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Loving and Hating School Mathematics

Why do so many school children and adults find mathematics intimidating and consider themselves hopelessly incapable of learning it? How do educators address this challenge?

Mathematical life is an immersion in a world of endlessly varied forms and relations. The mathematician is challenged and tempted to commit all her energy and enthusiasm to learn and to understand. Mathematical thinking is also enjoyed by people working puzzles, playing chess, or doing recreational problem solving. Engagement with and enjoyment of mathematics is the primary topic of this book.

But there’s also another thing called mathematics. It’s the thing people are talking about when they say:

“I hate math! I couldn’t learn it, and I can’t teach it!”

“I’m bad at math. It’s always been my least-liked subject.”

“I hated math in school . . . and my feelings haven’t changed since.”

These comments about school mathematics come from graduate students, who were actually preparing to become elementary school teachers by taking a seminar on math instruction. “The students were nearly evenly divided between those who liked and those who disliked math. In nearly all the cases, a correlation existed between attitude and success.”¹ What emotions about mathematics will children absorb from such teachers?

There is a large literature on negative attitudes toward mathematics. The best known is Sheila Tobias’ book on “math phobia.”² The issue of math phobia as experienced by individual students has been extensively explored. We wish to address related issues. One of these is the conflict between society’s need for mathematical engineers and scientists and the difficulties of many individual students who are not preparing for such careers. How can these two concerns be reconciled? Does everybody need to be a math expert in order to adequately respond to the demands of the information age?

A related question is, How much mathematics is actually needed for a career such as medicine? Today, studying mathematics isn’t just an option a student can choose according to her interest. It’s compulsory. Proficiency in algebra, and in some instances calculus, is considered essential for many professions. Math serves as a filter to screen applicants for college and for professional schools. How realistic is this requirement? Should your admission to medical school be contingent on your grade in calculus? This is another issue we explore in this chapter.

Reluctant Learners

It’s a common observation that a large proportion of students are profoundly alienated from mathematics. And this problem of avoiding and rejecting math isn’t disappearing. Newspaper headlines bemoan the poor standing of the United States in international comparisons of mathematical achievement and blame schools, television, and parents’ unavailability when their children need help with homework. With the passage of the federal No Child Left Behind Act, the pressure has become more intense. Schools that don’t test within the mandated range are penalized. Children whose performance lags behind the standards are stigmatized. These punishments contribute to math phobia. Popular surveys as well as more systematic studies find that mathematics is the school subject that provokes the strongest reactions, both negative and positive.

The following article had the headline, “Hate mathematics? You are not alone.”

People in this country have a love-hate relationship with math, a favorite school subject for some but just a bad memory for many others, especially women. In an AP-AOL News poll as students head back to school almost four in 10 adults surveyed said they hated math in school, a widespread disdain that complicates efforts today to catch up with Asian and European students. Twice as many people said they hated math as said that about any other subject. Some people, like Stewart Fletcher, a homemaker from Suwannee, Georgia, are fairly good at math but never learned to like it. “It was cold and calculating,” she said. “There was no gray. It was black and white.” Still, many people, about a quarter of the population, said math was their favorite school subject.³

Notice that while 40 percent hate it, there are 25 percent who like it better than any other subject! A lot of people do hate math, but a lot of other people love a chance to challenge their brains with math problems. There are many people who enjoy doing something where there is just one right answer and every other answer is wrong.

In his Apology, Hardy wrote, “The fact is that there are few more ‘popular’ subjects than mathematics . . . there are probably more people really interested in mathematics than in music. . . . There are masses of chess players in every civilized country. . . . Chess problems are the hymn-tunes of mathematics. We may learn the same lesson, at a lower level but for a wider public, from bridge, or descending further, from the puzzle columns of the popular newspapers. Nearly all their immense popularity is a tribute to the drawing power of rudimentary mathematics . . . nothing else has quite the kick of mathematics.⁴

A puzzle that was a great craze over a hundred years ago is still popular today. The Fifteen Puzzle is about sliding numbered blocks inside a square frame. Back in the spring of 1880, the New York Times wrote: “No pestilence has ever visited this or any other country which has spread with the awful celerity of what is called the ‘Fifteen Puzzle.’ It has spread over the entire country. Nothing arrests it. It now threatens our free institutions, inasmuch as from every town and hamlet there is coming up a cry for a ‘strong man who will stamp out this terrible puzzle at any cost of Constitution or freedom.’ ”⁵

Today, Su Do Ku, a puzzle about arranging numbers in a square box, is millions of people’s favorite game. In 1997 a retired Hong Kong judge, Wayne Gould, saw a partly completed puzzle in a Japanese bookshop. Over 6 years he developed a computer program to produce puzzles quickly. He promoted Su Do Ku to the Times in Britain, which launched it in 2004. By April and May 2005 the puzzle was part of several other national British newspapers. The world’s first live TV sudoku show, “Sudoku Live,” was broadcast on July 1, 2005. Nine teams of nine players (with one celebrity on each team) representing geographical regions competed to solve a problem. Addiction to math games and puzzles by one population coexists with the rejection of anything mathematical by many others. It’s likely that among those who say they don’t like math, or even hate it, are many who enjoy mathematical challenges in the form of games and puzzles.

When we hear somebody say he or she doesn’t like math or avoids math, we ask, “When did it start?” The answer is either “4th grade” or “6th grade” or “8th grade.” Recently, at dinner, our friend Claire answered, “6th grade.” She elaborated: “The teacher just called on boys, he didn’t think girls could really do math. And also, my friends would get on me, for being ‘too smart.’ ”

I said, “So your teacher thought you were too dumb, and your friends thought you were too smart?”

“Yes, that’s what happened.”

For some who dislike mathematics, it starts with fractions. (In fact, most adults in the United States have serious trouble adding 1/3 + 1/4.) For many others, it’s algebra, working with x and y. And for others, who thought they did all right in arithmetic and algebra, it’s their college calculus course that convinces them they’re “not a math type” or are even “bad at math.” People aren’t born disliking math. They learn to dislike it in school.

The first meeting with algebra in middle school—grades 6 through 8, ages 12 to 14—seems to be critical for many students. Unfortunately, this stage has received less study by educational researchers than have the early childhood and beginning primary school years. As Kristin Umland⁶ has pointed out, this stage of school is where the transition must be made from the “premathematical” to the “fully mathematical.” Roughly speaking, from the concrete to the abstract, or, as Bertrand Russell put it, from thinking about a particular thing to thinking about an unspecified member of a whole class of things. This leap is easy for some, but more difficult for others. We teachers need to understand better how to help children overcome this difficulty.

The current emphasis is on test scores, punishing individuals or schools that fall short of a prescribed level of tested mastery. To meet international competition, we increased the number of required algebra and trigonometry courses in high school.

Some have questioned this approach. In the Washington Post, columnist Richard Cohen criticized a new requirement of a year of algebra and trigonometry for graduation. He argued that this contributes to a higher dropout rate. Cohen recalled his own terror: “There are those of us who know the sweat, the pain, the trembling, cold fear that comes from the teacher casting an eye in your direction and calling you to the blackboard. It is like being summoned to your own execution”⁷

In an earlier article, Colman McCarthy, the founder of the Center for Teaching Peace, made a similar point: “Too many of us were forced to take algebra when the time and energy could have been devoted to subjects that were truly beneficial individually and nationally. Algebra isn’t essential to much of anything. Once adding, subtracting, multiplying and dividing are mastered—by eighth grade usually—why insist on more? Algebra . . . is a language, a way of symbolic communication that a few people find fascinating and practical, while most don’t. Would millions of high school students trudge into their algebra classes if there weren’t a gate through which they were forced to pass to enter college?”⁸

He further argued: “The world is crying out for peacemakers. We are not teaching the kids how to be the essential thing. We have conflicts all our lives.”⁹ In response, some people would argue that mathematics that is connected to daily life also addresses issues of conflict.

To these two contemporary voices, we can add one more from a century ago. The famous philosopher Bertrand Russell was coauthor, with Alfred North Whitehead, of the monumental Principia Mathematica, an epochal work that attempted to reduce all mathematics to symbolic expressions from formal logic. Yet this master of rigorous formal mathematics had misgivings about school algebra. In 1902 he wrote:

In the beginning of algebra, even the most intelligent child finds, as a rule, very great difficulty. The use of letters is a mystery, which seems to have no purpose except mystification. It’s almost impossible, at first, not to think that every letter stands for some particular number, if only the teacher would reveal what number it stands for. The fact is, that in algebra the mind is first taught to consider general truths, truths which are not asserted to hold only of this or that particular thing but of any one of a whole group of things. . . . Usually the method that has been adopted in arithmetic is continued: rules are set forth, with no adequate explanation of their grounds; the pupil learns to use the rules blindly, and presently, when he is able to obtain the answer that the teacher desires, he feels that he has mastered the difficulties of the subject. But of inner comprehension of the processes employed he has probably acquired almost nothing.¹⁰

Does it have to be this way? Some educators are trying different ways to teach mathematics in school, in the United States, as well as in many other countries, as described in the following section.

School Mathematics and Everyday Mathematics

Imparting basic mathematical knowledge to children and young people who are uninterested or fearful is a serious undertaking. They must learn to add, subtract, multiply, and divide fractions and also acquire some basic geometry and algebra. These are challenging tasks. They require strong teaching, with an emphasis on conceptual understanding, and most important, with connections to activities that are relevant to children’s lives.

Children enter school with a variety of experiences with shapes, categories of objects, estimation of areas, and some counting. According to the Swiss psychologist Jean Piaget, young school-age children are in the process of mastering conservation of quantity, seriating, and equivalence of corresponding sets, based on both visual alignments and counting. But these notions are acquired slowly. Many 5-year-olds can recite numbers but have not yet mastered the concept that counting means amount, something that remains the same even if the objects being counted are rearranged.

The context in which such mastery takes place varies greatly. Walkerdine (1997) suggests that even simple conceptual pairs such as “more” and “less” need to be rethought. Many children (particularly those raised in poverty) hear “more” as paired, not with “less,” but with “no more.”¹¹ The operations of addition, multiplication, and subtraction are embedded differently in different languages. In French, 90 is quatre-vingt-dix (four twenties and ten); its name uses both multiplication and addition. In the West African language Yoruba, 35 is named as “five from two twenties,” using multiplication and subtraction.¹² The Masai of Kenya signal the number 8 by raising four fingers of the right hand and waving them twice. ¹³

Some children enter the world of mathematical patterns more comfortably through visual experiences than through language. The Hopi of the American Southwest grow 24 varieties of corn. Children start acquiring basic mathematical concepts by helping to sort the corn according to color and size. The challenge is to build on these concepts in the classroom.¹⁴

Informal knowledge of geometric concepts is contained in traditional crafts and construction. The Mozambican mathematician Paulus Gerdes describes how in some African communities bamboo sticks are joined with ropes and shaped into rectangles to make components for a house. Mathematics teachers can use this familiar activity of artisans and house builders to introduce geometry to young learners.¹⁵

Young street vendors in Brazil accurately perform complex mental calculations, way beyond what they can achieve by school methods.¹⁶ Rather than multiply, they “perform successive additions of the price of one item, as many times as the number of items to be sold.”¹⁷ They use concrete referents and operations with which they are very familiar. “At the same time, while their everyday mathematics provides them the anchoring of specificity, it limits their flexibility.”¹⁸ Because they use repeated adding rather than multiplying, the street kids don’t learn the commutative law of multiplication. On the other hand, children in school who do know the laws of arithmetic may make careless mistakes; their errors will not cost them money.

Educators are looking for ways to connect everyday mathematics and school mathematics by introducing contexts that are meaningful to children. Jere Confrey, a well-known Piagetian mathematics educator, uses the metaphor of “splitting,” which includes sharing and mixing activities that are part of children’s daily lives. For example, mixing concentrated liquids to make lemonade is a way to teach proportions. As the children shift from one kind of drink to another, and change the quantities to be produced, they learn about ratios. They work in groups and then discuss their different approaches, providing insights to the teacher-researcher listening to the students’ voices. “It seems clear that . . . children can operate intelligently with ratios, especially if they are provided access to appropriate representations (data tables, ratio boxes and two dimensional plane) within interesting and familiar contexts.”¹⁹

Research on everyday mathematics is carried on in varied contexts, such as shopping, farming, sewing, and marketing. The anthropologist Jean Lave believes that knowledge about human problem solving is best pursued in “the experienced, lived-in world as the site and source of further investigations of cognitive activity.”²⁰ She studied the use of arithmetic by adults who focused on best-buy strategies while shopping for groceries. The shoppers compared prices, occasionally using hand calculators. They also were concerned with other issues, such as storage space and trying out new recipes. Participants sometimes used direct manipulation. One dieter needed to make a serving of three-quarters as much cottage cheese as the two-thirds of a cup allowed by her diet. If she had been in a classroom, she would have been expected to multiply 3/4 times 2/3 and cancel the 3’s to get the answer, 2/4 = 1/2. Instead, she solved the problem physically. She “filled a measuring cup two-thirds full of cottage cheese, dumped it out on a cutting board, patted it in a circle, marked a cross on it, scooped away one quadrant, and served the rest.”²¹ Thus, algorithms taught in school are not always transferred directly to everyday uses. But skills that are decontextualized in school can become alive and useful when applied to life experiences.

Math Reform

Studies like Confrey’s and Lave’s lead to new approaches to math instruction, connecting learning to real life experiences. There have been many efforts to reform math education. Some of them focus primarily on cognitive approaches, developing children’s number sense, mental mathematics, and understanding of patterns. One of these was begun by the famous Dutch mathematician Hans Freudenthal. Freudenthal was born to a Jewish family in Luckenwalde, Germany, in 1905. He received a Ph.D. in 1931 at the University of Berlin, under Heinz Hopf. He was then already in Amsterdam, having been invited there in 1930 to become Brouwer’s assistant. He soon mastered the Dutch language and even won first prize in 1944 for a novel! At that time, Holland was under Nazi occupation, and Freudenthal and his family were in hiding. A friend played the dangerous role of prizewinner at interviews, dinners, and speeches. But the badly needed prize money did reach Freudenthal and was a great help in surviving the last year of the war. When Holland was liberated, Freudenthal was appointed “professor of pure and applied mathematics and the foundations of mathematics” at the University of Utrecht. He became famous for contributions to topology and algebra, especially on the characters of the semisimple Lie groups.²² He also was a prolific contributor to the history of mathematics.

In 1971 he founded the Institute for the Development of Mathematical Education in Utrecht. (In September 1991, after his death, it was renamed the Freudenthal Institute.) He is credited with “single-handedly” saving the Netherlands from following the worldwide “new math” trend in the 1970s. His institute developed “realistic mathematics education,” which is based on problems taken from day-to-day experience. He taught that you learn mathematics best by reinventing it. He died quietly in October 1990, while sitting on a park bench, where he was found by playing children.

Freudenthal’s approach is being developed in New York by Catherine Fosnot and her collaborators. They describe their attitude to standardized calculating procedures, or “algorithms”: “Exploring them, figuring out why they work, may deepen children’s thinking. . . . They should not be the primary goal of computation instruction, however. . . . Children who learn to think, rather than apply the same procedures by rote regardless of the numbers, will be empowered.”²³

Everyday Mathematics is a popular program developed at the University of Chicago that uses similar constructivist principles. It emphasizes real-world situations like Confrey’s lemonade example and combines activities with the whole class, small groups, partners, and individuals. There are many opportunities for students to discuss their insights with each other and to compare their strategies. Pupils are encouraged to use calculators selectively, without making them into simple crutches.

Most reform efforts are based on constructivist theories that use counting as the basis of their instruction in arithmetic. A somewhat different approach is proposed by the Russian psychologist Vladimir V. Davydov. He was deeply influenced by the ideas of Lev S. Vygotsky who has been called the “Mozart of psychology.” A cultural historical theorist, Vygotsky was familiar with Piaget’s work, and the two theories do have some commonalities. One of their differences is the emphasis on “scientific concepts”. Vygotsky presents the notion that teachers should introduce broad concepts that cannot be acquired solely through everyday experience. They require carefully planned teaching.

In contrast with Piaget’s constructivist approach and its reliance on counting, Davydov emphasizes measurement as the basis of mathematical generalization. Both programs begin with concrete experiences such as the comparison of weights, areas, and children’s heights. However, these actions are represented schematically in the Russian approach. With increasingly complex problems, children invent new forms of representation. One of the advantages of this approach is that it provides a means for reconstructing a problem. By inventing multiple schemata, children overcome the challenge of fractions and square roots.

The American educator Jean Schmittau²⁴ replicated Davydov’s program in a northeastern school and found that children starting first grade used inventive methods in comparing quantities. They were able to engage in theoretical generalizations and were not troubled, as many of their American peers are, by multiplying ratios. Some of Davydov’s methods parallel historical inventions in mathematics. To date, these methods are not well known in the West, but a recent article about the program in MAA On-Line²⁵ may result in wider application.

Some reform programs include parents as well. One of these is the Family Math group of activities developed at Lawrence Berkeley National Laboratory in California. It, too, emphasizes manipulation, games, and everyday experience. “There is a growing sense that one of the goals of school mathematics is to help students make sense of both standard and nonstandard algorithms.”²⁶

The primary emphasis of reform programs is making mathematics cognitively accessible to learners. But reform can go beyond intellectual challenge; it can also include the emotional aspects of learning mathematics. This is important in overcoming the severe underrepresentation of minorities in mathematically oriented careers. “Blacks make up perhaps 15 percent of this country’s population, yet in 1995 they earned 1.8 percent of the Ph.D.’s in computer science, 2.1 percent of those in engineering, 1.5 percent in the physical sciences, and 0.6 percent in mathematics”.²⁷ Of the programs attempting to change this situation, the most influential is Robert Moses’ Algebra Project.

Moses was an outstanding leader in the struggle for voting rights in Mississippi during the 1960s. Now he is bringing proficiency in algebra to middle-school students in America’s Black communities. “Our aim is to change the situation that currently exists, where large percentages of minority students who go through a high school and get admitted to a college have to take remedial math in order to get to the place where they can even get college credit mathematics courses.”²⁸ He aims to make algebra enjoyable, by linking it to spatial concepts including travel, and by using multiple representations of mathematical notions. He is committed to giving young people a voice, to “involve the youth in all aspects of decision making”.²⁹ He views competence in mathematics, taught and acquired in middle school, as one of the civil rights of Black students. His program emphasizes community participation, peer instruction, and development of a stronger self-concept by learners. It is innovative in the way that mathematics is taught, and its particular strength is that it mobilizes all of a learner’s resources, including his or her emotions. It provides the balance of thought and emotion that so many mathematicians see as central to their enjoyment of their profession.

Figure 9-1. Bob Moses during his youth as a civil rights leader. © 1978 George Ballis/Take Stock

The participants in the Algebra Project view themselves not only as individual learners struggling with difficult ideas but also as members of a larger community. Teachers and coordinators share productive ideas, older students tutor younger ones, and all of them value the worth and potential of each individual learner. Using trips and directions as a key metaphor, they move from concrete experiences to increasingly more sophisticated algebraic expressions. Students receive tutoring from college student volunteers, older graduates of the Algebra Project, and peers helping each other. “The Mississippi kids could be taught in part by their own generation and learn more easily than the older generation being taught in the same workshop.”³⁰ “There is a way that young people reach young people, are able to touch each other that in my view is central to the future shape of the Algebra Project.”³¹ Young kids like to hang out with older kids. “And this hanging out didn’t have to be on street corners.” Moses and Cobb include many quotes from kids describing their experiences. For instance, Heather in Jackson, Mississippi, said, “My friends question me a lot about what I do. I don’t think they understand when I tell them that I leave school and go to work in the math lab. They say, ‘What do you mean you are going to work? That’s not working. You are just going over there and [play] with those computers.’ Working is McDonalds or Jitney Jungle, to them. They feel like I’m just learning, you know. And most people don’t put work and learning together.”³²

Young tutors like Heather are modeling what they themselves have learned. They have changed by being listened to, encouraged, and given responsibility. This is the emotional aspect that makes the project so effective. Even while being forced to slip under the rigid requirements of the present federal legislation affecting schools, the Algebra Project is contributing to greater mastery, self confidence, and self-respect in students who might otherwise have turned off and dropped out of school. But as Moses warns, “A network of tradition for this, involving teachers, students, schools, and community, isn’t established in one fell swoop. You go around it and around it, and you keep going around it and deepening it. You keep returning to it until all of the implications of what you are doing become clear and sink in.”³³

For college students, the positive impact of group interaction was revealed in a famous study that Uri Treisman conducted with Rose Asera at Berkeley in the late 1970s and early 1980s. They were trying to understand why many Black students who had performed well in high school dropped out of the calculus sequence after arriving at Berkeley. They noticed that Chinese students did much better than Black students. Treisman discovered the key difference: while the Black students studied alone, the Chinese students worked together in group sessions. They asked each other many questions, critiqued each other’s approaches, and helped each other with homework. “They might make a meal together and then sit and eat and go over the homework assignment. They would check each other’s answers and each other’s English. . . . A cousin or older student would come in and test them. They would regularly work problems from old exams, which are kept in the library.”³⁴

Based on their findings of this contrast with Chinese students, Treisman and his colleagues developed a new kind of intervention to assist minority students. They organized workshop communities where students met in addition to regular classes. These communities were not labeled “remedial classes” but rather were special opportunities for hard-working students. They were given challenging problem sets and were approached as members of an honors program, rather than as needing remediation. The program was an outstanding success, and over the last couple of decades it has been adopted at a variety of institutions.

The Treisman model bears some resemblance to the informal communities of research mathematicians, who enjoy exploring new problems and solutions with each other. Earlier in this book, we have described the crucial, supportive role of the Anonymous Group in Budapest, which served as an important setting for Paul Erdős, the best known collaborator in 20th century mathematics. In Indiscrete Thoughts, Gian-Carlo Rota writes of the Massachusetts Institute of Technology (MIT) commons room, where “at frequent intervals during the day, one could find Paul Cohen, Eli Stein, and later Gene Rodemich excitedly engaged in aggressive problem solving sessions and other mutual challenges to their mathematical knowledge and competence.”³⁵ In recognizing this reliance on conversations, advice, and arguments in parks, in cafes, or on the streets of Princeton and Göttingen, we challenge the notion that mathematics is the creation of isolated individuals. It is a socially created human endeavor for mature mathematicians as well as for students.

The reform movement has met strong opposition. The antireform people seemed to focus on textbooks, and wanted to move back to basics. Their position was publicized in the mass media. For example, this article appeared in the New York Times:

In Seattle, Gov. Chris Gregoire has asked the state Board of Education to develop new math standards by the end of next year to bring teaching in line with international competition. . . . Grass-roots groups in many cities are agitating for a return to basics. . . . Schools in New York City use a reform math curriculum, Everyday Mathematics, but some parents there, too, would like to see that changed. . . . A spokesman for the New York City Department of Education said that Everyday Mathematics covered both reform and traditional approaches, emphasizing knowledge of basic algorithms along with conceptual understanding. He added that research gathered recently by the federal Department of Education had found the program to be one of the few in the country for which there was evidence of positive effects on student math achievement. . . . The [antireform] frenzy has been prompted in part by the growing awareness that, at a time of increasing globalization, the math skills of children in the United States simply do not measure up: American eighth-graders lag far behind those from Singapore, South Korea, Hong Kong, Taiwan, Japan and elsewhere on the Trends in International Mathematics and Science Study, an international test. Many parents and teachers remain committed to the goals of reform math, having children understand what they are doing rather than simply memorizing and parroting answers. Traditional math instruction did not work for most students, say reform math proponents like Virginia Warfield, a professor at the University of Washington. “It produces people who hate math, who can’t connect the math they are doing with anything in their lives,” Dr. Warfield said. “That’s why we have so many parents who see their children having trouble with math and say, ‘Honey, don’t worry. I never could do math either.’ ”³⁶

The argument of the traditionalists concentrates on test scores. They claim that it’s the reform textbooks and curricula that result in the poor ranking of the United States in international evaluations. But it was the traditional curriculum and teaching that gave us adults who can’t add fractions, and 40 percent of whom say they “hate” math.

The critics of reform favor the Singapore textbook. Students in Singapore score highest on the international comparison of math test scores. The Singapore text doesn’t waste much time and space on needless motivation and explanation. It provides explicit directions and plenty of exercises, both easy and hard. It fits with the goals of the critics (see the web site Mathematically Correct, for example) who support the clear and direct teaching of algorithms and who stress competence in basic skills. What is good about this curriculum needs to be understood and explained, and ultimately integrated into curricula with more varied problem types and broader learning goals.

There can be many reasons for the low ranking the United States has received in international tests. Teachers in this country receive little respect, limited time for preparation, poor pay, and few opportunities to work with each other to develop stronger programs. In poor communities schools are badly neglected and convey the message, “You do not matter, and we do not expect you to succeed.” In some other cultures, learning is held in higher respect than wealth. If we want to strengthen the competitive position of American learners, we need more than curricular reform. We need to reform our economics, politics, and culture.

In the meantime, while some mathematical skill is needed for everyday life, a great deal of it is needed for science and technology. Computers and their software are the central nervous system of our society. On the one hand, the development, manufacture, and use of computers absolutely require workers with advanced mathematical training, but on the other hand, the universal use of computers at home and at work makes even elementary arithmetic unnecessary for nearly everybody else. These two opposite effects of the computer revolution put math education under acute tension. There is a wrenching strain between opposing pressures: a continuing demand for enough sophisticated math specialists, with a shrinking need for traditional math skills in the general population. Math reform has to strengthen the training of those who want and need advanced mathematical skill without alienating the large population that thinks they don’t need it.

In addressing this tension, we propose an approach that is developmental rather than oppositional. A long-term solution could realize some aspects of what each is advocating. Learning mathematics requires sufficient drill and practice, and it also requires challenging ideas. But it doesn’t require unchanging universal standards or compulsive reliance on test scores.

A Different Perspective

Basic arithmetic is necessary to survive in a postindustrialized society. Studying it should continue to be required, but not in such a manner that students remember it with antagonism and loathing. Is the solution to make school math more like real math, the math enjoyed by people who love math? The more it is taught with the goal of understanding, with willingness to engage in playful exploration, and with connections to job-related mathematics in adulthood, the more chance it has to succeed.

Why do so many people at some point in elementary or middle school “hit a wall” and give up on math? On one hand, there is a continuing demand for all children to master the basics: become skilful at timed tests of arithmetic, geometry, and algebra computations. It is demanded that their test scores rise to compare better with children in Bulgaria or Singapore.

On the other hand, there is the plain fact that once out of school, hardly anybody ever has to solve a quadratic equation or prove a geometry theorem. Yes, they have to do it to get into college or to get into many graduate or professional schools. But for many, once school is over, much of mathematical learning is forgotten.

Politicians and spokespersons for academia regularly issue statements decrying the low mathematical competence of our children. (The mathematical competence of adults is seldom mentioned as a problem.) It is argued that you have to be good at math if you want to make a good living and that a more mathematically competent workforce is needed in order for our country to compete in the world economy.

But do you know a doctor, lawyer, or businessperson who uses calculus, or even an algebraic equation or a theorem from geometry? When our country’s economic troubles are discussed in the business section of the New York Times (as opposed to the education supplement), the mathematical competence of the American population is never an issue. The U.S. steel industry collapsed because the costs of production and modernization of facilities are much higher here than in Brazil or China. General Motors and Ford are collapsing because of their high pension obligations and their weak competition in price and design against Japanese manufacturers—not because of poor algebra skills among the members of the United Auto Workers union in Detroit. America’s “back-office” computing is being outsourced to India because Indian computer techs work much cheaper than Americans—not because American computer techs know less arithmetic, algebra, or calculus.

In 1997 Underwood Dudley, the incoming editor of the College Mathematics Journal, derided the claim (in “Everybody Counts,” a document of the National Research Council) that “over 75% of all jobs require proficiency in simple algebra and geometry, either as a prerequisite to a training program or as part of a licensure examination.”³⁷ Dudley commented: “. . . this is silly. Just look at the next eight workers that you see and ask yourself if at least six of them require proficiency in algebra to do their jobs. . . . Almost all jobs, I counter-assert, require no knowledge of algebra and geometry at all. You need none to be President of the United States, nor to be a clerk at Walmart, nor to be a professor of philosophy. . . . You might think that engineers, of all people, would need and use calculus, but this seems not to be so.”³⁸ Dudley quotes Robert S. Pearson: “My work has brought me into contact with thousands of engineers, but at this moment I cannot recall, on average more than three out of ten who were well versed enough in calculus and ordinary differential equations to use either in their daily work.”³⁹

Professor Dudley concluded, “It is time to stop claiming that mathematics is necessary for jobs. It is time to stop asserting that students must master algebra to be able to solve problems that rise every day, at home or at work. It is time to stop telling students that the main reason they should learn mathematics is that it has applications. We should not tell our students lies. They will find us out, sooner or later.”⁴⁰ (Dudley has been teaching calculus at DePauw University for almost 40 years.)

The inability of most Americans to add 1 /4 and 1/3 correctly is embarrassing because they should have learned that in the 4th or 5th grade. If they ever need that sum, their calculator will give an answer close enough for practical purposes. But the principle involved in adding fractions could be taught more effectively than it is now. It was a real problem to some inmates at the New Mexico State Penitentiary Minimum Security Facility, where R. Hersh volunteered as a tutor for 5 years. When they got out of prison they would need a high school diploma in order to be employable, and they needed to add 1/3 + 1/4 in order to get a high school diploma.

It’s not that these adults hadn’t been drilled on adding fractions. They had been drilled, and drilled, and drilled again. The current No Child Left Behind Act intensifies testing and drilling by penalizing schools that don’t get the demanded test score results. As a result, math education in the United States nowadays is dominated by teaching to the test.

No wonder some people hate this! Suppose you couldn’t get a high school diploma without being able to sing, on key, the “Star-Spangled Banner” and half a dozen other “basic” numbers. We would produce a lot of people who hate to sing.

Fortunately, Hersh’s junior high school music teacher separated her class into “high boys,” “high girls,” “low boys,” “low girls,” and “listeners.” I (R.H.) was a happy listener. But my physical education instructor demanded over and over that I learn to climb a rope. That repeated humiliation of course intensified my dislike of physical education.

Many teachers, math educators, and mathematicians are trying to humanize school mathematics. They provide opportunities to work with real problems, cooperatively with schoolmates and teachers, where one learns by one’s own efforts, together with others, that 1/3 + 1/4 equals 7/12. In these contexts, many students develop some self-confidence and skill in thinking about numbers. We described several such programs above in which children do not hate mathematics because they know that mathematics is just thinking carefully about questions involving quantity. It also contributes to students approaching wide-ranging problems with reasoning and persistence. A constructive, experiential approach to learning arithmetic and algebra can be combined with practice, drill, and mastery of algorithms. But the reform curricula are not perfect. They are an important first step.

The greatest problem for math education in the United States is that there are nowhere nearly enough qualified math teachers. Moreover, those who are qualified don’t usually work in the less affluent school districts. In order to provide quality math education for all public school students, teachers with math qualifications must be paid salaries comparable to those in business and industry. And learning math need not be limited to schoolrooms; afterschool programs and community organizing efforts can develop a new consciousness and pride, as illustrated by the Algebra Project.

But isn’t there a contradiction between two different attitudes here? The Algebra Project of Bob Moses, which we admire and support, elevates the mastery of basic algebra to the level of a fundamental civil right, an actual demand on behalf of all children, especially the dispossessed inner-city children now so badly served in U.S. public schools. On the other hand, here we are arguing that algebra is not important or necessary for all people. Which is right?

It is unrealistic and unnecessary to guarantee that every child pass 10th grade algebra, and have a good facility with quadratic equations or with systems of two and three linear equations. What is necessary and realistic is that every child have the opportunity to learn algebra, in a well-equipped classroom, from a qualified, highly motivated teacher. All learners need to be able to think critically, with enough self-confidence to venture into the mathematical topics that are alienating when taught mechanically. Math phobia, which is frequently the result of the humiliating treatment of students, limits their ability to manage their financial lives and makes them vulnerable to deceptive lending practices.

In addition, it is essential to explicitly acknowledge the big difference between the school system in the United States and in many other countries. Here in the United States, vocational or industrial education in the public schools has almost been discredited. It was perceived as intrinsically discriminatory, a dumping ground for children in disadvantaged ethnic groups, especially Blacks and Latinos. There is an emerging goal that every child should be encouraged to have a college education. This goal is made hard to achieve by the discriminatory treatment at the primary and secondary levels. Such discrimination comes about largely as a consequence of the system of financing public education in the United States. Funding comes mainly from locally imposed property taxes. Educational districts in depopulated rural areas or in economically depressed urban areas have a much smaller tax base than those in affluent urban or suburban neighborhoods. As a consequence, less money is available to maintain the schools, and less money is available to attract the best teachers. So there is a huge disparity in the quality of public education between the inner city, for example, and the exclusive expensive suburbs of New York or Washington. A massive change in this system is urgently needed, and there is hope that it can be achieved under the new president, despite the present conditions of economic depression.

As we confront the many conflicting pressures that impact the U.S. educational system, we need to go beyond the polarity of reform and antireform issues. We need to find ways to make mathematics accessible and interesting to the greatest number of students while refraining from using it as a means of separating promising learners from alienated ones. By making mathematics more accessible and relevant to daily life and by having teachers who are passionate about the subject, we may succeed in decreasing the number of individuals who see themselves as incapable of dealing with numbers and numerical abstractions and help all learners to practice careful reasoning.

Math in College

At the university level, we believe that no one can be considered educated who doesn’t have some appreciation of mathematical thinking and its importance to science. However, the math requirements in college usually serve no such purpose. Most students are put into what is called a precalculus sequence—namely, a review of high school algebra and trigonometry—even though most of them stop there and never take calculus. Of those who do take calculus, most do so only because it is required for admission to medical school or business school. These prerequisites for schools of law, medicine, or architecture should be reconsidered. When we ask the faculty of those schools, “What do you want your students to know about calculus?” we are repeatedly told, “It doesn’t matter!” This “math filter” may have the advantage of being objective, easier to defend against charges of bias or favoritism. Still, it would be more rational and equitable to assess applicants to medical or business school according to abilities and commitments that are actually appropriate to their potential profession. For example, most doctors need to be able to make diagnoses. For that purpose, they need to elicit information from patients, to absorb and apply the results of medical research, and to develop and monitor adequate treatment. Thus, it seems that verbal and interpersonal intelligences and interest in science are more important for success in medicine than a knowledge of calculus. To select medical students we should look at portfolios, early internships, content-specific reasoning tasks, and judgments of their motivation for their chosen profession. The calculus filter is counterproductive. The kind of mathematics that physicians or entrepreneurs really are likely to use—basic statistics, and calculating with ratios and proportions—could be provided in a course specifically designed for them, either at the undergraduate or the professional level.

The following anecdote may be apocryphal, but it is enlightening:

An American mathematician of some note was returning from a trip abroad and had to go through customs. The U.S. customs officer asked him what he had been doing during his one week sojourn. The reply was that he had been at a mathematics conference. The customs officer then took this man aside and detained him for some time with a great many tedious questions about exactly where he had been and what he had been doing during his travels. The mathematician kept glancing nervously at his watch, worried that he would miss his connecting flight. The customs officer finally got to a point of asking our friend what he had for dinner each day. Finally the mathematician threw up his hands and exclaimed, “Why are you doing this to me?” The customs officer smiled and said, “Ah. Now you know how I felt when I took calculus.”⁴¹

Maybe the professor finally knew how the trapped student felt. But did the former student know how the professor felt? Does any mathematician enjoy forcing his subject down the throats of passive victims who want only a passing grade and to escape from the endless lectures? Paul Halmos wrote:

A class of students who take a course because of such requirements is a sad and discouraging class . . . the first prerequisite for the learning process to be both pleasant and effective—namely curiosity—is lacking, and that ruins it all. It ruins the teaching, it ruins the learning, it ruins the fun. I dream of the ideal university, full of students who are full of intellectual curiosity. The subset of those among them who take the mathematics course do so because they want to know mathematics . . . they come to me free, willing, and ask me to teach it to them. Oh, joy! If that really happened, I’d jump at the chance.⁴²

Does it have to be this way? Could it be different? Nel Noddings dares to have such a dream. After 23 years as an elementary and high school teacher and an administrator in New Jersey public schools, Noddings earned a Ph.D. and became professor of education at Stanford from 1977 to 1998. She also served as acting dean of the School of Education. She is a former president of both the Philosophy of Education Society and the John Dewey Society. She has raised 10 children. She presents an alternative approach that recognizes diversity in interests, talents, plans, and hopes among learners.

She writes, “We are overly reluctant to face the fact that human interests vary widely and that many highly intelligent people are just not attracted to mathematics. . . . I don’t know what talents and interests are lost under such coercion, what levels of confidence are eroded, what nervous habits develop, what rationalizations are concocted, or what evils are visited on the next generation as a result of our benevolent insistence.”⁴³ We agree with her, that “tracks” should not be lower and higher, just different and that all honest work should receive respect and dignity.

Figure 9-2. Nel Noddings, American educational philosopher. Courtesy of Nel Noddings.

Nor should we push all students to “think like a mathematician.” Better let them learn, as she says, to use math for their own purposes.⁴⁴ We should reject any assumptions about shared universal capabilities and recognize the diversity of intellectual strength. Noddings even wrote, “I do not think that children who are poor at math, who may never—no matter how hard they try—understand algebra and geometry, are in any important sense, handicapped, inferior, or in need of heroic intervention.”⁴⁵ She asks, “Why should a student who wants to major in literature, art, drama, law enforcement, history, or social work ‘learn’ algebra and geometry? . . . I have come to suspect that teaching everyone algebra and geometry is both wasteful and inconsiderate.”

We could then be more attentive and helpful to students genuinely interested in math. With them, she writes, we could even “work for deeper self-understanding, discuss the loneliness that sometimes accompanies extended intellectual work and the joy that emerges from successful encounters with mathematics.”⁴⁶ We agree with her, that such students should understand that their gifts are not higher than others—just different.

What Noddings is asking for on behalf of the student is exactly what Halmos is asking for on behalf of the teacher! She asks us to let students follow their interests and intellectual strengths, in addition to acquiring the mathematical skills needed for daily existence and effective reasoning. We agree with her position. Letting students deepen their knowledge in diverse contexts, using their different interests, is basic to an education that “counts.”

Today, algebra in high school and calculus in college are the primary “filters” for entering higher education. To justify this, it is argued that citizens must be able to reason logically, as voters and as consumers. Is it really true that passing algebra or trigonometry proves that the student can reason logically about politics or purchasing? We know of no evidence that high school algebra or trigonometry increase the ability to criticize phony advertising or political slogans. Some have a better chance to learn critical thinking in a domain where they have an interest, be it empirical science, literary analysis, or law. In non-school settings logic is needed for carefully researching major purchases, predicting the winners of sports events, or making important life decisions.

We are relying on the view of intelligence advocated by Howard Gardner, which recognizes the variety of human cognitive strengths and weaknesses. Gardner’s theory of multiple intelligences describes the human mind as “a series of relatively separate faculties, with only loose and non-predictable relations with one another.”⁴⁷ This opposes the traditional idea of intelligence as a unitary quality measured by a biologically predetermined IQ score. Gardner’s theory draws evidence from two research areas. One is stroke victims, who lose some cognitive skills but retain others. The second area is patients with Williams’ syndrome, children who excel in performing music but lack any ability to recognize other people’s emotions. In his book Frames of Mind, Gardner includes on his list of intelligences: linguistic intelligence, which is revealed in expressive written and oral speech; logical mathematical intelligence, shown in detecting patterns, thinking logically, and carrying out mathematical operations; spatial intelligence, in recognizing and manipulating patterns both in wide open and confined spaces; musical intelligence, in identifying pitches, tones, and rhythms and using them for performance or composition; and bodily kinesthetic intelligence, which is possessed by athletes, dancers, and surgeons. In addition to all these, he specifies two emotional intelligences: “interpersonal” intelligence, shown by someone such as a teacher or counselor who works well with others; and “intrapersonal” intelligence, shown by someone whose self-knowledge guides his or her own life.

Most activities combine two or more modes of intelligence. In addition to bodily kinesthetic capability, a surgeon needs reasoning skills and a strong ability for visual representation. Mathematicians differ in how much they rely on logic and linguistic processes versus kinesthetic and visual ones. Most mathematics teaching neglects this diversity and thereby adds to the fear of failure felt by so many learners. At present, students are called “good at math” if they are good in the narrowly restricted ways it is presently taught. “Maybe if we really understood how different people use different cognitive capacities to solve problems we could design instruction so that many people were good at math.”⁴⁸

Gardner’s different intelligences might be called abilities. Effective schooling needs to link students’ learning to their diverse predispositions, interests, and cultures. If mathematics beyond the elementary level were studied only by students who are interested and motivated—perhaps only one out of every four—there would still be enough graduates for the jobs that need such knowledge. Those motivated to study mathematics in depth could then receive more sustained and effective instruction.

The reform efforts now going on in elementary schools do lessen math phobia. Such programs as the Algebra Project increase access to advanced courses and professional careers for minorities. These programs view learners as active participants in their education. They take time to achieve results; they require material resources, and a great investment of time in teacher training and individual tutoring.

Learning requires passion, joy, surprise, sustained interest, and the ability to get help from teachers and mentors. A student whose skill in mathematics is limited in early years may later be better able to persevere after he gains self-confidence in domains better matched to his ability.

The current dominant model of math instruction is mechanical and inflexible and results for many in lifelong avoidance of mathematics. More positive attitudes, and slow, cumulative gains in mathematical achievement, could come from long-term applications of innovative curricula. Instead of stark choices between traditional and reform approaches, we support the sustained improvement of mathematics teaching. The greatest likelihood of success is where local leadership is supportive and university assistance is available. University faculty and students in mathematics departments and math education programs can make important contributions to local public education.

Improving the teaching of mathematics is not limited to curriculum. Even in the early grades and junior high school, it can benefit from communitywide participation. Teaching by older students, as well as adult tutors, can be crucial in giving children the personal attention they need when they struggle with the abstract concepts of mathematics. Sharing everyday uses of arithmetic with parents and community members makes these concepts more accessible to young learners. If we focus on slow, cumulative gains, rather than on international competition and punitive, standardized testing, we can create more confidence and comfort at the early levels and provide more choices for students once they enter high school. For those who avoid mathematics classes in high school and college that don’t seem relevant to their interests, we should provide opportunities to acquire relevant mathematics at a later stage. More mature individuals are more ready to take risks and to see the pragmatic value of math in the growing fields of technology.

Conclusions

We have presented many reasons for the widespread avoidance of mathematics by school children and by adults. Foremost among these is the formulaic way in which most teachers present mathematical abstractions. Many students become insecure and may avoid math for the rest of their lives. While some of these problems are being addressed by reform efforts in the United States and elsewhere, the pace of reform is slow and limited to curriculum. In our view, broader changes are needed to address this problem. Among these is discontinuing the use of mathematics as an academic filter.

Instead, the goal is to treasure diversity in talent and interest; to provide advanced mathematics teaching/learning to motivated students, while decreasing the number who suffer from math phobia. The challenge is to develop a systematic, societywide perspective, rather than imposing the same values and approaches on both enthusiastic and reluctant learners. Because we love mathematics, we want to minimize the number of those who hate it. Our purpose in these proposals is to shift the premise of the current debate. It is to create a humanistic role for mathematics and its teaching in our culture, a way of teaching mathematics that focuses on the needs and abilities of students as well as society.

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