The notions of "abstract" and "concrete" are central to the conceptualization of mathematical knowing and learning. Much of the literature takes a dualist approach, leading to the privileging of the former term at the expense of the latter. In this article, we provide a concrete analysis of a scientist interpreting an unfamiliar graph to show how…

This article was written as a result of the author reading "MT177," a special issue dedicated to the teaching of "proof" in mathematics. He used the ideas in this special issue for planning his session "mathematical reasoning and proof," which was part of a weekend course for primary trainees. It consisted of three activities: (1) How many…

Early studies of intuitive judgment and decision making conducted with the late Amos Tversky are reviewed in the context of two related concepts: an analysis of accessibility, the ease with which thoughts come to mind; a distinction between effortless intuition and deliberate reasoning. Intuitive thoughts, like percepts, are highly accessible.…

In 3 experiments, affirmative and hypothetical probes were presented after narrative texts containing conditional arguments. According to the data, readers represented modus ponens deductions as certain, except when it was only a weakly necessary cause of a given effect. They represented any logically invalid inferences resulting from affirming…

Certain definitions introduce appropriate concepts, among which are the definitions of the counting functions of the primes and twin primes, along with definitions of the correlation coefficient in a bivariate sample space. It is argued conjecturally that the characteristic functions of the prime "p" and of the quantity "p"+2 are highly…

This research emerged from the Social Science Research Council's collaborative partnership with the Pathways for College Network, with technical assistance in data collection provided by the Council for Aid to Education. The project has followed over 2,300 students at 24 institutions over time to examine what factors are associated with learning…

The present study investigated the effects of deductive and inductive teaching approaches to the acquisition of pragmatic competence on learners of English as a foreign language. In this study, 60 adult native speakers of Japanese with intermediate-level proficiency in English were each randomly assigned to 1 of 4 groups, which consisted of 3…

Abstraction is a major concept in computer science and serves as a powerful tool in software development. Pattern-oriented instruction (POI) is a pedagogical approach that incorporates patterns in an introductory computer science course in order to structure the learning of algorithmic problem solving. This paper examines abstraction processes in…

Over the past few decades there has been a rise in the number of federally funded large scientific research initiatives, with increased calls to evaluate their processes and outcomes. This article describes efforts to evaluate such initiatives in one agency within the U.S. federal government. The authors introduce the Evaluation of Large…

To reflect on the mathematics achievement of children engaging "Algebra and Analogies for Kids," the teacher researcher will explore the following inquiry: What happens when kids display analogical thinking in an algebra project? Specifically, the proposed study will examine the reasoning of five children in portfolios of best works and selected…

It has often been claimed that children's mathematical understanding is based on their ability to reason logically, but there is no good evidence for this causal link. We tested the causal hypothesis about logic and mathematical development in two related studies. In a longitudinal study, we showed that (a) 6-year-old children's logical abilities…

The social transformations in Russian society in the past two decades have made relevant the problem of paradoxality, including its application to young people. The results of many years of sociological studies by the authors investigating the social problems of young people completely confirm Toshchenko's conclusion that "paradoxality of…

This study examines the role of writing in learning scientific principles through analogy. Seventy-two university students observed two demonstrations concerning one of three topics: buoyant force of a fluid, projectile motion or forces internal to a system. Each composed an analogy on one of the topics through speaking-only, writing-only, or…

Electronic game technologies can prepare novice learners for future learning of complex concepts. This paper describes the underlying instructional design, learning science, cognitive science, and game theory. A structural, or syntactic mapping (structure mapping), approach to game design can produce a game world relationally isomorphic to a…

The mathematical performance of Chinese students, from mainland China, Hong Kong and Taiwan, is widely acclaimed in international comparisons of mathematics achievement. However, in the eyes of the Western educators, the environments established in Chinese schools are deemed relatively unfavourable for mathematics learning. This paper reports on a…