A 'New Agenda' for Mathematics Education
Education, it seems, has become the rhetorical sinecure of political policy: Conservatives support liberals and Republicans echo Democrats in identifying school reform as a major national priority.
Much of the concern with education stems from worries about international competitiveness. In a world of multinational corporations and international markets, inventiveness in science and technology--rather than abundance of labor or natural resources--is the key asset available to American industries.
Our leadership in these areas, however, is rapidly diminishing. As a nation, we are not renewing our intellectual capital.
A key predictor of scientific competitiveness is quality of education in mathematics. But recent international studies--such as the aptly titled report, The Underachieving Curriculum, based on the Second International Assessment of Mathematics Education--show that the current level of mathematics achievement in American schools lags significantly behind that of other industrialized countries. Without improvement, the United States will not be able to sustain into the next century its present position of leadership.
Moreover, the impact of computing compels re-examination of the objectives for school mathematics. Not only has computing changed the way math is used--thus altering the balance between essential and peripheral topics in math education--but even more important, it has mathematicized science and technology.
No longer is it sufficient for theoretical scientists alone to have a working knowledge of mathematics; now all scientists--indeed, virtually all professionals--encounter mathematical models in everything they do. Our schools, however, have yet to recognize the profound change that this development implies for the world which their graduates will enter: Because of the ubiquity of computers in the workplace, students need more rather than less mathematics.
In the context of recent proposals for reforming public education, scientists, mathematicians, and educators have been working to define new goals for school mathematics. Due to appear this year or next are national reports sponsored by the National Council of Teachers of Mathematics, the Mathematical Sciences Education Board of the National Research Council, and the American Association for the Advancement of Science. Although these undertakings differ greatly in detail and purpose, they generally agree that schools must increase students' involvement in learning by fostering a broad view of the mathematical sciences, by increasing use of computers, by linking math to science, and by stimulating student-led projects.
Since it takes a generation to complete the mathematical education of a single individual, we must move quickly to make the necessary improvements.
As we enter a major election season, candidates at every level will tell us that they believe in education. Let them back their words with platforms on mathematics education appropriate for the 21st century.
Proposals for change must address five areas: expectations for student achievement, testing, attitudes and environments for learning, approaches to teaching, and teacher preparation.
Relief in local control of education--and of expectations for students--is deeply embedded in the American body politic. Yet for mathematics education, this independence is largely a myth.
Effective control is exercised not by Washington bureaucrats, but by invisible state committees that approve textbooks and anonymous officials who administer standardized tests. Though few conclusions stand undisputed in educational research, one rule is rarely challenged: Teachers teach only what is in the textbook, and students learn only what will be on the test.
In fact if not in law, we have a national course of study in math. It is an "underachieving'' curriculum that follows a spiral of constant radius, each year reviewing so much of the past little new learning takes place.
In states that have recently raised their expectations, the consequences of change have often been surprising and revealing. The establishment of new standards in California, for example, led to initial rejection of all textbook series submitted for adoption, principally because the texts failed adequately to develop student capabilities to address complex, unpredictable problems.
To focus attention on the national need for high standards, the N.C.T.M. is preparing a report that details important new expectations for school mathematics, both in content (by grade level) and in instructional practice. According to these standards--intended as statements of what all children should learn--students should gain confidence in their ability to communicate and reason about mathematics: They should become mathematical problem solvers. They should learn not only arithmetic, but estimation, measurement, geometry, statistics, and probability--all the ways in which mathematics occurs in everyday life.
According to the NCTM's guidelines, for example, children in elementary school should not simply develop reasonable proficiency in arithmetic, but also learn to apply estimation techniques, to use calculators, and to explore alternative strategies for solving problems.
And--because statistics permeate the society in which we live--statistical ideas permeate the standards from elementary grades ("use data to make predictions; experiment with concepts of chance'') to high school ("design a statistical experiment and communicate the outcome'').
We must judge schools not by remembrances of things past but by expectations for the future. Policymakers must insist that guidelines such as the N.C.T.M.'s become the standards by which districts evaluate the mathematics education they provide.
To raise expectations and evaluate programs, governors and legislators in all 50 states are talking about assessment.
Unfortunately, testing is rarely practiced properly in this discipline. Tests designed for diagnostic purposes are used for evaluation of programs, while scores from self-selected populations--for example, those who take the Scholastic Aptitude Test--are used to compare districts and states. And commonly used achievement tests stress simple skills rather than sophisticated tasks--not because such skills are more important, but because they are easier to measure.
As we need standards for curriculum, so we need standards for assessment. If we want students to investigate and discover, then testing must not measure mimicry mathematics alone. Rather than merely validate computation with multiple-choice, right-or-wrong questions, we must seek ways to assess such long-range mathematical goals as communication, flexibility, imagination, persistence, and skepticism.
Assessment should be an integral part of teaching--a means by which teachers learn not only what students are able to accomplish, but also how students think about mathematics.
Many will argue that if tests change rapidly, new data will be incompatible with past records. By confusing means and ends, by making testing more important than learning, this view holds today's students hostage to yesterday's mistakes; it is an argument for merely preserving the status quo, no matter how outdated it is.
Because of political concern about progress in education, the Council of Chief State School Officers has recommended to the Educational Testing Service a structure for the next national assessment of school mathematics that will make possible state-by-state comparisons. In the past, such assessments have emphasized computational-mimicry aspects of the curriculum, at the expense of such holistic expectations as the student's ability to formulate problems, invent strategies, develop lines of argument, and communicate reasons. It is as if we were to judge writing solely on spelling and grammar, without paying any attention to what the student was trying to say.
Assessment must be altered to match the entire breadth of curricular objectives. If tests do not change, nothing will change. Conversely, there is no more rapid way to improve mathematics education than to match standards for assessment with new standards for instruction.
In the classroom itself, new attitudes and environments for learning must be cultivated. Despite mountains of daily homework, mathematics remains, for most students and teachers, primarily a passive activity: Teachers prescribe, students transcribe.
Yet educational research demonstrates that students learn math well only when they construct their own understanding. To learn math, they must act out the verbs in the new curriculum standards: "examine,'' "represent,'' "transform,'' "apply,'' "prove,'' and "communicate.''
Teachers, however, almost always present mathematics as an established doctrine--to be learned just as it is taught. This "broadcast'' metaphor for learning suggests to students that math is about right answers rather than clear, creative thinking.
In the early grades, arithmetic becomes the stalking-horse for this authoritarian model of learning. And the attitudes this outlook inculcates dominate students all the way through college. What math teacher isn't plagued with the query, "Can't you just tell me the right answer?''
When instruction matches these expectations, only short-term learning occurs. When instruction diverges from these expectations--as it must whenever math is viewed as a process of realistic problem-solving--students' confidence slips.
Given the freedom to explore mathematics on their own, students will construct strategies that bear little resemblance to the canonical examples presented in standard textbooks. Young people need an environment for learning math that provides generous room for trial and error. In one approach to creating such an environment, the Chicago public-school system has recently decided to give all schoolchildren calculators.
Classes in which students are told how to solve a quadratic equation and then assigned a dozen homework problems to practice the approved method will rarely stimulate as much lasting knowledge as ones in which students encounter such equations in a natural context; explore approaches to solutions including estimation, graphing, computers, and algebra; then compare various approaches and argue about their merits.
In the long run, more important than the mathematical skills themselves--which,without constant use, rapidly fade--is the confidence that one knows how to find and use mathematical tools whenever they become necessary. Only through the process of creating and discovering mathematics can this confidence be built.
To encourage students to explore and construct as they learn, states must establish guidelines for the environment in which learning takes place. Schools should provide teacher aides, appropriate equipment, reasonable class sizes--and then insist that learning be active.
In truth, we know very little about teaching and learning, except that students learn by doing while teachers teach by talking. In some sense, no one can teach mathematics. What we hope is that a good teacher can stimulate a student to learn mathematics.
For this to happen, teachers need experience doing math: They themselves must know how to explore, to test, to estimate, and to prove. They must have confidence that they can contain and redirect conjectures that emerge as students also explore, guess, and prove. Too often, math teachers are afraid that someone will ask a question they can't answer. Insecurity breeds rigidity, the antithesis of mathematical power.
Since teachers teach much as they were taught, their university courses must exemplify the highest standards for instruction. Unfortunately, most of the mathematics that teachers have studied in college has been presented in an authoritarian framework; very few teachers have had the experience of constructing for themselves any of the math that they are asked to teach.
Here, then, is a task for universities: Teach teachers as we would have them teach. All prospective teachers--indeed, all students--should receive their undergraduate mathematics instruction from faculty who are able to lead them through the process of guessing, checking, and proving.
Indeed, sound education of teachers is crucial to revitalizing school mathematics. All prospective teachers need a broad background in the specific areas they will be teaching. And they must also understand the mathematics that their students will encounter as they move from one level to the next.
At the same time, standards for preparation must be appropriate to the grade level at which the teacher will be working. Elementary-school teaching requires familiarity with several subjects and knowledge of young children. Teachers at this level should understand not only arithmetic and numbers but also geometry, data analysis, and probability--and the application of these ideas to science and measurement. At the high-school level, where faculty members generally specialize in one subject, math teachers must know a great deal about advanced topics.
At every level, they should learn mathematics in a manner consistent with the style in which they will be expected to teach it--as a process of constructing and interpreting patterns, of discovering strategies for solving problems, and of exploring the beauty and applications of the discipline.
Too often elementary-school teachers take just one course in mathematics, approaching it with trepidation and leaving it with relief. Such teachers will never inspire children to have confidence in their own abilities to construct mathematics appropriate to their lives.
Indeed, experienced elementary-school teachers often move up to middle grades without ever learning more mathematics. Profiles like these--not universal, but not at all uncommon--compel thoughtful people to question the current system.
The United States is one of few countries in the world that continues to pretend--despite massive evidence to the contrary--that elementary-school teachers are able to teach all subjects equally well. It is time that we educate a cadre of specialists who will be well prepared to teach young children both math and science in an integrated, discovery-based environment.
To create a tradition of such specialists, states must alter certification requirements. And to help prospective teachers gain confidence through their studies, universities must develop new courses and encourage open, constructive methods of instruction in mathematics.
If we are to maintain our position of world leadership, it is not enough simply to call school-reform our highest national priority. We must design and implement a new agenda for mathematics education.
Piecemeal approaches--changing some parts of the system while others remain fixed--will inevitably fail.
But since mathematics education involves millions of people, the system cannot change quickly. Rather, the path to reform must be one of punctuated evolution. Strategies for growth must be orchestrated so that everyone moves in the same direction at the same time. To ensure steady progress, those who lead must maintain a clear vision of long-range objectives and avoid tempting quick fixes or short-term strategies.
Given a sound understanding of the issues, leadership to improve
mathematics education must move beyond academe to the broader arena of
public policy. Unless we act now, America will surely forfeit its
competitive edge in the world economy.
Vol. 7, Issue 33, Page 28Published in Print: May 11, 1988, as A 'New Agenda' for Mathematics Education