On Knowing and Not Knowing Math
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This page contains a 1978 article "On Knowing and Not Knowing" written by Gene Maier.
The article is Copyright © 2003 by Eugene A. Maier, Ph.D. Used by permission. The article is protected against editing changes by readers. However, readers can comment about the article and contribute their own ideas through use of the discussion choice in the top menu bar.
This article is one of a collection of three closely related articles:
Mathematics and Visual Thinking.
Other writings by Eugene Maier that are available online can be found at:
http://www.mathlearningcenter.org/resources/gene/archive
http://www.mathlearningcenter.org/resources/gene/play-on-numbers/
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The Math Learning Center
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Salem, Oregon 97309-0929
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Cite this article as:
Maier, Eugene (1978). On knowing and not knowing, in school and out—a theory as to why so many of us are “bad” at math. Gene's Corner and Other Nooks & Crannies: Perspectives on Math Education. Salem, OR: The Math Learning Center. Retrieved from http://iae-pedia.org/Knowing_and_Not_Knowing_Math.
Contents |
Introduction
Many people understand mathematics, yet don’t understand that they understand. They know, but they don’t know. It is this, I believe, that inspires much of the needless apprehension and unease many people feel towards mathematics.
Examples
Let me explain by means of a couple of examples.
A student once asked me to help her with an arithmetic problem based on an advertisement. A dress was offered for sale at 20 percent off the list price. The sale price was $60. The problem was to find the list price.
I asked this student to tell me her difficulty in solving the problem. I was surprised when she promptly told me the solution and explained how she found it.
“The sale price of $60 was 80 percent of the list price,” she said. “So 20 percent of the list price was one-fourth of $60, or $15. Adding $15 to the sale price of $60 gives the list price of $75.”
“That’s right,” I said, a little baffled. “What is it you want to know?”
“How do you do the problem?”
“I don’t understand. You just did the problem.”
“No, I didn’t,” she insisted. “What’s the formula?”
As we talked further, the realization dawned on me: She believed she didn’t know how to do the problem unless she plugged numbers into a formula. She had correctly solved the problem, but couldn’t believe she had. She knew, but thought she didn’t.
Another example. Once when I was cashing a check, the bank teller noticed that it was issued by the Oregon Mathematics Education Council.
“Oh, mathematics,” she said with a grimace. I asked about her reaction. “Mathematics was my worst subject,” she said. “I never was any good at it.”
She paused, then was struck by the incongruity: a bank teller bad at mathematics. She quickly tried to reassure me that she was a capable teller and actually quite good at mathematics.
She told me that she didn’t know mathematics, and that she did. Almost in the same breath. “I know, but I don’t know.”
An Observation
I observe this phenomenon often. I see it in well-educated people who handle the mathematics of their daily lives without difficulty. Yet they disclaim any mathematical ability, vowing it was their worst subject in school. I hear about this phenomenon from parents who tell me their children are hesitant and unsure when they do school arithmetic assignments. Yet in playing board games at home, these same children handle the necessary arithmetic with aplomb and dispatch.
Observations like these have led me to conclude that when people say they don’t know mathematics, though they demonstrably do, they’re referring to some special kind of math—to school math. For these people, the math they learned (or failed to learn) in school is completely unrelated to the math they use in their everyday lives—the math I’ve dubbed “folk math.”
Some Brain Theory
I want to suggest some reasons many people see little or no connection between these two brands of mathematics. My conjectures are based on recent research into how people learn, particularly into how the two hemispheres of the brain play different roles in the process of learning.
The human cerebrum—the large front portion of the brain, the seat of conscious mental activity—is divided into halves or hemispheres. The two hemispheres, the right and the left, are joined only by a band of nerve fibers known as the corpus callosum.
Over the past century, psychiatrists and psychologists have noticed that when one of the cerebral hemispheres is damaged, or when the corpus callosum is severed, a person’s mental abilities will change in fairly predictable ways. Damage to the left brain is likely to impair the ability to speak and write. Damage to the right brain may affect spatial reasoning, the ability to sketch objects or physically manipulate them.
The corpus callosum has been severed surgically to control seizures in some persons suffering from severe epilepsy. In these persons, the two halves of the brain appear to work in isolation, the right brain guiding the left side of the body, the left brain guiding the right side. Such persons are able to perform visual, spatial tasks only with their left eye and hand, tied to the right brain. Conversely, they can name or give verbal description only to those objects seen or touched with the right eye or hand, tied to the left brain.
Thus the left brain seems predominantly involved with rational, analytic, sequential, logical tasks. It arranges words into sentences, does mathematical computations, and puts quantitative measures on phenomena. It dissects knowledge into sequential bits and pieces.
The right brain appears to deal with information in intuitive, global, relational ways. It recognizes shapes and faces, is responsible for spatial orientation, and has creative flashes and insights. It forms overall impressions or “gestalts” from diverse bits and pieces of data.
How do the two hemispheres related in an undamaged brain?
“In most ordinary activities,” says Stanford psychologist Robert Ornstein, “we simple alternate between the two modes, selecting the appropriate one and inhibiting the other.” Ornstein’s research into the patterns of brain activity suggests that the complementary working of the two hemispheres, and of their two modes of thought, underlies our most creative accomplishments. Bob Samples, a humanistic psychologist, argues that our psychological health depends on our attaining an equilibrium between the two modes of thought, on not allowing one mode always to dominate.
Relating the Brain Theory Ideas to Folk Math
This theory about the two modes of thought and how they work together appeals to me. It provides a framework into which my own experiences, observations, beliefs, and hunches about the teaching and learning of mathematics fit nicely. And perhaps this theory explains the phenomenon of “I know, but I don’t know.” I have an idea why many people find little connection between school math and folk math.
I suspect these people do their folk math using both hemispheres, both modes of thought, in a complementary way. They shift back and forth between hemispheres as needed. They possess a good, intuitive, right-brain sense of how numbers work. They translate that sense into appropriately analytic, symbolic, left-brain computations. The problem is solved.
But for these people, school math is not intuitively sensible. In school, right-brain thinking seems not to apply. School math instead is a sequence of abstract, isolated, linear tasks, tasks momentarily memorized, tasks whose meaning or significance is never directly experienced and never fully understood. School math is for the left brain only.
These people—they may constitute a majority of humankind—have two divergent understandings of mathematics, two concepts that are at odds. Their left-brain knowledge of math, mostly acquired in school, doesn’t fit and perhaps doesn’t even connect with their right-brain knowledge, acquired mostly outside school.
This may be why, if the National Assessment of Educational Progress can be believed, many U.S. schoolchildren can perform basic arithmetic as taught in school, but cannot apply that knowledge to simulations of “real-world” mathematical problems. This also may be why many people find mathematics in school an ugly, painful, frightening thing, to be avoided at any cost.
Gap Between School and Life
How does this happen? Why this gap between school and life?
Not long ago I visited a first-grade classroom during arithmetic time. The teacher asked someone to write the numeral “five” on the chalkboard. A little girl volunteered. She began writing it legitimately, but not precisely in the way taught in class. I thought she did the task in an acceptable way, but the teacher told her she was wrong, without explanation. Another child was called on, who used the “right” method and received the teacher’s approval.
I wonder about that little girl. I’m sure she since has learned the standard school method, which is fine. But I wonder what else she has learned. Her method appeared as efficient and reliable as the teacher’s, and apparently it had emerged from her own experience, her own sense of how numbers work. Yet she was told her method is illegitimate, that it doesn’t apply in school. She was told—or at least her right brain was told—that school math is a miscellany of arbitrary, capricious rules, rules that don’t make sense.
The teacher was not at fault. Teachers have long been taught that mathematics—and almost everything else—is most easily learned in little bits and pieces. This is the one belief shared by proponents of various educational fads, from “competency-based education” to “individualized instruction.” The smaller the bits, the easier learning is thought to be. Mathematics is this reduced to a vast number of rules for pushing symbols about with pencil and paper.
But the parts are less than the whole. Teachers and students become so immersed in procedures that they lose sight of what is important. The ability to analyze problems, to grasp relationships, to intuit possible solutions—all are forgotten as students and teachers slog through a rigid hierarchy of largely meaningless skills.
School math thus becomes a wholly left-brain activity, clean and logical, but sterile, abstract, and uncreative. Students come to regard math as painful, and teachers learn to tolerate it as dull.
Outside school, confronted by an actual, practical problem requiring some mathematical thinking, teachers and students alike may find themselves helpless, unable to perform the right-brain task of “seeing” how bits of school math might be combined into a solution. Their ability to cope with living is diminished.
Or else they may solve the problem by shifting to their right brain, their folk math, developed through experience and maintained despite the almost constant efforts of educators to repress it. Then they merely are cut off from a large part of our cultural heritage, they are unaware that they are doing mathematics. They are unaware that the thinking involved in their folk math is actually closer to the well-springs of human knowledge than the school math they fear and despise.
Conclusions and Recommendations
I reflect on my own mathematical training. I’ve succeeded at lots of school math, which I pursued through to a doctorate. But much more of the math I studied I comprehended only in a left-brain way. I mastered logical connections, analytic methods, and technical language. I passed tests. But I had no intuition, no feeling, no “big picture.” Only those mathematical topics that I messed with and mulled over, that I could envision graphically, that I had an intuitive sense for, a sense of the whole—only those topics did I truly understand, only those topics could I see in new or creative ways, only those topics could I make new discoveries about. I knew them in both left and right hemispheres. I knew them in my head and I knew them in my bones.
If making confident and creative use of mathematics depends on both right- and left-brain thinking, then school ought to encourage both. How might this be done?
First, school could honor intuitive and visual thinking as valid and useful accompaniments to rational and analytic thought. Appearances can be deceiving, but so too, unsuspected errors may lie hidden even in the most elegant deductive proof. In solving a problem, mathematical or otherwise, I find that mental images, intuitive hunches, and just plain guesses often suggest a likely strategy for finding a solution. Schools could be more concerned with providing children experiences likely to nurture their abilities to think.
Second, school could allow mathematics to emerge from students’ own experience. Too much of school math is tied to some imagined future event, and introduced because someone thinks students will need it someday. Such prophecies often turn out badly. But in any case, children cannot intuitively understand mathematical situations or problems they have never encountered themselves. Schools could draw on children’s mathematical experiences, and foster further experiences, as a basis for mathematical teaching.
Finally, school could encourage children to use the valid arithmetical methods they have figured out for themselves, based on their own experience. If these methods are slow or awkward, then quicker, more elegant ones may be offered as alternatives. Certainly children should not be made to think that a workable method is invalid merely because it is different. Above all, the importance and integrity of a child’s own intuitive and reflective efforts to make sense of numbers and the world ought to be respected.
About the Author
Quoting from The Math Learning Center Website:
- Dr. Eugene Maier is past president and cofounder of The Math Learning Center, and professor emeritus of mathematical sciences at Portland State University. Earlier in his career, he was chair of the Department of Mathematics at Pacific Lutheran University and, later, professor of mathematics at the University of Oregon.
- He has a particular interest in visual thinking as it relates to the teaching and learning of mathematics. He's coauthor of the Math and the Mind's Eye series and has developed many of the mathematical models and manipulatives that appear in Math Learning Center curriculum materials.
- He has directed fourteen projects in mathematics education supported by the National Science Foundation and other agencies, has made numerous conference and inservice presentations, and has conducted inservice workshops and courses for mathematics teachers throughout the United States and in Tanzania.
- Gene Maier was born in Tillamook, Oregon and is a lifelong resident of the Pacific Northwest.
