The Isis problem, which has a link with the Isis cult of ancient Egypt, asks: "Find which rectangles with sides of integral length (in some unit) have area and perimeter (numerically) equal, and prove the result." Since the solution requires minimal technical mathematics, the problem is accessible to a wide range of students. Further, it is…

Generalising arithmetic structures is seen as a key to developing algebraic understanding. Many adolescent students begin secondary school with a poor understanding of the structure of arithmetic. This paper presents a theory for a teaching/learning trajectory designed to build mathematical understanding and abstraction in the elementary school…

A group of teachers involved in embedding NRICH tasks into their everyday practice decided they needed to address the (im)balance between teacher and student activity in their classrooms. In this article they share the issues they identified, and what they have been doing to address them.

While others have focused on gender-based language and the workplace, this research study explores gender-based language, in this case women's language, and the classroom. The study specifically examines examples of women's language and how this language affects student response in the classroom. Five variables have been identified as…

Many exercises in mathematical induction require the student to prove a divisibility property of a function of the integers. Such problems are generally presented as being independent of each other. However, many of these problems can be presented in terms of difference equations, and the theory of difference equations can be used to provide a…

The K-W-L method of teaching is a simple method that actively engages students in their own learning. It has been used with kindergarten and elementary grades to teach other subjects. The authors have successfully used it to teach physics at the college level. In their introductory physics labs, the K-W-L method helped students think about what…

If you take a circle with a horizontal diameter and mark off any two points on the circumference above the diameter, then these two points together with the end points of the diameter form the vertices of a cyclic quadrilateral with the diameter as one of the sides. We refer to the quadrilaterals in question as diametric. In this note we consider…

This article describes how to multiply binomials so that middle school students produce the equivalence (a + b)[superscript 2] = a[superscript 2] + 2ab + b[superscript 2] after reasoning about and representing problems. First, a two-dimensional array representation is used so that students can become comfortable with multiplication embedded in…

In this note, we present an improved Heaviside approach to compute the partial fraction expansions of proper rational functions. This method uses synthetic divisions to determine the unknown partial fraction coefficients successively, without the need to use differentiation or to solve a system of linear equations. Examples of its applications in…

Advanced mathematics courses have been developed and refined by the first author, using an action research methodology, for more than a decade. These courses use the computer algebra system (CAS) Maple in an "immersion mode" where all presentations and student work are done using Maple. Assignments and examinations are Maple files downloaded from…

Based on a sequence of points and a particular linear transformation generalized from this sequence, two recent papers (E. Mauch and Y. Shi, "Using a sequence of number pairs as an example in teaching mathematics". Math. Comput. Educ., 39 (2005), pp. 198-205; Y. Shi, "Case study projects for college mathematics courses based on a particular…

In addition to data on overall performance since 1973, this article presents NAEP data on items that were recently released to the public. They are presented as examples of current student achievement in relation to achievement in the 1980s. Performance was stable or improving on most of the released items, with greatest improvement on items…

The introduction of the concept of the variable represents a critical point in the arithmetic-algebraic transition. This concept is complex because it is used with different meanings in different situations. Its management depends on the particular way of using it in problem-solving. The aim of this paper was to analyse whether the notion of…

Using a unique dataset, this dissertation analyzes the relationship between observable teacher characteristics and teacher effectiveness. Effectiveness is measured as a teacher's ability to improve student achievement on a standardized test. This analysis focuses on teachers of math and science at the elementary and middle school levels, as well…

The purpose of this research is to study the long-run impacts (i.e. on educational attainment) of a freshman math intervention called "double-dose algebra". The intervention was conducted in 2003 and 2004 within the Chicago Public Schools (CPS), a large, poor urban school district. In response to low passing rates in 9th grade algebra,…