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The New New Math

Focus on Algebra: An Integrated Approach

by Randall I. Charles, by Alba González Thompson
Addison-Wesley, 843 pp., $56.00

Life by the Numbers: Math As You’ve Never Seen It Before

narrated by Danny Glover. Seven boxed videotapes produced by WQED, Pittsburgh
$129.00

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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 better-paying jobs. Something clearly is wrong with how math is being taught in pre-college 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 pre-college 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 twenty-eight 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 mind-numbing jargon. “Multiculturalism” and “equity” are the book’s most-used 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 gender-equitable 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 self-image.”

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 fuzzy-math teachers is Marcia Ascher’s Ethnomathematics: A Multicultural View of Mathematical Ideas (1991).1 “Critical-mathematical literacy” is an even longer jawbreaker. It appears in the NCTM yearbook as a term for the ability to interpret statistics correctly.

Knowing how pre-industrial 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.

Fuzzy-math 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 “open-ended” 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.

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. “Hands-on” learning first, then rote learning. Unfortunately, some far-out 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 tick-tack-toe, 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 sixty-two squares be covered by thirty-one dominoes? After a group finds it impossible, see how long it takes for someone to come up with the beautiful parity (odd-even) 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 best-selling 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 Op-Ed page of The New York Times on August 11, 1997,2 I found myself cheering. As Cheney points out, at the heart of fuzzy-math 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 So-Called 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 so-called 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 fifth-graders 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.

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.

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    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 Barrios-Chacon and others (Addison-Wesley, 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.

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    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).

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