"The root of 18 is closer to 4 than it is to 5 because 18 is closer to 16 than it is to 25". Is this statement, voiced in an 8th grade class, valid? We suggest hypothetical arguments upon which this statement might be based, and analyze them from two complementary perspectives--epistemic and pedagogical--drawing on Toulmin's notion of…

Supporting students' mathematical reasoning is an important goal of mathematics instruction, but can be challenging for many teachers .We report the results of a study aimed at better understanding and identifying the ways in which teachers support student reasoning when provided with conceptually rich tasks. This study resulted in the Teacher…

In this paper we focus on variation of the design and the implementation of a specific task during three mathematics lessons in the 8th grade in a learning study (Marton & Tsui, 2004; Runesson, 2008). The theme of the lesson was division, with a denominator between 0 and 1. The teachers wanted their students to understand that when dividing…

Understanding frames of reference is critical in describing planetary motion and learning astronomy. Historically, the geocentric and heliocentric models were defended and advocated against each other. Today, there are still many people who do not understand the relationship between the two models. This topic is not adequately treated in astronomy…

The No Child Left Behind Act of 2001 calls for improving test scores of minority students in third through eighth grades, making middle level schools a key venue for closing the racial gap in achievement. This study, by analyzing data on the 15,000 eighth graders who took the 2000 National Assessment of Educational Progress (NAEP) in mathematics,…