This in-depth examination of a line explores several ways that the infinite nature and finite representation of a line can be perceived. These attempts to understand the nature of the line give insights into the nature of understanding itself. (MDH)

Discussed is the importance of establishing the link between students' understanding of mathematics in real world settings and the symbolism used to represent that mathematics. Examples provide evidence that students and sometimes teachers fail to establish that link. References are given for resources providing strategies and contexts to assist…

This paper discusses the various aspects of mathematics teaching and learning at the elementary school level. The National Council of Teachers of Mathematics (NCTM) standards are reviewed by giving examples and a list of principles of learning in mathematics is offered. Principles include the following: 1) the need for pupils to perceive learning…

Several common errors reflecting difficulties in probabilistic reasoning are identified, relating to ambiguity, previous outcomes, sampling, unusual events, and estimating. Knowledge of these mistakes and interpretations may help mathematics teachers understand the thought processes of their students. (MNS)

Objectives are defined in a manner that eliminates many of their attributes that have been criticized as being harmful or unappealing. Examples of objectives appropriate for secondary school mathematics are given and the relationship between classroom processes and objectives is discussed. (MK)

Use of manipulatives is neither a sufficient nor a necessary condition for meaningful learning. Provides some incidents in support of the argument. Lists 14 references. (YP)

Discusses the organization and retrieval of information. Describes the tip-of-the-tongue state during mathematics problem solving. Provides five rules for a deep level of processing of new concepts. (YP)

Presents young children's concepts related to probability grouped by definite, possible, and definitely not. Discusses the teaching methods of the probability concepts. (YP)

It is difficult for students to unlearn misconceptions that have been unknowingly reinforced by teachers. The examples "multiplication makes bigger,""pi equals 22/7," and the use of counter examples to demonstrate the numerical property of closure are discussed as potential areas where misconceptions are fostered. (MDH)

What is meant by meaningful instruction is discussed in terms of interest and relevance, pseudo-real problems, unconscious reinforcement, stories for symbols, and mathematical mirrors. That mathematics must be talked about with and by children is emphasized. (MNS)

Discusses research findings that relate to teaching fractions for conceptual understanding. Gives teacher/student dialogues that illustrate the thought processes of students as they form part-whole and improper fraction concepts. (MDH)

This book introduces journal-writing prompts in mathematics for Algebra I. Content prompts, attitudinal or affective prompts, and process prompts are presented. Each of the content prompts relates or connects topics within and outside of mathematics, targeting important or meaningful concepts and skills. The content prompts also provide situations…

This book introduces journal-writing prompts in mathematics for geometry. Content prompts, attitudinal or affective prompts, and process prompts are presented. Each of the content prompts relates or connects topics within and outside of mathematics, targeting important or meaningful concepts and skills. The content prompts also provide situations…

Presents a historical overview of learning theories in mathematics education applied to the teaching of integers. Theories discussed are meaning vs. rote learning; direct vs. incidental; discovery vs. exposition; concrete vs. abstract; and construction vs. transmission. Includes a reproducible activity sheet and 19 references. (MKR)

This paper presents a theoretical framework for investigating the role of algorithms in mathematical thinking using a combined ontological-psychological outlook. The intent is to demonstrate that the processes of learning and of problem solving incorporate an elaborate interplay between operational and structural conceptualizations of the same…