In this study, we present a mathematical analysis distinguishing two conceptions of equivalence: "proportional equivalence" and "unit equivalence." These two conceptions have distinct meanings in relation to equivalent fractions: one is grounded in proportionality, while the other is grounded in equal wholes. We argue that (a)…

Fractions are an imperative skill for students to master to achieve success in future mathematics concepts and classes. Yet many students, especially those experiencing math difficulties and/or characteristics of emotional and behavioral disorders, continue to exit elementary school without a concrete foundation of fractions skills required to…

In this article, I argue that the common practice across many school mathematics curricula of using a variety of different representations of number may diminish the coherence of mathematics for students. Instead, I advocate prioritising a single representation of number (the number line) and applying this repeatedly across diverse content areas.…

Learning the concept of fractions is among the most challenging topics in school mathematics. One of the main sources of difficulties in learning fractions is related to "natural number bias" (Van Hoof, Verschaffel & Van Dooren, 2015). Applying properties of the natural numbers incorrectly in situations involving rational numbers can…

Nat Banting, argues that the "invert and multiply" algorithm for division of fractions is, in fact, not an algorithm. He suggests that Liping Ma's (1999) algorithm, which involves writing the fractions with a common denominator and then dividing the numerator of the first fraction by the numerator of the second fraction, is better as it…

In this paper, we explore theoretical approaches to Japanese teachers' lesson designs involving the adaptation of mathematics textbooks for instructional change. In Japan, although most teachers use textbooks as the main resource for lesson planning, called "kyouzai kenkyuu," the ways in which they use textbooks can differ. In a…

Many students face difficulties with fractions. Research in mathematics education and cognitive psychology aims at understanding where and why students struggle with fractions and how to make teaching of fractions more effective. Additionally, neuroscience research is beginning to explore how the human brain processes fractions. Yet, attempts to…

The proliferation of computers, tablets, and internet access has brought the use of virtual manipulatives into the majority of classrooms in the developed world. In responding to the needs of today's students, many of whom are adept at accessing and manipulating technology devices, virtual manipulatives provide a variety of classroom…

Existing frameworks of teachers' knowledge required to teach mathematics do not adequately capture the dynamic aspects of knowledge manifested in teaching practice. In this paper, we examine the knowledge demands that arise in situ, in the course of a teacher listening and responding to students' thinking, while teaching the topic of decimal…

Originally written 30 years ago, this paper is an analysis of the central challenge of schooling--that of engaging fully the powers of students' minds in classroom learning. This challenge maintains its relevance today. The work of engaging what John Dewey referred to as students' "inner attention" becomes the focus of an investigation…

This paper addresses the continuing need for mathematics teachers to enrich their mathematical knowledge beyond the school curriculum, in order to effectively engage students in creative and imaginative thinking, particularly, but not exclusively, students who show exceptional promise. The author, a retired university professor, works staff and…

This study reports the results of a study examining how easily students are able to transfer frequency solutions to conditional probability problems to novel situations. University students studied either a problem solved using the traditional Bayes formula format or using a natural frequency (tree diagram) format. In addition, the example problem…

Mathematical creativity is rooted in the intellectual abilities and personality traits of each individual, in which the direct influence of education is only moderate. However, education could have more influence on three important components of creativity: expertise, original thinking, and intrinsic motivation, which underlie individual creative…

There are many continuous quantitative dimensions in the physical world. Philosophical, psychological and neural work has focused mostly on space and number. However, there are other important continuous dimensions (e.g., time, mass). Moreover, space can be broken down into more specific dimensions (e.g., length, area, density) and number can be…

Examines the effectiveness of Direct Instruction for teaching fraction and decimal skills to four seventh-graders with learning disabilities. Provides results which show the Direct Instruction group outperformed their peers on both standardized and informal assessments. Discusses implications for teachers, administrators, researchers, teacher…