Issues and Options in the Math Wars
Phi Delta Kappan, February 1999, pp.444-453.
By  Harold L. Schoen, James T. Fey, Christian R. Hirsch, and Arthur F. Coxford
 
The authors reassess the case for change in mathematics education and examine the objections of critics in light of recent research and evaluation evidence.
FOR MORE than 15 years mathematics education has been at the center of discussion and action aimed at reforming curricula, teaching, and assessment in American schools and universities.  Prodded by a series of critical national advisory reports(1) and by disappointing results from international comparisons of mathematics achievement(2), the National Council of Teachers of Mathematics (NCTM) formulated an agenda for reform in three volumes of professional Standards.(3) Extensive deliberations in the collegiate mathematics community led to proposals for the reform of undergraduate mathematics as well.(4) The National Science Foundation provided funding for mathematics curriculum development projects at all levels and for dozens of large-scale systemic change projects to enhance teacher knowledge and skills and to prepare the way for the implementation of proposed reforms.

But just as the new curricula, teaching methods, and assessment strategies are beginning to be tested in schools and universities across the country and are beginning to show promise of reaching the objectives of reform, critics have challenged the content goals, the pedagogical principles, and the assessment practices that are at the heart of the reform agenda.  What seemed to be an overwhelming national consensus on directions for change in mathematics education is now facing passionate resistance from some dissenting mathematicians, teachers, and other citizens.  Wide dissemination of the criticisms - through reports in the media, through Internet mailings, and through debates in the meetings and journals of mathematics professional societies - has shaken public confidence in the reform process.  Consequently, there seems to be a genuine risk that many schools will reject opportunities for much-needed improvement in mathematics education and will continue with comfortable and conventional, though demonstrably inadequate, curricula, teaching, and testing practices.

The spirited debates about the reform of school and undergraduate mathematics have led some proponents and opponents of change to indulge in such angry rhetoric that the controversy has come to be referred to as the "math wars." In this emotionally charged atmosphere, it is very difficult to make informed and balanced judgments about the key issues.  It is easy to lose sight of the reasons that major reforms were called for in the first place.  It is easy to forget the rationale and supporting evidence for dominant reform proposals and to overlook the evidence that recent reform initiatives improve the effectiveness of school and university mathematics for most students.  Since the critics have gotten most of the attention in recent public discourse about school mathematics, it seems appropriate to review the situation from a balanced perspective to reassess the case for change and the objections of critics in light of recent research and evaluation evidence.

The Reform Consensus K-12

When the NCTM formulated its Standards for curriculum and evaluation, teaching and teacher education, and assessment of students and programs, there was a broad consensus that major change was needed in each aspect of school mathematics, and there was agreement on the specific reforms that should be carried out.  That consensus was shaped by insight from the study of practices in other countries with more effective mathematics education; by broad consultation with people who use mathematics in the workplace; by the results of recent research on teaching and learning; by analyses of prospects for new technologies in teaching, learning, and doing mathematics; and by the experience-based wisdom of practice of many outstanding teachers.  The following analysis of reform proposals focuses in turn on issues of mathematical content, teaching strategy, and testing practice.  However, it is important to keep in mind that those issues are often interrelated and that effective reform must take a systemic perspective.

Content and process goals of the curriculum.  The U.S. does not have an official national curriculum in mathematics at any grade level.  However, strong informal traditions give remarkable consistency to the topics and organization of curricula from kindergarten through the end of high school.  The elementary grades (K-8) emphasize arithmetic - especially the algorithms for operations on whole numbers, common fractions, and decimals - with modest attention to topics in geometry, measurement and descriptive statistics.  The high school curriculum for most students is shaped by the goal of preparing large numbers of students for collegiate study.  It includes at least two full years of work in algebra - especially the procedures for manipulating symbolic expressions and equations - and a year of geometry that has traditionally included an introduction to logical reasoning and proof.  As these curricular traditions were reconsidered over the past decade, several major changes seemed in order.

First, the emergence of powerful technologies for numeric and graphic mathematical calculation suggested changes in the traditional focus of school curricula on procedural skills in arithmetic and algebra.  It now seems quite feasible to reduce the time devoted to training students in paper-and-pencil operations that are executed rapidly and accurately by low-cost calculators and computers.  Second, analysis of the ways that mathematics is used in the workplace and in personal problem solving suggests that topics in probability, statistics, and new areas of discrete mathematics deserve more substantial treatment than traditional curricula provide.  Third, comparisons of U.S. curricular traditions with those of other countries suggest that it might be more appropriate to present mathematical topics through integrated curricula that develop all major content strands in each year of secondary school, rather than in separate yearlong courses.  Fourth, disappointing experiences with the abstract and formal style of curricula in the "new math" era suggest that most students would learn better from a curriculum that develops key ideas from work on concrete problems in meaningful real-life contexts.  This emphasis on learning through problem solving is consistent with widespread concern that students should acquire the ability to apply mathematical ideas and techniques to problem solving and decision making in other fields of work.

Finally, reports from people in business, industry, and government suggest that, along with mathematical understanding and skills, students need to have well-developed abilities to analyze problem situations and to communicate ideas for solving those problems.  Thus the mathematics curriculum ought to focus on broad reasoning and communication goals as well as on specific content topics.

In addition to the list of new content objectives and emphases recommended for K-12 mathematics curricula, the Standards made a strong case for expanding access to significant mathematics for all students.  In particular, they discouraged the traditional American practice of curriculum tracking that provides our most able and interested students with challenging courses in algebra and geometry and relegates other students to endless repetition of topics in general mathematics.  The Standards argued for a common three-year core of topics to be studied by all students in high school, with extensions for students preparing to study in mathematically intensive fields in college.

Approaches to teaching and learning.  The predominant goal of traditional K-12 mathematics curricula has been training students in computational procedures of arithmetic and algebra.  The predominant method of reaching that goal has been a direct classroom instructional routine in which teachers explain and illustrate procedures and students practice those procedures on a host of similar exercises.  The typical product of this kind of instruction has been students with modest computational skills but very limited skills in problem solving.(5)

The NCTM Standards documents argue that with new approaches to classroom instruction we ought to be able to achieve better student learning, and they make recommendations for change in several aspects of traditional practice.  The Standards propose that instruction should be focused on student investigation of substantial mathematical problems; that the classroom teacher should act as a stimulant, sounding board, and guide in that student problem solving; that students should be encouraged to discuss mathematical ideas and discoveries with classmates and with the teacher; that the classroom activity should include frequent challenges to students to develop justifications for their ideas and discoveries; and that students should be encouraged to use calculators and computers in their mathematical explorations.

These recommendations drew from models of instruction used in other countries with effective mathematics education systems(6), from cognitive research evidence revealing the fundamental importance of students' constructing their understanding of mathematical ideas(7), from research and practical experience confirming the efficacy of cooperative learning environments(8), and from the strong experience-based commitment of the Standards authors to help students develop mathematical habits of mind by engaging in activities that involve exploring, inventing, conjecturing, proving, and problem solving.

Assessment of student learning.  Just as traditional school mathematics curricula have emphasized the acquisition of basic facts and computational skills in arithmetic and algebra, typical classroom and standardized tests of student achievement have emphasized short-answer questions and computational exercises presented in formats that can be scored quickly and "objectively." This typically American style of testing is quite different from traditions in other countries, where more complex problem solving is the norm on both classroom and external examinations.  Reflecting insights from international comparisons and the desire to focus testing on conceptual understanding and problem solving, the authors of the NCTM Standards documents proposed significant changes in the practice of assessment and evaluation.

The main themes in the assessment proposals from NCTM included using a variety of assessment tools, from classroom observation and journal writing to extended projects, portfolios, open-ended problems, and conventional tests; improving the alignment of assessment practices with curricular goals; and embedding assessment in everyday instructional activities so that it contributes to the improvement of teaching, not merely to the assignment of grades.

Prospects for reform.  Charting an ambitious and enticing reform agenda is the easy part of school improvement.  The real challenge lies in converting the ideas into operational school programs that faithfully reflect reform principles and lead to improved learning for students.  It is a demanding task to create curriculum materials that support effective implementation of those ideas - and to help teachers develop the new skills for implementing the curricula in the vast and varied American school system.

To make informed judgments about reform, one needs to study the curriculum materials, to see classrooms in action, and to analyze evaluation data.  That kind of detailed study is beyond the scope of a Kappan article.  However, we describe below the flavor of one typical reform mathematics curriculum at the high school level, sketch some of the concerns of critics, and present evidence that can help in sorting through the charges and countercharges of the "math wars" that have been sparked by the NCTM Standards movement.
 

Realizing the Standards Agenda

The Core-Plus Mathematics Project (CPMP) is one of several National Science Foundation-supported efforts to design, prepare, evaluate, and disseminate curricular options for a Standards-based high school mathematics program. In contrast to the traditional practice of tracked programs that offer advanced mathematics to a few and minimal mathematics to the majority of students(9), the CPMP curriculum is designed to make important and broadly useful mathematics meaningful and accessible to all students.

The curriculum consists of a single core sequence for both college-bound and employment-bound students during the first three years.  This organization is intended to keep postsecondary education and career options open for all students.  A flexible fourth-year course continues the preparation of students for college mathematics.  Recognizing that increasing numbers of college programs involve the study of mathematics, though not necessarily calculus, CPMP Course 4 consists of a core of four units for all college-bound students along with sequences supporting a variety of collegiate majors.  The sequence of units in the CPMP curriculum is presented in Table 1.

TABLE 1
Structure of Curriculum Units in the Core-Plus Mathematics Project
 
 
Course 1 Course 2 
Patterns in data Matrix models
Patterns of change Patterns in location, shape and size
Linear models Patterns of association
Graph models Power Models
Patterns in space and visualization Network optimization
Exponential models Geometric form and its function
Simulation models Patterns in chance
Capstone: Planning a benefit concert Capstone: Forests, the environment, & mathematics
   
Course 3 Course 4
Multiple-variable models Modeling motion
Modeling public opinion Rates of change
Symbol sense and algebraic reasoning Counting models
Shapes and geometric reasoning Composite, inverse, and logarithmic functions
Patterns in variation  
Families of functions  
Discrete models of change  
Capstone: Making the best of it - optimal forms & strategies  
   
Course 4A - Path to Mathematics and Physical Sciences Course 4B: Path to Management and Social Sciences
Polynomial and rational functions Binomial distributions
Functions and symbolic reasoning Problem solving, algorithms, & spreadsheets
Space geometry  Informatics
Mathematical structures  Statistical inference in surveys & experiments
Capstone: Building mathematical bridges Capstone: Analyzing published reports
 
 
A balanced curriculum.  In each year of the CPMP curriculum, mathematics is developed along four interwoven strands: algebra and functions, geometry and trigonometry, statistics and probability, and discrete mathematics.  This curriculum organization breaks down the artificial compartmentalization of the traditional "layer cake" curricula in this country and addresses weaknesses identified in the recent TIMSS findings.(10) It also elevates statistics, probability, and discrete mathematics - mathematics essential to the Information Age - to a central position in the curriculum for all students.  Finally, developing mathematics each year along these multiple strands capitalizes on the differing interests and talents of students and helps them to achieve diverse mathematical insights.(11)

The unit titles and sequence employed in the CPMP curriculum reveal several key principles that guided its design.  First and most important is the belief that mathematics is a vibrant and broadly useful subject that should be explored and understood as an active science of patterns.(12) As a consequence, CPMP students explore patterns of gender distribution of juries and multichild families as a way of beginning to work with the concepts and techniques of probability and statistics.  They conduct experiments that simulate bungee jumping, and they analyze patterns in the relation between jumper weight and bungee cord stretch as a prelude to the study of algebraic expressions and equations.  They study patterns in decorative designs and in computer graphic images and then the related geometric ideas of symmetry, congruence, and transformations.  Analysis of patterns in road and communication networks leads to important concepts in graph theory that are widely used in computer and management sciences.

The second curriculum design principle is commitment to the inclusion of topics that are demonstrably important for students to learn, not simply part of long-standing school tradition.  The content and organization of traditional mathematics curricula are driven largely by an intricate structure of topics presumed to be prerequisites for collegiate calculus.  Emerging technologies have changed the nature of those prerequisites, and the mathematics that is useful in advanced studies has broadened to include topics in statistics and discrete mathematics.  Thus, although sequential considerations and prerequisites were not ignored in developing the CPMP curriculum, the selection of mathematical content for each year was determined by answering the question, Is this the most important mathematics a student should know if it is the last mathematics he or she will study?

This "zero-based" development principle resulted in the elimination or de-emphasis of some topics found in traditional curricula, a reordering of other topics, and the inclusion of the most applicable and important mathematical ideas.  For example, in the algebra and functions strand in Course 1, the big ideas are patterns of change, linear functions and related equations, and exponential functions and related equations.  This last topic was chosen because of its importance in modeling real-world situations involving multiplicative growth or decay, and it replaces a traditional study of quadratic equations in Course 1. Other examples of reordering are: 1) the inclusion of work with matrices and coordinate geometry in Course 2 to develop, at an early point, the useful connections between algebra and geometry through the concept of transformation; 2) the movement of formal reasoning in algebra and proof in geometry to Course 3, by which point students have developed experience-based conceptual understanding of key ideas as well as the mathematical maturity required by more abstract and deductive arguments; and 3) the inclusion of units on probability, statistics, and discrete mathematics in each course because of their clear application to questions of immediate interest to high school students and to society at large.
 

A third underlying principle of the CPMP curriculum is that problems provide a context for developing student understanding of mathematics.(13) The curriculum is organized around the investigation of rich applied problem situations.  As suggested by many of the unit titles, mathematical modeling and related concepts of data collection, representation, interpretation, prediction, and simulation are emphasized.  Consistent with our view of mathematics as a science of patterns, exploration and experimentation necessarily precede and complement theory.  Investigations are always accompanied by opportunities for students to analyze and bring to the surface underlying mathematical structures that can be applied in other contexts and that can themselves be the subjects of further investigation.

A fourth underlying principle of the CPMP curriculum is the incorporation of graphing calculators - more aptly called hand-held computers - as tools for learning and doing mathematics.  The use of graphing calculators permits the CPMP curriculum to emphasize multiple representations (numerical, graphical, and symbolic) and to focus on goals in which mathematical thinking is central. Their use in the curriculum enables students to deal with realistic problem situations. Moreover, using graphing calculators removes the "skill filter" that paper-and-pencil symbol manipulation has become for some students and thereby enables them to study significant mathematics.

Integrated instruction and assessment. In providing an existence proof for a high school program based on the NCTM's Standards, the CPMP curriculum materials were developed not only to reshape what mathematics all students have the opportunity to learn, but also to influence the manner in which that learning occurs and is assessed. Each unit in the curriculum was built around a series of five or six multi-day lessons in which major ideas are developed through student investigations of applied problem situations. The lessons focus on several interrelated mathematical concepts and often span four or five days. The CPMP instructional materials recognize the pivotal roles played by small-group collaborative learning, social interaction, and communication in the construction of mathematical ideas.(14)

Each CPMP lesson is introduced as a whole-class activity in which students are asked to think about a context. Once a lesson is launched, students usually work together collaboratively in small groups or pairs as they investigate more focused problems and questions that are related to the launching situation. This investigative work is followed by a teacher-moderated whole-class discussion in which students share mathematical ideas developed in their groups and together construct a shared understanding of important mathematical concepts, methods, and approaches. Sharing and agreeing as a class on the mathematical ideas that the groups are developing are prompted by "checkpoints" in the instructional materials. Each checkpoint is followed by a related "On Your Own" assessment task that is to be completed individually by students.

Each lesson is also accompanied by a set of additional tasks designed to engage students in "Modeling, Organizing, Reflecting on, and Extending" (MORE) the mathematical understanding they have developed through the investigations.  These MORE tasks are intended primarily for individual work outside of class.

Assessment is embedded in the CPMP curriculum materials and is an integral part of instruction.  The instructional materials support continuous assessment of group and individual progress through observing and listening to students during the exploratory and summarization phases of instruction.  In addition, there are individual assessments at the end of each lesson that are used to measure understanding of mathematical concepts, methods, and skills, and there are similar individual and group assessments at the end of each unit, semester, and course.

One additional unique feature of each CPMP course is the inclusion of a thematic capstone, as seen in Table I. These project-oriented capstones provide individuals and groups with rich mathematical problems that require for their solution the use of mathematics from each of the four strands studied in the year.  This is a good opportunity for students to review and consolidate their learning and to demonstrate their mathematical growth over the year.  We summarize the impact of the CPMP curriculum and its instructional and assessment practices on student performance later in this article, following a brief discussion of the main concerns of the critics of reform.
 

What Is the Problem?

In broad terms it is hard to argue with any of the proposals that became the agenda of NCTM's Standards reform.  However, the curricula developed to bring about the reforms provide more concrete targets for criticism.  Although the innovative programs vary widely in their objectives and strategies, the criticisms share some common themes.  For example, many critics would reply no to some (or perhaps all) of the following questions.
 

Most aspects of these important questions could be informed by careful empirical study.  Some of the contentiousness surrounding them, on the other hand, reflects differences of opinion within the mathematical community about what is valued, as can be seen by the long-standing tension between mathematicians who prefer pure or applied approaches to the subject.  Unfortunately, when concerns are raised about new curricula, teaching, and assessment in school mathematics, the discourse is typically more like the adversarial style of American courtrooms than like the dispassionate search for "truth" that one associates with scientific work.

Public critics of mathematics programs based on the NCTM Standards often suggest that there are only two choices - radical reform or back-to-basics traditional programs.  They frequently express their contrary opinions in harsh rhetorical style.  They may support their judgments with carefully selected illustrations of errors in specific curriculum materials, often from early drafts of those materials, or with anecdotal evidence of problems experienced by individual students or teachers.  But seldom are the critics' attacks supported by a careful analysis of the complete curriculum and evaluation evidence.  The critics also usually fail to acknowledge that the motivation for the whole reform movement is deep concern about the inadequate effects of long-standing American traditions in curriculum, teaching, and testing.

Apparently acknowledging that concerns like the ones just listed are largely empirical questions, critics of reform have often complained about the absence of evaluation evidence to compare learning of students in reform curricula with that of students in more traditional curricula.  Yet the critics also make the tacit assumption that such evidence would support the continuation of traditional practices.  Since a new curriculum must be developed and implemented before any summative evaluation is possible, evidence of this type has only recently become available.  In the next section, we report such evidence from the CPMP.
 

What Is the Evaluation Evidence?

Following two years of careful initial development, pilot testing, and revision, each CPMP course is field-tested for a full school year.  This national field test is being conducted in 36 high schools in Alaska, California, Colorado, Georgia, Idaho, Iowa, Kentucky, Michigan, Ohio, South Carolina, and Texas.  More than 4,000 students are or have been involved in the field test.  A broad cross section of students from urban, suburban, and rural communities, reflecting ethnic and cultural diversity, is included.  Course I was field-tested in 1994-95, Course 2 in 1995-96, and Course 3 in 1996-97.  The field test of Course 4 is being conducted in 1998-99.  To supplement the large-scale field test, other focused research and evaluation studies have been and continue to be conducted in subsets of the schools that took part in the initial field test.  An overview of results from this program of evaluation and research follows.  More complete reports are available elsewhere.(15)

Mathematical thinking.  One achievement measure administered to students in the field test schools is the Ability to Do Quantitative Thinking (ATDQT), which is the mathematical subtest of the nationally standardized high school test battery, the Iowa Tests of Educational Development.  The ATDQT is a 40-item, multiple-choice test with the primary objective of measuring students' ability to employ appropriate mathematical reasoning in situations that require the interpretation of numerical data and charts or graphs that represent information related to business, social and political issues, medicine, and science.  Different forms of the ATDQT for appropriate grade levels were administered to CPMP students as a pretest at the beginning of Course I (mainly grade 9) and as posttests at the end of all three courses.  The main findings follow.
 

  • CPMP students significantly outperformed students in the nationally representative ATDQT norm group, which had been tested in 1992 before any curricula based on the NCTM Standards were available.  CPMP students' average pretest to posttest growth was nearly twice that of the norm group during the first year, and this improved level of performance was maintained over the three-year field test. 

  • The pattern of significantly better performance by CPMP students than by the ATDQT norm group was consistent for subgroups of students when the data set was disaggregated by rural, urban, and suburban schools; by gender of students; by whether students were native English speakers; by minority group status of students; and by the type of grouping used in the school (i.e., all students grouped heterogeneously, a wide range but not the best students, a wide range but not the academically strongest and weakest, college preparatory only, and work-preparatory only). 

  • At the same times that CPMP Course 1 and Course 2 students in the field test took the ATDQT, it was administered to comparison students studying traditional curricula in some of the same schools.  On the posttests, CPMP Course 1 and Course 2 students outperformed comparison students (mainly students in first-year algebra and geometry, respectively) with similar pretest means.  Furthermore, this advantage held for students in the bottom, middle, and top thirds of the pretest national norm distribution; CPMP students' posttest means in Course 1 and Course 2 were equal to or greater than those of comparison students in each pretest third. 

  • The effects of the CPMP curriculum with students who had a strong interest in and aptitude for mathematics were examined further by the administration of the ATDQT to students in a specialized mathematics and science center that also was using the CPMP curricula.  The 90 CPMP students at this magnet school had very high pretest scores (median score at the 97th percentile nationally), but they still demonstrated gains during the year that were approximately double those of comparable students in the national norm group.

  •  

    Other mathematical achievement outcomes.  Several other measures of mathematical achievement have been used in the CPMP field test with the following results.
     

  • On a test consisting of 30 released 12th-grade items from the 1990 and 1992 administration of the National Assessment of Educational Progress (NAEP), which was administered to CPMP students at the end of Course 3, the CPMP students' means were higher on each of the six content and three process subtests than those of a nationally representative sample of 12th-grade students tested in October of 1990 or 1992.  CPMP students who completed this test had pretest ATDQT scores suggesting that they were comparable in average mathematical aptitude to a typical group of students at their grade level.  Across the subtests, the mean percentage advantages for CPMP students over the NAEP sample were highest on the data, statistics, and probability subtest (22.5% higher) and lowest on the procedural subtest (10.2% higher). 

  • On CPMP-developed posttests given at the end of Course I and Course 2, CPMP students demonstrated significantly better understanding than comparison students of algebraic and geometric concepts that both groups had had the opportunity to learn.  The CPMP students were also better able to reason with and apply methods of algebra and geometry. 

  • In addition to algebra and geometry, CPMP students showed a clear pattern of growth in learning across the three courses on posttests of probability, statistics, and discrete mathematics. 

  • On procedural algebra tasks with no applied contexts (e.g., solving equations and simplifying or factoring algebraic expressions), traditional students at the end of a first-year algebra course scored higher than CPMP students at the end of Course 1. By the end of Course 2, when the traditional students had just completed geometry, the means were virtually equal. 

  • At the end of Course 3, CPMP students and students who had taken advanced algebra and were matched by school and by eighth-grade mathematics achievement test scores completed a researcher-developed test of reasoning with algebra and functions.  On the three subtests, CPMP students scored significantly better on open-ended problem solving and on concepts and applications, while the advanced algebra students scored significantly better on paper-and-pencil algebraic procedures. 

  • Preliminary data on college entrance examinations show no significant differences in the mean scores of CPMP students and comparison students in traditional curricula (in the same schools and of similar prior school achievement).  We will continue to monitor scores on college entrance tests for at least two more years.

  •  

    Supplementing written achievement tests. In an attempt to better describe the nature of the differences in algebraic performance between students in the CPMP curriculum and those in traditional high school mathematics programs, the evaluation team interviewed individuals or small groups of students during the 1996-97 Course 3 field test site visits.  Fifteen CPMP students and a similar number of students in traditional college-preparatory mathematics courses - both groups with a wide range of achievement levels - were interviewed.

    Across all the interview-based problems, many students in traditional mathematics showed basic misunderstandings about the connection between a verbally stated problem situation and its mathematical model as well as misunderstandings of the various representations of that model.  Consistent with a large body of research on mathematical thinking and problem solving(16), many of the students in traditional mathematics curricula did not seem to have - or even to think it useful to strive for - a holistic understanding of a verbally stated problem situation.  Rather they scanned the problem statement for the numbers and a few cue words, such as "feet per second" and "seconds" in an effort to guide their often erroneous choice of mathematical procedures.

    For example, two students who had completed first-year algebra and advanced algebra with mainly "A" grades both groaned when presented with the task in Figure 3. One said, "I can't do word problems." The other added, "This is why I dropped out of Core [after the first semester of Course 1]." Concerning performance on each part of this task, none of the 15 traditional students we interviewed could correctly predict the shape of the graph in part a; they indicated that it was linear or sometimes only noted that time would be on the horizontal axis and height on the vertical.  Part b called for computing h when t is 3 seconds, which some traditional students were able to do.  Others were not.  Like the two students described above, they focused on the numbers.  They reread the stem of the problem and noted that it said "128 feet per second." Since 3 seconds was given in part b, these students wrote 128 x 3, multiplied with their calculators, and wrote the product 384 feet.  When quizzed about their solution, they were sure that they were right because "rate times time gives distance."
     

    FIGURE 3.
    Algebra Concepts Interview Task
     
    While warming up before a game, the Pirates right fielder and center fielder were standing together on the outfield grass. The right fielder threw a baseball straight up with a velocity of 128 feet per second for the center to catch.  The height h of the baseball after t seconds is represented by
    h = -16t2+128t
     
    a. Describe the patterns that you would expect to find or graph of the time (time, height) relationship.

    b. Find the predicted height of the baseball after 3 seconds.

    c. Write an equation whose solution is the number of seconds after the baseball is thrown that it is 160 feet above the ground. Solve the equation, check it, and explain how you found the solution.

     
     
    In part c, some traditional students wrote the equation correctly, that is, 160 = -16t2 + 128t, but none could solve it, typically resorting to some variation of methods for solving linear equations, such as dividing both sides by t. For example, the two traditional students (described above) incorrectly wrote:
      h= -16t2 + 160t
            t

    They decided that the equation could be solved by dividing by t. They immediately canceled both t2 and t and wrote: h = -16 + 160 = 144.  Prompts by the interviewer to help them think of other methods to solve the equation, including using a graphing calculator, were not fruitful.

    In verbally presented situations that can be modeled by an algebraic function, such as the example in Figure 3, CPMP Course 3 students who were interviewed showed fair to good understanding.  They thought of the context as something real about which it was possible to gain meaning by reasoning from their own experience, and they expected the output of the model to agree with their experience.  For example, nearly all CPMP students who were interviewed showed an understanding of each part of the problem in Figure 3, and they usually used the table-building capability of their calculators to solve the quadratic equation in part c. For the most part, they clearly related the problem situation to real-life experience.  Consider the reactions of three CPMP Course 3 students (with A, B, and C grades) to the interviewer's follow-up question, after they had solved for one root of the quadratic equation, "Do you think that's the only answer?" Two students immediately and almost simultaneously said, "No." The third said, "The down side." When asked to explain, the third student said, "Because it's a parabola, up and down." Another student added, "Throw the ball up. It's going to go up past 160, and then it's going to come right back down."

    This is not to suggest that CPMP students' levels of mathematical understanding and execution were beyond the need for improvement.  Rather, we wish to suggest that CPMP students exhibited a more solid understanding than students in traditional classes of the connection between an applied context and its mathematical model.  And they had a better grasp of the connections between the tables, graphs, and equations and the verbal representations of the model.

    When faced with algebraic procedures, such as solving a linear equation or simplifying an algebraic expression with no context given, traditional students were more automatic than CPMP students. When the traditional students knew and recalled the appropriate algorithm, they were able to generate solutions to problems efficiently. However, when their recall was faulty (as in the two traditional students' attempts to solve their equation in part c above), they seemed to have no inclination, or perhaps no means, to check their work or to reflect on their reasoning.  CPMP students, on the other hand, took more time trying to recall previous, related learning.  Their paper-and-pencil algorithmic procedures were less efficient than those of many traditional students, and they made about the same number of errors in execution.  Unlike traditional students, though, most CPMP students were able to use such alternative solution methods as the graphing and tablebuilding capabilities of their calculators.

    In summary, the interview data support and help to explain the pattern of differences between students in CPMP and those in traditional curricula that we found in the written test results.  Generally, CPMP students demonstrated better conceptual understanding and problem-solving skills, less automatic proficiency with paper-and-pencil procedural algorithms, and better proficiency with graphing calculators.

    Student perceptions and attitudes.  A written survey of students' perceptions and attitudes about various aspects of their mathematics course, using a Likert-type scale, was administered at the end of each school year.  Some of the pertinent findings follow.  These are mainly taken from the responses of CPMP students at the end of Course 2 as compared with traditional geometry students at the end of the same year. (We present Course 2 results since any effect of the novelty of the CPMP approach is likely to have worn off by the end of the second year.) Each of the following findings was consistent across levels of student achievement.
     

  • CPMP students agreed (79.2%) that cooperative group work helped them learn mathematics.  The advantages of learning in groups most often cited by the students were seeing how other people attack problems and the support of group members during problem-solving efforts. 

  • Students perceive the CPMP curriculum to be quite difficult, at least as challenging as traditional college-prep mathematics courses.  A common perception of students is that CPMP is challenging and makes them think, but they also believe that, with effort, it is possible for them to understand the mathematical ideas and their applications. 

  • A significantly higher percentage of CPMP students than of traditional students agreed that their mathematics course contained realistic problems, made the mathematical ideas interesting, and increased their ability to talk and write about mathematics.  The levels of agreement for CPMP students on these survey items ranged from 66.5% to 76.5%, compared with 40.6% to 47.8% for traditional students. 

  • CPMP students were much more likely than geometry students (75% versus 42%) to want to take a mathematics course taught in the same way in the coming year.  And at the end of Course 3, 27% of CPMP students agreed that it was mainly because of CPMP that they took a third year of mathematics.  These findings, coupled with substantial increases in enrollments in junior and senior mathematics courses in many field test schools, provide strong evidence that the CPMP curriculum is a factor in keeping more students in mathematics courses longer.

  •  

    To Change or Not to Change

    The CPMP evaluation results show that students experiencing the Core-Plus Mathematics Project's interpretation of a curriculum and pedagogy based on the NCTM Standards were, on average, more positively disposed toward mathematics and understood and were able to apply many important mathematical ideas significantly better than the traditional students to whom they were compared.  These findings are encouraging and bode well for the potential of curricula based on the Standards to improve the overall mathematical literacy of a broad range of U.S. high school students and to increase enrollments in mathematics courses in the later years of high school - two of the main goals of the reform effort.

    The present evidence on paper-and-pencil skill proficiency is mixed, with deficits for CPMP students after Course I and Course 3. There is evidence that these deficits may disappear when the testing is conducted some time after students have practiced the tested skills, as was the case in comparisons involving CPMP Course 2 and geometry students.  In response to these results from the field tests, the published version of the CPMP curriculum has been adjusted to better address algebraic paper-and-pencil skills.  More testing of procedural skills, both immediate and remote to skill practice, is planned for the Course 4 field test.  On the positive side for CPMP students, when they have difficulty recalling a paper-and-pencil algorithm for a given task, their flexibility and skill with a graphing calculator give them powerful alternative methods that are now unavailable to traditional students.

    The CPMP evaluation results bear on the concerns of the critics of reform that we outlined above, because they were obtained using a curriculum that has many of the characteristics questioned by the critics.  We highlight here some of the facets of our evaluation data that are directly relevant to the concerns of the critics.
     

  • Cooperative investigations were a prominent, though not exclusive, feature of CPMP classrooms.  Teachers reported using cooperative groups, on average, between 40% and 50% of the class time, and nearly 80% of students reported that the cooperative group work was an aid to their learning. 

  • The CPMP curriculum integrates four strands of content - algebra and functions, geometry and trigonometry, statistics and probability, and discrete mathematics.  The reported achievement results suggest that content integration does not necessarily lead to the watering down of important ideas and skills, as some critics of reform fear. 

  • The improvements in understanding and in problem-solving skills of CPMP students were consistent across entering levels of mathematical background and achievement and in a mathematics and science magnet school.  So too were the positive perceptions and attitudes about the CPNV program, including the perception that it is a challenging curriculum.  There is no evidence of a generally detrimental effect on the performance of the most talented students.  On the contrary, the better students in the CPMP program improved more in quantitative thinking skills than did their equally talented peers in traditional curricula.  And the size of the difference in favor of the CPMP students was about the same as that for other comparison groups of CPMP and traditional students. 

  • One of the most positive aspects of the CPMP curriculum for students was working with real-life problem contexts; they made for added interest and helped students gain a concrete understanding of mathematical ideas.  The understanding of underlying mathematical structures and the levels of excitement about pure mathematics have not been measured directly in CPMP studies to date.  It should be remembered, though, that when these were the focus of the curriculum in the "New Math" era of the 1960s, the results were disappointing. 

  • Whether or not the lower levels of paper-and-pencil procedural skills prove to be persistent outcomes for CPMIP students, in today's technological world the improved mathematical literacy and the better attitude toward mathematics might make it worth the tradeoff.  Moreover, these slightly lower skills have not been shown to hamper students in further mathematics learning or applications. 

  • A study of students' facility with formal reasoning and deductive proof is currently under way in CPMP Course 3 classes. Preliminary results indicate that there is no significant difference in these areas between CPMP students and comparison students who have completed traditional first-year algebra, geometry, and advanced algebra. 

  • As we write, no students who have completed all four CPMP courses in at least a field test version have graduated from high school.  The first students will do so in the spring of 1999.  Thus it is too early to obtain evidence concerning the transition from the CPMP curriculum to college.  Preliminary data from college entrance tests suggest that CPMP students' scores do not differ significantly from those of students who have taken the traditional curricula.

  •  

    More research is needed and is being done.  In particular, we need research that studies the effects of the final version of reform curricula after teachers and students have adjusted to the new content and instructional expectations.  When complete K-12 curricula based on the NCTM Standards documents are in place and when careful documentation of the effects of specific implementations of these curricula have been compared to traditional alternatives, schools and the public will be in a better position to choose wisely among the curricular options.  As evidence from research and evaluation studies accumulates, we can hope that the findings will help to raise the level of discourse about mathematics reform.  Perhaps such evidence will become a standard of truth that will allow us to move beyond this current version of the "math wars." 

    End Notes

    1. See, for example, National Commission on Excellence in Education, A Nation at Risk: The Imperative for Educational Reform (Washington, D.C.: U.S. Government Printing Office, 1983); and National Science Board Commission on Pre-College Education in Mathematics, Science, and Technology, Educating Americans for the 21st Century (Washington, D.C.: National Science Foundation, 1983).

    2. Curtis C. McKnight et al., The Underachieving Curriculum: Assessing U.S. Mathematics from an International Perspective (Champaign, Ill.: Stipes, 1987).

    3. Curriculum and Evaluation Standards for School Mathematics (Reston, Va.: National Council of Teachers of Mathematics, 1989); Professional Standards for Teaching School Mathematics (Reston, Va.: National Council of Teachers of Mathematics, 1991); and Assessment Standards for School Mathematics (Reston, Va.: National Council of Teachers of Mathematics, 1995).

    4. See, for example, Anthony Ralston and Bail S. Young, eds., The Future of College Mathematics (New York: Springer-Verlag, 1983); Ronald G. Douglas, ed., Toward a Lean and Lively Calculus (Washington, D.C.: Mathematical Association of America, MAA Notes Number 6, 1986); National Research Council Committee on the Mathematical Sciences in the Year 2000, Moving Beyond Myths: Revitalizing Undergraduate Mathematics (Washington, D.C.: National Academy Press, 1991); and Crossroads in Mathematics: Standards for Introductory College Mathematics Before Calculus (Memphis: American Mathematical Association of Two-Year Colleges, 1995).

    5. See, for example, Edward A. Silver, ed., Teaching and Learning Mathematical Problem Solving: Multiple Research Perspectives (Hillsdale, N.J.: Erlbaum, 1985); and Randall I. Charles and Edward A. Silver, eds., The Teaching and Assessing of Mathematical Problem Solving (Reston, Va.: National Council of Teachers of Mathematics, 1988).

    6. High school mathematics curricula in nearly all other countries are integrated.  Some of the best recent research on teaching practices in Japanese classrooms comes from the TIMSS (Third International Mathematics and Science Study) videotape studies.  See, for example, James W. Stigler and James Hiebert, "Understanding and Improving Classroom Mathematics Instruction: An Overview of the TIMSS Video Study," Phi Delta Kappan, September 1997, pp. 14-21.

    7. Paul Cobb, "Where Is the Mind?  Constructivist and Sociocultural Perspectives on Mathematical Development," Educational Researcher, October 1994, pp. 13-20.

    8. One of several comprehensive reviews of research to appear in the 1980s on the effects of cooperative learning activities is Steven T. Bossert, "Cooperative Activities in the Classroom," in Ernst Z. Rothkopf, ed., Review of Research in Education (Washington, D.C.: American Educational Research Association, 1988), pp. 225-50.

    9. National Research Council, Everybody Counts: A Report to the Nation on the Future of Mathematics Education (Washington, D.C.: National Academy Press, 1989).

    10. William H. Schmidt et al., Facing the Consequences: Using TIMSS for a Closer Look at United States Mathematics Education (Boston: Kluwer Academic Publishers, 1998).

    11. Christian R. Hirsch and Arthur F. Coxford, "Mathematics for All: Perspectives and Promising Practices," School Science and Mathematics, vol.92, 1997, pp. 232-41.

    12. Lynn A. Steen, ed., On the Shoulders of Giants: New Approaches to Numeracy (Washington, D.C.: National Academy Press, 1990).

    13. See, for example, Alan H. Schoenfeld, "Problem Solving in Context(s)," in Charles and Silver, pp. 82-92; idem, "Learning to Think Mathematically: Problem Solving, Metacognition, and Sense Making in Mathematics," in Douglas A. Grouws, ed., Handbook of Research on Mathematics Teaching and Learning (Reston, Va.: National Council of Teachers of Mathematics, 1992), pp. 334-70; and James Hiebert et al., "Problem Solving as a Basis for Reform in Curriculum and Instruction: The Case of Mathematics," Educational Researcher, May 1996, pp. 12-21.

    14. For a general review of research on the effects of cooperative learning activities, see Bossert, op. cit.  For reviews of research on the value of cooperative learning, especially for women and certain minority groups, see Jeannie Oakes, "Opportunities, Achievement, and Choice: Women and Minority Students in Science and Mathematics," in Courtney B. Cazden, ed., Review of Research in Education, vol. 16 (Washington, D.C.: American Educational Research Association, 1990), pp. 153-222; Gilah C. Leder, "Mathematics and Gender: Changing Perspectives," in Grouws, pp. 597-622; and Wisconsin Center for Education Research, "Equity and Mathematics Reform," NCRMSE Research Review: The Teaching and Learning of Mathematics, Fall 1994, pp. 1-3.  For the importance of social interaction, see Cobb, op. cit.  And for the role of communication, see Edward A. Silver, "Moving Beyond Learning Alone and in Silence: Observations from the QUASAR Project Concerning Communication in Mathematics Classrooms," in Leona Schamble and Robert Glaser, eds., Innovations in Learning, New Environments for Education (Mahwah, N.J.: Erlbaum, 1996), pp. 127-59.

    15. Harold L. Schoen and Steven W. Ziebarth, Mathematical Achievement on Standardized Tests: A Core-Plus Mathematics Project Field Test Progress Report (Iowa City: University of Iowa, 1998); idem, Assessment of Students' Mathematical Performance: A Core-Plus Mathematics Project Field Test Progress Report (Iowa City: University of Iowa, 1998); Harold L. Schoen, Christian R. Hirsch, and Steven W. Ziebarth, "An Emerging Profile of the Mathematical Achievement of Students in the Core-Plus Mathematics Project," paper presented at the annual meeting of the American Educational Research Association, San Diego, 1998; and Harold L. Schoen and Johnette Pritchett, "Students' Perceptions and Attitudes in a Standards-Based High School Mathematics Curriculum," paper presented at the annual meeting of the American Educational Research Association, San Diego, 1998.

    16. For one review of this research, see Schoenfeld, "Learning to Think Mathematically."
     

    HAROLD L. SCHOEN is a professor of mathematics education at the University of Iowa, Iowa City (e-mail: harold-schoen@uiowa.edu). JAMES T. FEY is a professor of mathematics education at the University of Maryland, College Park.  CHRISTIAN R. HIRSCH is a professor of mathematics education at Western Michigan University, Kalamazoo.  ARTHUR F. COXFORD is a professor of mathematics education at the University of Michigan, Ann Arbor. The authors are co-directors of the Core-Plus Mathematics Project, which is supported, in part, by a grant from the National Science Foundation (MDR-9255257).  But the opinions and conclusions reported are solely those of the authors.

     


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