This article reports on students' problem-solving approaches across three representations--number lines, coordinate planes, and function graphs--the axes of which conventional mathematics treats in terms of consistent geometric and numeric coordinations. I consider these representations to be a part of a "hierarchical representational…

Master narratives frame students' historical knowledge, possibly hindering access to more historical representations. A detailed analysis of students' historical narratives about the origins of their own nation is presented in terms of four master narrative characteristics related to the historical subject, national identification, the…

We examine how extended engagement in argumentation with peers leads to enhanced metalevel understanding of argumentive discourse. Following such engagement, young adolescents demonstrated enhanced understanding of counterargument and use of evidence as objectives of skilled argumentation, relative to a nonparticipating comparison group. Among the…

This article aims to account for students' assessments of the plausibility and applicability of analogical explanations, and individual differences in these assessments, by analyzing properties of students' underlying knowledge systems. We developed a model of explanation and change in explanation focusing on knowledge elements that provide a…

Using data collected in 125 seventh-grade and 56 eighth-grade Texas classrooms in the context of the "Scaling Up SimCalc" research project in 2005-07, we examined relationships between teachers' mathematics knowledge, teachers' classroom decision making, and student achievement outcomes on topics of rate, proportionality, and linear…

This paper reports the mathematical generalizations of two groups of algebra students, one which focused primarily on quantitative relationships, and one which focused primarily on number patterns disconnected from quantities. Results indicate that instruction encouraging a focus on number patterns supported generalizations about patterns,…

This research analyzes students' conceptual change across a semester in an 8th-grade thermodynamics curriculum. Fifty students were interviewed 5 times during their 8th-grade semester and then again preceding their 10th- and 12th-grade years to follow their subsequent progress. The interview questions probed students' understanding of…

Previous research (e.g., De Bock, 2002) has shown that--due to the extensive attention paid to proportional reasoning in elementary and secondary mathematics education--many students tend to overrely on proportional methods in diverse domains of mathematics (e.g., geometry, probability). We investigated the development of misapplication of…

In this study, we examined why observational learning positively affects learning outcomes of new writing tasks. In this study, we focused on the effects of observational learning on the temporal organization (i.e., orchestration) of writing processes and on the subsequent influence on text quality. An experiment was set up in which participants…

In this article, I present a study in which 12 pairs of 8th-grade students solved problems about a physical device with algebra. The device, called a winch, instantiates motions that can be modeled by pairs of simultaneous linear functions. The following question motivated the study: How can students generate algebraic models without direct…

Analogical reasoning has long been believed to play a central role in mathematics learning and problem solving (see Genter, Holyoak, & Kokinov, 2001); however, little is known about how analogy is used in everyday instructional contexts. This article examines analogies produced in naturally occurring U.S. mathematics lessons to explore patterns…