This study reports findings from a teaching experiment in which 4th grade children engaged in solving multiplication tasks with rectangular arrays in a computer environment. The environment provided flexible task-solving through dynamic virtual manipulatives (VMs) so children could use their existing knowledge of multiplication to complete array…

In this study, we considered how four teachers reasoned about proportional situations involving geometry. We used pilot interviews conducted with in-service middle grades teachers around a set of proportional tasks. Analysis relied on a knowledge in pieces lens to uncover the resources that each teacher used in solving the tasks and which…

We examine how rectangular array model can support students' spatial structuring in learning multiplication. To begin, we define what we mean by spatial structuring as the mental operation of constructing an organization or form for an object or set of objects. For that reason, the eggs problem was chosen as the starting point in which the…

Specialists in the field of mathematical education consider that the learning of multiplication has an extremely important role not only in the area of mathematics, but also in day to day life. We can distinguish two different stages in the process of teaching-learning multiplication: a first step implies the discovery of the results within the…

This qualitative research study explores teachers' understandings of multiplicative reasoning as a key developmental understanding (KDU). A KDU entails knowingly applying the same mathematical concepts within different contexts. A KDU supports an individual to build a connected understanding of mathematics as opposed to only understanding…

Learning mathematics is challenging. It requires discipline, logic, precision, perseverance, and accuracy. It can also be fun. When mathematics is set in a context that inspires students to want to solve interesting problems, students will have an intrinsic desire to learn the necessary skills to accomplish a specific goal. The game of Crypto! was…

Previous research has consistently shown that the left parietal cortex is critical for numerical processing, but the role of the right parietal lobe has been much less clear. This study used the intraoperative cortical electrical stimulation approach to investigate neural dissociation in the right parietal cortex for subtraction and…

Estimation is more than a skill or an isolated topic. It is a thinking tool that needs to be emphasized during instruction so that students will learn to develop algorithmic procedures and meaning for fraction operations. For students to realize when fractions should be added, subtracted, multiplied, or divided, they need to develop a sense of…

A branch of mathematics commonly used in cryptography is Galois Fields GF(p[superscript n]). Two basic operations performed in GF(p[superscript n]) are the addition and the multiplication. While the addition is generally easy to compute, the multiplication requires a special treatment. A well-known method to compute the multiplication is based on…

The purpose of this study was to determine if a typical third-grade boy and fifth-grade girl and a boy with learning disabilities could benefit from the combined use of Direct Instruction (DI) flashcard and math racetrack procedures in an after-school program. The dependent variable was accuracy and fluency of saying basic multiplication facts. A…

This study examined the impact of a commercially available math fluency intervention program on the basic multiplication fluency of three classrooms of fifth grade students. Baseline assessment and progress monitoring of thirty-three regular- and eleven special education students occurred over nine weeks. Within-group analyses revealed significant…

The purpose of this study was to replicate and extend a previous study by Burns ("Education and Treatment of Children" 28: 237-249, 2005) examining the effectiveness of incremental rehearsal on computation performance. A multiple-probe design across multiplication problem sets was employed for one participant to examine digits correct per minute…

The purpose the present research was to evaluate the efficacy of DI flashcards on the math performance of two students with behavior disorders. The number of correct digits per minute was assessed. The outcomes indicated the DI flashcards were somewhat effective in improving the number of math facts that each participant could correctly write.…

This quantitative study investigated relationships between higher level mathematics learning and multiplication fact fluency, multiplication fact speed-recall, and reading grade equivalency of eighth grade students in Algebra I and Pre-Algebra. Higher level mathematics learning was indicated by an average score of 80% or higher on first and second…

Recent reform efforts in mathematics education call for the development of early algebraic reasoning, and for the provision of rich mathematical experiences for all students. In spite of such emphasis, many elementary school students, especially those with mathematical learning disabilities (MLD), often experience great difficulty in early algebra…