By working through well-designed tasks, students can expand their thinking about mathematical ideas and their approaches to solving mathematical problems. They can come to see the value of looking at tasks from different perspectives and of using different representations. This article discusses four tasks that encourage high school students and…

A multiple-baseline design across math-fact sets was used to evaluate the effects of a taped-problems intervention on students' performance with addition facts and their inverses in an intact, rural, second-grade classroom. Results suggested that the procedure was effective in improving fluency on math facts as well as their inverses. Across 3…

The mathematical topic of inverse functions is an important element of algebra courses at the high school and college levels. The inverse function concept is best understood by students when it is presented in a familiar, real-world context. In this article, the authors discuss some misconceptions about inverse functions and suggest some…

This article introduces a trigonometric field (TF) that extends the field of real numbers by adding two new elements: sin and cos--satisfying an axiom sin[superscript 2] + cos[superscript 2] = 1. It is shown that by assigning meaningful names to particular elements of the field, all known trigonometric identities may be introduced and proved. Two…

This study identifies the kinds of problems teachers pose when they are asked to (a) generate problems from given information and (b) create new problems from ones given to them. To investigate teachers' problem posting, preservice and inservice teachers completed background questionnaires and four problem-posing instruments. Based on previous…

One of the most fundamental math skills for students to learn is basic computational skills. Students who have difficulty learning math skills may benefit from representational approaches, such as TouchMath. A multiprobe, multiple baseline, across-participants design was used in this study to investigate the use of TouchMath with students with…

Teaching mathematics using graphic calculators has been an issue of didactic discussions for years. Finding ways in which graphic calculators can enrich the development process of creative activity in mathematically gifted students between the ages of 16-17 is the focus of this article. Research was conducted using graphic calculators with…

In what unique ways can mathematics tasks contribute to pre-service teachers' understanding of subject matter and pedagogy? And what school mathematics tasks can usefully be included in a pre-service program? To contribute to answering these questions, we report on the selection and choice criteria for mathematics tasks that we use in an…

In an elementary undergraduate abstract algebra or group theory course, a student is introduced to a variety of methods for constructing and deconstructing groups. What seems to be missing from contemporary texts and syllabi is a theorem, first proved by Edouard Jean-Baptiste Goursat (1858-1936) in 1889, which completely describes the subgroups of…

An undergraduate seminar was designed to help preservice teachers focus on students' learning. Preservice teachers planned and conducted weekly tutoring sessions with fourth graders and discussed their experiences in weekly discussions. The author investigated what preservice teachers learned about teaching mathematics from their focus on…

Researchers have documented the positive effects of classwide peer tutoring on academic performance, engagement, and other social behaviors of students with and without disabilities. Commonly, in classwide peer tutoring, students are paired and the class is divided in half. Points are awarded for tutoring behavior and academic responding during…

Researchers who make tutoring systems would like to know which sequences of educational content lead to the most effective learning by their students. The majority of data collected in many ITS systems consist of answers to a group of questions of a given skill often presented in a random sequence. Following work that identifies which items…

The "Cognitive Tutor[R] Algebra I" curriculum, published by Carnegie Learning, is an approach that combines algebra textbooks with interactive software. The software is developed around an artificial intelligence model that identifies strengths and weaknesses in each individual student's mastery of mathematical concepts. It then…

Several decades ago, V. A. Krutetskii conducted a multiyear study to investigate the various types of thinking that academically advanced, or as he called them, gifted mathematicians used. Following an in-depth look at Krutetskii's nine ways of thinking, a model is proposed that will provide direction for teachers in selecting problems. The model…

We present a general formula for a triple product involving four real numbers. As a particular case, we get the sum of a triple product of four odd integers. Some interesting results are recovered. We derive a general formula for more than four odd numbers.