In my Keynote Address to PME-NA 45, I offer an embodied framework for naming what makes mathematics powerful for mathematicians and scientists, yet intractable for many learners. The essential claim is this: Students reside in the Real World, where math is grounded, embodied and meaningful, while mathematics resides in the ungrounded, disembodied…

People spontaneously discover new representations during problem solving. Discovery of a mathematical representation is of special interest, because it shows that the underlying structure of the problem has been extracted. In the current study, participants solved gear-system problems as part of a game. Although none of the participants initially…

Seven principles are presented for improving creative thinking, based on assumptions of creativity as a perceptual shift resulting from a metamorphic mental image. Principles include (1) the future initiates and pulls creative thought; (2) initial fact finding is best postponed; (3) problem redefinition is often retrospective; and (4) metaphors…

Pullout pages provide suggestions for teaching elementary students at all levels about the wonders of human ingenuity. The suggestions help students see that great ideas come from the need to solve real-life problems and that one great idea leads to another, so current inventions help predict future inventions. (SM)

The book "Productive Thinking", by Gestalt psychologist Max Wertheimer, was published over 50 years ago, yet it continues to wield influence. The book's argument on productive thinking, and why it is still relevant today, are the focus of this paper. Productive thinking involves going from a situation of bewilderment or confusion about some issue…

A Discovery Unit was designed to develop problem-finding behavior in 67 gifted children (grades 3-6). Students identified a problem and its solution using topics of their choice. The results of this two-year pilot project support the need to instruct and practice problem-finding skills at the elementary level. (DB)

Describes grade-appropriate weekly activities designed to appeal directly to students by giving them open-ended questions. Activities may be worked on individually, in pairs, or in small groups. This particular set of activities focuses on stories from literature involving measurement, earning money, marketing a product, and setting up a…

This document reports on two 3-day workshops designed to train three vocational education instructors or curriculum developers from every Wisconsin school district in specific critical thinking skills strategies that they could implement in their districts. The document includes a three-page narrative description of the workshops that includes…

Describes three activities that the teacher can employ to help students develop thinking skills through mathematics instruction: (1) memorization using the technique of chunking; (2) higher order thinking with magic squares; and (3) predicting games. Identifies eight facets of the teacher's role in promoting thinking skills. (MDH)

Highlights combinatorial problems appropriate for students aged 4 to 12 years that utilize manipulatives in a hands-on approach. Examines and identifies students' strategies and self-monitoring techniques that produce effective problem solving. (MDH)

Describes an activity that demonstrates a teaching-learning model in which students ask yes or no questions to determine an unknown number on a number line. Provides a sequence of five steps to carry out the activity and suggestions for extending and expanding the activity for different grade levels. (MDH)

Discusses how a single problem can be utilized to discuss mathematical skills and concepts at several different levels. Presents a problem involving fractions examined at the following four levels according to the abilities of the students: (1) finding solutions; (2) identifying methods for finding solutions; (3) exploring similar examples; and…

Preservice elementary teachers worked cooperatively to explore the relationship between the measure of the central angle and the number of sides of regular polygons using the reflection of a line in a hinged mirror. The experience involved conjecture and experimentation, communication, and building connections between science, geometry, and…

This document examines how a unit on model rockets, designed to be a "hands-on" activity within the "Mission to Mars" curriculum that was implemented in the Nashville (Tennessee) area middle schools, has been used to investigate children's understanding of experimentation. A literature review explores some of the traditional…

This book attempts to introduce students to the creative aspects of mathematics through exploratory problems. The introduction presents the criteria for the selection of the problems in the book. Criteria indicate that problems should: be immediately attractive, require data to be generated or gathered, appeal to students from junior high school…