This paper discusses findings from an ongoing study investigating mental mechanisms involved in the conceptualisation of linear transformations from the perspective of Action (A), Process (P), Object (O), and Schema (S) (APOS) theory. Data reported in this paper came from 44 first-year linear algebra students' responses on a task regarding the…

For many students, developing mathematical reasoning can prove to be challenging. Such difficulty may be explained by a deficit in the core understanding of many arithmetical concepts taught in early school years. Multiplicative reasoning is one such concept that produces an essential foundation upon which higher-level mathematical thinking skills…

Children (n=336) in grades 1-5 were interviewed individually using a Piagetian task to study development from additive to multiplicative thinking. Multiplicative thinking was found to appear early (in 45% of second graders) but to develop slowly (only 48% of fifth graders used consistently solid multiplicative thinking). (Author/MKR)

In this study, we investigated the development of basic effects that have been found in single-digit multiplication arithmetic: the problem size, five, and tie effects. Participants (9-,10-, and 11-year-olds and adults) performed a production task on simple multiplication. The procedure replicated Campbell and Graham's (1985) study ["Canadian…

Describes two studies that analyzed the impact of problem type on the understanding of commutativity of multiplication of (n=20) children ages 8-10. The concept seemed poorly understood, but the children did significantly better in the context of rotation than of rearrangement of elements. (Author/MKR)

Describes environments for concretely enacting multiplication and division. Discusses difficulties occurring when students use one of the concrete environments to model situations involving modified environments. (YP)

Studied were the misconceptions that preservice elementary teachers have about multiplication and division. Results indicated that they are influenced by the same primitive models as students; the most common errors made by both groups are quite similar. (MNS)

Structure of mathematical concept of multiplication and its integration into conceptual system, with respect to formal and informal aspects of understanding for 10 university students, are investigated through linear scale assessment of examples for prototypical effects and through follow-up interview which included direct explanations of…

Describes the characteristics of progressive schematization with regard to column multiplication and column division. Contrasts this with column arithmetic based on progressive complexity. Presents a summary of research data concerning column arithmetic. (TW)

A mapping procedure based on the SOLO Taxonomy developmental model was used to classify the problem-solving strategies of students (n=34) in grades K-2. Only one multiplication problem was used to isolate three components of the problem-solving procedure. (MDH)

Discussed are preservice elementary teachers' misconceptions and inconsistent beliefs about multiplication and division with decimals. Sources of inconsistencies and recommendations for overcoming inconsistencies are included. (KR)

Five- to six-year-old children estimated the size of Euclidian objects using an addition rule of Height plus Width, rather than a multiplying rule. Within the framework of information integration theory, tested whether intensive handling of objects would facilitate shift from addition rule to multiplication rule. Found that following handling,…