The Circumference of a Circle
Can students be led to find that math is not boring?
Successful 5th graders learn that the circumference of a circle is equal to pi times the circle's diameter: c = pi x d. That is a fairly useless lesson—a computer can do it more quickly and accurately than the best student. On the other hand, finding that pi is equal to the circumference divided by the diameter (pi = c ÷ d) can be exciting. Some (most likely Greek) human noticed that the ratio stays constant irrespective of the circle's size. New patterns can be discovered by thoughtful observation. Even 4th grade students can be given pieces of string and circles of various sizes and discover for themselves that three diameter lengths plus a little extra goes around a circle, regardless of how big or little the circle is. Wow!
The fact that most math pedagogy is of the c = pi x d type may be the reason that many students find math so hard. The average National Assessment of Educational Progress score on the mathematics test for 17-year-olds in recent years is just above 300. This means that the average American 17-year-old can compute with decimals, use fractions and percents, recognize geometric figures, solve simple equations, and use moderately complex reasoning. The averages among African-Americans (286) and Hispanics (292) were below 300, meaning that many of those students could do no more than the four arithmetic operations with whole numbers and solve one-step problems. Over half the students entering California's state college system need to take a "developmental" mathematics course. Over one-fourth of college freshmen feel that they will need tutoring or remedial work in math. This compares to one in 10 for English, science, and foreign languages.
Math phobia and the resulting deficiencies have always had deleterious effects; but these soon will become even more serious. The reauthorization of the Elementary and Secondary Education Act signed by President Bush early this year mandates annual tests in grades 3-8. Business and other organizations that have fought to bring about these high-standard tests will now apply pressure to extend testing to the 12th grade. Students in a number of states will be facing high school exit exams in the next few years (Massachusetts in 2003).
Students who fail will be denied their high school diplomas. Many, if not most, of the denied will have failed their math exam. (In this trial year, 18 percent of Massachusetts 10th graders failed English/language arts, while 25 percent failed mathematics.) Even today, many interested in college programs leading to degrees in a technology field are denied entrance because of their weakness in mathematics. Current practices serve half our students poorly, and the National Adult Literacy Survey, or NALS, documents the difficulty beyond adolescence. By some estimates, less than a fourth of American adults are at the upper-end, literate levels of 4 and 5.
The evidence suggests that students will not be able to graduate from high school if the high math standards are part of the exit exams and the current prevalent math pedagogy and assessments persist. If that is so, then the policy-relevant hypothesis is: Can changes in pedagogy and assessment make it possible to have both high graduation rates and rigorous exit exams in high school mathematics?
We have been trying to test this hypothesis in Baltimore's high schools. Can students be led to find that math is not boring? But wow! The technique we are testing in Baltimore is project-based learning.
Students in these high schools must use mathematics to complete a project. They also make use of technology and tools such as spreadsheets to develop, for example, a business plan. The math is rigorous: Probability determines inventory, lines intersect at the break-even point, charts and graphs must be made and explained.
Algebraic equations are used because students find it easier to use symbols like r and c, rather than type in the words "revenue" and "cost" in their spreadsheets. In this way, they can visualize the problem, something impossible to do when the abstractions x and y are used. Indeed, we eschew x and y throughout high school so that visualization is possible.
Many math educators object. "The power of mathematics is its generalizablity," they insist. The empirical fact that most students never get it does not faze them. We believe that students would do better if they began with the specific and moved to the general later—maybe introduce x and y as variable names in college algebra.
What are the results of this approach? Three "zone" (neighborhood) high schools, among the city's most-challenged secondary schools, used this project approach in math, English, and other courses. Baltimore's zone high schools have an almost incredible 29 percent graduation rate—a 71 percent dropout rate. Only 64 percent of the students who made it through 11th grade in the comparison group drawn from these schools took and passed Algebra 2. Their average math grade point average was 1.46.
About 90 percent of the students in our project-based group (a total of 84 in two graduating classes) graduated. About 90 percent of these students took and passed Algebra 2. Their average GPA, although only 2.12, was 45 percent better than the comparison sample.
A note about the comparison is in order. The youngsters in the test group were not randomly chosen, but were deemed to be in the middle 50 percent of the class—neither the best nor the worst. The comparison group, however, was taken from those who had made it past 11th grade—a small minority and the better students in these schools.
These students have not been tested in any standardized way, because there is no standard assessment that tests what is often called "quantitative literacy" (a term coined by Lynn Arthur Steen in Mathematics and Democracy). What would the high school exit assessment look like if this pedagogy were adopted? Certainly, the assessment would not look like those current exams containing x and y equations, factoring, and the old algebra problems of canoes going upstream and trains crossing in the night. On the contrary, pedagogy and assessment should be directed to making students quantitatively literate—that is, able to use math in their likely adult roles.
What sort of assessments would test their capacity to successfully fill their likely adult roles as producer (worker), consumer, and citizen? One set of questions would see if students could handle budgets: For their future worker role, using a spreadsheet with algebraic formulas, can they develop a budget for a retail store, construction project, manufacturing operation, or personal-services office, such as a dentist's? As a consumer, using pencil and paper and a set of criteria and prices, can they develop a monthly budget for a family of four or put together a budget for a party? As a citizen, can they explain an agency or organization budget for the last five years, demonstrating understanding of the growth or decline of budget components themselves and as shares of the total? Can they relate these changes to other variables, such as inflation and population growth?
Scheduling provides another example. Again, in the worker role and using a spreadsheet with algebraic formulas, can students develop a schedule for a construction project, advertising campaign, conference, medical regime, or software project? The solution should require conversions from hours to workweeks. Students should understand the difference between activities done in sequence and simultaneously and be able to use PERT and Gantt charts.
As consumers, and using pencil and paper without a calculator, can they plan a party or a meal when a solution requires converting hours to minutes? A quantitatively literate citizen, on the other hand, should be able to understand why it takes so long to build a road or school.
Students should be able to use statistics in each of these three roles. For example, as a worker, can they use techniques of statistical process control to monitor a manufacturing process, or patient or customer complaints? As a consumer, can they understand statements about the quality of the products or services purchased? As a citizen, can they understand debates about environmental safeguards or AIDS?
Students should also know how to use mathematical models of systems. Can they, in their worker role, develop an information (or traffic, or other) system flowchart and build a mathematical model to simulate its operation? They should be able to think on their feet, including manipulating numbers mentally to negotiate about quantitative matters. For example, a worker should be able to participate in a labor-management negotiation. Similarly, an educated consumer should be able to understand a construction contractor's or mechanic's proposal and negotiate a fair agreement. An educated citizen should understand government negotiations.
Teachers, test developers, and administrators will have to invest in a substantial effort to make the suggested changes in pedagogy and assessment. Some members of each group will undoubtedly complain loudly about "losing" the power and beauty of mathematics. Unfortunately, most American students never find mathematics powerful, interesting, or useful under current practices.
Arnold Packer chairs the Scans 2000 Center within the Institute for Policy Studies at Johns Hopkins University in Baltimore. The center is conducting the study mentioned in the essay.
Vol. 21, Issue 22, Pages 44, 47Published in Print: February 13, 2002, as The Circumference of a Circle