Observed four fifth-graders who collaboratively solved problems integrating concepts related to fractions and area. Compared features of learning through peer collaboration with teacher-directed learning, including whole solution versus component parts, appropriation versus scaffolding, analysis versus synthesis, conflict versus errorless…

Examined aspects of early childhood environment related to developing mathematical concepts in three studies. Found that family day care providers rated math skills as less important than others, but believed skills were acquired the same way. Parents and providers reported similar frequencies for engaging children in math activities. Activity…

Sfard and Thompson (1994) state that what matters most is that educators develop ways of thinking, teaching, and learning mathematics. This study introduced constructive habituation, a new strategy developed to aid both students and teachers in the thinking, teaching, and learning of mathematics. Constructive habituation attempts to unite…

For many decades Americans have been concerned about the effective teaching of mathematics, and educational and political leaders have often advocated reforms such as a return to the basics and strict accountability systems as the way to improve mathematical achievement. International studies, however, suggest that such reforms may not be the best…

We articulate and explicate a mechanism for mathematics conceptual learning that can serve as a basis for the design of mathematics lessons. The mechanism, reflection on activity-effect relationships, addresses the learning paradox (Pascual-Leone, 1976), a paradox that derives from careful attention to the construct of assimilation (Piaget, 1970).…

This article describes a middle-school project that challenged students to design scale models of three-dimensional blocks used in theater programs. Students applied skills such as measurement, proportionality, and spatial reasoning in a cooperative setting. (Contains 1 table and 9 figures.)

This article discusses the importance of measurement in providing a foundation for mathematical study and how measurement can be integrated with other mathematical strands. (Contains 1 table.)

During the preliminary stages of classroom-based research, this author conducted a sequence of interviews with six seventh graders regarding their understanding of fractions, decimals, ratios, and proportions. The students represented a variety of ability levels and were all in their first semester of prealgebra. The interviews began with notions…

This article showcases four Shel Silverstein works and how they can be used to launch mathematics investigations across the curriculum. Each selection focused on a different mathematical strand; for example, a poem titled "Where the Sidewalk Ends" demonstrates measurement and proportional reasoning, and another poem, "Falling Up," demonstrates…

Investigations with the computer can have different functions in the mathematical learning process, such as to let students explore a subject domain, to guide the process of reinvention, or to give them the opportunity to apply what they have learned. Which function has most effect on mathematical level raising? We investigated that question in…

Based on studies carried out on qualitative data an instrument was constructed for investigating how larger numbers of students handle graphics. This test, consisting of 18 pages, each with its own graphic display(s) and a set of tasks, was distributed to 363 students, 15-16 years of age, from five different schools. The format of the questions…

This article describes Cryptography, defined as the science of encrypting and deciphering messages written in secret codes, it has played a vital role in securing information since ancient times. There are several cryptographic techniques and many make extensive use of mathematics to secure information. The author discusses an activity built…

The Departments of Mathematics and Statistics at Grand Valley State University are engaged in a project to address the mathematical preparation of all its majors, including prospective elementary and secondary mathematics teachers. In this article, we describe how the mathematics and mathematics education faculty in our mathematics department…

Generalizing or making claims that extend beyond particular situations is a central mathematical practice and a focus of classroom mathematics instruction. This study examines how aspects of generality are produced through the situated activities of a group of middle school mathematics students working on an 8-week population-modeling project. The…

Concerns about students' difficulties in statistical reasoning led to a study which explored form five (14- to 16-year-olds) students' ideas in this area. The study focussed on descriptive statistics, graphical representations, and probability. This paper presents and discusses the ways in which students made sense of information in graphical…