Mathematics Education Reform Research Articles

Lenses on Learning: Administrators' Views on Reform and the Professional Development of Teachers

If the intellectual norms and values embedded in the mathematics education reform movement are to move beyond individual classrooms and significantly influence entire schools and districts, school and district administrators will need to become centrally, rather than peripherally, involved. This paper discusses the ways administrators' ideas about the nature of mathematics, learning, teaching, and school culture affect their interpretations of the nature and intent of the elementary mathematics reform movement and their thoughts about of how they might support it. In particular, administrators' views of parents' concerns, professional development for teachers, and of how new ideas move around in a school are discussed. I argue that administrators have well-formed ideas about mathematics, learning, and teaching, which influence their views of reform and their ideas of how to provide support. These ideas need to be taken into account if administrators are to be central actors in reform.

Learning Mathematics for Teaching: From a Teachers' Seminar to the Classroom

A successful practice grounded in the principles that guide the current mathematics education reform effort requires a qualitatively different and significantly richer understanding of mathematics than most teachers currently possess. However, it is not as clear how teachers' mathematical understandings develop and how those understandings affect instruction. This paper explores two avenues for K-6 teachers' mathematical development, (a) engagement in inquiry into mathematics itself, and (b) investigation of children's mathematical thinking, illustrating how the need for these two kinds of investigations arises in classroom situations and how they can be pursued in a professional development setting.

Reforming the NCTM standards in light of historical perspective: Premature changes

The mathematics education community finds itself in strange times; times of transition and continual change. The constant evolution of mathematics education reform is upon us. A full nine years into the publishing and partial implementation of the NCTM Curriculum and Evaluation Standards mathematics educators eagerly anticipate the evaluation, rewriting, and publication of reform Standards. However, a din of opposition has arisen: one, a call for reform from within, seeking to rewrite the current Standards, and call from without, calling for a return to “Back to Basics.” This parallels the reform antagonism and Back to Basics movement which followed the unsuccessful reform effort of the New Math Movement more than three decades ago. Therefore, since historical precedence exists, this present phenomenon must be thoroughly considered in light of past reform efforts. This paper defines some of the historically consistent events and situations surrounding both the New Math Movement and the NCTM Standards. These commonalities include: overestimating teachers; overly ambitious suggestions in curriculum modifications; delayed teacher training; philosophically inconsistent evaluatory methodologies; and insufficient programmatic assessment. Arguing from these commonalities, this paper reports that any immediate reform of the NCTM Standards is premature and not supported by empirical evidence.

The Mathematics Education Reform: Why You Should Be Concerned and What You Can Do

In 1962, when the New Math was still on its ascent, 75 leading mathematicians published an open letter in this Monthly to chide the New Math for its excesses [11]. It decried empty abstraction and rigid formalism, and made a strong case for learning the fundamentals of traditional mathematics: “elementary algebra, plane and solid geometry, trigonometry, analytic geometry and the calculus”. It also affirmed the importance for students to be able to use mathematical language with some fluency, . . . to find proofs and, what may be the most important activity, to recognize a mathematical concept in, or to extract it from, a given concrete situation. Unfortunately, this forceful statement is now all but forgotten in the mathematical research community. Thirty-five years later, we are faced with another mathematics education reform. In the usual way in which this term is understood, it refers to both the K-12 mathematics education reform led by the National Council of Teachers of Mathematics (NCTM) and the calculus reform. This reform once again raises questions about the values of a mathematics education, this time not by imposing empty abstractions and rigid formalisms, but by redefining what constitutes mathematics and by advocating pedagogical practices based on opinions rather than research data of large-scale studies from cognitive psychology. The reform has the potential to change completely the undergraduate mathematics curriculum and to throttle the normal process of producing a competent corps of scientists, engineers, and mathematicians. In some institutions, this potential is already a reality. The purpose of this article is to discuss briefly some of the salient features of the reform, explain why the stakes are so high this time around, and finally point out some possible avenues for individual and collective action by mathematicians. Real progress in changing the direction of the reform will come only when reasoned arguments are heard from the whole mathematical community.

It Doesn't Add Up: African American Students' Mathematics Achievement

Mathematics education has been heralded for its leadership role in the U.S. school reform effort (Stein, Grover,&Henningsen, 1996; Grant, Peterson, & Shojgreen- Downer, 1996). Prominent in the reform of mathematics education is the call for students not merely to memorize formulas and rules and apply procedures but rather to engage in the processes of mathematical thinking, that is, to do what mathematicians and other professional users of mathematics do. The revamped mathematics education program is based on engaging students in problem posing and problem solving rather than on expecting rote memorization and convergent thinking. These changes in mathematics education suggest that mathematics teaching must build on students' learning and on their ability to pose and solve problems previously considered too difficult for their age-grade levels (Carpenter&Fennema, 1988; Fennema, Franke, Carpenter,&Carey, 1993).

The Mathematics Education Reform: Why You Should be Concerned and What You Can Do

The Mathematics Education Reform: Why You Should be Concerned and What You Can Do

The Challenges of Implementation: Supporting Teachers

Reform in mathematics education has been stimulated and propelled by the publication of standards documents by the National Council of Teachers of Mathematics. This article examines the vision of teacher decision making that is portrayed in NCTM Professional Teaching Standards: choosing worthwhile mathematical tasks, orchestrating and monitoring classroom discourse, creating an environment for learning, and analyzing one's practice. The philosophical orientation and the set of commitments to teaching and learning on which the standards are based include stances on equity, curriculum, teaching, and learning. These stances are summarized under the following headings: inclusiveness, depth over coverage, teaching for understanding, active engagement of students, curriculum investigations, applications, and connections.

The Technological Revolution and the Reform of School Mathematics

Technology and technology-intensive mathematics curricula are catalysts for the mathematics education reform movement. Students can understand mathematics more deeply when they assume responsibility for their own learning as they engage in and reflect on authentic mathematical activity. Analyzing technology in mathematics classrooms as cognitive technologies with differing degrees of transparency and different abilities to externalize representations, viewing technology used to enhance computational skills as amplifiers, and viewing technology used to encourage students to investigate mathematical phenomena and real-world situations as reorganizers, I evaluate the ways in which empirical research supports this vision or calls it into question.

Parental Involvement in the Reform of Mathematics Education

High Schools across the nation have been, and are, engaged in efforts to implement the recommendations in the three Standards documents of the National Council of Teachers of Mathematics (1989, 1991, 1995). These efforts continue in the context of a larger educational-reform movement that spans all content areas (Fullan 1991; U.S. DOE 1994a). Many of the more recent reform documents, which point the direction for this movement, call for increaes in parental involvement and the promotion of partnerships between schools and communities (National Parent-Teacher Association 1994; U.S. DOE 1994a, 1994b). These aims are apparent in the U.S. DOE's national goals for the year 2000 (U.S. DOE 1994a):

Lecturing at the "Bored"

Even a superficial review of the literature on collegiate mathematics education reform reveals a common thread: the imperative nature of the college mathematics student's active participation in the learning of mathematics. Active participation in the classroom may take the form of working cooperatively with classmates in small groups, spending time in the computer lab using class-related software, presenting problems or concepts on the board to classmates, or other activities germane to the subject bestdes passively listening to a lecture presented by the instructor. There is a wide range of lectures styles, but the stle to which I refer is one characterized by little expectation of student engagement besides perhaps following the logic of the lesson. Because active learning in the mathematics classroom is advocated by several professional groups ([1], [4], [5], [6], [8]), it is certainly reasonable for mathematics instructors to reflect on their rationale for maintaining a lecture style of teaching, even part of the time. During the 1995-1996 academic year, that is exactly what I did. I was teaching a section of reformed calculus, with a class size of forty. I had planned on extensive use of cooperative groupwork, and had accordingly divided the class into ten groups of four people each. I attempted to make the groups heterogeneous (in gender and ability) by collecting quantitative measures of the students' previous mathematical performance. Several times during the first two sveeks of class, I set aside time for group work. However, due to the physical layout of the classroom (five rows of auditorium seating with twelve seats per row and an aisle up the middle) my students were having a difficult time actually communicating in their groups. Classtime initially devoted to guided discovery in the groups degenerated into work time for forty individuals, certainly not the vision I had foreseen when planning earlier that summer. To restructure this time, I tried to view my entire class as a large cooperative group, with me as the group leader. Instead of breaking the class into four-person groups as before, I would present examples of concepts we were currently studying, and pose open-ended questions to the class. For example, while developing a geometrical approach to Newton's method for finding the roots of an equation, I asked my students which solution they thought the graphical process would yield when the initial guess for the x-value was directly between two of the roots. Was there a pattern that allowed us to predict the general result? Another time wheh discussing inverse functions, we did an impromptu in-class exploration of which Itnear functions, and later, which general functions, are their own inverses. Such discussions involved much conjecturing, arglling, and scratchpaper seatwork, individually and in pairs, among the students, and were opportunities for algebraic, geometric, and numerical exploration. l Because of my class's general enthusiasm and willingness to participate, If felt we had developed a classroom atmosphere free of criticism, and open to conjecturing. The students often presented problems for each other, furthering their own mathematical communication. Although I was somewhat disappointed not to be able to watch my students develop cooperative skills, I felt that we had made the

Mathematics and Science Teacher Education and School Reform: A Statewide Process in Georgia

The author describes an ongoing statewide effort in Georgia to improve the quality of preservice teacher education programs in middle school mathematics and science. This effort is called POET (Principles of Educating Teachers) and is one of several statewide initiatives to reform K-12 mathematics and science education curriculum and teaching currently underway in Georgia. The goals, foundations, processes, and activities of POET are described along with what the POET staff has learned about statewide teacher education reform. The complexities and challenges of making long-term improvements in teacher education programs are discussed.

Results of Third-Grade Students in a Reform Curriculum on the Illinois State Mathematics Test

Over the past decade, there has been a call for major reforms in mathematics education, from classrooms where students memorize facts and practice algorithms to classrooms in which reasoning and understanding are given more emphasis (National Council of Teachers of Mathematics [NCTM], 1989, 1991, 1995). In response, a number of curricula have been developed that attempt to meet this vision. One reform curriculum in widespread usage is the University of Chicago School Mathematics Project's (UCSMP) elementary curriculum, Everyday Mathematics. In the UCSMP curriculum, students work in small groups exploring mathematics in real-life contexts, using calculators, manipulatives, and other mathematical tools from kindergarten onward. In contrast to traditional curricula, students are encouraged to use these tools or to invent their own computational algorithms to solve problems, and the sharing of their alternative solution methods is a regular part of class discussions. Additionally, problems are nearly always application-based and never presented as sets of symbolic problems. For example, in a second-grade UCSMP lesson, students are given a picture depicting various animals and their heights or lengths. During this activity, students work in small groups to construct number stories that compare the animal heights and then to find a solution method. In the follow-up discussion, students share their stories and solution procedures and offer alternative methods.

Brief Report: Results of Third-Grade Students in a Reform Curriculum on the Illinois State Mathematics Test

Over the past decade, there has been a call for major reforms in mathematics education, from classrooms where students memorize facts and practice algorithms to classrooms in which reasoning and understanding are given more emphasis (National Council of Teachers of Mathematics [NCTM], 1989, 1991, 1995). In response, a number of curricula have been developed that attempt to meet this vision. One reform curriculum in widespread usage is the University of Chicago School Mathematics Project's (UCSMP) elementary curriculum, Everyday Mathematics. In the UCSMP curriculum, students work in small groups exploring mathematics in real-life contexts, using calculators, manipulatives, and other mathematical tools from kindergarten onward. In contrast to traditional curricula, students are encouraged to use these tools or to “invent” their own computational algorithms to solve problems, and the sharing of their alternative solution methods is a regular part of class discussions. Additionally, problems are nearly always application-based and never presented as sets of symbolic problems. For example, in a second-grade UCSMP lesson, students are given a picture depicting various animals and their heights or lengths. During this activity, students work in small groups to construct number stories that compare the animal heights and then to find a solution method. In the follow-up discussion, students share their stories and solution procedures and offer alternative methods.

Response: On Claims That Answer the Wrong Questions

Response: On Claims That Answer the Wrong Questions

Mythmaking and mythbreaking in the mathematics classroom

Constructivism has become a major focus of recent pedagogical reform in mathematics education. However, epistemological reform that is based on the constructivist referent of learning as conceptual change has a very limited viability in traditional mathematics classrooms because of its cultural insensitivity. By contrast, the social epistemology of critical constructivism addresses the socio-cultural contexts of knowledge construction and serves as a powerful referent for cultural reform. From this perspective, the social reality of traditional mathematics classrooms is governed by powerful cultural myths that restrain the discursive practices of teachers and students. The power of the repressive myths of cold reason and hard control is evident in the ways in which they act in concert to create a highly coherent and seemingly natural social reality. Epistemological reform of traditional mathematics classroom learning environments is, therefore, synonomous with cultural reconstruction. Critical constructivism, which has a central concern with discourse ethics and the moral agency of the teacher, draws on the social philosophy of Jurgen Habermas and argues for an alternative culture of communicative action to be established in mathematics classrooms. Teachers are expected to work collaboratively as agents of cultural change in forums beyond their classrooms.

Connections and confusion: teaching perimeter and area with a problem-solving oriented unit

Problem-solving-oriented mathematics curricula are viewed as important vehicles to help achieve K-12 mathematics education reform goals. Although mathematics curriculum projects are currently underway to develop such materials, little is known about how teachers actually use problem-solving-oriented curricula in their classrooms. This article profiles a middle-school mathematics teacher and examines her use of two problems from a pilot version of a sixth-grade unit developed by a mathematics curriculum project. The teacher's use of the two problems reveals that although problem-solving-oriented curricula can be used to yield rich opportunities for problem solving and making mathematical connections, such materials can also provide sites for student confusion and uncertainty. Examination of this variance suggests that further attention should be devoted to learning about teachers' use of problem-solving-oriented mathematics curricula. Such inquiry could inform the increasing development and use of problem-solving-oriented curricula.

The Clock Is Ticking: Time Constraint Issues in Mathematics Teaching Reform

Abstract Time issues raised by sixth and seventh-grade teachers involved in field testing an NSF-sponsored investigation-centered mathematics curriculum (the Connected Mathematics Project—CMP) for middle-grades students were examined in this study. Questions investigated included the following: How much scheduled time is actually available for mathematics instruction in elementary and middle schools and how is it configured? How do project teachers and students spend their time in class? What factors influence CMP teachers' pacing through this new curriculum? Findings indicate that teaching in the spirit of the current mathematics education reform movement may be highly dependent upon flexibility in class scheduling. Innovations in teaching mathematics (e.g., increased group work, writing, extended projects, and alternative forms of assessment) seem to require additional time, and new ways of thinking about using class time.

Reform of Mathematical Education in Primary Schools: the Experiment in Barking & Dagenham

Experimental reforms in the teaching of mathematics incorporating Continental teaching methods were begun in January 1995 in fifteen classes in six primary schools in the London Borough of Barking and Dagenham. The classes were visited in June 1996 by the Secretary of State for Education Mrs Gillian Shephard, by HM Chief Inspector of Schools Mr Chris Woodhead, and by the Opposition spokesman for education Mr David Blunkett; the media, including the BBC television programme, Panorama, provided accounts for the wider public. The reforms resulted from a wider research programme—comparing Continental and British productivity, education and vocational training—that has been under way at the National Institute for over a decade, by a research team led by SJ Prais; in recent years the research has benefited from close co-operation with the inspectorate and schools in Barking and Dagenham; this phase was funded by the Gatsby Charitable Foundation, to the Trustees of which—and especially Mr David Sainsbury for his personal encouragement—the Institute is much indebted. The commentary below outlines the background to these educational reforms, explains what has been done so far, and sets out for discussion some proposed next steps.

Changing Beliefs: Teaching and Learning Mathematics in Constructivist Preservice Classrooms

Abstract Constructivism is a learning theory which has emerged from Piagetian research. Reform in mathematics education frequently centers around constructivist principles. Many barriers prevent reform from occurring. The authors describe changes in instruction which have been implemented in their university mathematics education classrooms and relate these changes to their own constructivist philosophies. They examine the effects of change in instructional pedagogy and classroom environment on preservice teachers' beliefs about mathematics learning and teaching as well as the prospective teachers' feelings and attitudes toward mathematics.

Assessment of a Problem-Centered Mathematics Program: Third Grade

Six classes received problem-centered mathematics instruction for 2 years in second and third grade. This instruction was generally reflective of a socioconstructivist theory of knowing and compatible with recommendations for reform in mathematics education. A class-by-instruction factorial design was used to compare students in problem-centered classes for 2 years with students in problem-centered classrooms for 1 year, and with students in textbook classes for 2 years on a standardized achievement test. In addition, classes using problem-centered instruction for 2 years were compared with students in problem-centered classes for 1 year on an instrument designed to assess students' conceptual development in arithmetic and an instrument developed to examine personal goals and beliefs about reasons for success in mathematics. The results of the analyses indicate that significant differences exist in arithmetic learning for students in problem-centered classes for 2 years on the standardized achievement test and the arithmetic test. The results indicate that after 2 years of instruction in reform-based classes, students score significantly higher on standardized measures of computational proficiency as well as conceptual understanding. Additionally, these students hold stronger beliefs about the importance of finding their own or different ways to solve problems.