Student errors are springboards for analyzing, reasoning, and justifying. The mathematics education community recognizes the value of student errors, noting that "mistakes are seen not as dead ends but rather as potential avenues for learning." To induce specific errors and help students learn, choose tasks that might produce mistakes.…

This study examines high school students' views of mathematics as a tool for social inquiry in light of their classroom experiences using mathematics to explore social issues. A critical theoretical perspective on mathematics literacy is used to ascertain the ways in which their views challenge or affirm the dominant image of mathematics in…

Solving mathematics contextual problems is one way that can be used to enable students to have the skills needed to live in the 21st century. Completion contextual problem requires a series of steps in order to properly answer the questions that are asked. The purpose of this study was to determine the steps performed students in solving…

Sometimes, in the teaching and learning of mathematics, open-ended problems posed by teachers or students can lead to a fuller understanding of mathematical concepts--a depth of understanding that no one could have anticipated. Interesting solutions and ideas emerged unexpectedly when the authors asked prospective and in-service teachers an "old"…

When a two-dimensional (2D) traveling salesman problem (TSP) is presented on a computer screen, human subjects can produce near-optimal tours in linear time. In this study we tested human performance on a real and virtual floor, as well as in a three-dimensional (3D) virtual space. Human performance on the real floor is as good as that on a…

Students' cognitive differences in problem solving have been the focus of much research. One classification of these differences is related to whether visualisation is used. Like mathematical thinking differences, multiple representation preferences are important when considering individual differences. Choosing an appropriate representation is an…

Solving mathematics story problems requires text comprehension skills. However, previous studies have found few connections between traditional measures of text readability and performance on story problems. We hypothesized that recently developed measures of readability and topic incidence measured by text-mining tools may illuminate associations…

Current conceptualizations of knowing and learning tend to separate the knower from others, the world they know, and themselves. In this article, we offer "relationality" as an alternative to such conceptualizations of mathematical knowing. We begin with the perspective of Maturana and Varela to articulate some of its problems and our alternative.…

A facility with signed numbers forms the basis for effective problem solving throughout developmental mathematics. Most developmental mathematics textbooks explain signed number operations using absolute value, a method that involves considering the problem in several cases (same sign, opposite sign), and in the case of subtraction, rewriting the…

Both exploration and explicit instruction are thought to benefit learning in many ways, but much less is known about how the two can be combined. We tested the hypothesis that engaging in exploratory activities prior to receiving explicit instruction better prepares children to learn from the instruction. Children (159 second- to fourth-grade…

The art gallery problem asks for the maximum number of stationary guards required to protect the interior of a polygonal art gallery with "n" walls. This article explores solutions to this problem and several of its variants. In addition, some unsolved problems involving the guarding of geometric objects are presented.

This article describes the technique of introducing a new variable in some calculus problems to help students master the skills of integration and evaluation of limits. This technique is algorithmic and easy to apply.

This paper presents a comparative analysis of prospective elementary teachers' mathematical problem solving-related beliefs in Cyprus and England. Twenty-four participants, twelve from a well-regarded university in each country, were interviewed qualitatively at the exit point of their undergraduate teacher education studies. Analyses revealed…

This paper discusses some difficulties in teaching introductory courses to programming, paying particular attention to their mathematical nature. We consider some aspects, which have not been commented in detail in textbooks and often neglected by course outlines and schedules. Some of these are constructing complex conditions, exceeding array…

The article provides a review of recent research on human performance on the traveling salesman problem (TSP) and related combinatorial optimization problems. We discuss what combinatorial optimization problems are, why they are important, and why they may be of interest to cognitive scientists. We next describe the main characteristics of human…