses in mathematics (calculus, linear algebra) the curriculum is generally regarded as a “set piece” adequately rendered in any one of a number of “identical” text books. What you teach is the stuff that’s in those books and there’s not a lot more to be said.

But maybe there is a lot more to be said. Open a standard introductory calculus or linear algebra text. Do you find those books interesting, engaging or imaginative? And don’t excuse yourself with the apology: “I was never good at math.” Imaginative means imaginative, in any language.

Let me put the matter somewhat differently. In a literature course the students are asked to read and talk and think about works of literature: Some of these are “great,” others are not, but they are all works of art of some sophistication. And they are almost all “beyond” the student, not in the sense of her capacity to appreciate or gain some understanding of the art, but in the sense that she is not expected to be able to produce a work of comparable sophistication on her own. At least not yet.

Does this description apply at all to mathematics? What could be more artistic and sophisticated than calculus and linear algebra?

But somehow it doesn’t work in the same way. The bulk of the students who take these courses don’t tend to come away with a feeling of wonder for the greatness of the work, nor a feeling of pleasure in their capacity to engage it. The skills they learn are fragmentary and quickly fade. The ideas are abstract and set in a restricted, somewhat artificial context. So much injustice to the student, so much waste of a rich opportunity.

Not easy to put right. How are we to find imaginative, sophisticated examples and problems which engage the students, are faithful to the rich ideas of the subject, and which the students can handle technically? It is generally understood that the structure of mathematics is such that students really can’t move forward without a fairly comprehensive encounter with the prerequisite ideas and techniques. There’s certainly some truth in this, but for decades it has been used as a blunt axe to fragment the rich and beautiful constructs of the discipline.

In order to tackle this question properly, one must have a good understanding of my bible on how learning takes place, Alfred North Whitehead’s The Aims of Education. I recommend this short and transparent collection of essays to anyone engaged in teaching, particularly the essay on “the rhythm of education.” Whitehead identifies three stages in learning: Romance, Precision and Generalization. They tend to proceed in that order, but over many different time scales, so that each small learning task goes through the stages, as does each course, as does the entire life of the learner. So although education flows through these stages like a river opening into the sea, the stages also cycle like eddies, especially when the stream flows fast and early.

And the tragedy is that most university lectures are pitched at stage II-Precision. Romance is the stage of initiation, wonder, mystery, surprise; even outrage, games, crazy logic, intuition. Precision is the stage of care, close attention, slow patient mastery of technique. Generalization is that wonderful pulling together of many strands. The first and last stages are filled with a sense of freedom, but they are different freedoms. The first is chaotic, the unruly freedom of a child, but the last is a mature freedom informed by the insight and skill acquired at the second stage. The middle stage is ruled by discipline.

A point Whitehead makes again and again is that it is bad to short-change Romance. In a fast-paced world there is increasing pressure to cut to the chase and go after the skills before the student is ready. In doing so we incur a risk that the fruit will die on the vine before the harvest of stage III. In first year, I try to spend as much time as possible on the stage of Romance, even though it means omitting part of the “standard” skill set for the subject. I find my students are starved for this kind of activity.

A statement from one of my students, Joanna, addresses a number of these issues.

When I took his first-year linear algebra course, I noticed that he did a number of things to involve the class. His lecture notes were the textbook for the course. The great thing is that he wrote them in the first person, so when reading them I felt that he was writing to me. He also added personal exclamations throughout his notes, which highlighted concepts that were particularly fascinating. He would give the class a few minutes to figure out a problem and then take various solutions, even the wrong ones, and try them out to see which ones worked. It seemed that he was figuring out the problems in his mind at the same time we were. When we did come across solutions that worked, he would comment about how fascinating they were, and most of us couldn’t help but share his enthusiasm.

One of the wonderful things about his course was that few people felt pressured by the workload. It was not an easy one, his assignments were tough and for many people they took a lot of time. The key point, though, was that I trusted Dr. Taylor. He told us exactly the type of problems that were going to be on the exam, and those were the problems he put in his review. It was so refreshing for me to be able to focus on understanding the material rather than worrying too much about my mark.