This article focuses on the Thinking Through a Lesson Protocol (TTLP) that is intended to facilitate the design of lessons based on cognitively challenging mathematics tasks. Included is a discussion of the key features of the TTLP, suggestions on ways in which the protocol can be used as a tool for collaborative lesson planning, and a discussion…

This article describes the use of a case report, Multiplication as original sin (Corwin, R. B. (1989). "Multiplication as original sin." "Journal of Mathematical Behavior, 8", 223-225), as an assignment in a mathematics course for preservice elementary teachers. In this case study, Corwin described her experience as a 6th grader when she revealed…

This submission contains the Proceedings of the 2011 Annual Meeting of the Canadian Mathematics Education Study Group (CMESG), held at Memorial University of Newfoundland in St. John's, Newfoundland. The CMESG is a group of mathematicians and mathematics educators who meet annually to discuss mathematics education issues at all levels of learning.…

Although errors in reasoning about conditional probability have been the focus of interest of psychologists for a long time, the development of conditional reasoning in school students has received little attention. This paper considers the responses of 69 students across grades 3 to 13 in an attempt to model the development of appropriate…

This article suggests how teachers can use questions while students play games to develop their mathematical reasoning. (Contains 4 figures.)

This article explores how conceptualization of absolute value can start long before it is introduced. The manner in which absolute value is introduced to students in middle school has far-reaching consequences for their future mathematical understanding. It begins to lay the foundation for students' understanding of algebra, which can change…

This article, reprinted from "Arithmetic Teacher" in 1979, as part of the "MT" 100 celebration, discusses the importance of fractions.

Usiskin takes another look at fractions, years after writing the article "The Future of Fractions" for "Arithmetic Teacher."

In 1854 the German scientist Zeising claimed that the ratio of a person's height to the height of their navel is in the same ratio as the Golden Ratio ([phi] = 1.62). There is so much hype about the Golden Ratio that it is worth reading an article at http://www.maa.org/devlin/devlin_06_04.html. It explains why some of the more far-fetched ideas…

This paper is essentially a story that can be used to help students understand some of the definitions found in a standard introductory real analysis course.

The golden ratio appears in a classical mechanics coupled-oscillator problem that many undergraduates may not solve. Once the symmetry is broken in a more standard problem, the golden ratio appears. Several student exercises arise from the problem considered in this paper.

Assuming that dynamic features of Dynamic Geometry Software may provide a basic representation of both variation and functional dependency, and taking the Vygotskian perspective of semiotic mediation, a teaching experiment was designed with the aim of introducing students to the idea of function. This paper focuses on the use of the Trace tool and…

Algebra teachers find themselves overwhelmed by the sheer volume of information that they have to teach their students. One way to address this issue is to have teachers teach for understanding by focusing on concepts. In this article, I share how language can help uncover the core ideas that define the concept of function and share activities…

To be able to express generality one must first be able to "identify" the generality in a situation, and this is not something that comes naturally. Learning to recognise generality involves many steps. The first step is to write down a list of invariant and changeable aspects of the situation. In this article, the author describes her growing…

Children's natural curiosity about numbers, big and small can lead to exploring place-value ideas. But how can these abstract concepts be experienced more concretely? This article presents some practical approaches for conceptualising very small numbers using linear models, area models, volume models, and diagrams.