We analyze a classroom design experiment, conducted in a fourth grade classroom, that served to explore an instructional path in which the introduction of unit fractions and supporting proportional reasoning coincide. Central to this path is the use of means of support in which the objects that unit fractions quantify are not characterized as…

Proof serves many purposes in mathematics. In this qualitative study of 17 high school mathematics teachers, we found that these teachers perceived that two of the most important purposes for proof in school mathematics were (a) to enhance students' mathematical understanding and (b) to develop generalized thinking skills that were…

Rate is an important, but difficult, mathematical concept. Despite more than 20 years of research, especially with calculus students, difficulties are reported with this concept. This paper reports the results from analysis of data from 20 Australian Grade 10 students. Interviews targeted students' conceptions of rate, focussing on the influence…

This article presents data gathered from an investigation which focused on the experiences children have with measurement in the early years of schooling. The focus of this article is children's understandings of length at this early stage. 32 children aged 4-6 years at an Australian primary school were asked to draw a ruler and describe their…

The structure of two-tier testing is such that the first tier consists of a multiple-choice question and the second tier requires justifications for choices of answers made in the first tier. This study aims to evaluate two-tier tasks in "proportion" in terms of students' capacity to write and select justifications and to examine the effect of…

We have been investigating university student conceptions of mathematics over a number of years, with the goal of enhancing student learning and professional development. We developed an open-ended survey of three questions, on "What is mathematics" and two questions about the role of mathematics in the students' future. This questionnaire was…

This paper presents a research-based framework for analyzing and monitoring students' understanding of functions in equation form. The framework consists of "growth points" which describe "big ideas" of students' understanding of the concept. The data were collected from Grades 8, 9, and 10 students using a set of tasks involving linear and…

Recent educational research has turned increasing attention to the structural development of young students' mathematical thinking. Early algebra, multiplicative reasoning, and spatial structuring are three areas central to this research. There is increasing evidence that an awareness of mathematical structure is crucial to mathematical competence…

This article discusses students' perceptions as a source for understanding education. It addresses what didactically experienced children, called "didactikids", taught us about the empty number line as a didactical model for teaching whole number calculations. The article mainly reports on a student consultancy study carried out in the…

Studies such as the Effective Teachers of Numeracy Study (Askew, Brown, Rhodes, Johnson, & Wiliam (1997) have contributed much to our understanding of what constitutes effective teaching of numeracy. This paper aims to build on these findings and to contribute a model that could be used to understand teachers' numeracy practices and the factors…

This paper focuses on the changes in thinking involved in the transition from school mathematics to formal proof in pure mathematics at university. School mathematics is seen as a combination of visual representations, including geometry and graphs, together with symbolic calculations and manipulations. Pure mathematics in university shifts…

This paper considers mathematical abstraction as arising through a natural mechanism of the biological brain in which complicated phenomena are compressed into thinkable concepts. The neurons in the brain continually fire in parallel and the brain copes with the saturation of information by the simple expedient of suppressing irrelevant data and…

The basic theory of Rasch measurement applies to situations where a person has a certain level of a trait being investigated, and this level of ability is what determines (to within a measurement error) how well the person does on each item in a test. This paper responds to frequent suggestions from colleagues that the use of Rasch measurement…

This article examines students' ability to use the balance model to solve for unknowns. A teaching experiment was conducted in four Year 3 classrooms. This experiment focused on exploring the application of the balance model as an analogue for representing equations and solving for unknowns. The teaching experiment promoted a shift by students…

Presents findings from a research project in two high school mathematics classrooms. Focuses on the ways in which teachers' classroom interventions promote the growth of students' mathematical understanding. Describes 'blocking' as one of the major interventions of mathematics teachers. Discusses implications of the intervention strategy for…