1.
Surveys have shown for many decades that the mathematical skills of American high school students lag far behind those of their counterparts in Japan, Korea, Singapore, and many European countries. In the United States whites do better than blacks, Hispanics, and Native Americans. Males outscore females. Students from high socioeconomic backgrounds do better than those from lower strata.
These are troubling statistics because, in an advanced technological society such as ours, a firm grasp of basic mathematics is increasingly essential for betterpaying jobs. Something clearly is wrong with how math is being taught in precollege grades, but what?
In the late 1960s the National Council of Teachers of Mathematics (NCTM) began to promote a reform movement called the New Math. In an effort to give students insight into why arithmetic works, it placed a heavy emphasis on set theory, congruence arithmetic, and the use of number bases other than ten. Children were forbidden to call, say, 7 a “number.” It was a “numeral” that symbolized a number. The result was enormous confusion on the part of pupils, teachers, and parents. The New Math fad faded after strong attacks by the physicist Richard Feynman and others. The final blow was administered by the mathematician Morris Kline’s 1973 best seller Why Johnny Can’t Add: The Failure of the New Math.
Recently, the NCTM, having learned little from its New Math fiasco, has once more been backing another reform movement that goes by such names as the new new math, whole math, fuzzy math, standards math, and rain forest math. Like the old New Math, it is creating a ferment among teachers and parents, especially in California, where it first caught on. It is estimated that about half of all precollege mathematics in the United States is now being taught by teachers trained in fuzzy math. The new fad is heavily influenced by multiculturalism, environmentalism, and feminism. These trends get much attention in the twentyeight papers contributed to the NCTM’s 1997 yearbook, Multicultural and Gender Equity in the Mathematics Classroom: The Gift of Diversity.
It is hard to fault most of this book’s advice, even though most of the teachers who wrote its chapters express themselves in mindnumbing jargon. “Multiculturalism” and “equity” are the book’s mostused buzzwords. The word “equity,” which simply means treating all ethnic groups equally, and not favoring one gender over another, must appear in the book a thousand times. A typical sentence opens Chapter Eleven: “Feminist pedagogy can be an important part of building a genderequitable multicultural classroom environment.” Over and over again teachers are reminded that if they suspect blacks and females are less capable of understanding math than Caucasian males, their behavior is sure to subtly reinforce such beliefs among the students themselves, or what one teacher calls, in the prescribed jargon, a student’s “internalized selfimage.”
“Ethnomathematics” is another popular word. It refers to math as practiced by cultures other than Western, especially among primitive African tribes. A book much admired by fuzzymath teachers is Marcia Ascher’s Ethnomathematics: A Multicultural View of Mathematical Ideas (1991).^{1} “Criticalmathematical literacy” is an even longer jawbreaker. It appears in the NCTM yearbook as a term for the ability to interpret statistics correctly.
Knowing how preindustrial cultures, both ancient and modern, handled mathematical concepts may be of historical interest, but one must keep in mind that mathematics, like science, is a cumulative process that advances steadily by uncovering truths that are everywhere the same. Native tribes may symbolize numbers by using different base systems, but the numbers behind the symbols are identical. Two elephants plus two elephants makes four elephants in every African tribe, and the arithmetic of these cultures is a miniscule portion of the vast jungle of modern mathematics. A Chinese mathematician is no more concerned with ancient Chinese mathematics, remarkable though it was, than a Western physicist is concerned with the physics of Aristotle.
Fuzzymath teachers are urged by contributors to the yearbook to cut down on lecturing to passive listeners. No longer are they to play the role of “sage on stage.” They are the “guide on the side.” Classes are divided into small groups of students who cooperate in finding solutions to “openended” problems by trial and error. This is called “interactive learning.” The use of calculators is encouraged, along with such visual aids as counters, geometrical models, geoboards, wax paper (for folding conic section curves), tiles of different colors and shapes, and other devices. Getting a correct answer is considered less important than shrewd guesses based on insights, hence the term “fuzzy math.” Formal proofs are downgraded.
No one can deny the usefulness of visual aids. Teachers have known for centuries that the best way to teach arithmetic to small children is by letting them “interact” with counters. Each counter models anything that retains its identity—an apple, cow, person, star. What’s the sum of 5 and 2? A girl who knows how to count moves into a pile five counters, then two more, and counts the heap as seven. Suppose she first moves two, then five. Does it make a difference? Similar procedures teach subtraction, multiplication, and division.
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After a few days of counter playing it has been traditional for children to memorize the addition table to at least 9. Later they learn the multiplication table to at least 10. “Handson” learning first, then rote learning. Unfortunately, some farout enthusiasts of new new math reject anything resembling what they call “drill and kill” memorizing. The results, of course, are adults who can’t multiply 12 by 12 without reaching for a calculator.
Aside from its jargon, another objectionable feature of the yearbook is that its contributors seem wholly unaware that the best way to keep students awake is to introduce recreational material which they perceive as fun. Such material includes games, puzzles, magic tricks, fallacies, and paradoxes. For example, determining whether the first or second player can always win at ticktacktoe, or whether the game is a draw if each player makes the best moves, is an excellent way to introduce symmetry, combinatorics, graph theory, and game theory. Because all children know the game, it ties strongly into their experience.
For what the yearbook likes to call a “cognitively challenging” task, give each child a sheet with a checkerboard on it. Have each of them cut off two opposite corner squares. Can the remaining sixtytwo squares be covered by thirtyone dominoes? After a group finds it impossible, see how long it takes for someone to come up with the beautiful parity (oddeven) proof of impossibility.
If new new math teachers are aware of such elegant puzzles, and there are thousands, there is no hint of it in the yearbook. This is hard to understand in view of such bestselling textbooks as Harold Jacobs’s Mathematics: A Human Endeavor (1970; third edition, 1994), which has a great deal of recreational material; Mathematics: Problem Solving Through Recreational Mathematics, a textbook by Bonnie Averbach and Orin Chein (1980); and scores of recent books on entertaining math by eminent mathematicians.
I seldom agree with the conservative political views of Lynne Cheney, but when she criticized extreme aspects of the new new math on the OpEd page of The New York Times on August 11, 1997,^{2} I found myself cheering. As Cheney points out, at the heart of fuzzymath teaching is the practice of dividing students into small groups, then letting them discover answers to problems without being taught how to find them. For example, teachers traditionally introduced the Pythagorean theorem by drawing a right triangle on the blackboard, adding squares on its sides, and then explaining, perhaps even proving, that the area of the largest square exactly equals the combined areas of the two smaller squares.
According to fuzzy math, this is a terrible way to teach the theorem. Students must be allowed to discover it for themselves. As Cheney describes it, they cut from graph paper squares with sides ranging from two to fifteen units. (Such pieces are known as “manipulatives.”) Then they play the following “game.” Using the edges of the squares, they form triangles of various shapes. The “winner” is the first to discover that if the area of one square exactly equals the combined areas of the other two squares, the triangle must have a right angle with the largest square on its hypotenuse. For example, a triangle of sides 3,4,5. Students who never discover the theorem are said to have “lost” the game. In this manner, with no help from teacher, the children are supposed to discover that with right triangles a2+b2=c2.
“Constructivism” is the term for this kind of learning. It may take a group several days to “construct” the Pythagorean theorem. Even worse, the paper game may bore a group of students more than hearing a good teacher explain the theorem on the blackboard.
One of the harshest critics of fuzzy math is the writer John Leo, whose article on the subject, “That SoCalled Pythagoras,” was published last year in US News and World Report (May 26, 1997). (His title springs from a reference he found in a book on ethnomathematics to “the socalled Pythagorean theorem.”) Leo tells of Marianne Jennings, a professor at Arizona State University, whose daughter was getting an A in algebra but had no notion of how to solve an equation. After obtaining a copy of her daughter’s textbook, Jennings soon understood why.
Here is how Leo describes this book:
It includes Maya Angelou’s poetry, pictures of President Clinton and Mali wood carvings, lectures on what environmental sinners we all are and photos of students with names such as Taktuk and Esteban “who offer my daughter thoughts on life.” It also contains praise for the wife of Pythagoras, father of the Pythagorean theorem, and asks students such mathematical brain teasers as “What role should zoos play in our society?” However, equations don’t show up until Page 165, and the first solution of a linear equation, which comes on Page 218, is reached by guessing and checking.
Romesh Ratnesar’s article “This is Math?” (Time, August 25, 1997) also criticizes the new new math. It describes fifthgraders who were asked how many handshakes would occur if everyone in the class shook hands with everyone else. At the end of an hour, no group had the answer. Unfazed, the teacher said they would be trying again after lunch. Professor Jennings makes another appearance. She told Ratnesar that she became angry and worried when she saw her daughter use her calculator to determine 10 percent of 470.
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Curious about her daughter’s textbook, which is now widely used, I finally obtained a copy by paying a bookstore $59.12. Titled Focus on Algebra: An Integrated Approach, this huge text contains 843 pages and weighs close to four pounds. (In Japan, the average math textbook is two hundred pages.) It is impossible to imagine a sharper contrast with an algebra textbook of fifty years ago.
“Integrated” in the subtitle has two meanings:
(1) Instead of being limited to algebra, the book ranges all over the math scene with material on geometry, combinatorics, probability, statistics, number theory, functions, matrices, and scatter graphs, and of course the constant use of calculators and graphers. Fifty years ago high school math was given in two classes, one on algebra, one on geometry. Today’s classes are “integrated” mixtures.
(2) The book is carefully integrated with respect to gender and to ethnicity, with photographs of girls and women equal in number to photographs of boys and men. Faces of blacks and whites are similarly equal, though I noticed few faces of Asians.
On the positive side is the book’s lavish use of color. Only a few pages lack fullcolor photos and drawings, all with eyecatching layouts. When it comes to actual mathematics the text is for the most part clear and accurate, with a strong emphasis on understanding why procedures work, and on inducements to think creatively. “After all,” the text says on its first page, “what good is it to solve an equation if it is the wrong equation?” The trouble is that the book’s mathematical content is often hard to find in the midst of material that has no clear connection to mathematics.
Not having taught mathematics myself, I have no opinion about the value of students working in small groups as opposed to sitting and listening to a teacher talk. Nor have I found research studies that make a decisive case in favor of either method. Clearly a great deal depends on the qualities of particular teachers, and these would be hard to appraise in any survey. The authors justify the group approach by saying it anticipates the workplaces in which students will find themselves as adults. John Donne’s remark about how no man is an island is quoted. The book’s first “exercise” is a question: “In general, do you prefer to work alone or in groups?”
An emphasis on ethnic and gender equity is, of course, admirable, though in this textbook it seems overdone. For example, twelve faces of boys and girls of mixed ethnicity reappear in pairs throughout the pages. Each has something to say. “Taktuk thinks…” is followed with “Esteban thinks…,” “Kirti thinks…” is followed by what “Keisha thinks…,” and so on. These pairings become mechanical and predictable.
The book jumps all over the place, with transitions as abrupt as the dream episodes of Alice in Wonderland. I think most students would find this confusing. Eight full pages are devoted to statements by adult professionals, with their photographs. Each statement opens with a sentence about whether they liked or disliked math in high school, followed by generally banal remarks. For example, Diana GarciaPrichard, a chemist, writes: “I liked math in high school because all the problems had answers. Math is part of literacy and the framework of science. For instance, film speed depends on chemical reactions. I use math to model problems and design experiments. I like getting results that I can publish and share.” Presumably such statements are intended to convince students that math will be useful later on in life.
Many of the book’s exercises are trivial. For example, on page 20 students are asked to play forward and backward a VCR tape of a skier, then answer the question: “How will this affect the way the skier appears to move?” On page 11: “A circle graph represents 180 kittens. What does 1/4 of the circle represent?” (Answer, to be found in the back of the book: 45.) A chapter on “the language of algebra” opens with a page on the origin of such phrases as “the lion’s share,” “the boondocks,” and “not worth his salt.” It is not clear what this has to do with algebra.
Many pictures have only a slim relation to the text. Magritte’s painting of a green apple floating in front of a man’s face accompanies some problems about apples. Van Gogh’s selfportrait is alongside a problem about the heights and widths of canvases. A picture of the Beatles accompanies a problem about taxes only because of the Beatles’ song “Taxman.” My favorite irrelevant picture shows Maya Angelou talking to President Clinton. Beside it is the following extract from one of her prose poems:
Lift up your eyes upon
This day breaking for you
Give birth again
To the dream.
Women, Children, Men,
Take it into the palm of your hands.
Mold it into the shape of your most
Private need. Sculpt it into
The image of your most public self.
Why is this quoted? Because the “parallel” phrases shown underlined are similar to parallel lines in geometry! Is this intended to “integrate” geometry and poetry?
The book is much concerned with how the environment is being polluted. Protecting the environment is obviously a good cause, but here its connection with learning math is often oblique, if not arbitrary. A chapter on functions opens with a page headed “Unstable Domain.” Its first question is “What other kinds of pollution besides air pollution might threaten our planet?” Page 350 has a picture of crude oil being poured over a model of the earth. It accompanies a set of questions relating to the way improper disposals of oil are contaminating ground water.
A page headlined “Hot Stuff” shows three kinds of peppers to illustrate how they are used in cooking. Two of the “exercises” are: “The chili cookoff raises money for charity. Describe some ways the organizers could raise money in the cookoff,” and “How would you set up a hotness scale for peppers?” This page introduces a chapter on how to solve linear equations.
Another section on equations opens with pictures of zoo animals. It discusses what can be done to prevent species from becoming extinct. The first question is “What role should zoos play in today’s society?” The book’s index, under the entry “Animal study and care,” lists thirtytwo page references.
A section on mathematical inequalities is preceded by a page on how Mary Rodas became vicepresident of a toy company, and how Linda Johnson Rice found a creative way to market Eboné cosmetics for black women. Under a photo of a smiling Mary, the first questions are: “Would you like to own your own business someday? Why or why not?”
On page 67, a picture of Toni Morrison is used to illustrate a problem about how many ways four objects can be placed in a row. The text then introduces four students who each read an excerpt from something Morrison has written. In how many different orders, the text asks, can the four excerpts be read? A man from Mysore, India, who creates shadow pictures on the wall with his fingers is featured on page 421. What this has to with the following section on solving systems of inequalities is not evident. A photo of Alice Walker on page 469 illustrates the question: “Is the time it takes to read an Alice Walker novel always a function of the number of pages?” This and other such references give the impression that wellknown writers are being dragged into the text.
The most outrageous page—it opens a section on linear functions—concerns the Dogon culture of West Africa. Students are told that this primitive tribe, without the aid of telescopes, discovered that Jupiter has satellites, that Saturn has rings, and that an invisible star of great density orbits Sirius once every fifty years. Presumably the Dogon had supernormal powers. However, it has long been known that the Dogon made no such discoveries. They merely learned these astronomical facts from missionaries and other Western visitors.^{3}
Like the authors of the NCTM yearbook, those who fashioned this huge textbook seem wholly uninterested in recreational material. The book’s only magic trick (page 246) is a stale, utterly trivial way to guess a number. Although strongly favoring the use of calculators, the authors don’t seem aware that the hundreds of amazing number tricks that can be done with them provide excellent exercises. A child can learn a lot of significant number theory in discovering why they work. None is in the book.
An old brain teaser involves a glass of wine and a glass of water. A drop is taken from the wine and added to the water. The water is stirred, then a drop of the mixture goes back to the wine. Is there now more water in the wine than wine in the water, or vice versa? The surprising answer is that the two amounts are precisely equal.
Students will be fascinated by the way this principle can be modeled with a deck of cards. Divide the deck in half, one half consisting of all the red cards, the other half consisting of all the black cards. Take as many cards as you like from the red (wine) half and insert them anywhere among the blacks (water). Shuffle the black half. From it remove from anywhere the same number of cards you took from the reds and put them back among the reds. You’ll find the number of blacks among the reds is exactly the same as the number of reds among the blacks. Students will enjoy proving that this is always the case. But will it work if the two starting portions of the deck are unequal? (Yes. It doesn’t matter if the two glasses in the brain teaser are not the same size; nor does it matter how many cards are in the black and red piles.)
This secondary math textbook has an index that is not very helpful. What value are more than 180 page references for the entry “Science”? What use is a similar quantity of page numbers for the entry “Industry”?
WQED’s boxed set of seven videotapes, Life by the Numbers, was funded mainly by the National Science Foundation and the Alfred P. Sloan Foundation. The photography is excellent. There are scenes of men and women mathematicians seated at computer consoles, or driving a car, or walking down a street or through the woods. There are many closeups of their faces, dazzling glimpses of mountains and skyscrapers, baseball games, martial arts contests, blossoming flowers, wild animals, and everything imaginable that has little to do with math. The tapes rate high on special effects, low on mathematical content.
The seventh tape covers a typical new new math class. To discover that the longer a pendulum, the slower its swing, students tie weights to the ends of string and swing the weights back and forth while other students keep charts of string lengths and pendulum periods. After several days they learn that the period of a pendulum is a function of its length. This discovery enables them to calculate whether the victim in Poe’s horror story “The Pit and the Pendulum” has enough time to escape from the huge pendulum which threatens to cut him in half as it swings lower and lower over his reclining body. It is assumed that because students have fun swinging weights they will remember the function better than if a teacher takes a few minutes to demonstrate it by swinging a weight and slowly lengthening the string.
2.
One of the most telling attacks on new new math is Bernadette Kelly’s article “Déjà Vu? The New ‘New Math,”‘ in the professional journal Effective School Practices (Spring 1994). Kelly summarizes four case studies by four supporters of fuzzy math in which they report on four fifthgrade teachers.^{4} Two of the teachers, called Sandra and Valerie, are enthusiastic users of new new math techniques. The other two teachers use traditional methods.
Sandra was very good at getting students to cooperate in groups. However, in one exercise she told students that one could obtain the perimeter of a rectangular field by multiplying its length by its width! In another project she calculated the volume of a sandbox by multiplying together its length and width in yards, then multiplying the product by the box’s height in feet!
In an interview Sandra said that while working on the sandbox problem her pupils asked what a cubic foot was. “You know, the thing is that I couldn’t really answer that question. Then I thought and thought, then I remembered how to measure a cube.” Neither Sandra nor her students were ever aware of her two huge mistakes. In spite of these errors, the author of the article about her said she was an “exemplary teacher.” Sandra is praised for getting her students to enjoy their cooperative efforts to solve problems “in the context of real world situations.” Finding a correct answer was less important than having fun in working on the problem.
Valerie made an equally astonishing blunder. The task was to determine the average number of times her thirty students had eaten ice cream over a period of eight days. This was “solved,” by dividing 30 by 8, to get 3.75, which Valerie rounded up to 4!
As with Sandra, neither Valerie nor her students ever became aware that they obtained a totally wrong answer. Nevertheless, the author of the paper about her forgives her mistake on the grounds that she had succeeded so well in getting her students to work on a problem in the context of their experience. Moreover, the work had impressed on the students the “usefulness and relevance of averages.” No matter that they completely failed to find an average.
As for Jim and Karen, the two teachers who used more traditional methods, the authors of the case studies are unimpressed by their students having scored high on tests. Both are castigated for failing to appreciate the methods of the new new math. What is deplorable, as Bernadette Kelly’s article points out, is not so much that the case studies revealed the incompetence of two teachers, who come through as ignoramuses, as the authors’ praise of Sandra and Valerie for finding ways to get their pupils working joyfully on problems. Little wonder that new new math is called fuzzy. Insights are deemed significant even when they are wrong.
The mathematician Sherman Stein, in his 1996 book Strength in Numbers, devotes a chapter to a history of math reform movements. His hopes for the new new math are dim. “I am disturbed,” he writes,
that the authors of the [new new math books] do not cite any pilot project or any school district as a model to show that their goals can be achieved in the real world. That means that they are proposing to change the way an entire generation learns mathematics without checking the feasibility of their recommendations. A manufacturer introduces a new soap with more care, first testing its reception in a few stores or towns before committing to mass production.
But evaluating the efficacy of fuzzy math will not be easy. Too many variables are involved, including the skill of teachers and the educational background of parents, to mention only two. A glaring example of how research can be biased is provided by a recent testing of precollege math students around the world by the Third International Mathematics and Science Study. Results announced last February revealed that American students did better than students in just two other countries, Cyprus and South Africa. A cartoon in The New York Times (March 8) showed a car’s bumper sticker that said “My kid’s math scores beat kids in Cyprus and South Africa.” Inside the car a father is giving a thumbsup sign.
These statistics are worthless. In many cases the students in a foreign country were much older than students here at the same grade level. More significantly, in most foreign nations students in early grades who show no aptitude for math are sent off to trade schools or to jobs, if they can find them. In the US such students are required to continue attending high school. Obviously our high school students will do less well on math tests than students in countries where poor students are quickly moved out of the system.
Although we lack clear, systematic evidence that methods of fuzzy math are inferior to older methods, education officials in California, the nation’s largest customer for math textbooks, have suddenly turned against the new new math. The change in state policy was mainly in response to the outrage of parents who complained that their children were unable to do the simplest arithmetical calculations. Their outrage was backed by many top mathematicians and scientists. Michael McKeown, for example, a distinguished molecular biologist at the Salk Institute, heads a parental group called Mathematically Correct. “We’re not opposed to teaching concepts,” he told Newsweek (“Subtracting the New Math,” December 15, 1997). “I am opposed to failing to give a kid tools to solve a problem.”
In a vote of ten to zero (one person abstained) the eleven members of California’s Board of Education recommended this spring a broad return to basics in math teaching. The decision is sure to have an effect in other states. The board said students should learn the multiplication table by the end of the third grade, and that fourth graders should know how to do long division without consulting a calculator. It banned the use of calculators on state tests. Teachers were urged not to introduce calculators before grade six.
Defenders of fuzzy math are, of course, dismayed. They branded the board’s decisions a product of nostalgia, and a contribution to our country’s dumbing down. The National Science Foundation, which has given more than $50 million to California districts for research on new math teaching, is furious. It has threatened to withdraw further funding to any California district that adopts the board’s recommendations.
The conflict is bitter and far from over. It may be many years before it becomes clear how to sift out from the new new math what is valuable while retaining worthy aspects of older teaching methods.^{5} My own opinion is that the most important question concerning the teaching of math is not how big and colorful textbooks are, how many visual aids are used, how the classroom is physically arranged, or even what methods are used in it. The greatest threat to good math teaching is surely the low pay that keeps so many excellent teachers and potential teachers out of our schools. What matters more than anything else is having trained teachers who understand and love mathematics, and are capable of communicating its mystery and beauty to their pupils.
This Issue
September 24, 1998

1
Textbooks emphasizing multiculturalism are proliferating rapidly. Here are a few: Africa Counts: Number and Pattern in African Culture, by Claudia Zaslavsky (Lawrence Hill, 1997); Multiculturalism in Mathematics, Science and Technology, by Miriam BarriosChacon and others (AddisonWesley, 1993); Multicultural Mathematics: Teaching Mathematics from a Global Perspective, by David Nelson, George Gheverghese Joseph, and Julian Williams (Oxford University Press, 1993); Teaching with a Multicultural Perspective: A Practical Guide, by Leonard Davidman and Patricia T. Davidman (Perseus, 1996). Striking multicultural math posters are available from teaching supply houses. ↩

2
See also the letters in The New York Times of August 17, 1997, and an earlier article by Cheney in the Weekly Standard (August 4, 1997). ↩

3
On the myth of Dogon astronomy, see Carl Sagan, Broca’s Brain (Random House, 1979), pp. 6364, and Chapter Six; Ian Ridpath, “Investigating the Sirius ‘Mystery,”‘ in The Skeptical Inquirer, Vol. 3 (Fall 1978), pp. 5662; and Terence Hines, Pseudoscience and the Paranormal (Prometheus Books, 1988), pp. 216219. ↩

4
R.T. Putnam, R.M. Heaton, R.S. Prewat, and J. Remillard, “Teaching Mathematics For Understanding,” in Elementary School Journal, Vol. 93 (1992), pp. 213228. ↩

5
That the new new math has positive aspects goes without saying. It is important that students understand the basic concepts of math and not just memorize procedures that work; and to give students such conceptual understanding teachers themselves must have such understanding. This is the theme of a recent monograph, Middle Grade Teachers’ Mathematical Knowledge and Its Relationship to Instruction, by Judith Sowder, Randolph Philipp, Barbara Armstrong, and Bonnie Schappelle (State University of New York, 1998). ↩