This article describes 1 prevalent practice that a group of 6th- and 7th-grade students engaged in when they used fractions in the context of area and perimeter, decimal operations, similarity, and ratios and proportions. The study's results revealed that students did not simply take the concepts and skills learned in formal fractions units and use them in these other mathematical content areas. Their understanding of how to use fractions was tied to their understanding of situations in which they could be used. Students had to take into account both mathematical and situational contexts when making choices about how to use fractions. This led students to raise questions regarding what was appropriate when using fractions in these new contexts and how fractions and the new contexts were related.

### Anderson Norton, Jesse L. M. Wilkins and Cong ze Xu

Through their work on the Fractions Project, Steffe and Olive (2010) identified a progression of fraction schemes that describes students' development toward more and more sophisticated ways of operating with fractions. Although several quantitative studies have affirmed this progression, the question has remained open as to whether it is specific to the U.S. classrooms in which these studies were conducted or whether it describes a developmental progression that crosses international boundaries. The purpose of our replication study was to address that question using data gathered from written assessments of 76 5th- and 6th-grade students in China. Results indicate a remarkably similar progression among students in the United States and students in China.

### Dina Tirosh

In this article I present and discuss an attempt to promote development of prospective elementary teachers' own subject-matter knowledge of division of fractions as well as their awareness of the nature and the likely sources of related common misconceptions held by children. My data indicate that before the mathematics methods course described here most participants knew how to divide fractions but could not explain the procedure. The prospective teachers were unaware of major sources of students' incorrect responses in this domain. One conclusion is that teacher education programs should attempt to familiarize prospective teachers with common, sometimes erroneous, cognitive processes used by students in dividing fractions and the effects of use of such processes.

### Catherine Lewis and Rebecca Perry

An understanding of fractions eludes many U.S. students, and research-based knowledge about fractions, such as the utility of linear representation, has not broadly influenced instruction. This randomized trial of lesson study supported by mathematical resources assigned 39 educator teams across the United States to locally managed lesson study supported by a fractions lesson study resource kit or to 1 of 2 control conditions. Educators (87% of whom were elementary teachers) self-managed learning over a 3-month period. HLM analyses indicated significantly greater improvement of educators' and students' fractions knowledge for teams randomly assigned to lesson study with resource kits. Results suggest that integrating researchbased resources into lesson study offers a new approach to the problem of “scale-up” by combining the strengths of teacher leadership and research-based knowledge.

### Geoffrey B. Saxe, Edd V. Taylor, Clifton McIntosh and Maryl Gearhart

This study had two purposes: (a) to investigate the developmental relationship between students' uses of fractions notation and their understandings of part-whole relations; and (b) to produce an analysis of the role of fractions instruction in students' use of notation to represent parts of an area. Elementary students (*n* = 384) in 19 classes participated in the study. Pre- and posttests were administered before and after fractions instruction, and key lessons were recorded with videotape and field notes. Students' written responses were coded in two ways: for the forms of the notations (e.g., use of numerator, denominator, and separation line), and for the concepts captured by the notations (e.g., part-whole, part-part, or other kinds of relations). The lessons captured on videotapes and in field notes were rated with respect to their alignment with principles supported by reform frameworks in mathematics education (e.g., opportunity to build understanding of fractions concepts, ongoing assessment of student understanding). Our analyses indicated (a) notation and reference were acquired somewhat independently, and (b) classroom practices that built on students' thinking were more likely to support shifts toward normative uses of notation.

### Andrew Izsák;, Erik Tillema and Zelha Tunç-Pekkan

We present a case study of teaching and learning fraction addition on number lines in one 6th-grade classroom that used the Connected Mathematics Project *Bits and Pieces II* materials. Our main research questions were (1) What were the primary cognitive structures through which the teacher and students interpreted the lessons? and (2) Were the teacher's and her students' interpretations similar or different, and why? The data afforded particularly detailed analyses of cognitive structures used by the teacher and one student to interpret fractions and their representation on number lines. Our results demonstrate that subtle differences in methods for partitioning unit intervals did not seem important to the teacher but had significant consequences for this student's opportunities to learn. Our closing discussion addresses knowledge for teaching with drawn representations and methods for examining interactions between teachers' and students' interpretations of lessons in which they participate together.

### Susan B. Empson

This article presents an analysis of two low-performing students' experiences in a firstgrade classroom oriented toward teaching mathematics for understanding. Combining constructs from interactional sociolinguistics and developmental task analysis, I investigate the nature of these students' participation in classroom discourse about fractions. Pre- and post-instruction interviews documenting learning and analysis of classroom interactions suggest mechanisms of that learning. I propose that three main factors account for these two students' success: use of tasks that elicited the students' prior understanding, creation of a variety of participant frameworks (Goffman, 1981) in which the students were treated as mathematically competent, and frequency of opportunities for identity-enhancing interactions.

### Maryl Gearhart, Geoffrey B. Saxe, Michael Seltzer, Jonah Schlackman, Cynthia Carter Ching, Na'ilah Nasir, Randy Fall, Tom Bennett, Steven Rhine and Tine F. Sloan

In this study we addressed 2 questions: (a) How can we document opportunities to learn aligned with the NCTM *Standards*? (b) How can we support elementary teachers' efforts to provide such opportunities? We conducted a study of the effect of curriculum (problem solving vs. skills) and professional development (subject-matter focused vs. collegial support) on practices and learning. From analyses of videotapes and field notes, we created 3 scales for estimating students' opportunities to learn. Analyses of fractions instruction in 21 elementary classrooms provided evidence of the technical quality of the indicators and indicated that support for teachers' knowledge may be required for a problem-solving curriculum to be beneficial.

### Andrew Izsák, Erik Jacobson, Zandra de Araujo and Chandra Hawley Orrill

Researchers have recently used traditional item response theory (IRT) models to measure mathematical knowledge for teaching (MKT). Some studies (e.g., Hill, 2007; Izsák, Orrill, Cohen, & Brown, 2010), however, have reported subgroups when measuring middle-grades teachers' MKT, and such groups violate a key assumption of IRT models. This study investigated the utility of an alternative called the *mixture Rasch model* that allows for subgroups. The model was applied to middle-grades teachers' performance on pretests and posttests bracketing a 42-hour professional development course focused on drawn models for fraction arithmetic. Results from psychometric modeling and evidence from video-recorded interviews and professional development sessions suggested that there were 2 subgroups of middle-grades teachers, 1 better able to reason with 3-level unit structures and 1 constrained to 2-level unit structures. Some teachers, however, were easier to classify than others.

### Kathleen A. Cramer, Thomas R. Post and Robert C. delMas

This study contrasted the achievement of students using either commercial curricula (CC) for initial fraction learning with the achievement of students using the Rational Number Project (RNP) fraction curriculum. The RNP curriculum placed particular emphasis on the use of multiple physical models and translations within and between modes of representation—pictorial, manipulative, verbal, real-world, and symbolic. The instructional program lasted 28–30 days and involved over 1600 fourth and fifth graders in 66 classrooms that were randomly assigned to treatment groups. Students using RNP project materials had statistically higher mean scores on the posttest and retention test and on four (of six) subscales: concepts, order, transfer, and estimation. Interview data showed differences in the quality of students' thinking as they solved order and estimation tasks involving fractions. RNP students approached such tasks conceptually by building on their constructed mental images of fractions, whereas CC students relied more often on standard, often rote, procedures when solving identical fraction tasks. These results are consistent with earlier RNP work with smaller numbers of students in several teaching experiment settings.