In the transition years, Grades 7 to 9, the shift from natural numbers to rational numbers and the associated multiplicative concepts prove challenging for many learners. The new concepts, operations and notation must be mastered if the student is to thereafter rise to meet the challenges of algebra and more advanced and powerful mathematics. The…

Many arithmetic problems can be solved in two ways: by a calculation involving several steps, and by direct retrieval of the answer. With practice on particular problems, memory retrieval tends to supplant calculation--an important aspect of skill learning. We asked how the distribution of practice on particular problems affects this kind of…

We investigated the influence of self-efficacy beliefs and working memory capacity on mathematical problem-solving performance, response time, and efficiency (i.e., the ratio of problems solved correctly to time). Students completed a letter-recoding task (Experiment 1) or an operation span task (Experiment 2), rated their self-efficacy for…

Children's understanding of the inversion concept in multiplication and division problems (i.e., that on problems of the form "d multiplied by e/e" no calculations are required) was investigated. Children in Grades 6, 7, and 8 completed an inversion problem-solving task, an assessment of procedures task, and a factual knowledge task of simple…

We examine whether the array representation can support children's understanding and reasoning in multiplication. To begin, we define what we mean by understanding and reasoning. We adopt a "representational-reasoning" model of understanding, where understanding is seen as connections being made between mental representations of concepts, with…

In this article, the author presents a six by six array in which individuals can obtain 182 in total even if they use a different set of numbers. The author then explain why this is possible. The author uses the k-translation of a sequence for this equation. (Contains 8 figures, 2 tables and 6 footnotes.)

A grounded theory approach was employed to explore the variation in strategies used by students when solving mental multiplication problems. Metacognitive prompts, externally generated stimuli that activate thoughtful cognition, were automatically generated via computer during the solving of mental multiplication problems to elicit awareness,…

A regression design was used to test the unique and interactive effects of self-efficacy beliefs and metacognitive prompting on solving mental multiplication problems while controlling for mathematical background knowledge and problem complexity. Problem-solving accuracy, response time, and efficiency (i.e. the ratio of problems solved correctly…

People remember information better if they generate the information while studying rather than read the information. However, prior research has not investigated whether this generation effect extends to related but unstudied items and has not been conducted in classroom settings. We compared third graders' success on studied and unstudied…

For over a decade, research studies of mathematics education in high-performing countries have pointed to the conclusion that the mathematics curriculum in the United States must become substantially more focused and coherent in order to improve mathematics achievement in this country. To deliver on the promise of common standards, the standards…

This study investigated middle school students' conceptual understanding of algebraic equations. Participants in the study--257 sixth- and seventh-grade students--were asked to solve one set of algebraic equations and to generate story problems corresponding with another set of equations. Structural aspects of the equations, including the number…

This book is about playing with mathematics, teaching: (1) How to multiply and divide large numbers mentally; (2) Making addition and subtraction easy; (3) Tricks for understanding fractions and decimals; and (4) How to quickly check answers for each calculation. Described as a foolproof way to do multiplication, division, factoring estimating,…

The effectiveness of a pentop computer when teaching multiplication facts to three middle school students with mild intellectual disabilities was examined. A multiple probe design was used to assess the students learning of one and two digit multiplication facts over a 2-3 week intervention period. During the intervention phase, students used a…

It is not often that one can introduce primary school students to a problem at the forefront of mathematics research, and have any expectation of understanding; but with magic squares, one can do exactly that. Magic squares are an ideal tool for the effective illustration of many mathematical concepts. This paper assumes little prior knowledge on…

In this article, the author presents an innovative approach to teaching multiplication facts for children in middle school or younger. After introducing the dangers of some contradictions that persist, relating a brief summary of related research and guidelines regarding math facts and their relation to strategic approaches to more complex math,…