Contemporary research from a psychology of mathematics education perspective has turned increasing attention to the structural development of mathematics as an explanation for the wide differences in mathematical competence shown upon school entry and in the early school years. Patterning, multiplicative reasoning and spatial structuring are three…

This article introduces a mathematical entity, the varimeter, that can be used to engage future mathematics teachers in reflection about the structure of measurement systems, thus aiding in the development of insight and understanding which may be beneficial to the future teachers when they work with their own students on measurement. Furthermore,…

The present study investigated numerical magnitude processing in children with mild intellectual disabilities (MID) and examined whether these children have difficulties in the ability to represent numerical magnitudes and/or difficulties in the ability to access numerical magnitudes from formal symbols. We compared the performance of 26 children…

This paper is drawn from a doctoral thesis (Rosa, 2008) that examines the relations established between the construction of online identities and the teaching and learning of the definite integral concept in an online learning course. The role-playing game (RPG) is played out through chat and calls for the creation of characters (online…

Freely rising air bubbles in water sometimes assume the shape of a spherical cap, a shape also known as the "big bubble". Is it possible to find some objective function involving a combination of a bubble's attributes for which the big bubble is the optimal shape? Following the basic idea of the definite integral, we define a bubble's surface as…

Children learn from a very early age what it means to get their "fair share." Whether it is candy or birthday cake, many children successfully create equal-size groups or parts of a collection or whole but later struggle to create fair shares of multiple wholes, such as fairly sharing four pies among a family of seven. Recent research suggests…

The National Science Foundation (NSF) funded project "Mathematics, Science, and Technology Partnership" (MSTP) developed a multidisciplinary instructional model for connecting mathematics to science, technology and engineering content areas at the middle school level. Specifically, the model infused mathematics into middle school curriculum…

Probability knowledge and skills are needed in science and in making daily decisions that are sometimes made under uncertain conditions. Hence, there is the need to ensure that the pre-service teachers of our children are well prepared to teach probability. Pre-service teachers' conceptions of probability are identified, and ways of helping them…

This article examines the question of finding the best quadratic function to approximate a given function on an interval. The prototypical function considered is f(x) = e[superscript x]. Two approaches are considered, one based on Taylor polynomial approximations at various points in the interval under consideration, the other based on the fact…

A plane algebraic curve, the complete set of solutions to a polynomial equation: f(x, y) = 0, can in many cases be drawn using parametric equations: x = x(t), y = y(t). Using algebra, attempting to parametrize by means of rational functions of t, one discovers quickly that it is not the degree of f but the "relative degree," that describes how…

In building models of students' fractions knowledge, two prominent frameworks have arisen: Kieren's rational number subconstructs, and Steffe's fractions schemes. The purpose of this paper is to clarify and reconcile aspects of those frameworks through a quantitative analysis. In particular, we focus on the measurement subconstruct and the…

I argue for the inclusion of topics in high school mathematics curricula that are traditionally reserved for high achieving students preparing for mathematical contests. These include the arithmetic mean--geometric mean inequality which has many practical applications in mathematical modelling. The problem of extremalising functions of more than…

The fundamental theorems of the calculus describe the relationships between derivatives and integrals of functions. The value of any function at a particular location is the definite derivative of its integral and the definite integral of its derivative. Thus, any value is the magnitude of the slope of the tangent of its integral at that position,…

Many optimization problems can be solved without resorting to calculus. This article develops a new variational method for optimization that relies on inequalities. The method is illustrated by four examples, the last of which provides a completely algebraic solution to the problem of minimizing the time it takes a dog to retrieve a thrown ball,…

We consider the expected range of a random sample of points chosen from the interval [0, 1] according to some probability distribution. We then use the notion of convexity to derive an upper bound for this expected range which is valid for all possible choices of this distribution. Finally we show that there is only one distribution for which this…