Three issues concerning the relationship between research and practice are addressed. (1) A certain "prototype mathematics classroom" seems to dominate the research field, which in many cases seems selective with respect to what practices to address. I suggest challenging the dominance of the discourse created around the prototype mathematics…

The technique of using "seed problems" involves starting with a simple problem and seeing how far it leads. It may inspire different, related problems, or it may be approached with varied tools. Problems related to points and lines, pyramids, Pythagorean quadruples, and regular pentagons are discussed. (MNS)

Discussed is changing from the lecture-and-demonstration approach to teaching which treats learning as primarily the transfer of information. The approach used in a school in New Jersey is described in detail. (MNS)

Discussed are the range and balance of learning activities in the classroom, including discussion of the various roles of the people involved. The analysis is based on records of observations of 174 mathematics lessons with students aged 12-14; microcomputer use was integrated as a teaching aid. (MNS)

Discussed is the behavior displayed by good problem solvers of feeling free to restructure, revise, or redefine a problem. Examples of student work are included. (MNS)

Why students have difficulty with a proof (such as Cantor's) is discussed, with the focus on proof by contradiction. Methods may fail due to the difficulty of the concept and lack of understanding of how students are thinking. (MNS)

The reversal phenomenon was studied with nearly 1,000 Australian students in grades 9-12. The insertion of a multiplication sign in an equation tended to lower the incidence of reversals, but switching from initial letters to other letters made little difference. (MNS)

The status of mathematics instruction, especially in the elementary school, is discussed. A meaningful, holistic approach is advocated, rather than an emphasis on rules and procedures. (MNS)

Three schools with differing ways of promoting excellence are described. Background, distinctive features, curriculum features, teacher behaviors in class, and teacher activities out of class are each discussed. (MNS)

The range of problems associated with learned helplessness in mathematics is introduced through three hypothetical case studies. Then the basic theory of, the evidence for, and variables affecting learned helplessness are described. Issues of cure and prevention are discussed, and some suggestions are offered for teachers. (MNS)

Examples of relatively novel computer software environments from the representation perspective are described. Even more novel approaches to curriculum reform to cultivate higher-order thinking skills are then discussed. (MNS)

The discussions of the theme group on Technology and Cognitive Development at the Fifth International Congress on Mathematical Education are summarized. How computers can be used to engage students actively, to promote problem-solving skills, and to achieve better understanding is discussed. (MNS)

How multiplication is usually taught in school and how it could be taught are discussed. Development of understanding is illustrated through children's words and work. (MNS)

How the notion of limit can be developed through a meaningful approach is discussed. Selected portions of the high school calculus course are described, and errors on a test are analyzed. (MNS)

A novel approach to teaching mathematical induction was used, based on a Piagetian theory of learning abstract mathematical concepts in which the learner uses reflective abstraction to construct new schemas out of old ones. Computer experiences are used to induce students to make the appropriate reflective abstractions. (MNS)