Four proofs, designed for classroom use in varying levels of courses on abstract algebra, are given for the converse of the classical Chinese Remainder Theorem over the integers. In other words, it is proved that if m and n are integers greater than 1 such that the abelian groups [double-struck z][subscript m] [direct sum] [double-struck…

Theories of number concepts often suppose that the natural numbers are acquired as children learn to count and as they draw an induction based on their interpretation of the first few count words. In a bold critique of this general approach, Rips, Asmuth, Bloomfield [Rips, L., Asmuth, J. & Bloomfield, A. (2006). Giving the boot to the bootstrap:…

This study examines ways of approaching deductive reasoning of people involved in mathematics education and/or logic. The data source includes 21 individual semi-structured interviews. The data analysis reveals two different approaches. One approach refers to deductive reasoning as a systematic step-by-step manner for solving problems, both in…

Students in many teacher education programmes experience confusion about portfolios. This study investigated whether using an analogy in the introduction of a portfolio helps teacher education students understand both the purpose of a portfolio and how to compile it. It was also investigated whether these students' understanding correlates with…

Gary Greenberg and Charles Lambdin's review (in the summer 2007 issue) does an excellent job of summarizing the contents of Uttal's book "Neural Theories of Mind: Why the Mind-Brain Problem May Never Be Solved" (hereafter NTM). These authors make several insightful comments about the issues raised in NTM. In this response, the author disagrees,…

Science requires imagination nourished by knowledge, experience and sustained critical thinking. Science teaching has the same requirements, but metacognition is even more important to a teacher than it is to a practitoner of science. Critical thinking is essential to both science and science teaching: in either domain, imagination relies on…

Mathematical argumentation is an important skill. It leads to the process of proof, which is one area that students are being asked to master by the end of secondary school. Encouraging explanation and justification in math class allows this skill to develop. Explaining and justifying their ideas forces students to think deeply about mathematics…

Analogical reasoning is an important component of intelligent behavior, and a key test of any approach to human language and cognition. Only a limited amount of empirical work has been conducted from a behavior analytic point of view, most of that within Relational Frame Theory (RFT), which views analogy as a matter of deriving relations among…

The author reviewed a two-part critique of dissociative identity disorder published in the "Canadian Journal of Psychiatry". The two papers contain errors of logic and scholarship. Contrary to the conclusions in the critique, dissociative identity disorder has established diagnostic reliability and concurrent validity, the trauma histories of…

Mathematics is a discipline that has significantly advanced through the use of digital technologies with improved computational, graphical, and symbolic capabilities. Digital videos can be used to present challenging mathematical questions for students. Video clips offer instructional possibilities for moving students from a passive mode of…

This study investigated how different forms of tutor action influenced novice students' performance on the Wason selection task, and how the students perceived the situation. In the control condition, the tutor provided supportive feedback which was minimal in terms of content (CG), in contrast to help in the form of directives (IM) or questions…

Evaluations of gaming simulations and business games as teaching devices are typically end-state driven. This emphasis fails to detect how the simulation being evaluated does or does not bring about its desired consequences. This paper advances the use of a logic model approach, which possesses a holistic perspective that aims at including all…

Teachers were trained to use reinforcement and induction to increase prosocial behavior in a sample of 98 children in Head Start-affiliated preschools, using a peer coaching model. There was one control group and three intervention groups: reinforcement-only, induction-only, and reinforcement-and-induction. Results indicated that the intervention…

Middle school students make great learning gains when they participate in lessons that invite them to practice their developing scientific reasoning skills; however, designing developmentally appropriate, clear, and structured lessons about scientific thinking and reasoning can be difficult. This challenge can be met through lessons that teach…

Two studies investigated the role of children's moral motivation and sympathy in prosocial behavior. Study 1 measured other-reported prosocial behavior and self- and other-reported sympathy. Moral motivation was assessed by emotion attributions and moral reasoning following hypothetical transgressions in a representative longitudinal sample of…