Several studies have shown that children do not only erroneously use additive reasoning in proportional word problems, but also erroneously use proportional reasoning in additive word problems. Traditionally, these errors were contributed to a lack of calculation and discrimination skills. Recent research evidence puts forward an additional…

The reasons for teaching programming in school are indeed manifold. Programming can for example be utilized as a vehicle for understanding and learning particular mathematical subject matter, or as a tool for solving mathematical problems. In this paper, however, we propose to utilize programming as a vehicle for developing mathematical…

Although counting problems are easy to state and provide rich, accessible problem-solving situations, there is much evidence that students struggle with solving counting problems correctly. With combinatorics (and the study of counting problems) becoming increasingly prevalent in K-12 and undergraduate curricula, there is a need for researchers to…

From a Marxian/Vygotskian perspective, learning is social in origin and it happens in the presence of others that are more knowledgeable. Extending this view to the learning of mathematics, such learning also becomes inseparable from the presence of others (people and artefacts). Researchers over decades have studied different interactions to see…

In this paper we report the results of an experiment designed to test the hypothesis that when faced with a question involving the inverse direction of a reversible mathematical process, students solve a multiple-choice version by verifying the answers presented to them by the direct method, not by undertaking the actual inverse calculation.…

The introduction of the concept of the variable represents a critical point in the arithmetic-algebraic transition. This concept is complex because it is used with different meanings in different situations. Its management depends on the particular way of using it in problem-solving. The aim of this paper was to analyse whether the notion of…

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…

This study illustrates how a diagnostic testing model can be used to make detailed comparisons between student populations participating in international assessments. The performance of Turkish students on the TIMSS-R mathematics test was reanalyzed with a diagnostic testing model called the Rule Space Model. First, mathematical and cognitive…

Before the advent of symbolism, i.e. before the end of the 16th Century, algebraic calculations were made using natural language. Through a kind of metaphorical process, a few terms from everyday life (e.g. thing, root) acquired a technical mathematical status and constituted the specialized language of algebra. The introduction of letters and…

Discusses the limitations in the explain-practice and active learning methods of teaching mathematics. Proposes that problem-centered learning is a teaching method that encourages student reflection, and presents examples demonstrating that encouraging reflection results in greater mathematics achievement. (MDH)

This study covers 11-year-olds, school leavers, student nursery nurses, and bus conductor trainees, and their understanding of subtraction, and whether shortcomings in understanding and performance that occurred at ages six or seven would continue in later years and affect feelings toward the subject. (MP)

The purpose of this exploratory study was to investigate the impact of writing during mathematical problem solving. The study involved an analysis of ninth grade algebra students' written and verbal descriptions of their mathematical problem solving processes. Through this comparison, a better understanding of the connection between problem…

Studied for three years were eight Swedish classes with pupils aged 10-12 using calculators whenever they could be of use, along with experimental textbooks. The experimental classes were as competent as control classes in mental arithmetic and calculations with simple algorithms, and had better understanding of numbers and problem solving. (MNS)

Research conducted in several countries has shown consistent patterns of performance on "change,""combine," and "compare" word problems involving addition and subtraction. These findings are interpreted within a theoretical framework which emphasizes development of levels of word problem-solving ability related to…

Freudenthal was the founder of realistic mathematics education, in which reality serves as a source of applications and learning. Takes a newspaper article about reproducing a Van Gogh painting using plants in a field to exemplify a rich context problem containing elements of all areas of elementary school mathematics. (MDH)