The past ten years have seen a remarkable amount of progress in
improving mathematics education at all levels. The goal is to enable
all students, including those from all racial and ethnic backgrounds
and both genders, to master and appreciate mathematics. The emphasis
is on understanding mathematics rather than thoughtlessly grinding out
answers. For various reasons, there is now an increasing amount of
resistance to what is usually called "math reform," which reflects
some serious concerns that need to be addressed.
Pre-college math reform, based in large part on the NCTM Standards, and college math reform, usually labeled "calculus reform," are compatible in their goals and are now facing similar resistance. I believe that we are all in this together and that we need to work together to maintain momentum and establish better mathematics education for all. Parents, teachers, and the general public need to realize that the new approaches make sense and will empower the young people for the next century.
Unfortunately in the past, much of mathematics has been presented as a bunch of rules - rules for manipulating numbers and symbols. Underlying principles, general problem solving techniques, and serious quantitative thinking got lost. Certainly much of the interest, beauty, and fun vanished. A major thrust of the current reform movements is to present mathematics in a much broader context. It encompasses ideas and techniques that aren't even seen in traditional treatments of mathematics, and they are interconnected. Mathematics isn't just a sequence of isolated topics that are to be struggled with, learned (or not), and forgotten.
At the college level the emphasis has been on "calculus reform." An excellent overview of calculus reform can be obtained by reading the articles in the January 1995 issue of UME Trends. As a starter I especially recommend Alan Schoenfeld's article titled "A Brief Biography of Calculus Reform." A more formal and in-depth report can be found in the just published MAA report Assessing Calculus Reform Efforts, edited by J. R. C. Leitzel and Alan Tucker. This is a very readable and interesting account of the history and current status of calculus reform. Where there's hard data, these reform efforts have been largely successful.
Many people within the MAA and other mathematical organizations are working hard to improve teacher education programs, develop new curricula, and help collegiate mathematicians get involved in the schools.
The term "calculus reform" is misleadingly restrictive, because the changes at the post-secondary level extend far beyond calculus. A glance through the programs of the past few national mathematics meetings, especially the minicourses and sessions of contributed papers, shows that there are parallel changes in the way abstract algebra, linear algebra, differential equations, and precalculus are taught. More specialized courses - such as dynamical systems, Fourier series, and modeling - are also being taught in new, exciting ways that involve taking advantage of the new technology.
The current push for calculus reform got its jumpstart from the now famous Tulane Conference in January 1986. During the same period the NCTM Standards were being created. They were published in 1989 and have been very widely accepted and used. The current political and sociological climate has led to some backlash. The NCTM is aware of this serious threat and has appointed a task force that will seek appropriate responses. The MAA representative on this task force is Naomi Fisher; her email address is u37158@ uicvm.uic.edu.
The goals of the NCTM Standards, which address mathematics education for K-12, are to "Create a coherent vision of what it means to be mathematically literate both in a world that relies on calculators and computers...and in a world where mathematics is rapidly growing and is extensively being applied in diverse fields," and "Create a set of standards to guide the revision of the school mathematics curriculum and its associated evaluation toward this vision." The vision calls for changes in the curriculum, including new content such as probability, statistics, and discrete mathematics, as well as for different approaches to some of the topics in the existing curriculum. It is envisioned that students will (1) learn to value mathematics; (2) become confident in their own ability; (3) become mathematical problem solvers; (4) learn to communicate mathematically; and (5) learn to reason mathematically. Each of these goals is elaborated on. For example, (3) states that "students need to work on problems that may take hours, days, and even weeks to solve . . . some may be relatively simple...others should involve small groups or an entire class working cooperatively. Some problems also should be open-ended with no right answer...."
The most vivid changes in teaching have involved technology, but the real focus has been to improve the learning of students and to make sure that a wider group of students is able to benefit than has in the past. Changes in instructional practice include hands-on experiences using technology, increased focus on conceptual understanding, cooperative learning, student project activity, extensive writing, and less reliance on timed tests in assessment.
It's easy to detect flaws in any movement as broad as the reform movement and to overlook the progress. In February I attended an NSF/DOE conference on systemic reform in science and mathematics titled "Joining Forces: Spreading Successful Strategies." It became clear at this conference that a large number of people across the country provide excellent education in various creative ways. The focus of the conference, as its title suggests, was the daunting but vital task of identifying those programs that really can be duplicated throughout the country, without losing their effectiveness, and then implementing them nationwide.
Statistics from the Department of Education (the Condition of Education, 1994) show that we are making progress. For example, substantially more high school graduates in 1992 are taking mathematics courses at the level of algebra I or higher than their counterparts in 1982. Thus in 1992, 56.1% of the high school graduates took algebra II and 7O.4% took geometry, while only 36.9% and 48.4%, respectively, took these courses in 1982. During the same period, the percentage taking remedial or below-grade-level math dropped from 32.5% to 1 7.4%.Another table shows that these dramatic shifts are happening for all racial/ethnic groups. We don't hear much about such statistics hidden in dusty government tomes, even when they are positive! With such big changes nation-wide in ten years, something right must be happening.
So what are the concerns that are leading to resistance to these changes? One is that the laudable focus on understanding has led to some decline in mathematical skills. Since it is easier to measure and spot deficiencies in skills than understanding, this problem can easily be over-emphasized. on the other hand, this is a serious problem, especially since our future scientists, engineers, and mathematicians must obtain both substantial understanding and substantial skills. The reform movements need to address this issue.
For teachers who are following the NCTM Standards, there's no doubt that it is more difficult to determine (or at least quantify) students' knowledge, understanding, and skills. This is now leading to serious challenges as the mathematics community faces assessment issues. Similar challenges are faced by post-secondary faculty as they change their instructional practices. I have no wisdom here except to acknowledge the difficulties tempered with the belief that they can be overcome, though it won't be easy. To steal a quote, "Tests should measure what's worth learning, not just what's easy to measure."
Another concern is largely political. There is a natural American resistance to centralized control. Some fear that the NCTM Standards are subverting local control. Wide-spread reform is hard to accomplish in such an atmosphere. We are still suffering from the bad taste that so-called "New Math" left in America's mouth thirty years ago. That was an effort that focused entirely on the curriculum. An equally serious problem all along has been in the pedagogy. Finally, in the United States, we are suffering from a widespread case of anti-intellectualism wherein all of us are experts on the schools because we once attended schools. We need to continue to learn from the successes and failures of other countries where many of the same problems are being faced.
In the long term, the college and pre-college reform efforts are intimately linked. Students and parents expect pre-college mathematics education to be a preparation for college-level mathematics. This is an important endeavor and everyone needs to be involved . A future column will discuss the role of post-secondary faculty and the MAA in addressing this interface. Students' mathematics learning should be seen as seamless as they progress K-16. We've come a long way in ten years. We have a long way to go.
Home || The Math Library || Quick Reference || Search || Help