The mathematical model used to describe independence between two events in probability has a non-intuitive consequence called dependent spaces. The paper begins with a very brief history of the development of probability, then defines dependent spaces, and reviews what is known about finite spaces with uniform probability. The study of finite…

This documentary account situates teacher educator, prospective teacher, and elementary students' mathematical thinking in relation to one another, demonstrating shared challenges to learning mathematics. It highlights an important mathematics reasoning skill--creating and analyzing representations. The author examines responses of prospective…

In longitudinal studies data is collected in a series of waves. Each wave after the first suffers from attrition. Therefore it can be difficult to discriminate between changes in sample parameters due to a longitudinal process (e.g. ageing) and changes due to attrition. The problem is particularly vexing if one of the purposes is to compare…

When using linear models for cluster-correlated or longitudinal data, a common modeling practice is to begin by fitting a relatively simple model and then to increase the model complexity in steps. New predictors might be added to the model, or a more complex covariance structure might be specified for the observations. When fitting models for…

A linearization method of the nonlinear differential equation for conservative nonlinear oscillators is analysed and discussed. This scheme is based on the Chebyshev series expansion of the restoring force which allows us to obtain a frequency-amplitude relation which is valid not only for small but also for large amplitudes and, sometimes, for…

Electricity is regarded as one of the most challenging topics for students of all ages. Several researchers have suggested that naive misconceptions about electricity stem from a deep incommensurability (Slotta and Chi 2006; Chi 2005) or incompatibility (Chi et al. 1994) between naive and expert knowledge structures. In this paper we argue that…

Introduction: We present some aspects of a wider investigation (Canadas, 2007), whose main objective is to describe and characterize inductive reasoning used by Spanish students in years 9 and 10 when they work on problems that involved linear and quadratic sequences. Method: We produced a test composed of six problems with different…

Air cargo enjoys a special importance: together with maritime transport it is the backbone of global trade and is indispensable for contemporary globalization. Air transport is the only mode that combines worldwide reach with high speed. Nonetheless there is a dearth of geographic research that analyzes the current restructuring affecting the air…

In the present study, I examine a modelling strategy as employed by a teacher in the context of an algebra lesson. The actions of this teacher suggest that a modelling approach will have a greater impact on enriching student learning if we do not lose sight of the need to manage associated cognitive loads that could either aid or hinder the…

An educational priority of many nations is to enhance mathematical learning in early childhood. One area in need of special attention is that of statistics. This paper argues for a renewed focus on statistical reasoning in the beginning school years, with opportunities for children to engage in data modelling activities. Such modelling involves…

Two complementary processes involved in mathematical modelling are mathematising a realistic situation and applying a mathematical technique to a given realistic situation. We present and analyse work from two undergraduate students and two secondary school teachers who engaged in both processes during a mathematical modelling task that required…

Superitem test based on the SOLO model (Structure of the Observing Learning Outcome) has become a powerful alternative assessment tool for monitoring the growth of students' cognitive ability in solving mathematics problems. This article focused on developing a superitem test to assess students' algebraic solving ability through interview method.…

Achievement gap language has become associated with the observed disparities on a number of educational measures between the academic performances of Black and White students. This theoretical lens is problematic because it sends an unintended message that Black students are not worthy of study in their own right. Using a mixed-methodological…

To what extent does the use of computational tools offer teachers the possibility of constructing dynamic models to identify and explore diverse mathematical relations? What ways of reasoning or thinking about the problems emerge during the model construction process that involves the use of the tools? These research questions guided the…

The butterfly curve was introduced by Temple H. Fay in 1989 and defined by the polar curve r = e[superscript cos theta] minus 2 cos 4 theta plus sin[superscript 5] (theta divided by 12). In this article, we develop the mathematical model of the butterfly curve and analyse its geometric properties. In addition, we draw the butterfly curve and…