# Another Case of Swindling

By Dr. Eugene Maier

A few weeks ago I attended a meeting of the advisory board of the Interactive Mathematics Program (IMP), Rocky Mountain Region. This is a professional development and support program for secondary mathematics teachers who are implementing IMP curriculum, an integrated, problem-centered secondary curriculum that stresses the development of problem-solving skills and conceptual understanding. It requires changes in the role of students and teachers from that of the traditional mathematics classroom. The student becomes an active rather than passive learner, and the teacher becomes a mentor rather than a lecturer.

One session of the meeting was a panel of upper-division college students who had been IMP students in high school and were describing how their involvement in IPM had benefited their college educations. They also contrasted the nature of their college mathematics courses with what they had experienced in IMP. As an IMP student, understanding mathematics concepts and attaching meaning to procedures was important. In most of their college classes, the panel members reported, that wasn't important, at least not to the students - they just wanted to know how to do the problems. And often the professors' lectures were beyond comprehension. One bright young woman who was about to graduate in international business recounted in particular a statistics class she had taken in which she had no idea what was going on. I asked her what grade she got in the course. She replied, dismissing the question as if the answer should be obvious for, after all, she was a good student, "Oh, I don't remember, it was a B or a B+." Just as I expected, I thought to myself, another case of math swindling.

I borrow the phrase from Carl Jung. In his autobiography, Memories, Dreams, Reflections, Jung ponders why school mathematics was so trying to him when he had no doubts about his ability to calculate. He describes an algebra class in which he was completely confused: "From time to time the teacher would say, 'Here we put the expression so-and-so,' and then he would scribble a few letters on the blackboard. I had no idea where he got them and why he did it—the only reason I could see was that it enabled him to bring the procedure to what he felt was a satisfactory conclusion. I was so intimidated by my incomprehension that I did not dare ask any questions." However, Jung tells us, "I was able to get along, more or less, by copying out algebraic formulas whose meaning I did not understand, and by memorizing where a particular combination of letters had stood on the blackboard." "And as," he continues, "thanks to my good visual memory, I contrived for a long while to swindle my way through mathematics, I usually had good marks."

Swindling one's way through mathematics. I know exactly what Jung meant. I've done it, too—even as a graduate student in mathematics. There are courses I have taken, where if you would look at the scores recorded on test papers, I was near the top of the class when, in reality, I had no idea what was going on. I could state all the definitions and prove all the theorems, but, in so doing, I was relying on my memory and not my understanding.

Others, I have discovered, have also been in on the scam. On a number of occasions I have asked an audience of adults if anyone ever swindled their way through a math class, that is, taken a math class, done the required work and passed—perhaps with an above-average grade—and afterwards wondered what the class was all about. Hands go up all over the room. And if I ask about their experiences, tales of memorized procedures, rote learning, and repetitive drill abound. A lot of us have been math swindlers.

That's not to say we swindlers have done anything dishonorable. We played by the rules of the game. We figured out how to give our teachers the answers they wanted. Within us, we knew that our understanding was superficial. But that was of small concern compared to passing courses, earning diplomas, and getting scholarships.

While swindling may be prevalent, it's not universal. I have been in a number of school settings where swindling is not an issue. Concepts are introduced with concrete examples, students discuss their understandings with one another, meaningful questions are asked, procedures are developed and tested, problems are posed and readily tackled, and students willingly talk about what they know and don't know. And students are evaluated on other than their ability to successfully carry out routinized procedures.

The next time you look at a set of test scores, ask yourself what's been measured. The test taker's mathematical expertise or their skill at swindling.