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Presented is a report on a Research and Development program in the use of computer-assisted instruction to teach mathematics in grades 4 through 6. Approximately 60 computer terminals of the PLATO type were installed in elementary school classrooms, 4 terminals per classroom. Each student received one-half hour of mathematics lessons, via computer, each school day, plus whatever instruction the teachers chose to provide. The CAI in question was not the usual sort, using frame-by-frame presentations of content via natural language instruction. Instead, it was based upon the Madison Project strategy of paradigmatic learning experiences, and was presented to students via a computer system having a good audio-visual interface between computer and child. Several independent third-party evaluations of the computer-based curriculum have been carried out. These evaluations indicate positive results for the curriculum in terms of achievement and attitude. (Author/MK)

It is noted that it has long seemed possible to teach mathematics in two distinct ways: (1) learning with understanding, and (2) learning without understanding. The study discussed was designed to ascertain what might be lost if "understanding" was eliminated from instruction, as compared to teaching in which the "understanding" of students was emphasized. An experimental approach was not used in this investigation. Instead, a range of pupils were observed from third grade to calculus students, with episodes identified and categorized that seemed to indicate either a presence or lack of understanding in some particular form. Further, identified behaviors were related to some basic conceptualizations of human information processing that have emerged from recent "cognitive science" studies. (MP)

The ideas and techniques involved in learning about fractions were investigated with students in grades 1-12, in the first 3 years of colleges, in community college mathematics courses, and in graduate school. Also included were some high school mathematics teachers, some mathematicians, and some retired persons. Part I provides the rationale and an overview of the project. Part II presents the theoretical background, plans for interviews, and the findings, with excerpts from tape-recorded interviews. An appendix contains transcripts of the interviews. It was concluded that nearly all students have access to good processing capabilities when dealing with concrete materials, and a major task of school programs is to build on these capabilities. The importance of good mental representations is stressed. (MNS)

Discusses why improvements in teaching mathematics are needed, including discussion of students, human development, knowledge and learning, structure of schools and instructional programs, and mathematics. Also discusses a need for two theories: (1) how young people form self-concepts and (2) classification of approaches to curricula and demographics. (81 references) (MKR)

Research on mathematics instruction is reviewed in order to respond to two questions: (1) Has the influx of talented people who have entered the mathematics instruction field over the last three decades changed anything? and (2) Will any of the work being done actually improve mathematics instruction? The different ways in which parents, students, teachers, non-teaching professionals, and other groups perceive and use mathematics is discussed to illustrate the sometimes conflicting views of mathematics and how these views affect instruction. Ten statements, based on analysis of research on diagnosis and evaluation in mathematics, are discussed: (1) Schools mostly teach rote aspects of mathematics and neglect strategy, analysis, heuristics, decision making, intuition, flexibility, and creativity; (2) Typical students are creative and can be resourceful in mathematics if appropriately taught; (3) Most instructional programs do not recognize students' creativity; (4) Students are confused about mathematics; (5) Complex mathematical ideas can be taught through building upon previous learning; (6) Consistent errors signal misunderstanding; (7) Representations determine problem-solving ability; (8) Students solve problems through stimulus-response; (9) Mental tasks in mathematics are diverse; and (10) Many students can learn substantially more mathematics than they presently do. (CJ)