An understanding of fraction addition can be thought to involve two quantitative ideas: (1) the understanding that adding to an original quantity increases its size, and (2) a sense of how much increase occurs. Both of these ideas should underlie or inform a child's approach to problems involving fraction addition and thereby constrain the class…

The author defines and discusses the cognitive theory of constructivism as it relates to teaching mathematics. It is suggested that the philosophical and theoretical view of knowledge and learning embodied in constructivism offers hope that educational processes will be discovered enabling students to acquire deep understanding rather than…

Recent contributions from theory, research, and practice in second-language instruction are discussed in relation to mathematics education. Three methods of teaching and learning second languages are described--"Delayed Oral Production," the "Silent Way," and the "Counseling Learning/Community Language Learning." (KR)

Discussed are some situations students face that result in cognitive conflict, possible sources of these conflicts, and strategies which students use to resolve, remove, or circumvent them. A global account for observed systematic errors is offered based on a general problem-solving rule called the "Matching Rule." (KR)

Investigated the upper limits of (n=22) seventh-grade students' informal processes in solving first degree equations in one unknown prior to instruction. Results indicated the existence of a cognitive gap between arithmetic and algebra characterized as the students' inability to operate spontaneously with or on the unknown. (37 references)…

Described are inconsistencies, definitions, and examples and their complex relationship which can be used to interpret students' reactions to the geometric tasks used to investigate inconsistencies in student thinking. Discusses the nature of definitions, the value of precise vocabulary, the use and limitations of prototypes, and the power of…

After observing secondary school students having great difficulty learning geometry in their classes, Dutch educators Pierre van Hiele and Dina van Hiele-Geldof developed a theoretical model involving five levels of thought development in geometry. It is the purpose of this monograph to present English translations of some significant works of the…

Discussed are inconsistencies and cognitive conflict with respect to current mathematical knowledge of students and how that knowledge might be modified is discussed. The inconsistencies that students generate for themselves and those produced by the teacher are described. (KR)

This handbook has been structured to provide teachers (K-2) with the most recent research on problem solving. It contains information and practical applications which deal with the key concepts of problem solving in mathematics. The handbook consists of four sections. The first section establishes the background and gives the purposes of the…

Discussed are possible reasons behind the inconsistencies in the learning of calculus. Implicated are students' beliefs, mathematical paradigms including concept image and concept definition, language use, and curriculum sequencing. (KR)

Discussed are the issues of inconsistencies and compartmentalization and the use of inconsistencies in the classroom. Included are discussions on paradoxes, the naive student, logical reasoning ability, and the development of the students' cognitive structures. (KR)

Discusses a system of classification for students' inconsistent ideas in mathematics, the sources of students' mathematical inconsistency, and instructional strategies for helping students to resolve apparent learning inconsistencies. Language, curriculum, and instruction are included in the discussion. (KR)

This paper begins with a summary and analysis of Robert Glaser's arguments, presented in his paper "Education and Thinking: The Role of Knowledge," which contend there are strong interactions between structures of knowledge and cognitive processes, and that problem solving and reasoning are taught best in the context of the acquisition…