Teaching Quantitative Reasoning: What Counts
Broeklundian Professor of Psychology, Brooklyn College
Building Bridges: the third annual high school / college conference
May 8, 2001
For three years now, colleagues and I have been working on an NSF- sponsored project to increase the teaching of quantitative reasoning in many classes here at Brooklyn College. "Quantitative reasoning" or sometimes "Numeracy" is not just the ability to do mathematics although some basic skills in mathematics are a pre-requisite. However, it extends to the ability to understand quantitative arguments
and reason about quantitative concepts in all areas of the curriculum, the arts and humanities, social sciences, and the natural and life sciences. In the time we've been involved in our project, we've more than once heard highly educated, competent people, when they find out what we are doing, confess without evident embarrassment or discomfort that they really never could do mathematics. This is a disclaimer you'll hear from lots of people when the use of quantitative concepts is raised in polite company, people who often spent a lot of time and money to be well-educated. At times, it almost seems like a statement of honor or comradeship that distinguishes the normal people from the "dweebs".
As George Brinton has mentioned, I am a psychologist by training. One of the major things that psychologists do is try to understand why people act in the intriguing ways they do. So I have been interested in understanding why people who would never admit in public that they just can't master English grammar or can't marshal a coherent argument feel only slightly uncomfortable admitting that they just don't "get" applications of mathematical concepts. There must be some explanation for this interesting behavior, and that's the part of the QR issue that I want to put on the table for discussion today.
I am credentialed in the field of Psychology by virtue of my advanced training. But in a way, all of you, regardless of your education, are psychologists too. Every day, whether conscious of it or not, we try to explain the behavior of our friends, our families, and the person making strange noises across from us on the subway. Depending on how introspective we are, we may also try to analyze why we ourselves are acting as we do. The relevance of this to the topic at hand, quantitative reasoning, is, that some of the barriers to more effective use of quantitative reasoning are not related only to how quantitative reasoning is taught. They are internal and relate to how we explain ourselves to ourselves.
In school and in other situations involving performance and competition, we formulate beliefs about our competence. In a given situation, we make judgements about what causes success and failure, and how likely our efforts are to succeed and be rewarded. More globally, we make judgements about how competent we are in all sorts of domains, judgements that are not necessarily conscious. Research has shown that people who believe that they lack competence and have low expectations of success in a given area, or who attribute past failures to factors they don't control, tend to avoid these tasks when they can. And when they can't avoid them, they exert less effort and persist less than people with more positive beliefs. In the context of quantitative reasoning, this may explain the fact that people who really could understand quantitative concepts at a reasonable level regard themselves as inept and really don't put much effort into mastering them.
We also have research that shows that people explain failure in different ways. Whether they are aware of them or not, these explanations have important implications for expectations and behavior in achievement situations. When individuals succeed or fail at a goal or a stated standard of performance, the psychologist in them makes judgements about the cause of the outcome. Did I fail because I'm incompetent? Or because I was tired, or anxious, or unlucky? Or because the person next to me distracted me, or bumped my elbow, or made me nervous, or sabotaged me? Many words have been written about math anxiety, and there certainly are people who get themselves worked into such a frenzy in the face of numbers that they can't process anything. But perhaps math anxiety is just an extreme example of negative attributions of personal competence.
We can see the causes of things that happen to us as inside ourselves (so-called Internal Locus of Control) or we can believe that we are passive victims of factors outside of ourselves (External Locus of Control). Our attributions can be based on factors that we believe are relatively stable, such as a trait or ability, or varying from time to time, like effort, mood , or luck.
In listening to students and my colleagues talk about quantitative reasoning, I have noticed patterns that I think reflect a cultural alibi about quantitative reasoning. There seems to be a tendency for our society to categorize some intellectual abilities as learned and others as just "there" or "not there". Many in our society view mathematical ability as a stable trait that you either have or don't have. It's seen as different from abilities that you develop by hard work and study, like a large vocabulary; "ganas" (desire or motivation) as Mr. Escalante emphasizes, plays a key role in mastering the relevant concepts. Yet as a culture, there's a tendency for us to see it more like musical or athletic ability - natural gifts that some were given but that have been withheld from others. If you listen carefully, there are parallels in the way people say "I just can't carry a tune" and the common "I never could do math".
Acknowledging that there are people who are gifted in areas like mathematics, music, and athletics isn't equivalent however to accepting that lacking those gifts, the rest of us can't develop a modicum of competence, even if we may have to work harder to do it than some others. But let's not equate extraordinary skill with functional competence. Sure, not everyone can be an Einstein. Not everyone can be a Shakespeare either. When we examine these gifted individuals closely, we often find that while they may start with an advantage, they also devote many, many hours of hard work and practice to achieve their high level of skill. In almost all cases, it's not "given", but "attained".
But if we convince ourselves that we lack ability, we almost certainly won't take the steps, do the work, required to develop that ability, so it becomes a self-fulfilling prophesy. So part of the problem we are facing, eloquently described by Mr. Escalante, is that our students, and ourselves if we're in the "I never could to math" group, need to be confronted constantly with evidence that this belief is false. Society might have to be confronted too. It's true that not everyone can solve Fermat's last theorem, but that's not the level of quantitative skill that most of us need to function in daily life. The proposition in play today is that every cognitively-intact person can learn to reason effectively with numerical concepts at a level appropriate to the demands of modern life. Part of our job as educators is to find ways to end the mind games that prevent people from realizing that fact.