This commentary raises and discusses questions based on some of the agreements, disagreements, and themes found in the four chapters on fractions. It considers (1) the importance of tasks that are based in perception and readily available activity in light of an emphasis on problem solving in mathematics education, and the role that theories about…

The present study is motivated by a significant body of research documenting teachers' perennial difficulties with a critical swath of topics related to multiplication. In response, we track how Nina, a future middle grades mathematics teacher, made progress constructing explanations across topics by reasoning with measurement-based definitions of…

We examine opportunities and challenges of applying a single, explicit definition of multiplication when modeling situations across an important swathe of school mathematics. In so doing, we review two interrelated conversations within multiplication research. The first has to do with identifying and classifying situations that can be modeled by…

Findings are presented from an analysis of how six future middle-grade teachers reasoned with strip diagrams and a variable parts perspective on proportional relationships to develop and explain equations in two variables. One equation was for two quantities varying together and one was for a line through the origin in a coordinate plane. Both…

This article explores teaching practices described in NCTM's Principles to Actions: Ensuring Mathematical Success for All. Common factors, common multiples, strip diagrams, and double number lines are discussed in this, the third installment in the series. [For the second installment in this series see "Connecting the Dots: Building…

In this article, we present a mathematical analysis that distinguishes two distinct quantitative perspectives on ratios and proportional relationships: variable number of fixed quantities and fixed numbers of variable parts. This parallels the distinction between measurement and partitive meanings for division and between two meanings for…

The present study extends recent advances coordinating research on cognition and psychometric modeling around fractions. Recent research has demonstrated that the Diagnosing Teachers' Multiplicative Reasoning Fractions survey provides information about distinct components necessary for reasoning in terms of quantities when solving fraction…

We present a mathematical analysis that distinguishes two quantitative perspectives on ratios and proportional relationships: Multiple Batches and Variable Parts. We argue that (a) existing research on proportional relationships has addressed Multiple Batches but has largely overlooked Variable Parts, (b) Multiple Batches makes the co-variation…

We report results from a mathematics content course intended to help future teachers form a coherent perspective on topics related to multiplication, including whole-number multiplication and division, fraction arithmetic, proportional relationships, and linear functions. We used one meaning of multiplication, based in measurement and expressed as…

We have extended two perspectives of proportional reasoning to solve problems based in probability. Four future middle grade teachers were enrolled in a mathematics content course that emphasized reasoning about multiplication with quantities. The course expected future teachers to generate and explain methods for solving proportions. Probability…

"Focus in Grade 1: Teaching with Curriculum Focal Points" describes and illustrates learning paths for the mathematical concepts and skills of each grade 1 Focal Point as presented in Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics. It includes representational supports for teaching and learning that can facilitate…

The twofold purpose of this study was to trace prospective and practicing mathematics teachers' understandings of content area reading instruction in relation to domain knowledge in mathematics, and to examine the extent to which online pedagogical mentoring supported the integration of such instruction and knowledge. The design called for two…

Are sequences functions? What can't the popular "vertical line test" be applied in some cases to determine if a relation is a function? How does the idea of rate of change connect with simpler ideas about proportionality as well as more advanced topics in calculus? Helping high school students develop a robust understanding of functions requires…

The Institute of Education Sciences (IES) publishes practice guides in education to bring the best available evidence and expertise to bear on current challenges in education. Authors of practice guides combine their expertise with the findings of rigorous research, when available, to develop specific recommendations for addressing these…

Instructional tips help educators carry out recommendations contained in IES Educator's Practice Guides. The tips translate these recommendations into actionable approaches that educators can try in their classrooms. These tips are based on a practice guide authored by Sybilla Beckmann, Mark Driscoll, Megan Franke, Patricia Herzig, Asha Jitendra,…