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Wilson, P. S. (Ed.)(1993). New York: MacMillan.
Educational research is conducted within a variety of constraints  isolation of variables, availability of subjects, limitations of research procedures, availability of resources, and balancing of priorities. Various research methodologies are used in mathematics education research including a clinical approach that is frequently used to study problem solving. Typically, mathematical tasks or problem situations are devised, and students are studied as they perform the tasks. Often they are asked to talk aloud while working or they are interviewed and asked to reflect on their experience and especially their thinking processes. Waters (48) discusses the advantages and disadvantages of four different methods of measuring strategy use involving a clinical approach. Schoenfeld (32) describes how a clinical approach may be used with pairs of students in an interview. He indicates that "dialog between students often serves to make managerial decisions overt, whereas such decisions are rarely overt in single student protocols."
Percentages of U.S. High School Students Enrolled in Various Courses
The basis for most mathematics problem solving research for secondary school students in the past 31 years can be found in the writings of Polya (26,27,28), the field of cognitive psychology, and specifically in cognitive science. Cognitive psychologists and cognitive scientists seek to develop or validate theories of human learning (9) whereas mathematics educators seek to understand how their students interact with mathematics (33,40). The area of cognitive science has particularly relied on computer simulations of problem solving (25,50). If a computer program generates a sequence of behaviors similar to the sequence for human subjects, then that program is a model or theory of the behavior. Newell and Simon (25), Larkin (18), and Bobrow (2) have provided simulations of mathematical problem solving. These simulations may be used to better understand mathematics problem solving.
The 1980s: Prelude to National Standards
Constructivist theories have received considerable acceptance in mathematics education in recent years. In the constructivist perspective, the learner must be actively involved in the construction of one's own knowledge rather than passively receiving knowledge. The teacher's responsibility is to arrange situations and contexts within which the learner constructs appropriate knowledge (45,48). Even though the constructivist view of mathematics learning is appealing and the theory has formed the basis for many studies at the elementary level, research at the secondary level is lacking. Our review has not uncovered problem solving research at the secondary level that has its basis in a constructivist perspective. However, constructivism is consistent with current cognitive theories of problem solving and mathematical views of problem solving involving exploration, pattern finding, and mathematical thinking (36,15,20); thus we urge that teachers and teacher educators become familiar with constructivist views and evaluate these views for restructuring their approaches to teaching, learning, and research dealing with problem solving.
The National Science Foundation
Progressive education was forced into retreat in the 1950s, and evenbecame the butt of jokes and vitriol.^{29} During the previoushalf century, enrollment in advanced high school mathematics courses,andother academic subjects, had steadily decreased, thanks at least inpartto progressive education. From 1933 to 1954 not only did the percentageof students taking high school geometry decrease, even the actualnumbersof students decreased in spite of soaring enrollments. The followingtablegives percentages of high school students enrolled in high school mathcourses.^{30}
There are 39 articles in this section.
The University of Illinois Committee on School Mathematics headed byMax Beberman began in 1951 and was the first major project associatedwiththe New Math era. Beberman's group published a series of high schoolmathtextbooks, and drew financial support from the Carnegie Corporation andthe U.S. Office of Education. In 1955, the College Entrance ExaminationBoard established a Commission on Mathematics to investigate the"mathematicsneeds of today's American youth." The Commission, consisting of highschoolteachers, math educators, and mathematicians, issued a report withrecommendationsfor a curriculum to better prepare students for college, and produced asample textbook for twelfth grade on probability and statistics.^{33}The efforts of these and other early groups received little attentionuntilthe U.S.S.R launched , the first space satellite, in thefall of 1957. The American press treated as a majorhumiliation,and called attention to the low quality of math and science instructionin the public schools. Congress responded by passing the 1958 NationalDefense Education Act to increase the number of science, math, andforeignlanguage majors, and to contribute to school construction.
TERC's Investigations in Number, Data, and Space (K5)
The National Council of Teachers of Mathematics set up its owncurriculumcommittee, the Secondary School Curriculum Committee, which came outwithits recommendations in 1959. Many other groups emerged during thisperiodincluding, the Ball State Project, the University of MarylandMathematicsProject, the Minnesota School Science and Mathematics Center, and theGreaterCleveland Mathematics Program. In the late 1950s, individual highschooland college teachers started to write their own texts along the linessuggestedby the major curriculum groups.^{35}
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