Folk Math
From IAE-Pedia
This page contains a 1976 article on Folk Math written by Gene Maier. It also contains an email letter from Gene Maier written in April 2008.
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/
The Folk Math 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.
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Cite this article as:
Maier, Eugene A. (1976). Folk Math. Gene's Corner and Other Nooks & Crannies: Perspectives on Math Education. Salem, OR: The Math Learning Center. Retrieved from http://iae-pedia.org/Folkmath.
The April 2008 email message to David Moursund is reproduced with the permission of Gene Maier.
Contents |
Folk Math (1976 Article)
One of my first realizations that something was awry in school mathematics was the chasm that existed between people's everyday encounters with mathematics and what they learned in school. “Folk Math” addresses the nature of that gap and offers some suggestions for narrowing it.
“Folk Math” first appeared in the November 1976 issue of The Math Learning Center Report (later the Continuum), published and distributed with the support of a National Science Foundation Dissemination grant. It was reprinted in the February 1977 issue of Instructor magazine and in the December 1980 issue of Mathematics Teaching, a publication of Great Britain's Association of Teachers of Mathematics.
Consider the following question from the “consumer mathematics section of the first National Assessment of Educational Progress, the nation-wide testing program that has surveyed the “educational attainments” of more than 90,000 Americans: “A parking lot charges 35 cents for the first hour and 25 cents for each additional hour or fraction of an hour. For a car parked from 10:45 in the morning until 3:05 in the afternoon, how much money should be charged?”
The question was answered correctly by only 47 percent of the 34,000 17-year-olds tested, a result widely cited as an example of Americans’ poor mathematical skills. But does the “parking-lot” exercise have any validity? Does it actually measure ability to handle real parking-lot arithmetic?
Obviously, in a parking lot the problem of figuring one’s bill is never so clearly or explicitly stated. One is not handed a paper on which all necessary data are neatly arranged. Instead, information must be gathered from a variety of sources—a sign, a wristwatch, a parking-lot attendant. Paper and pencil are seldom available for doing computations. One is unlikely to go through the laborious arithmetical algorithms or procedures taught in schools and used in tests. One is more likely to do some quick mental figuring.
And few people compute an exact bill, even mentally. It is easier and more efficient to figure an approximate answer: “I’ve parked here less than six hours and the rates are slightly higher than 25 cents an hour on average, so my bill shouldn’t be more than $1.50.” Unfortunately, approximation is little taught in schools, and most people do not feel comfortable enough with numbers to try it. Most people might have a very rough notion of what the bill should be, intuitively or based on prior experience. They rely on the parking attendant to compute it. The attendant, of course, almost certainly relies on some mechanical or electronic computation device.
MR. BROWN’S GARDEN SPRAY
There may be some mathematics done in parking lots, but probably very little of the sort measured by the National Assessment. The situations are radically different. People do parking-lot arithmetic in parking lots using methods appropriate to parking-lots, not in classrooms using paper-and pencil methods. But testers are not alone in producing specious measures of “realworld” mathematical skills. The average elementary mathematics textbook is full of “story problems” like the following: “Mr. Brown made 3 gallons of garden spray. He put the spray into bottles holding 1 quart each. How many bottles did he fill?” It is hard to imagine anyone facing such a problem outside a classroom. Certainly few elementary students have ever manufactured garden spray, or even witnessed adults doing what Mr. Brown was purported to do.
Yet such “problems” give the appearance of being from the “realworld.” Supposedly they are intended to relate school experiences to life outside school. But they have little in common with that life. They are school problems, coated with a thin veneer of “real-world” associations. The mathematics involved in solving them is school mathematics, of little use anywhere but in school. Much school mathematics consists of abstract exercises unrelated to anything outside school. This is why school math is disliked and rejected by many. Schoolchildren recognize that school math is not a part of the world outside school, the world most important to most people.
Yet if school is to be preparatory for life outside school, the school world ought to be as much like the nonschool world as possible. In particular, young people in classrooms ought to do mathematics as it is done by folk in other parts of the world. School math ought to emulate folk math.
MATH FOLKS DO
Woody Guthrie defined folk music as “music that folks sing.” In that same way, folk math is math that folks do. Like folklore, folk math is largely ignored by the purveyors of academic culture—professors and teachers— yet it is the repository of much useful and ingenious popular wisdom. Folk math is the way people handle the math-related problems arising in everyday life. Folk math consists of a wide and probably infinite variety of problem-solving strategies and computation techniques that people use. I believe the first goal of mathematics education should be to assist students to cultivate and enlarge their inherent affinities and abilities for folk math.
Attempts have been made in recent years to teach “real-world” mathematics, a phrase betraying the curious notion that school is somehow unreal. These attempts usually have failed, producing jumbles of “story problems” like “Mr. Brown made 3 gallons of garden spray…” Such “real-world” mathematics describes no place I have ever been or wanted to be. Such curricula only serve to make school math seem more meaningless and absurd.
Surveys “in the field” to determine what mathematics is used in various occupations have produced lists of topics that read much like the table of contents of an arithmetic text: addition, subtraction, multiplication, division, fractions and decimals, ratio, proportion, and percentages. Teachers conclude they are already teaching those things, and return to the security of the text. What is overlooked is how and why mathematics is done outside school. School math and folk math range over much the same mathematical topics.
But folk mathematicians—that is, all of us when we’re facing a math-related problem in everyday life—do mathematics for reasons and with methods different from those commonly involved in school math.
SOME EXAMPLES
I’ve heard a college math teacher tell me how she became aware of the vast differences between school mathematics and the mathematics in her kitchen. She noted that in halving a recipe that called for 1⁄3 cup shortening, she simply filled a 1⁄3 cup measure until it looked half full. In school this would be presented as a “story problem” with the correct answer “(1⁄3) ÷ 2 = 1⁄6.” Solving the problem would require paper and pencil, certain reading and writing skills, and the ability to divide fractions. But in the kitchen paper and pencil are seldom handy. Neither the problem nor the solution is written. And dividing fractions is not really necessary.
I remember eavesdropping on a friend of mine, a building contractor, as he related his train of thought in computing 85 percent of 26. “Ten percent of 26 is 2.6, and half of that is 1.3,” he said. “So that’s 3.9, and 3.9 from 26 is—let’s see, 4 from 26 is 22—22.1 is 85 percent of 26.” He computed 15% of 26 and then subtracted, using some slick mental arithmetic in the process. I was struck by what I heard, knowing that the method of finding percentages taught in school involves a complicated algorithm requiring paper and pencil. I asked him whether his school experience had anything to do with how he handled the percentage problem. “Didn’t you know? I quit school in the sixth grade to help out on the farm,” he said.
I once watched a crew of workmen replace the metal gutters and downspouts in an old three-story building. With only a few metal-working tools and measuring tapes, they quickly cut, bent, and fitted the gutters and downspouts, probably unaware of how nicely they were dealing with three-dimensional space. I wondered what would happen if a crew of mathematics educators wrote a textbook on mathematics for gutter- and downspout-installers. I imagined all the standard school geometry and trigonometry the textbook would contain, and how the crew I was watching would see no relationship between what they were doing and the contents of the text. Such are the differences between school math and folk math.
The differences don’t only exist in the adult world. Watch children playing Monopoly. One lands on Pacific Avenue and owes rental on two houses. “That’s $390,” demands the owner. A $500 bill is offered, correct change made, and the game proceeds. But translate the same problem into school mathematics. “Mr. Jones sent Acme Realty a check for $500. However, he owed them only $390. How much would be refunded?” The same children scurry for pencil and paper, ask a flurry of questions, worry over correct procedures, and hurry to get on to other things.
Consider an incident the father of a third-grader related to me. His son brought home a teacher-made drill sheet on subtraction. The child had completed all the exercises, save one which asked for the difference 8 − 13. That one the child had crossed out. The father knew his son had computed such differences in trinominoes, a game in which it is possible to “go in the hole.” The father asked the boy if he knew the answer.
- “Yes,” he said, “it’s –5.”
- “Well, why did you cross that exercise out?” the father asked.
- “In school, we can’t do that problem, so the teacher said to cross it out.”
Further discussion revealed that since negative numbers had not yet been introduced in class, the teacher had said that one can’t subtract a larger number from a smaller number. Hearing this, the father’s older son, a sixth grader, asked, “Why do teachers lie?”
For the third-grader, school math had already become different from folk math. Knowledge and skills acquired outside school no longer seemed to apply inside, a most confusing development. To the sixth grader, teachers were no longer trustworthy. The older boy had begun to realize that school math is less authentic and reliable than folk math.
SOME GENERALIZATIONS
Some of the general differences between school math and folk math are clear. One is that school math is largely paper-and-pencil mathematics, while folk mathematics is not. Folk mathematicians rely more on mental computations and estimations and on algorithms that lend themselves to mental use. When computation becomes too difficult or complicated to perform mentally, more and more folk mathematicians are turning to calculators and computers. In folk math, paper and pencil are a last resort. Yet they are the mainstay of school math.
Another difference is in the way problems are formulated. In school almost all problems are presented to students preformulated and accompanied by the requisite data. For folk outside school, problems are seldom clearly defined to begin with, and the information necessary for solving them must be actively sought from a variety of sources. While talking with a trainer of apprentice electricians, I realized that an industrial electrician is much more likely to be asked “What’s the problem?” than be told “Do this problem.” Yet technical mathematics courses for electricians are filled with the latter statement, whereas the former question seldom, if ever, occurs.
And the problems themselves differ between school math and folk math. Many so-called “problems” in school math are nothing more than computation exercises. They focus on correct procedures for pushing symbols around on paper. Folk mathematicians compute too, but for them it is not an end in itself. Pages of long division exercises are not part of folk math. Problems in folk math deal with what it will cost, how long it will take, what the score is, how much is needed. Problems in folk math deal with a part of one’s world in a mathematical way.
SOME SOLUTIONS
How can school math be made more like folk math? School math should provide schoolchildren the opportunity to deal with the mathematics in their own environments in the same way proficient folk mathematicians do. Schoolchildren should be encouraged to formulate, attempt to solve, and communicate their discoveries about mathematical questions arising in their classrooms, their play yards, their homes. All are rich with questions to explore: How many tile are in the ceiling? How big is the playground? How old are you—in seconds? Children should be encouraged to develop their own solutions and ways of computing, building on their previous knowledge. Schools should be mathematically rich environments providing many opportunities to develop and exercise mathematical talent.
In this setting, the role of the teacher is to bring mathematical questions to the attention of students, encourage them to seek answers, and talk with them about possible solutions while allowing them to grope, err, and discover for themselves.
All this is terribly idealistic and difficult to achieve. But teachers, administrators, curriculum developers, and especially those of us who call ourselves mathematics educators should be seeking ways of making school math more like folk math. There may be something inherent in schools, in the constraints and demands placed upon them, that will prevent school math and folk math from ever being the same. But the gap between the two need not be a chasm.
SOME SUGGESTIONS
A few suggestions follow that I believe are feasible for any teacher to try.
One is to extend math activities beyond textbook and drill work-sheets. Develop activities out of what’s going on around school and in other subject areas. Encourage students to look for what is mathematical in their environment, and ask them to formulate math problems related to their discoveries.
Second, stress mental arithmetic and ways of computing that lend themselves to mental use. Good folk mathematicians are good at mental computation and estimation. Other computation tools may not always be available, but folk mathematicians always carry their brains with them.
Third, use games and puzzles to develop skills and friendliness with numbers. Many parents and educators feel that games and puzzles are out of place in school and should be permitted only when regular lessons are finished. I shared this view until I realized how much of my own early mathematical development and that of my children was enhanced by playing games. Outside school, little folk mathematicians use numbers mainly, perhaps solely, to play games. And if school math is to emulate their folk math, games should be included.
Fourth, take advantage of the innate fascination and aesthetic appeal of mathematics. Often students are urged to learn mathematics because it will be “useful” someday. Often that is a lie, for no one can predict precisely what mathematical skills any one child will need twenty or thirty years from now. Such urging fails to motivate most children anyway, since for them the future, beyond the next school holiday, is a blur. But most people can find immediate enjoyment in some aspects of mathematics, and schools should not overlook that appeal. In fact, I think it reasonable to include in the curriculum math activities whose major value is simply that they are fun to think about and do.
Fifth, accept and use the electronic calculator the way folk mathematicians do. Folk mathematicians use the calculator as it was intended, as an efficient, economical machine for performing calculations. I don’t see folk mathematicians using the calculator as a device for checking paper-and pencil computations, or as a toy for playing games. Good folk mathematicians use calculators at will. They make mental estimates as a check, and they don’t use the calculator when mental computation is more efficient. When they use the calculator, it is as a replacement for paper and pencil.
Finally, refuse to let standardized tests determine the curriculum. Folk math skills can no more be measured by paper-and-pencil multiple-choice tests than can the ability to play the violin. I believe that good folk mathematicians can do well at these tests, and at school math generally. But many persons who are good at school math are poor at folk math. They have had their folk math abilities stifled and blunted by school math. They are unable to do simple addition or subtraction without resort to paper and pencil. They are enslaved to the slow and awkward procedures learned in school.
April 2008 Email from Gene Maier
Dave Moursund wrote the following email message to Gene Maier:
Hi Gene:
The more I think about it, the more that I believe that your Folk Math work is truly brilliant. Did it make a visible dent in the world of math education?
Dave Moursund
Here is Gene Maier’s response, reprinted with his permission given on 4/2/08.
Dave, I don't know if it made a dent or not. It was printed in several different publications including Instructor Magazine. As evidenced by your remark, which was very pleasing to me, I would venture that it did cause some folks to think about the issues it raises. Your remark did trigger the following thoughts about how school math differs from folk math, or better, how school math hinders the natural development of mathematical expertise.
1. School focuses on training (dragging behind) rather than educating (drawing out); “Here’s how you do it” rather than “Figure out how to do it.” The consequence is that the child’s natural mathematician, that Stanislas Dehaene and others talk about, isn’t nourished and in some cases, is starved or destroyed, resulting in math anxiety or avoidance.
2. School math promotes the artificial rather than the authentic, in a number of ways:
- a. Students are lied to. They are told “You must study this because you’ll need it sometime,” implying that at some point in one’s post-school life, what’s being taught will be critical to their future well-being and success. In actuality, for many topics in math, the only need to know them is to pass tests and courses. When you tell a class, “You’ll need this sometime,”— other than to pass school— you are likely to be lying to someone in the class, and for some topics, such as the division of fractions or the long division algorithm, you’re lying to almost everyone. I told students it was impossible for me to know what mathematics they might need in the future; what I wanted for them is to have the confidence and capability to learn whatever math they might want to use in the future, and so the point of school math, as I saw it, was to develop and nurture their inner mathematician.
- b. School, and school math, is cast as being unreal—not “real world,” whatever that means. We talk about the “real world” and “real world applications,” as if there were some other world. I prefer to think of school as being very much a part of the world, it’s a universal experience for almost everyone. As often used, the phrase “the real world” excludes academia, thereby suggesting that most mathematicians live in some kind of non-real atmosphere. I wanted my students to know that academic mathematics was a very real world to many folk who found it fascinating, even though some of them might view it as esoteric and irrelevant.
- In our attempts to connect the school world with other parts of the world, math books include “applications” which are supposedly ways in which the topic at hand is used outside of school. Most of these are phony—how many of us are going to mix peanuts and cashews to get a mixture that sells at $4.98 a pound? Also, that isn’t much of a story. In writing materials, I used the phrase “puzzle problem” instead of “story problem” or “application” conveying the notion that this was something you were to use your mathematical know-how to puzzle out as an exercise in developing one’s mathematical expertise.
- c. Many questions asked in the classroom aren’t authentic. To me, an authentic question is one you don’t know the answer to. Often, a person asks a question instead of making a statement (“Don’t you think it would be better if…?” rather than “I think it would be better if …”). Instead of engaging the class in a guessing game as to what something is called, just tell them. (Bob Samples maintained that questions were acts of aggression. “How do you work this problem?” has a sense of judgment about it. “I’m interested in how you worked this problem.” conveys a broader concern than judgment.)
- d. The emphasis on tests and grades turns school into a game in which students try to figure out what the teacher wants and what they need to know to pass tests rather than focus on their learning, thereby “swindling” their way through math classes. It also causes teachers to teach to tests rather than do what they think best for their students learning.
- e. School math ignores or misuses the technology of the day. You know more about this than I do. What I do know is that many schools disallow the use of calculators, and they drill on algorithms and techniques that one has little use for outside of school. Sometimes they even use computers to teach these techniques. On the other hand, I don’t think all uses currently made of calculators and computers are helpful. For example, I have a gut feeling that it’s better to manipulate base ten pieces with one’s hands rather than moving facsimiles thereof around a computer screen with a mouse. With all the brain imaging going on these days, I think it would be interesting to compare the brain activity when one manipulates base ten pieces, or tile, by hand compared to that when facsimiles are moved with a mouse. Perhaps you can get the brain research people at Oregon to do an experiment of this sort.
3. School is competitive rather than cooperative. Tests and grades foster this, especially when grading is done on a curve. It’s also fostered by parents who want to get their kids in TAG classes and the best colleges, and win Little League championships, besides.
4. Math education is not multi-sensory; it focuses on the left-brain. In so doing, it eliminates many strengths of the human intellect. Math and the Mind’s Eye was aimed at involving all the senses in learning.
5. There's no story line in math classes (stories are a time-honored mode for conveying knowledge). Someone once commented that a certain algebra textbook had about as much plot as a New York City phone book. There are, at least to me, some fascinating story lines in mathematics: how one invents all the numbers one needs to solve any quadratic equation (this is closely related to the fundamental theorem of algebra); the remarkable connection between variable rates of change and the areas of irregular shapes (this is closely related to the fundamental theorem of calculus). A class needn't have a single story line; it can consist of a number of related short stories. I disliked teaching problem-solving courses because I had a hard time developing a story line, so I would teach it in the context of some mathematical short stories.
That's a rather long answer to a question I don’t know the answer to. Much of what I've mentioned above, I incorporated into my teaching over the years. From time to time, I think about expanding on these points in a small book reflecting on my teaching career and how it evolved.
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.
References
Servín, Agüero (2001). Mathematizing: Every day strategies in problem solving. Retrieved 8/13/08: http://www.umanitoba.ca/unevoc/conference/papers/deaguero.pdf. A good paper for those who are interest in Folk Math.
