Efforts to meet the needs of children's learning in arithmetic has led to an increased emphasis on the teaching of mental calculation strategies in England. This has included the adoption of didactical tools such as the empty number line (ENL) that was developed as part of the realistic mathematics movement in the Netherlands. It has been claimed…

Students of Architecture and Building Engineering Degrees work with Computer Aided Design systems daily in order to design and model architectonic constructions. Since this kind of software is based on the creation and transformation of geometrical objects, it seems to be a useful tool in Maths classes in order to capture the attention of the…

The main purpose of this note is to present and justify proof via iteration as an intuitive, creative and empowering method that is often available and preferable as an alternative to proofs via either mathematical induction or the well-ordering principle. The method of iteration depends only on the fact that any strictly decreasing sequence of…

The general purpose of this article is to shed some light on the understanding of real numbers, particularly with regard to two issues: the real number as the limit of a sequence of rational numbers and the development of irrational numbers as a continued fraction. By generalizing the expression of the golden ratio in the form of the limit of two…

Based on relativistic velocity addition and the conservation of momentum and energy, I present simple derivations of the expressions for the relativistic momentum and kinetic energy of a particle, and for the formula E = mc[superscript 2]. (Contains 5 footnotes and 2 figures.)

The reactions by Ridley and Shaw-Ridley (EJ832451) and Lichtenberg (EJ832452) to the authors' meta-analysis on the effects of experience on judgment accuracy add positively to what is hoped will become an ever more focused discourse on this most basic question: How can mental health clinical decision making be improved? In this rejoinder, the…

This article explores the notion of collateral learning in the context of classic ideas about the summation of powers of the first "n" counting numbers. Proceeding from the well-known legend about young Gauss, this article demonstrates the value of reflection under the guidance of "the more knowledgeable other" as a pedagogical method of making…

The study explored the effects of a computer-assisted COnceptual Model-based Problem-Solving (COMPS) program on multiplicative word-problem-solving performance of students with learning disabilities or difficulties. The COMPS program emphasizes mathematical modeling with algebraic expressions of relations. Participants were eight fourth and fifth…

The topic of coupled oscillations is rich in physical content which is both interesting and complex. The study of the time evolution of coupled oscillator systems involves a mathematical formalization beyond the level of the upper secondary school student's competence. Here, we present an original approach, suitable even for secondary students, to…

The derivative of a composite function, taken with the chain rule is one of the important notions in calculus. This paper describes a study conducted in Turkey that shows that the chain rule was given with the formula in function notation and/or the Leibniz notation without relating these formulas to life-related problem situations in the…

Questions are used to obtain information, stimulate thinking, and redirect reasoning. Academics in higher education use questions on a daily basis to stimulate thinking and reasoning in students. Final examination papers are used by academics to assess the retention and application skills of students. The assumption, however, exists that questions…

This paper suggests two new, related methods for estimating a test-score equating relationship from small samples of test takers. These methods do not require the estimated equating transformation to be linear. Instead, they constrain the estimated equating curve to pass through 2 prespecified end-points and a middle point determined from the…

Communicating and collaborating online are becoming common requirements in education. The specialized notations and symbols necessary for some content areas where mathematical expressions are part of the conversation require that users select appropriate communication tools. The purpose of this article is to describe tools that can be used for…

The assessment of knowledge and learning progress in the context of game-based learning requires novel, noninvasive, and embedded approaches. In the present paper, we introduce a mathematical framework which relates the (problem solution) behavior of a learner in the game context to the learner's available and lacking competencies. We argue that a…

One of the difficulties in any teaching of mathematics is to bridge the divide between the abstract and the intuitive. Throughout school one encounters increasingly abstract notions, which are more and more difficult to relate to everyday experiences. This article examines a familiar approach to thinking about negative numbers, that is an…