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Mathematics Opinion

Fuzzy Math, Old Math, and Dewey

By Richard S. Prawat — December 10, 1997 8 min read
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Richard S. Prawat is a professor of education at Michigan State University in East Lansing, Mich., and the chairman of the college of education’s department of counseling, educational psychology, and special education.

Increasing criticism is being directed at what was considered the Cadillac of curricular standards--those developed in 1989 by the highly respected National Council of Teachers of Mathematics. Critics now apply pejorative labels like “fuzzy math” or “new new math” to this sensible effort to ratchet up expectations in mathematics classrooms. More emphasis on problem-solving, less on rote learning, the argument went, would decrease the likelihood that students will be taken in by questions like the following: “There are 26 sheep and 10 goats on a ship. How old is the captain?” Nearly 80 percent of the elementary students responding to this query in a 1986 study simply added the two numbers together. Research suggests that the situation has improved in the past decade, but there is still much work to be done. It is tempting, therefore, for math reformers to dismiss critics as narrow-minded and ideological. Those who hearken back to a rule-following view of mathematics, reformers argue, do so because it fits with their rule-following view of the world. For them, it is not just mathematics that has been fuzzed up but the whole moral order. Thus, critics of the critics maintain, the attack on mathematics reform is a futile effort to turn back the clock.

Despite this pro-reform rhetoric, something does seem to have gone wrong. As in previous reforms, many well-intentioned educators have abandoned the fixed curriculum in favor of the activity-oriented, “hands on” approach. Tom Loveless, in a recent Commentary in these pages, at least makes this claim. (“The Second Great Math Rebellion,” Oct. 15, 1997.) Given the opportunity to open up mathematics instruction, he argues, reform-minded teachers have gone overboard, seeing mathematics as an occasion to promote interpersonal skills and--through the careful selection of problems--an opportunity to pursue other agendas like heightened social awareness. Mr. Loveless cites as an example a model question taken from the California math framework that asks students to contrast the tax burden of the poorest families in California (14.1 percent of income) with that of the middle 20 percent (8.8 percent). Students are asked to reflect on this difference, indicating whether or not they think it is fair. Examples like this portray “new new math” as ideological and ineffective.

Educational reformers contribute to the concerns raised by critics like Mr. Loveless. An example is found in another recent Commentary, titled “Polishing the Progressive Approach.” In this piece, Elaine Winter charts her own journey toward progressive education. Predictably, she credits powerful advocates like John Holt and Jonathan Kozol with opening her eyes to the possibilities of learner-centered education. Equally eye-opening, she writes, were her own positive experiences with experiential education. One that stands out in her mind occurred in her own sons’ school. “One of the richest activities I remember from that time was the class replicating the fall vendemmia, or wine-making process,” she writes. “Parents brought in plastic bags to cover children’s feet, then the children stomped big buckets of grapes to makes themselves grape juice. It was exciting because it was so relevant to the children’s lives and cultures.”

Must we pit child against curriculum, what is enjoyable against what is educative?

Undoubtedly the outcome Ms. Winter celebrates is just what those who formulated the tax-disparity item had in mind: To make math exciting by making it relevant to children’s lives and culture. Exciting and relevant it may be, but reformers who expose students to the fuzzy-math equivalent of grape stomping must be prepared to answer questions about the educative value of such experiences. Reform-minded teachers have stumbled when asked this question in the past, creating a real Hobson’s choice for parents and policymakers: Must we pit child against curriculum, what is enjoyable against what is educative?

John Dewey spent his entire adult life devising a solution to this dilemma, a solution not found, interestingly enough, in his education writings. The education work, especially the early work, was most explicit about what was wrong with the traditional approach. Dewey believed that a lasting solution to the issues that divide educators (for example, the child-vs.-curriculum dilemma) lies in the philosophical domain rather than in the educational, in how we think about meaning and knowledge rather than in what we do to develop it in students. It is worth elaborating on Dewey’s solution to this philosophical problem because it affords a way out of the Hobson’s choice that confronts educators in this country.

Dewey presided over one huge swing of the educational pendulum in the 1930s. His criticism of traditional education contributed to the swing toward reform. His subsequent criticism of the reform effort contributed to the swing back in the other direction: “What is needed in the new education,” Dewey wrote in 1940, “is more attention, not less, to subject matter.” Contrary to popular opinion, Dewey was not an advocate of activity- or project-oriented curricula. He believed that both subject and child could be highlighted. Content knowledge need not be lifeless or inert. Conceptualized a certain way, it could be a tool that opens up the child’s world. What Dewey proposed was a whole new way of thinking about content: subject-matter knowledge as a set of powerful ideas, developed in disciplines, available to everyone, used as a means to illuminate aspects of the world that otherwise would remain closed off to the individual forever. Subject matter, for Dewey, was literally a way to “be” in the world, at least in a perceptual sense. The concept of photosynthesis allows one to see green leafy plants in a new way. The same is true of negative number in mathematics; a person who has grasped this notion views realities like temperature or financial indebtedness with new eyes. Or additive composition, the idea that number can be taken apart and put back together in different ways; one looks at number differently having grasped this notion.


Dewey realized that swings of the education pendulum, signifying child- or learner-centered instruction in one decade and a return to fact- and skill-based learning in another, are symptomatic of a deeper philosophical problem: the separation of mind and body and mind and world. His solution to these two problems offers a way out of the box mathematics reformers find themselves in today. It is important for educators to understand why this is the case. Dewey was one of the first to grasp the dangers associated with walling the mind off from the world. Educators, for one, interpret this to mean that “subjects,” with their unique needs and interests, are separate or distinct from the objects they seek to understand. Subject and object, process and content, child and curriculum all flood in, once the door is opened a crack. Teachers take sides and students are caught in the middle, an unfortunate occurrence that is likely to happen again in the current math war.

Dewey's solution to the problem: Teach the big ideas in mathematics and other subjects.

Dewey’s solution to the problem is elegant in its simplicity, representing a happy middle-ground solution for reformers and traditionalists alike: Teach the big ideas in mathematics and other subjects, Dewey argued. Ideas, because they represent embodied ways of knowing, offer a way around the mind-body problem. Ideas are embodied because they originate as rich, experientially based images or metaphors. Thus, the notion of photosynthesis has its origins in an image or metaphor of the leaf as “food factory.” Positive and negative number is understood, in part, by grasping the role that “zero” plays as a point of symmetry on the number line. Fortunately, we experience countless instances of symmetry in our daily lives, the way that the right half of our body mirrors the left half being but one example. The private notes of great scientists are replete with illustrations of how rich, metaphoric imagery contributes to the conjuring up of big ideas. Darwin, one scholar argues, hit on the notion of “nature selecting” by reflecting on the process of animal breeding. If my neighbor, through selective breeding, can produce unique characteristics in dogs or sheep, why can’t nature do the same? Darwin reasoned. Ideas join mind and body, Dewey believed, after some give and take on both sides.

Of even greater importance from a reform perspective, ideas integrate mind and world. They literally move outside the boundaries of the skin to engage with the objects or events they were developed to illuminate. Dewey was deadly serious about this attribute of ideas. Ideas put the child in touch with the world; the world, in turn, gives life to ideas. The experiential richness of this type of learning exceeds even that of grape stomping. Appreciating the role of green leafy things as food producers is not only enjoyable, it is immensely educative as well. The solution Dewey proposes thus offers a way out of the conundrum that has plagued efforts to reform U.S. education.

If this is true, one may ask, why has Dewey’s solution been such a well-guarded secret? Perhaps that is not the case when viewed from an international perspective. I’ll pass on one example of what I mean, relayed to me by Bill Schmidt when he tried to recruit the Japanese to participate in the Third International Mathematics and Science Study. It is fair to say that Japanese educators were somewhat reluctant at first to enter what they viewed as yet another international “horse race,” one that their students do very well in, especially in mathematics, which invites a certain amount of envy and resentment from the rest of the world. This attitude changed when it became clear that an effort would be made to determine how teachers in countries around the world handle important topics like fractions and decimals: “We have developed a number of ways of teaching ‘part-whole’ relations in mathematics,” a Japanese ministry official said, “but we think we may be able to discover a few that we haven’t thought of if we participate in the study.”

As this comment suggests, the Japanese do indeed focus on big ideas in mathematics and science, at least at the elementary school level. The comparative study by Harold Stevenson and James W. Stigler confirms this notion. Dewey would be tempted to go further, arguing that this fact explains why the Japanese have avoided the kind of counterproductive debate in this country that pits math reformer against math traditionalist.

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A version of this article appeared in the December 10, 1997 edition of Education Week as Fuzzy Math, Old Math, and Dewey

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