Left to their own devices, many students are happy to work within their comfort zone of skill and understanding, a level where they are confident that they will achieve regular success. The job of the classroom teacher is to help students reach beyond this and to help them make this level their new comfort zone. Clearly, teachers need to employ a…

Proportional thinking is the mathematical basis of a wide range of topics in the middle-school mathematics curriculum. While the concept is obvious in the traditionally-named ratio and proportion sections, proportional thinking is also the key to such diverse topics as rate, gradient of a linear function, similarity, trigonometry and percentage,…

As a result of conversations with students, previous teaching experiences, and through reading relevant literature, this author was aware that many students struggle with understanding fraction concepts. In particular, she knew that students had difficulty with reading, renaming, ordering, interpreting, and applying common fractions, fraction…

We often look for a best-fit function to a set of data. This article describes how a "pretty good" fit might be better than a "best" fit when it comes to promoting conceptual understanding of functions. In a pretty good fit, students design the function themselves rather than choosing it from a menu; they use appropriate variable names; and they…

This article describes instructional activities designed to help students develop an understanding of the multiple relationships between fractions and percents and unit price as they develop their mental-math abilities. (Contains 4 figures.)

This is a study of David, a senior enrolled in a high school precalculus course. David's understandings of functions and periodicity was explored, through clinical interviews and contextualized through classroom observations. Although David's precalculus class was traditional his understanding of periodic functions was unconventional David engaged…

Previous work has investigated adults' knowledge of principles for arithmetic with positive numbers (Dixon, Deets, & Bangert, 2001). The current study extends this past work to address adults' knowledge of principles of arithmetic with a negative number, and also investigates links between knowledge of principles and problem representation.…

In this study, the authors wish to explore the use of visual representations in facilitating the understanding of multiplication. In doing so, they examine the different aspects of multiplication that they can access through different representations. In addition, they draw on a study that they have been carrying out looking at pupils' actual use…

In his 1976 book, "Proofs and Refutations," Lakatos presents a collection of case studies to illustrate methods of mathematical discovery in the history of mathematics. In this paper, we reframe these methods in ways that we have found make them more amenable for use as a framework for research on learning and teaching mathematics. We present an…

What happens if chords a drawn in a circle? The interior of the circle will be partitioned into regions. This article discusses two special partitions of a circular region.

Calculus is a mathematical concept that is fundamental to how we understand the world around us. Whether it is in the world of technology, finance, astronomy, sociology, medicine, calculus in one form or another can be found. This brief article describes the origins of calculus in Greece, further developments by Newton and Leibniz, and the…

This paper describes the PD program that was delivered during the first year of the study. The main goal of the intervention was to increase teachers' capability to teach positive rational number topics effectively. The program included a 3-day summer institute (18 hours per teacher), five 1-day seminars held during the school year (30 hours per…

This paper presents a grade 10 mathematics teacher's in-depth description of the process she uses to determine the final grade for the report card within a standards-based context. Her case was part of a three-year comparative study of grading practices of teachers from two Canadian provinces that differ in their level of standardization of…

Many developed nations have a serious problem with a shortage in the supply of numerate graduates, fuelled by their school students' negative attitudes towards their future study of mathematics. At the same time, the smart phone and other personal sensing technological devices are becoming commonplace amongst students in schools and universities.…