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Every Teacher Is a Teacher of MATHEMATICS

January 14, 2007

By Steen, Lynn Arthur

Even under the best of circumstances, teaching mathematics is a daunting challenge. But in today’s society, it is increasingly important. Basic numeracy is one of the few essential skills that students absolutely must master, both for their own good and for the benefit of the nation’s democracy and economic well-being. Abstract mathematical ideas that took centuries to develop are now routinely employed as powerful tools in all walks of life.

But today’s circumstances are far from the best. Many adults- most of whom are parents-proudly proclaim their ignorance of things mathematical. Phobias about mathematics create avoidance behaviors, which children pick up from their parents, friends, and even some teachers. The many students who have been conditioned to disdain, fear, and avoid quantitative reasoning confront mathematics teachers, especially at the secondary level, with the dual challenge of motivating as well as teaching. As difficult as it is to teach mathematics well, overcoming society’s quarantine of quantitative thinking is a far greater challenge.

For lasting learning to take place, mathematics must be seen by students as both meaningful and useful. The traditional curriculum serves this goal reasonably well for the modest number of students who anticipate pursuing scientific or technical careers. In this approach, mathematics is taught as a sequence of separate subjects to be learned first and applied later. Because the topics that dominate school mathematics are prerequisites for college courses in science and engineering, most students with these aspirations can imagine a path connecting the mathematics they are studying with the careers they hope to pursue.

But the majority of students cannot foresee such a path. For them, mathematics is learned better when it arises as a natural tool for understanding issues or solving problems that are relevant to their other interests. This is the essence of numeracy: using quantitative methods in context. The essence of mathematics, in contrast, is independent of external contexts. Whereas numeracy is concrete and tangible, mathematics is abstract and Platonic. Both are important, and both employ the same core elements of numbers and ratios, circles and triangles, deduction and inference, percentages and graphs, variables and equations. Yet their architecture and ambiance are very different. To serve all students well, schools must offer both mathematics and numeracy.

Upstairs, Downstairs

The need to understand and be able to use mathematics in daily life and work has never been greater. I would argue that the mathematics underlying elections, polls, surveys, consumer finance, home construction, and clinical trials is at least as important, if not more so, for students’ futures as most of what now constitutes the standard curriculum of high school mathematics-algebra I and II, geometry, and trigonometry. This is not to say that the latter are unimportant, only that for most students they are less important. The survival of the nation’s democracy depends more on its citizens’ ability to understand the implications of different voting systems than on their ability to factor polynomials or solve quadratic equations (Steen, 2001).

Unfortunately, numeracy is often characterized as watered-down mathematics-a minor league curriculum that schools offer to those who are unable to compete in the major league of algebra, trigonometry, and calculus. This view of the relation between mathematics and numeracy is a profound misunderstanding-some might call it a mathematicians’ conceit-that is based on several plausible but misleading assumptions:

* That abstract mathematics is more profound than contextual mathematics, thus on a higher intellectual plane

* That the mathematics curriculum designed in the 1950s for physics and engineering is still the best curriculum for the 21st century

* That the importance of mathematics reflects the caste system of science, so mathematics used in the physical sciences (calculus) stands above that of the life sciences (combinatorics), with social sciences (statistics) at the bottom

* That numeracy problems are much easier than those encountered in traditional mathematics, thus less rigorous and less deserving of high marks.

Because of this “upstairs/downstairs” view of mathematics, most high schools offer two mathematics tracks that one might loosely call “mathematics” and “numeracy? (Usually they are called something like “academic” and “general”-or “consumer” -mathematics.) Strong students take a rigorous calculus-prep program of traditional mathematics, and weak students take a potpourri of vocational, commercial, and remedial courses designed more to fill time than actually teach mathematics.

Beginning in 1989, when the National Council of Teachers of Mathematics (NCTM) published the first national standards for school mathematics and continuing through the development of state standards and the No Child Left Behind Act, schools have been under considerable pressure to eliminate the downstairs track The argument is rather straightforward, albeit not airtight: Because most good jobs require some kind of postsecondary education, every student should leave high school prepared for college mathematics. Given the nature of college mathematics courses, only the upstairs track- traditional algebra and geometry-can meet that standard.

Those who urge that all students clear the traditional hurdles of algebra, geometry, and trigonometry correctly assert that calculus, the gateway to higher mathematics, requires proficiency in the skills taught in these courses. Too many students-about one in three- enter college without proficiency in these subjects; these students are required to take noncredit courses to remove this deficiency, thus delaying their education and adding to its cost.

Notwithstanding the simple logic of expecting all students to just do what has worked well for a minority, evidence suggests that increasing the number of students taking more advanced traditional mathematics courses may not yield the desired result of greater proficiency. For example, the proportion of students taking algebra II has more than doubled in the last 10-15 years, yet scores on the 12th-grade NAEP have hardly budged during that same period. Neither has the proportion of remedial mathematics in colleges declined. Only one in five students who enter ninth grade ever successfully completes calculus, the ostensible goal of high school mathematics. So although it might be theoretically desirable for every high school graduate to be proficient in algebra and trigonometry, it is not at all clear that today’s strategy of requiring all students to take these mathematics courses will lead to that result.

Why doesn’t more teaching yield more learning? Because the mathematics in these courses reinforces the current and future career interests of only some students-typically about a third. Most of the other two-thirds have grown up in home and community contexts in which mathematics skills are not something that people aspire to attain. If high schools are to shift from the old model of mathematics for the few to the new mantra of mathematics for all (“no” child left behind), they must offer mathematics of sufficient variety to appeal to all students. That’s where numeracy enters the picture.

Numeracy

Several factors have contributed to the increased importance of numeracy in the modern world. One is the trend, roughly a century old, for decisionmakers to expect numerical evidence in support of policy proposals. Public enterprises in the mid20th century-such as building dams and highways, supplying troops in World War II, and mass vaccination campaigns-led to extensive analysis of benefits versus costs. Another factor, one that is more recent, is the unprecedented capability of computers to store, organize, and analyze vast quantities of data that can then be brought to bear on decisions in government, business, and everyday life. Because analysis and argument in so many areas of life now make use of numerical data, it is no longer possible to be literate without also being numerate.

As alert citizens, high school graduates should understand:

* How different voting procedures (e.g., runoff, approval, plurality, preferential, instant runoff) can influence the result of elections

* How small samples can accurately predict public opinion, as well as how sampling bias can distort results

* How the way data is summarized may make perfectly fair processes appear biased, and vice versa

* How unusual events, such as cancer clusters, can easily occur by chance alone

* How proposals for school district budgets and tax law changes will work in practice.

As careful consumers, graduates should be able to:

* Compare credit card, mortgage, and automobile lease offers that have different interest rates and different durations

* Estimate the long-term costs of making very low monthly credit card payments

* Interpret conflicting reports of medical studies

* Understand the investment benefits of diversification and income averaging

* Understand terms and conditions of different health insurance policies

* Understand tax implications of financial decisions.

At work, people should be able to:

* Review budgets and identify relevant trends

* Determine the break-even point for development and sale of a new service or product

* Deve\lop a schedule to improve work flow on a complicated project

* Interpret and prepare graphs that illustrate relations among different factors

* Maintain and use quality control charts.

I call attention to three implications of these common examples of numeracy: First, numeracy in society is pervasive. second, numeracy in schools is AWOL. Virtually none of the quantitative tools that are central to the previous examples are central parts of the traditional school curriculum (although some have been added as “if there is time” supplements to traditional courses).

Third, virtually all of the contexts for these examples arise more naturally in subjects other than mathematics. This helps explain why they are generally absent from mathematics courses, but it also highlights the anomaly of their absence from other courses. As long as teachers sidestep quantitative methods in their own classes and leave such tools for mathematics teachers to cover in mathematics class, all the problems of poor test scores and excessive remediation in mathematics will continue. Not that many students will ever be motivated to master something that appears to live only in “math class.”

By its very nature, numeracy is contextual mathematics. Although mathematics teachers can (and do) offer students contexts for the mathematics they study, it is too much to expect mathematics teachers to know all the relevant contexts. But it is not too much to expect teachers of other subjects to employ quantitative tools in their own areas whenever they are relevant to give students a chance to reinforce skills learned in mathematics class. If the real-world contexts that mathematics teachers talk about are invisible in other courses, students will very soon suspect that they are caught up in a con game that they cannot win.

Fortunately, the rapid expansion of quantitative tools into all sectors of society opens up innumerable opportunities for teachers of all subjects to help students recognize and develop their expertise in quantitative methods. Like reading and writing, the third R belongs in every subject. The analogy with writing is especially apt. When only English teachers focused on writing, students had no incentive to express ideas clearly in other subjects. So it is with numeracy: if it is only in mathematics classes that numerical arguments and logical deductions are emphasized, the widespread culture of “mathophobia” is reinforced by the very educational system that is supposed to overcome it Teachers teach mathematics both when they use it and when they avoid it If educators expect all students to value quantitative methods, all teachers need to demonstrate its value.

PREVIEW

The false dichotomy between mathematics and numeracy must be eliminated.

Many mathematical problems arise naturally in other disciplines and should be addressed in context.

All teachers must become comfortable using quantitative methods in their classrooms to achieve mathematics literacy for all.

Numeracy in Careers

This sidebar illustrates the extensive spread of quantitative tools in the workplace. At best, these examples suggest but do not fully describe the landscape of numeracy in relation to contemporary lives. They show, however, why numeracy is best learned “across the curriculum.”

Architects use geometry and computer graphics for design, statistics and probability to model usage, and calculus to understand engineering principles.

Cinematographers use computer graphics to enhance scenes, as well as to create digital images.

Doctors need the ability to understand statistical evidence and to explain risks with sufficient clarity to develop “informed consent.”

Emergency medical personnel need to interpret dynamic graphs from cardiac monitors to distinguish significant irregularities from regular patterns and common pathologies.

Farmers use satellite imaging data, soil samples, geographic information systems, and algorithms for geometric projections to create terrain maps that reflect soil chemistry and moisture levels.

Financial advisers use mathematical models to balance risk and return and to determine the value of intangible assets.

Insurance agents need to understand and be able to explain to their customers the complicated conditions under which policies pay out and how such payments are calculated.

Journalists need a sophisticated understanding of risks, rates, samples, surveys, and statistical evidence to develop an informed and skeptical understanding of claims and studies.

Lawyers rely on careful logic to build their cases and on subtle arguments about probability to establish or refute reasonable doubt.

Manufacturing technicians use statistical process control to analyze production data to detect patterns that might signal an impending reduction in quality before it actually happens.

Musicians use symmetry and ratios to transform compositions, as well as digital codes to compose and record.

Nurses need to understand how to calculate dosages of medicines to match available medicines with prescribed levels.

Politicians mine census and other data from government agencies to forecast the needs of society and the costs of proposed policies.

Social workers need to understand complex regulations about income and expenses to explain and verify their clients’ personal budgets.

Teachers use computer spreadsheets to organize assessment data into forms suitable for gauging student learning and diagnosing strengths and weaknesses.

Reference

* Steen, L. A. (Ed.). (2001). Mathematics and democracy: The case for quantitative literacy. Princeton, N.}.: National Council on Education and the Disciplines, the Woodrow Wilson National Fellowship Foundation. Retrieved from www.maa.org/ql/ mathanddemocracy.html

Lynn Arthur Stem is a professor of mathematics and the special assistant to the provost at St. Olaf College in Northfield, MN.

Copyright National Association of Secondary School Principals Jan 2007

(c) 2007 Principal Leadership; High School ed.. Provided by ProQuest Information and Learning. All rights Reserved.




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