This inquiry raises questions about the manner in which the No Child Left Behind Act aims to improve mathematics education through continued reliance on standardized testing and mandated use of scientifically based teaching practices. Specifically, it is argued that this approach is tied to assumptions about intellectual ability and achievement that precipitated the dividing practices used to justify differential access to mathematics learning almost a century ago. An examination of so-called objective and scientific approaches to school mathematics suggests the need for more earnest reflection about the particular path toward educational progress privileged by this legislation.
Mathematics enables us to fly to the moon, track our genetic codes, create beautiful music, design our cars, build our houses, and contact others around the world almost instantaneously. However, mathematics, that abstract language which helps us to access the relationships in our physical universe(s), is rarely invoked in the service of preparing young people for democratic participation. Deborah Ball and Hyman Bass take on the challenge of situating the highly revered, somewhat mystical discipline of mathematics as a key contributor to concepts of democracy.
The article describes results of a qualitative study of an elementary school as it enacted a variety of strategies to help students raise scores on a newly required state test designed to serve as an accountability tool.
Observations of the first days of school in eight sixth-grade classrooms identified three different classroom environments.
This study examined the effects of early acceleration of students in mathematics on the development of their attitude and anxiety toward mathematics across junior and senior high school, using data from the Longitudinal Study of American Youth (LSAY).
The first main section of the chapter, The evolving content(s) of mathematics instruction, discusses changes in the mathematics curriculum over the course of the century. The second main section, On research: psychological, epistemological, and methodological issues, describes the development of the discipline of research in mathematics education. A concluding discussion, And next?, discusses some of the evolutionary needs and pressures that may shape mathematics education in the early parts of the twenty-first century.
Our goal in this chapter is to provide an accurate characterization
of the current state of elementary and middle school mathematics education
in the United States and to highlight crossroads points where
crucial reevaluations are needed.
As one constructivist remarked, In summary
then the term constructivism appears to be fashionable, mostly
used loosely with no clear definition of the term, and is used without
clear links to an epistemological base. Although there are countless
thousands of constructivist articles, it is rare to find ones with fully
worked out epistemology, learning theory, educational theory, or ethical
and political positions. This makes appraisal difficult.
In this chapter, we examine the construction of mathematical knowledge in classroom teaching and learning.
Drawing upon a teacher survey, this article proposes that successful instructional policies are themselves instructional: teachers opportunities to learn about and from policy influence both their practice and, at least indirectly, student achievement.
Drawing on the Longitudinal Study of American Youth, this study examines the effects of individual characteristics and different types of parental involvement on student participation in advanced mathematics.
The author evaluates and extends the literature demonstrating that gender differences on standardized tests of quantitative reasoning may reflect underlying differences in cognitive processing that might be explained in part by socialization patterns inherent in American culture.
Background to the key notions underlying the National Council of Teachers of Mathematics (NCTM) Curriculum and Evaluation Standards for School Mathematics
One mathematician's view of the NCTM standards document
A discussion of the danger of losing the essence of mathematics
The author explores further the areas of agreement and disagreement across the articles about the general issue of active learning, teaching for understanding, or, at the risk of raising a red flag, constructivism.
The authors have selected three trends as points of discussion for this special section of the Record featuring expert commentary on the condition of education in the nation. The trends address the following issues: (1) appraising the state of academic achievement among American elementary and secondary students, (2) improving the quality of the nations K-12 faculty, and (3) the march toward the expansion of higher education for all students.
One of two articles on academic achievement, this article discusses the performance of U.S. students on mathematics and science achievement tests compared to students in other countries. Recent data show they are performing better. This commentary examines the importance of meeting certain research conditions before drawing conclusions about achievement trends.
This article presents data on trends in mathematics and science achievement among U.S. students according to the National Assessment of Educational Progress and various international studies.
An examination of curriculum, instruction, and organizational formats in multi-grade and single-grade mathematics classes
This article focuses on opportunity-to-learn standards, which define a set of conditions that schools, districts, and states must meet in order to ensure students an equal opportunity to meet expectations for their performance.
Some mathematics should be taught to all students, but an adequate
presentation of a "common curriculum" for mathematics cannot
consist of a list of topics to be covered, however extensive and
carefully prepared. I use the word "curriculum" as a course of study,
its contents, and its organization, and my task in this chapter is to
consider four questions which shape an outline for a common curriculum
for mathematics. The questions to be examined are:
1. What does it mean to know mathematics?
2. Who decides on the mathematical tasks for students and for what
3. What should be the principles from which a common curriculum
can be built?
4. For this yearbook, how should individual differences be considered?
One of the most valuable types of intellectual talent for both
society and the individual is mathematical reasoning ability. It
undergirds much of current achievement in technology, science,
and social science. Usually this ability is poorly assessed by in-school
mathematics tests, because often they consist of a mixture of computation,
learned concepts, and reasoning. Also, it is difficult to
measure mathematical reasoning ability until the young student has
acquired enough knowledge of elementary general mathematics
with which to reason. The basic content of the test items must be
fairly well known so that reasoning can be the chief trait measured.
Both mathematicians and historians are familiar with the fact that
the evolution of counting was an extremely slow process. This has
been ascertained by piecing together remains of ancient cultures
turned up by archeologists and by observing the counting habits of
those primitive societies that still existed during the last century.
This chapter will not attempt a comprehensive review of literature
in the psychology of mathematics learning. It will instead examine
some of the current theoretical issues in psychology that have
relevance to education in mathematics, citing empirical literature
only for illustrative purposes.
This is a brief description of the development of the number system.
This presentation requires very little mathematical background; it is an
attempt to provide, for nonspecialists, an overall view of the successive
enlargements of the concept of number which take place in the early
school years. The approach we use is very near that of the best currently
available commercial textbooks. These comprise the so-called second
generation new math programs. But we do not discuss current and projected
In this chapter, the content of the geometry and measurement program
at the various grade levels will be examined, and a rationale will be
offered for what is being done and for what is being proposed for the
The purpose of the present article is to focus on the function concept,
to describe several of its aspects, and to chart some of its appearances in the mathematics curriculum. At the start, we shall rely upon examples;
the modem concept of function will not emerge in its final form until
the end of the chapter, where we enter more deeply into the technical
In human affairs, there seems to be a tendency to swing from one
extreme to another. Often, having swung violently between two
extremes, man finds himself coming to rest somewhere in the middle,
like a pendulum that has stopped oscillating. In mathematics education,
this pendulum-like action has occurred with respect to many
issues. In this chapter we will examine some of those issues.
The main part of our discussion appears under the next four major
headings of this chapter and will be concerned with the undergraduate
training of teachers at Levels I-III with particular attention
to Level III. The first of these four sections, the one immediately
following the Introduction, will describe briefly the situation prior to 1961; the next section will outline the CUPM recommendations;
and the following two will respectively (a) survey the present situation
and (b) discuss deficiencies in the present program and attempt
some predictions as to future developments.