Describes and analyzes a learning episode in which two children in year 4 interact with each other and their teacher while playing a probability game involving chocolate M&Ms. Children developed key ideas in probability from a game that was designed to produce cognitive conflict. (ASK)

This study compared the spatial-geometrical abilities of 85 students (ages 5-13) with learning disabilities (LD) and 94 children without LD, matched for IQ and age. Generally, students with LD did not perform as well as same-aged students without LD, suggesting that LD students experience delayed development in this general spatial-cognitive…

Presents a theoretical framework based on interviews with undergraduate students and written questionnaires to study how undergraduate students cope with abstract algebra concepts. Indicates that students' responses can be interpreted as a result of reducing the level of abstraction. Examines the theme of reducing abstraction based on three…

Presents the first of two articles examining Alfred North Whitehead's notion of the rhythm of the university, discussing the rhythm of teaching and learning, the importance of academic freedom to an imaginative faculty, and the relationship between Whitehead's own pedagogy and his considerable administrative practice. The paper's purpose is to…

We comment on the paper "The co-emergence of machine techniques, paper-and-pencil techniques, and theoretical reflection: A study of CAS use in secondary school algebra" by Carolyn Kieran and Paul Drijvers. We look at aspects of Kieran and Drijvers' analysis with regard to "task-technique-theory" in the light of a model of abstraction in context…

As engineers gain experience and become experts in their domain, the structure and content of their knowledge changes. Two studies are presented that examine differences in knowledge representation among freshman and senior engineering students. The first study examines recall of mechanical devices and chunking of components, and the second…

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…

The author believes that very young children are able to understand the abstract concept of peace. In her primary classroom she introduces the concept of peace to the children in a low energy environment with low lights, and soft music. When children feel at peace in their hearts, they relate peacefully to those around them. She begins with the…

A number of thought experiments are cited, some well-known, some not. These illustrate the power of thought experiments. Other examples are given that show some of the dangers. As well as examples from the science, some examples of visual reasoning from mathematics are also presented, again with an eye to illustrating their promise and perils.…

We applied a mixed-method research design to investigate the patterns of reasoning used by novice undergraduate chemistry students to classify chemical substances as elements, compounds, or mixtures based on their particulate representations. We were interested in the identification of the representational features that students use to build a…

Science classes are full of abstract or challenging concepts that are easier to understand if an analogy is used to illustrate the points. Effective analogies motivate students, clarify students' thinking, help students overcome misconceptions, and give students ways to visualize abstract concepts. When they are used appropriately, analogies can…

In his influential book "Dialogues with Children" (1984), Gareth Matthews presents strong evidence that children as young as six can and should engage in philosophical discourse. His findings form the basis of this article, a theoretical and practical guide for parents and teachers seeking respectful and creative ways to encourage…

Findings from an on-going design experiment within a year-long graduate course for middle school teachers of mathematics are reported. The purpose of the course was to help teachers assist students in transitioning from arithmetic to algebraic reasoning. Goals included developing teachers' ability to interpret, compare, and generalize across…

A model for processes of abstraction, based on epistemic actions, has been proposed elsewhere. Here we apply this model to processes in which groups of individual students construct shared knowledge and consolidate it. The data emphasizes the interactive flow of knowledge from one student to the others in the group, until they reach a shared…

Cross-sectional and longitudinal data from students as they advance through the middle school years (grades 6-8) reveal insights into the development of students' pattern generalization abilities. As expected, students show a preference for lower-level tasks such as "reading the data," over more distant predictions and generation of abstractions.…