Getting students to think about the relationships between area and perimeter beyond the formulas for these measurements is never easy. An interesting, nonroutine, and accessible problem that will stimulate such thoughts is the Lattice Octagon problem. A "lattice polygon" is a polygon whose vertices are points of a regularly spaced array.…

Teaching computer concepts to individuals with visual impairments (that is, those who are blind or visually impaired) presents some unique challenges. Students often have difficulty remembering to perform certain steps or have difficulty remembering specific keystrokes when using computers. Many cannot visualize the way in which complex computing…

If we want to make something concrete in mathematics education, we are inclined introduce, what we call, "manipulatives", in the form of tactile objects or visual representations. If we want to make something concrete in a everyday-life conversation, we look for an example. In the former, we try to make a concrete model of our own,…

We describe a set of manipulatives developed to help students learn three-dimensional Cartesian, cylindrical, and spherical coordinate systems. (Contains 6 figures.)

This study is about prospective secondary mathematics teachers' understanding and sense making of representational quantities generated by algebra tiles, the quantitative units (linear vs. areal) inherent in the nature of these quantities, and the quantitative addition and multiplication operations--referent preserving versus referent…

A picture is worth a thousand words. This statement is especially true in mathematics teaching and learning. Visual representations such as pictures, diagrams, charts, and tables can illuminate ideas that can be elusive when displayed in symbolic form only. The prevalence of representation as a mathematical process in such documents as…

This design-based research study was conducted to identify what importance of a tangible user interface (TUI) can add to teaching and learning. Over a 2-year period, teachers ("n"?=?39) and students ("n"?=?145) participated in the study. The identified problem for investigation was how students, including those with low fine…

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…

We used population-based, longitudinal data to investigate the relation between mathematics instructional practices used by first-grade teachers in the United States and the mathematics achievement of their students. Factor analysis identified four types of instructional activities (i.e., teacher-directed, student-centered,…

How can children's natural perceptuo-motor skills be harnessed for teaching and learning mathematical structure? We address this question in the case of the integers. Existing research suggests that adult mental representations of integers recruit perceptuo-motor functionalities involving symmetry. Building on these findings, we designed a…

Students can use geometric representations of numbers as a way to explore algebraic ideas. With the help of these representations, students can think about the relations among the numbers, express them using their own words, and represent them with letters. The activities discussed here can stimulate students to try to find various ways of solving…

This is an account of events in a real classroom when the notion that there is a distinction between calculation by counting, and calculation by structuring is a focus for learning. Pupils have "hands-on" experiences that support their understanding of "number". The teaching "aid" might be unfashionable in…

The purpose of this study was to determine variables that predict performance when virtual manipulatives are used for mathematics instruction. This study used a quasi-experimental design. This design was used to determine variables that predict student performance on tests of fraction knowledge for third- and fourth-grade students in two treatment…

Engaging students while providing them with the necessary linguistic and critical skills as a foundation for further exploration are the principle challenges for those of us who teach disciplinary introductory courses. My own response to this challenge has been to develop and implement what I refer to as the "pedagogy of play." Informed…

In this article, the author discusses the creativity of origami and discusses a model she designed in May 2008 in Freiburg at the 20th International Origami Convention of Origami Deutschland. The model resulted from her investigation of a geometric model that exposes the centre part of a square paper sheet. The base model of the series called…