Our study concerns the analysis of teacher and student activities. Secondary school 6th grade students were confronted, for the first time, with solving equations. We used our cognitive models of students and experts (in algebraic calculations) for analyzing the teaching process and the cognitive functioning of students. Our model led us to…

This paper introduces technology-based teaching ideas that facilitate the development of qualitative reasoning techniques in the context of quadratic equations with parameters. It reflects on activities designed for and used with prospective secondary mathematics teachers in accord with standards for teaching and recommendations for teachers in…

Pattern work is now undertaken as early as kindergarten, and both researchers and teachers have discovered that children engage in pattern work with great enthusiasm and innate ability. Having some flexibility in pattern perception and selecting mathematically useful patterns require some training, although there is nothing particularly algebraic…

One of the most interesting and important proofs in the history of mathematics is the Pythagorean school's proof of the "irrationality" of the square root of 2. After a brief look at G. H. Hardy (1941) thoughts regarding it, two versions of the classic Pythagorean proof are examined and discussed in this article, one written by an American…

The new representations enabled by the educational computer game the "Juggler" can place vectors in a central role both for controlling and measuring the behaviours of objects in a virtual environment simulating motion in three-dimensional spaces. The mathematical meanings constructed by 13 year-old students in relation to vectors as…

This paper reports on a design experiment in the domain of number sequences conducted in the course of the "WebLabs" project. We iteratively designed and tested a set of activities and tools in which 10-14 year old students used the "ToonTalk" programming environment to construct models of sequences and series, and then shared…

Preparing for an oral examination is an alternative way for students to learn mathematics by working cooperatively in small teams to solve problems. Reformed college mathematics courses are emphasizing the ability to communicate ideas clearly, both orally and in writing. The instructor is able to ask probing questions that are not usually possible…

Rhymes are often useful teaching devices for small children, and can even become a creative resource for teaching mathematics. The students and their teachers can change many of them to better fit their environment and/or extend the rhyme so that higher numbers are involved. A number of rhymes and folk tales lend themselves to more detailed…

This article illustrates how a computer algebra system, such as Maple[R], can assist in the study of theoretical fluid mechanics, for both Newtonian and non-Newtonian fluids. The continuity equation, the stress equations of motion, the Navier-Stokes equations, and various constitutive equations are treated, using a full, but straightforward,…

This classroom scholarship report is based on the teaching experience using Davydov's mathematics curriculum, which was developed in the former Soviet Union. While "from arithmetic to algebra" is the normally accepted instructional sequence in school mathematics, Davydov's curriculum is laid out "from algebra to arithmetic," focusing on algebraic…

A study was undertaken at the College of the Canyons, in California, to assess the variation of grades given by mathematics instructors teaching intermediate algebra. A sample was drawn from all instructors who taught Math 070 repeatedly from fall 1990 to fall 1994, resulting in the examination of the grades of at least 200 students from classes…

This paper has two goals. The first is to present a model of the acquisition of a concept image of function. Theories describing the objectification of function are outlined through two different but related paths, and both stem from the conception of function as a process. The first path to objectification involves the generalization of the…

This paper presents two examples in which technology, in this case a fairly sophisticated authoring system--ToolBook, was used as a tool to construct student understandings in mathematics. In doing so, students were able: (1) to successfully identify the variables (unknowns) and the information given (data) in the problem; and (2) to create…

Dispositions of eighth graders accelerated into first-year algebra were described in this study. Data were collected through surveys, observations, interviews, and cumulative academic files. The most frequently reported reasons for enrolling in algebra were for acceleration of course-taking and preparation for high school. Males demonstrated a…

A study of 29 students enrolled in Algebra II in an 8-week summer instructional program for minority students was conducted to reveal the skills that students have in translating among representational forms in mathematics problems where the representational form is varied in the given problem and in the required form of the answer. All students…