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The Center for Public School Renewal

NOTE: Published in slightly different form by the NCSM Newsletter, October, 1995. [Special thanks are due to Bill Rosenthal for his helpful comments on a draft of this essay.]

Can It Be True That All Children Can Learn Mathematics?
by Barry McGhan

"Everybody Counts."
"Algebra for Everyone."
"Making Mathematics Accessible to All."

These titles(1) express a wonderfully egalitarian educational ideal. But is it just an ideal? What is the reality behind them? Consider the view of Robert Nielsen (1990):

"The assertion that 'all kids can learn math' is probably the most radical idea associated with the education reform movement. It is likely that the majority of mathematics teachers don't believe it."

The belief that all children can't learn mathematics--to about the same extent--creates differential expectations for performance, leads to tracking practices, and in the upper grades, supports a widely differentiated curriculum. These practices deliver larger or smaller doses of mathematics content for learners who are judged, perhaps unfairly, to have different capabilities.

Contrast Nielsen's position with the view of Zalman Usiskin (1994), director of the University of Chicago School Mathematics Project. Usiskin says, "This belief [that students should not be distributed among a variety of classes] is common among mathematics educators today." In a thought-provoking article, Usiskin reprises several models of ability that he believes collectively represent the current thinking of most mathematics educators, and offers other student characteristics also deserving of consideration.

The phrase "all kids can learn math" implies that we are trying to change the common view of mathematics as an exclusive activity for the few, to a new view as an inclusive opportunity for the many. This is a most ambitious change to seek. What most teachers believe about the phrase is a matter of great concern as we move into this new era of mathematics instruction. There is no simple way to prove whether Nielsen or Usiskin is right about the degree to which most mathematics teachers believe this new philosophy. A growing body of research supports the ideal that "everybody counts," and so Usiskin's view may be gaining strength. Certainly, it is in accordance with the National Council of Teachers of Mathematics' call for a common core curriculum (NCTM 1989, NCTM 1991). In spite of this, I find Nielsen's view of the thinking of most teachers--that they don't believe all children can learn mathematics--is likely to be most accurate.

It would be a grave mistake to assume that the transformation to an egalitarian vision of mathematics education will be easy. Teachers' belief in natural mathematics ability is not an idea that will go quietly away, once criticized. It is going to take a while to form a consensus favoring the idea that everyone can learn mathematics. If you believe it, you need to gird yourself for a long and difficult struggle.

A Paradigm Shift

Beliefs about natural ability underlie all other beliefs about why people are different from each other. This includes the belief in natural intellectual ability, and its corollary, mathematical ability. In fact, it seems that mathematics is the academic subject most marked by a belief in natural ability. The "all children can learn" view will be difficult for teachers to appreciate unless their faith in the "natural" model exemplified by the normal curve is reduced.

I used to believe that human capacity for learning fit the natural ability model. However, my thinking on this subject has changed. I experienced what is often called a paradigm shift. I became a believer in the "all children can learn" philosophy--and others can too. Promoting the shift to the "everybody can learn mathematics" paradigm needs to be placed at the center of our efforts to transform mathematics education for the 21st century.

The Challenge

Back in the 70's, when I became interested in this general topic, the "nature/nurture" controversy was in full swing. Arthur Jensen (1969) had published an article that sparked a whole series of responses supporting or challenging his position on the importance of IQ. After reading most of these treatises, and studying the history of the controversy (which can be traced back to Plato and Aristotle), I decided that neither side had convincing proof that it was right and the other wrong. The tools used to study intelligence in modern times--largely statistical analyses of standardized test data--are simply too weak to give us irrefutable knowledge about this topic. Statistics is not, as some think, the scalpel that dissects genetic from environmental influences. Instead, it is more of a blunt instrument, often used to intimidate people with opposing viewpoints. As Usiskin says " . . . we know virtually nothing about the ability of people to learn anything, let alone mathematics, because we still know very little about how the brain operates."

The belief in natural ability is a fundamental ingredient in teachers' thinking about their students and what they can do with them. It even has a kind of pernicious utility for teachers. The utility is this: if achievement is mostly a function of heredity, some will feel excused from doing less than their best with all children. Such negative beliefs, like the threads the Lilliputians used to bind Gulliver, set limits--limits on who gets the more advanced mathematics instruction, who goes to college, and so forth.

In order to make any progress with the idea that all children can learn mathematics it seems to me that we must make a direct frontal assault on the belief in natural ability. The objective of this assault would not be to destroy the belief--since no incontrovertible evidence for that view exists, either--but to weaken it and allow room for alternate paradigms to grow.

Science and Politics

An important point to make early in this quest to weaken the belief in innate ability is that this is as much a political issue as it is a scientific one. The best recent example of this comes from The Bell Curve: Intelligence and Class Structure in American Life, by Herrnstein and Murray (1994). There are at least two lessons to be learned from the excitement that has surrounded the publication of this book. First, it underscores the appeal of the traditional ability paradigm for many people. Second, the book seems to be a fairly blatant attempt to press a politically conservative social agenda. For example, its authors claim that Head Start and other similar programs should be curtailed because they waste money on people who are not smart enough to benefit from them. As the 18th Century French philosopher Condorcet said, such approaches "make nature herself an accomplice in the crime of political inequality."

Mathematics educators must be concerned with the politics that surround their support of the idea that everyone can learn mathematics. According to Michael Apple (1992), "the very real elements of democratic potential within the Standards can be washed away in a context of rightist reaction. An understanding of this context is an essential precondition for transforming mathematics education."

Choosing Against "Nature"

There are several tools that can be used to take the hot air out of the belief in natural ability. I will provide a brief description of some of them, as well as a few references for further study.

First, given the lack of proof for either nature or nurture, we can argue that we are free to choose what to believe. I choose to favor the nurture/environment side because it seems more optimistic. That is, believing in the power of the environment to improve the human condition gives me reason for hope--not just for my students, but for myself and everyone I know. We can all look forward to becoming more competent, knowledgeable, understanding, caring, intelligent people. Also, it provides a reason for trying to improve, for working at problems--it values effort. If the environment can make a difference, then I can make a difference by helping to change it for the better. Some people will find this line of reasoning convincing.

Many will still wonder how this viewpoint squares with reality. They will think something like, "You can't make a cow jump over the Moon just because you want her to." And we can't, because it is certainly true that there are some people who are profoundly mentally handicapped. Estimates usually peg this proportion of the population at about 2% or less(2). At the other end of the human spectrum, there are people like Einstein, DaVinci, Mozart, Gauss, and those children who graduate from college at age 12. So, if necessary, we might say there is another 2% or less at the other end of the intellectual spectrum who seem profoundly different from most people. That leaves 96% or more of us who might be just about the same, insofar as innate intellectual capacity is concerned. A graph of this kind of intellectual ability might look like this:

Whether this model of human capability (it's similar to a model Usiskin calls the squeezed normal distribution) is accurate is unknown. Here again we can make an existential choice:

I choose to believe that there are a few mentally handicapped people in the world, and a few truly gifted people, but most of us are very much alike in intellectual capacity.

There also is a partly mathematical reason not to believe in the normal distribution of ability. It has to do with the Central Limit Theorem (CLT) in statistics. It is possible to show that, even when a measurement has a non-normal distribution in a population, the process of repeated sampling and calculating a sum or an average value for the measure produces a normal distribution of sample calculations. With respect to IQ scores this means that any given individual's score may represent the average or sum of many genetic, physical and social factors residing with that person. So, by the CLT, IQ should be normally distributed. Yet, that fact tells us nothing at all about the underlying "components" that compose it. For an explanation of this phenomena see Paulos (1991). If as some believe, intelligence is a complex phenomena composed of multiple parts which may not all be measured by IQ tests (Gardner 1993), then the normal distribution of IQ scores may be nothing more than an artifact of the Central Limit Theorem.

One of the most complete challenges to the idea of fixed and limited intelligence comes from paleontologist Stephen J. Gould (1981). Gould offers a comprehensive approach to debunking what he calls "hereditarian" views of human intelligence. He attacks the historical underpinnings of hereditarianism and notes the self-serving motives of hereditarians. He then moves on to demolish their logic, in particular that of reification. By reification he means the conversion of an abstract concept, namely intelligence, into a measurable entity. His most basic premise is that every scientific study is conducted according to the rules of a particular paradigm that "colors" the conclusions reached. His review of the research of hereditarians consistently uncovers this lack of objectivity. Ultimately, he concludes that the empiricism of the hereditarians is simply out of touch with the principles of modern evolutionary biology.

Another approach to attacking the meaningfulness of the concept of intelligence comes from F. Allan Hanson (1993). He argues that tests actually produce many of the qualities that they are supposed to measure, and "nowhere is this clearer than with the concept of 'intelligence.'" His argument, very briefly, is that by defining intelligence as the thing that is measured by scores on intelligence tests, a condition is set up where people strive to master those areas the tests assess. In other words, they set out to show how intelligent they are by practicing just those things that the test will reward with a high score.

A related argument is that IQ tests were originally devised to measure school success, and that's all they measure--school achievement, rather than any innate qualities. It is well known that social status strongly influences school achievement, so there would be little reason to conclude anything about innate intelligence based on IQ tests.

The issue that seems most clearly to differentiate between hereditarian and anti-hereditarian views is that of the validity of the IQ tests themselves. Do these tests measure "intelligence" or not? Paulos, Gould, and Hanson all seem to say, in one way or another, that they do not--that there is an artificiality to the tests that makes any conclusions based on them suspect.

Choosing For "Nurture"

It is difficult to destroy a belief without offering a reasonable replacement for it. So, a key strategy to produce a paradigm shift away from nature toward nurture lies in offering an alternate view of human achievement. There are several good candidates for replacement paradigms.

One is found in the work of Benjamin Bloom (1984) and his study of mastery learning. Bloom finds that mastery learning significantly reduces the variation in performance among students. In fact, there is a growing body of research in the field of cognitive psychology that supports the position that new teaching strategies can significantly expand the learning opportunities of many currently underserved students (Resnick 1986, Bruer 1993). If, as seems likely to be the case, success in traditional schooling defines what we mean by IQ, then a definition of schooling that incorporates these new strategies will change what we mean by IQ.

Another replacement paradigm can be found in the field of social psychology. Brookover and Erickson (1969), offer a view of learning based on the idea that children learn to behave in ways that people with whom they associate behave. Specifically, they say, the individual acquires conceptions of his ability to learn through interaction with significant others. The expectations of those significant others--especially low expectations--play an important part in determining what children ultimately accomplish. For example,

  • Teacher expectations can have an impact on students' IQ scores (Rosenthal and Jacobsen 1968).

  • Peer expectations for appropriate school behavior can severely curtail academic performance (Ogbu 1992).
  • Social class based parental expectations for valid forms of schooling can open up or limit educational opportunities (Pearlin and Kohn 1966).

Also, consider the cross-cultural studies of school achievement in American and Asian elementary schools by Stevenson and Stigler (1992). They found that Asian teachers have a strong belief in the importance of effort over ability. In America, they found that teachers believe innate ability is the key to success, and there is little hope that students can improve much once they start to struggle. Stevenson and Stigler also report that

"within individual classrooms the variability in levels of academic achievement differs little between the United States and Japan, Taiwan, or China. Where . . . Asian countries have succeeded in reducing variability, they have done so by reducing the discrepancies in educational outcomes among different elementary schools."

In other words, Asian children are as individually different from each other as are American children. But, not being treated as different makes their performance similar.

Finally, alternate views of how one can work with students in ways that distribute knowledge of mathematics to a much wider proportion of the population include

  • the use of cooperative learning strategies,

  • attention to different learning styles,
  • an emphasis on helping students learn to communicate their understanding of mathematics,
  • an emphasis on how to apply mathematics to meaningful uses.

Several of these items are key recommendations of the NCTM's Curriculum and Evaluation Standards and Professional Standards for Teaching.

Conclusion

If you believe that all children can learn mathematics, you're going to have to fight for it. Learn more about the arguments and ideas mentioned above. Remember to offer people an alternate perspective to believe in. Don't expect this transformation to happen quickly because people give up basic beliefs only reluctantly. On the one hand, giving up the belief in natural ability provides an opportunity for teachers to make a difference with most kids. On the other hand, few people (as Harry Truman suggested) are comfortable with the notion that the buck stops with them. This is especially true in this era when teacher-bashing seems to be on the rise. If other segments of society abrogate their responsibility for what goes on in school, and place the whole burden on the educational system, then the notion that all children can learn mathematics may be very hard to sell to teachers.

Footnotes

(1) Everybody Counts, Mathematical Sciences Education Board, National Academy Press, 1989. Algebra For Everyone, NCTM, 1991. "Making Mathematics Accessible to All," A professional development project for high schools sponsored by Western Michigan University Department of Mathematics and Statistics, 1992.
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(2) This percent can actually increase beyond those with genetic disorders since prenatal conditions (e.g., fetal alcohol syndrome, poor nutrition) and conditions in life (e.g., accidents, abuse) may cause profound mental deficiencies.
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Bibliography

  1. Apple, Michael. "Do the Standards Go Far Enough? Power, Policy, and Practice In Mathematics Education." Journal for Research in Mathematics Education, vol 23, no. 5, 412-431, 1992.

  2. Bloom, Benjamin. "The Search for Methods of Group Instruction as Effective as One-to-one Tutoring." Educational Leadership, May, 1984, pp. 4-17.
  3. Bruer, John T. Schools for Thought: A Science of Learning in the Classroom. MIT Press: A Bradford Book, 1993.
  4. Brookover, Wilbur B. and Erickson, Edsel L. Society, Schools And Learning, Allyn and Bacon, Inc. 1969.
  5. Gardner, Howard. Multiple Intelligences: The Theory in Practice, Basic Books, 1993.
  6. Gould, Stephen J. The Mismeasure of Man, W.W. Norton & Company, 1981.
  7. Hanson, F. Allan. Testing Testing: Social Consequences of the Examined Life, U. of California Press, 1993.
  8. Herrnstein, and Murray, The Bell Curve: Intelligence and Class Structure in American Life, The Free Press, 1994.
  9. Jensen, Arthur. "How Much Can We Boost IQ and Scholastic Achievement?" Harvard Educational Review, Winter, 1969.
  10. NCTM. Curriculum and Evaluation Standards For School Mathematics. National Council of Teachers of Mathematics, 1989.
  11. NCTM. Professional Standards For Teaching Mathematics. National Council of Teachers of Mathematics, 1991.
  12. Nielsen, Robert. "Anyone Can Learn Math". American Educator. Spring, 1990, pgs 29-34.
  13. Ogbu, John. "Understanding Cultural Diversity and Learning." Educational Researcher, November, 1992, pgs 5-14,24.
  14. Paulos, John. "Statistics - Two Theorems" in Beyond Numeracy, A Borzoi Book: Alfred A. Knopf, 1991.
  15. Pearlin, Leonard I. and Kohn, Melvin L. "Social Class Occupations and Parental Values: A Cross-National Study." American Sociological Review, August, 1966, p. 466ff.
  16. Resnick, Lauren B. "Cognition and Instruction: Recent Theories of Human Competence and How It Is Acquired." Learning Research and Development Center, University of Pittsburgh. June, 1986.
  17. Rosenthal, Robert and Jacobsen, Lenore. Pygmalion in the Classroom: Teacher Expectation and Pupil's Intellectual Development. Holt, Rinehart and Winston, Inc., 1968.
  18. Stevenson, Harold W. and Stigler, James W. The Learning Gap: Why Our Schools are Failing and What We Can Learn From Japanese and Chinese Education, Summit Books, 1992.
  19. Usiskin, Zalman. "Individual Differences in the Teaching and Learning of Mathematics." UCSMP Newsletter, #14, Winter, 1994. pg. 9ff.

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