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

Examining Pinterest as a Curriculum Resource for Negative Integers: An Initial Investigation

“…To this end, we use the term mathematical model (i.e., model) to refer to "material, visual sketches, paradigmatic situations, schemes, diagrams, and even symbols" that help students manage, document, communicate, or interpret mathematical ideas and phenomena [27] (p. 13). Much of the research with integers supports the use of the number line model [10,28] and chip model [29], though some researchers have also described models such as elevators [13,22,24]. Research examining textbooks and curricula in the domain of integer operations [30] has shown the use of number lines, chip models, and contexts as models for dealing with integer operations.…”

“…A further obstacle in teaching about integers is that although curricula and tasks exist that support conceptual understandings of integers (e.g., [28,32,33]), it is a challenge to support teachers in developing the ability to discern between materials that are conceptually focused versus those that are procedurally focused. This is particularly the case for integer subtraction, where different number sentences support different types of reasoning [13].…”

Studying Cognition through Time in a Classroom Community: The Interplay between “Everyday” and “Scientific Concepts”

“…First, the curriculum provided a useful arena for longitudinal analysis. As we noted previously, in an experimental study contrasting LMR classrooms with controls, LMR classrooms achieved strong learning gains [Saxe et al, 2013a] that were reflected in Mr. K's classroom. Second, the curriculum supported an analysis of top-down developments since it privileged the introduction of discipline-linked mathematical definitions throughout the lessons that cover mathematical content often considered "hard to learn" and "hard to teach."…”

“…8 The teacher, whom we refer to as Mr. K, had more than 20 years of upper elementary teaching experience; he had some familiarity with LMR lessons prior to the year of data collection, having piloted selected lessons in the year prior to data collection. Noteworthy also is that the video and other forms of data collected in Mr. K's classroom were part of an efficacy study of the LMR curriculum; like other LMR classrooms in the study, the students in Mr. K's classroom made strong gains relative to control classrooms on measures of integers and fractions knowledge [Saxe et al, 2013a].…”

“…Our goal is to understand what it might look like for a productive interplay to be supported in classroom instruction as lesson topics shift over time. To leverage our analytic efforts, we focus on a fifthgrade case study classroom in which a teacher is implementing a curriculum called Learning Mathematics through Representations (LMR) [Gearhart & Saxe, 2014;Saxe, de Kirby, Le, Sitabkhan, & Kang, 2014;Saxe, Diakow, & Gearhart, 2013a]. The LMR curriculum is a 19-lesson sequence that provides a fruitful context for study, a context in which a potential for an interplay stands out in remarkably clear relief.…”