The current study examined the effectiveness of self-explanation prompts, visual signaling cues, and a combination of the two features on middle school students' (N = 202) algebra learning. Also explored were the differential effects of features for students with faulty conceptual knowledge (evidenced by a higher prevalence of making errors during…

The struggle with fraction learning in kindergarten through Grade 12 in the United States is a persistent problem and one of the major stumbling blocks to succeeding in higher mathematics. Research into this problem has identified several areas where students commonly struggle with fractions. While there are many theories of fraction learning,…

This article investigates why students reported liking a student-driven learning design better than a highly guided design despite equivalent gains in knowledge assessments in both conditions. We created two learning designs based on the distinction in the literature between student-driven and teacher-led approaches. One teacher assigned each of…

A total of 136 eighth-grade math students from 2 Singapore schools learned from either productive failure (PF) or vicarious failure (VF). PF students "generated" solutions to a complex problem targeting the concept of variance that they had not learned yet before receiving instruction on the targeted concept. VF students…

Some intensive quantities, such as slope, velocity, or likelihood, are perceptually privileged in the sense that they are experienced as holistic, irreducible sensations. However, the formal expression of these quantities uses "a/b" analytic metrics; for example, the slope of a line is the quotient of its rise and run. Thus, whereas students'…

Gestures are often taken as evidence that the body is involved in thinking and speaking about the ideas expressed in those gestures. In this article, we present evidence drawn from teachers' and learners' gestures to make the case that mathematical knowledge is embodied. We argue that mathematical cognition is embodied in 2 key senses: It is based…

This article examines how conversations during design reviews in which 8th-grade mathematics students shared population models with visiting specialists expanded the disciplinary expertise of the classroom. "Re-contextualizing" is a conversational exchange that visiting specialists initiated to invite groups to consider their models in novel…

Describes the refinement of a conception in the domain of linear functions. Examples show how students refine their use of the x-intercept in equations of the form "y=mx+b" by narrowing problem contexts, making connections between conceptions, using a mathematical procedure, and refining verbal descriptions. Resources that teachers can draw on to…