Sybilla Beckmann and Andrew Izsák

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 multiplication—one rooted in counting equal-sized groups, the other in scaling the size of the groups. We argue that (a) the distinction in perspectives is independent from other distinctions in the literature on proportional relationships, including the within measure space versus between measure space ratio distinction; (b) the psychological roots for multiplication suggest the accessibility of the two perspectives to learners; and (c) the fixed numbers of variable parts perspective, though largely overlooked in past research, may provide an important foundation for central topics that build on proportional relationships. We also suggest directions for future empirical research.

Maria Blanton, Ana Stephens, Eric Knuth, Angela Murphy Gardiner, Isil Isler, & Jee-Seon Kim

This article reports results from a study investigating the impact of a sustained, comprehensive early algebra intervention in third grade. Participants included 106 students; 39 received the early algebra intervention, and 67 received their district’s regularly planned mathematics instruction. We share and discuss students’ responses to a written pre- and post-assessment that addressed their understanding of several big ideas in the area of early algebra, including mathematical equivalence and equations, generalizing arithmetic, and functional thinking. We found that the intervention group significantly outperformed the nonintervention group and was more apt by posttest to use algebraic strategies to solve problems. Given the multitude of studies among adolescents documenting students’ difficulties with algebra and the serious consequences of these difficulties, an important contribution of this research is the finding that—provided the appropriate instruction—children are capable of engaging successfully with a broad and diverse set of big algebraic ideas.

Keith R. Leatham, Blake E. Peterson, Shari L. Stockero, & Laura R. Van Zoest

The mathematics education community values using student thinking to develop mathematical concepts, but the nuances of this practice are not clearly understood. We conceptualize an important group of instances in classroom lessons that occur at the intersection of student thinking, significant mathematics, and pedagogical opportunities—what we call Mathematically Significant Pedagogical Opportunities to Build on Student Thinking. We analyze dialogue to illustrate a process for determining whether a classroom instance offers such an opportunity and to demonstrate the usefulness of the construct in examining classroom discourse. This construct contributes to research and professional development related to teachers’ mathematically productive use of student thinking by providing a lens and generating a common language for recognizing and agreeing on a critical core of student mathematical thinking that researchers can attend to as they study classroom practice and that teachers can aspire to notice and build upon when it occurs in their classrooms.

Cynthia W. Langrall

December is typically the time for year-in-review reflections, but given the publication schedule for JRME, a report of the status of the journal falls to the January issue.

Reviewed by M. Kathleen Heid

Review of *The SimCalc Vision and Contributions: Democratizing Access to Important Mathematics*. Stephen J. Hegedus & Jeremy Roschelle (Eds.) (2013).

Randall E. Groth

Statistics education has begun to mature as a discipline distinct from mathematics education, creating new perspectives on the teaching and learning of statistics. This commentary emphasizes the importance of coordinating perspectives from statistics education and mathematics education through boundary interactions between the two communities of practice. I argue that such interactions are particularly vital in shared problem spaces related to the teaching and learning of measurement, variability, and contextualized problems. Collaborative work within these shared problem spaces can contribute to the vitality of each discipline. Neglect of the shared problem spaces may contribute to insularity and have negative consequences for research and school curricula. Challenges of working at the boundaries are considered, and strategies for overcoming the challenges are proposed.