In 1989 the National Council of Teachers of Mathematics (NCTM) published the landmark volume, Curriculumand Evaluation Standards for School Mathematics. Several companion pieceshave followed, including Principlesand Standards for School Mathematics, a clarification and amplificationof the original work, published in 2000. Collectively, these works are oftenreferred to simply as the Standards. What vision of school mathematics ledto the Standards, and to what extent are we fulfilling that vision?
A Brief History
Concerns about mathematics education and reform movements are certainly nothing new. As long ago as 1894 The Committee of Ten, a group sponsored by the National Education Association and commissioned to explore many educational issues, published its report. This report included a number of recommendations for the teaching of mathematics that, over a hundred years later, sound hauntingly familiar. Willoughby summarized their findings in the following way:
- We should decompartmentalize mathematics so that arithmetic, geometry, algebra, and so on are not taught as isolated subjects.
- We should have more emphasis on realistic problem solving.
- We should have more emphasis on intuition and thinking (9).
One hundred years later, in the 1993 NCTM Yearbook, Diana Lambdin continued to explore the historical unfolding of mathematics education. Consider the following five quotations that she includes in her study, all of which advocate assessment techniques that were recommended in the Standards of 1989:
- In general, observation, discussion, and interview serve better than paper-and-pencil tests in evaluating a pupil's ability to understand the principles and procedures he uses.
- Information is best collected through informal observation of students as they participate in class discussions, attempt to solve problems, and work on various assignments individually or in groups.
- Evaluation of the thinking and procedures employed by students usually is better done by careful observation and interview than by objective testing.
- From the standpoint of the classroom teacher in particular, frequent informal observations of student behavior have a vital role to play in the evaluation process. They neither replace nor are replaced by the more formal observations of student behavior that are made on the basis of tests and the like.
- Observation of the pupil's oral and written work ... [is] a very important testing procedure and should be encouraged. Closely associated with the observation technique is the interview with the pupil regarding his daily work or his solution or attempted solution of items of a test.
All five of these points speak to issues dealt with in the AssessmentStandards for School Mathematics published in 1995, but, in fact, they come from a period of time between 1940 and 1990. Perhaps the use of language in some gives them away, but it is remarkable that even today mathematics educators continue to deal with the same issues. The quotes are from, respectively, 1946, 1989, 1961, 1970, and 1951 (8).
The fact that the mathematics reform song has had the same chorus and practically the same verses, decade after decade, is indeed discouraging. That is not to say, however, that mathematics education has not changed over the years. The most famous, or infamous, change was spurred by the launching of Sputnik in 1957 and the perceived need for the United States to catapult our educational system to the forefront. In the 1960s there was considerable belief in the "reasoned" man and the notion that intellectualism could lead society toward the solution of its educational, economic, and societal problems. Thus, we had the pedagogical philosophy which has given the term "new math" a bad taste in the mouths of the better part of a generation: Teach the structure of mathematics and all else will fall into place.
By the late '60s, societal and political problems loomed large, and the ideal that intellectualism might be the answer crashed and burned. In society, the notion of accountability emerged. In education, the buzzword became "relevant." And so in the 1970s we saw the proliferation of competency-based educational programs that were mired in the framework of behavioral objectives. We were in an era of anti-intellectualism with the educational mantra of "back to basics."
As is usually the case, the pendulum kept swinging until it had swung too far. In 1983, then U.S. Secretary of Education Dr. Terel Bell released A Nation at Risk. That dramatic report captured public attention when it declared, "The rising tide of educational mediocrity...threatens our very future as a Nation and as a people" (National Commission on Excellence in Education 1). Massive media coverage triggered a national "wake-up call," resulting in hundreds of school reform studies during the following decade. The report challenged the United States to strengthen requirements, set rigorous standards, and extend higher expectations for all students. At a time when the Information Society was developing and new societal goals of mathematically literate workers and opportunity for all were emerging, we were reading report after report that our educational system was failing us.
Out of this climate of dissatisfaction with the present and fear for the future came the Curriculumand Evaluation Standards for School Mathematics in 1989. Soon afterwardscame the ProfessionalStandards and the Assessment Standards. To capture the general thrustof the Standards, consider the following general goals for studentsthat are articulated in that document.
New Goals for Students
- that they learn to value mathematics
- that they become confident in their ability to do mathematics
- that they become mathematical problem solvers
- that they learn to communicate mathematically
- that they learn to reason mathematically (6)
This was a groundbreaking document as curriculum reform goes. It was developed with the broad-based support of the educational community as well as advice and suggestions from major corporations. It has served as the model for reform in other subject areas as well. Most notably, whereas the life-span of most reform movements is at most ten years, the Standards is still at the forefront of discussion about the content and pedagogy of school mathematics. Although the document of 1989 has been revised, it is a revision mostly in form, not in content or direction. In other words, the education community still overwhelmingly feels that we are moving in the right direction.
That is not to say that everyone is enamored with the Standards. The dissenting voices have been many and loud. Just as proponents of the Standards cite the broad-based support for the reform movement as a positive force for furthering their ideals, opponents enjoy broad-based support as well. Much of the dissatisfaction with the Standards stems from the discrepancy between what the Standards say and how they are being put into practice. For example, many opponents decry the fact that the basic facts of arithmetic are no longer stressed. In fact, the K-4 standards call for a decreased attention to "isolated treatment of paper-and-pencil computations" but increased attention to both "mental computation" and "thinking strategies for basic facts." Such misinterpretations in the public and even among some in the education community were among the driving forces that resulted in the revision released in 2000, Principles and Standards for School Mathematics.
The Mandate for Staying the Course
Despite the opposition to the Standards that has arisen here and there,it remains the watchword for mathematics reform. What is it about this currentreform movement that has given it its staying power? Three factors can be citedto suggest that this time it does not make sense for the pendulum to swing backto where we were before.
First of all, the technology that is available to our students is not going to go away, nor should it. The availability of graphing calculators has allowed more math to be accessible to more students by providing a mode of visualization that enriches their understanding. The increasingly easy accessibility of calculators with built-in computer algebra systems is certain to have a profound effect on the curriculum. What and how do we teach, and what and how do we assess when our students have in their hands relatively inexpensive machines that can perform in closed-form much of the content of the traditional mathematics curriculum? We cannot afford to ignore these questions that arise from the use of this technology.
Secondly, we live in an increasingly technological world where an increasing proportion of jobs are skilled ones. We need to be graduating more and more students who can think, analyze, and problem-solve. In 1991, the U.S. Department of Labor released the SCANS report What Work Requires of Schools. The Secretary's Commission on Achieving Necessary Skills spent twelve months talking to business owners, to public employers, to the people who manage employees daily, to union officials, and to workers on the line and at their desks. The message was uniform across the country and across job types: "Good jobs depend on people who can put knowledge to work. New workers must be creative and responsible problem solvers and have the skills and attitudes on which employers can build. Traditional jobs are changing and new jobs are created every day. High paying but unskilled jobs are disappearing. Employers and employees share the belief that all workplaces must 'work smarter' " (United States Department of Labor).
Thirdly, we are leaving a legacy of gargantuan societal problems to be solvedby the youth we are teaching. In his speech to the European Council of IndependentSchools in The Hague in 2001, John Abbott presented a frightening list of problemsthat must be solved:
- World over-population. There are two and a half times as many people on the planet now as the day he was born.
- Depletion of resources, and environmental collapse.
- Global warming. Scientists do not deny it, yet politicians won't touch it.
- The ethical problems inherent in human cloning.
- AIDS in Namibia, where, "for every teacher they'll need in ten years time, they have to employ three now because two out of the three will be dead of AIDS in ten years."
- International terrorism (8).
None of these problems has an easy solution, and they all need to be solved urgently. We need to be training students with all the possible tools at their disposal to grapple with them.
The Good News
In the fifteen years since NCTM issued its challenge to all of mathematics education, many new programs have been designed, generally purporting to advance the goals set forth in the Standards . A search of the "Mathematics Education Research and Reform: Curriculum and Materials Development" page of the Math Forum Internet Mathematics Library revealed 184 links to curricular programs and projects in the spirit of the reform movement. Included among them is the following bewildering alphabet soup of options: IMP, CPM, CSMP, CPMP, COMPASS, MegaMath, Everyday Math, Math Thematics, Math Land, Math Connections, Mathematics in Context, Middle School Math Curriculum, and Middle School Math Through Applications.
Yet, many schools have navigated the morass of acronyms and similar sounding names to adopt reform curricula. There are programs at all levels of school mathematics to choose from, and some are catching hold. To help schools identify those programs that are most promising, the U.S. Department of Education's Math & Science Education Expert Panel issued a report in 1999 of Exemplary and Promising Mathematics Programs. The panel identified five exemplary programs: Cognitive Tutor TM Algebra, College Preparatory Mathematics (CPM), Connected Mathematics, Core-Plus Mathematics Project, and Interactive Mathematics Program (IMP). They also have designated five other programs as promising: Everyday Mathematics, MathLand, Middle-School Mathematics through Applications Project (MMAP), Number Power, and The University of Chicago School Mathematics Project (UCSMP).
In the 2000 National Survey of Science and MathematicsEducation, more than half of high school mathematics teachers around thecountry stated that they were fairly familiar with the NCTM Standards.Roughly 75 percent of those teachers familiar with them agreed with the visionof the Standards and indicated that they were implementing them in their classrooms(Whittington 9).
The Challenges to Overcome
On the other hand, in his report on the Status of High SchoolMathematics Teaching, based on that same National Survey of 2000, DawayneWhittington notes that:
"High school mathematics teachers reported placing heavy emphasis on mathematics concepts and reasoning. However, the typical high school mathematics class spends the highest percentages of time solving worksheet or textbook problems, reviewing homework and worksheet assignments, and practicing routine computation and algorithms, which do not seem to strongly support the development of mathematics concepts and reasoning. In addition, instructional activities that focus on strengthening students' ability to effectively communicate mathematical ideas or carry out investigations are used infrequently." (26)
These survey results are validated by the video study of classroom teaching completed as a part of the Third International Mathematics and Science Study (TIMSS). In 78% of the topics covered during the U.S. lessons, teachers stated procedures and ideas but did not develop them (Stigler & Hiebert, 61). To what extent are the goals of the NCTM Standards really taking hold? What seem to be the barriers to progress?
First and foremost, under the best of circumstances, change is difficult. In TheHuman Side of School Change, Robert Evans explains that change results inpersonal loss, challenges competence, creates confusion, and causes conflict(28-37). As we know, it seems that change never does come "under the bestof circumstances." Tom Bird, reflecting on the factors that affect anyschool change project, observes: "At any time, about half of the personsneeded to pursue a solution are getting married or divorced; tending a sick orwell relative; going bankrupt or coming into money; just starting, getting readyto leave, or near retirement; taking care of babies or putting children throughcollege; making up or breaking up; getting sick, getting well; getting chronicor dying" (45). As a result, it is important that those involved in orchestratingchange acknowledge the complex feelings that accompany transitions and that theygive people the time and space to adapt.
For meaningful change in pedagogy and content to occur, teachers need significant hours of professional development. In the vast majority of schools, these hours have not been made available. For example, a survey of professional development opportunities in California, the site of a great deal of reform activity, reported that fewer than 5% of elementary school teachers participated in more than two weeks of special in-service education during 1998, the year of the survey (Cohen & Hill 4).
The lack of learning opportunities for teachers is especially striking when compared with some other occupations. General Motors, for example, provides 92 hours per year for continuing education for every Saturn employee to improve design and productions skills (Hiebert, 19). In the Cohen & Hill survey, only 4% of California elementary school teachers had this much time for improving their craft (Hiebert, 19).
Contrast the typical professional development situation in schools in this country with kounaikenshuu, the process of continuous school-based professional development engaged in by Japanese teachers. Teachers meet in groups regularly and over long periods of time on a variety of activities that constitute a comprehensive program of school improvement with set goals and focus areas for each year's efforts (Stigler & Hiebert, 110). Commitment at this level to improving the art of teaching is seldom the purpose of the professional development experiences of most American teachers.
Another challenge is the quality of resources often available to teachers. Many mathematics textbooks, even though they may purport to support the Standards, are woefully inadequate for promoting conceptual understanding and real problem solving. We currently live in an "MTV world."?Open almost any Algebra II textbook to almost any page and your eye will be bombarded with pictures and graphics, very little of which really adds to the understanding of the content. (By the way, this content is apt to be what was traditionally Algebra I content.)?We need textbooks that promote "NPR" learning and thinking skills, not "MTV" ones.
Larry Suter, as deputy director of the division of research, evaluation, and communication at the National Science Foundation, described the United States math curriculum with the oft-quoted phrase "a mile wide and an inch deep" (Checkley, 1). The challenge of disowning this indictment is a difficult one. The Standards calls for new topics, more activities to build understanding, more reflection and discussion, and all of this takes more time. At the same time, over the past several decades, the American curriculum has expanded, eroding our available class time. We cannot provide richer learning experiences for our students with greater depth of coverage while maintaining the current breadth of topics. Yet every topic in the curriculum has a horde of staunch advocates with really good reasons why it would be a travesty to drop that topic from the curriculum. Resolving the dilemma of breadth versus depth will require frank conversations and difficult choices, but we must resolve the dilemma.
The increased emphasis on standardized testing and the high stakes associated with so many of those tests have left many teachers feeling hamstrung. In many classrooms an inordinate amount of time is devoted to preparing students for tests. In some cases these tests are actually good tests, at least of the material they are testing.?But conceptual understanding and problem solving ability cannot be tested in a multiple choice format, and it is an unfortunate fact of educational life that that which is not tested is often not valued. Courageous teachers need to set the example and prove that "teaching to understanding" can successfully replace "teaching to the test" without jeopardizing test scores. In fact, if done correctly, it is almost certain to enhance those scores.
Math as described in the Standards is harder to teach at a time when good teachers are harder to find and harder still to keep. While student enrollments are rising rapidly, more than a million veteran teachers are nearing retirement. Experts predict that overall we will need more than 2 million new teachers in the next decade. To meet the increasing challenge, schools are turning to more and more innovative strategies, from signing bonuses to down payments for housing.
Any adoption of a reform curriculum must be accompanied by a proactive parent education campaign. When parents don't understand our purposes or our curriculum well enough to help their third grader with math, it is no wonder that they become uncomfortable. When they encounter contradicting viewpoints about what constitutes good math education, they are understandably confused. There is much safety in what is familiar to them, in the way it was done when they were in school, even if the painful experiences sometimes associated with it have been, like the pains of childbirth, forgotten over the years. Schools undergoing change need to do a better job of promoting their cause, in explaining their rationale, and in publicizing their successes.
The road to full implementation of the dream set forth by NCTM in the Standards has proven to be long and arduous. Many bumps and turns have been encountered along the way. The progress has been slow and not always steady. Much of the road still stretches out before us. But the journey is a worthwhile one, even a necessary one. We need to stay the course.
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