ORIGINAL ARTICLE Towards curricular coherence in integers and fractions: a study of the efficacy of a lesson sequence that uses the number line as the principal representational context Geoffrey B. Saxe • Ronli Diakow • Maryl Gearhart Accepted: 21 September 2012 / Published online: 13 October 2012 Ó FIZ Karlsruhe 2012 Abstract This paper reports a study of the efficacy of Learning Mathematics through Representations (LMR), an innovative curriculum unit designed to support upper ele- mentary students’ understandings of integers and fractions. The unit supports an integrated treatment of integers and fractions through (a) the use of the number line as a cross- domain representational context, and (b) the building of mathematical definitions in classroom communities that become resources to support student argumentation, gen- eralization, and problem solving. In the efficacy study, fourth and fifth grade teachers employing the same district curriculum (Everyday Mathematics) were matched on background indicators and then assigned to either the LMR experimental classrooms (n = 11) or the comparison group (n = 8 with 10 classrooms). During the fall semester, LMR teachers implemented the LMR unit on 19 days and district curriculum on other days of mathematics instruction. HLM analyses documented greater achievement for LMR stu- dents than Comparison students on both the end-of-unit and the end-of year assessments of integers and fractions knowledge; the growth rates of LMR students were similar regardless of entering ability level, and gains for LMR students occurred on item types that included number line representations and those that did not. The findings point to the efficacy of the LMR sequence in supporting teaching and learning in the domains of integers and fractions. Keywords Elementary mathematics education  Curriculum  Fractions  Integers 1 Introduction There is consensus in the U.S. that many U.S. students have inadequate knowledge of fractions. Citing findings from myriad sources, the National Mathematics Advisory Panel (2008) and the U.S. Department of Education’s What Works Clearinghouse (2010a) have highlighted patterns of weak student performance on fractions assessments, pat- terns that raise concerns about students’ preparation for mathematics topics like algebra that require solid under- standing of rational numbers. Part of the problem is cur- ricular. Fractions is one of many topics that are deeply connected within the domain of rational numbers, yet in textbooks and in instruction these topics are often treated as unrelated (Schmidt, Whan, & McKnight, 2005; Stigler, Gonzales, Kawanaka, Knoll, & Serrano, 1999). Clearly there is a critical need to strengthen the U.S. elementary curriculum and teacher professional development in the domain of rational numbers. In this article we report results from a study of the efficacy of a new elementary curriculum designed to address national concerns. Entitled Learning Mathematics through Representations (LMR), the curriculum integrates the topics of integers and fractions, two strands of the elementary curriculum that have deep connections and yet are often separated in textbooks (Schmidt, et al., 2005). In the LMR curriculum, the number line serves as the prin- cipal representational context, and students’ initial work with the representation of integers on the line provides a foundation for students’ subsequent work with the repre- sentation of fractions on the line. Across LMR lessons, teachers guide students to construct, apply, and investigate the entailments of mathematical definitions for core ideas such as order, unit interval, and subunit interval as they solve challenging problems. While LMR’s focus on the G. B. Saxe (&)  R. Diakow  M. Gearhart University of California, Berkeley, CA, USA e-mail: saxe@berkeley.edu 123 ZDM Mathematics Education (2013) 45:343–364 DOI 10.1007/s11858-012-0466-2