In most integer-related research, researchers (a) document students’ errors, misconceptions, or ways of approaching certain types of problems (Chiu, 2001; Gallardo, 2002; Peled (1991); Thomaidis & Tzanakis, 2007; Vlassis, 2008) or (b) propose instructional models or contexts for teaching integers and operations on integers (Janvier, 1983; Liebeck, 1990; Linchevski & Williams, 1999; Streefland, 1996; Thompson & Dreyfus, 1988). The majority of integer research falls into the latter category, with description of a variety of instructional models, tools, and contexts for teaching integers (e.g., scores and forfeits, number lines, microworlds). Much of the former category of integer-related research documents the significant challenges middle and high school students have in either understanding or correctly responding to integer-related tasks (see, for example, Gallardo, 2002; NAEP, 1996; Thomaidis & Tzanakis, 2007; Vlassis, 2008).

Integers and integer operations are building blocks for success with algebra. They mark a transition from arithmetic to algebra because of their abstract nature (Hefendehl-Hebeker, 1991; Linchevski & Williams, 1999); and to navigate algebraic equations, students must perform algebraic procedures using additive inverses, which first come into play with the introduction of integers. Although research shows that students struggle to make sense of integers, to date, few researchers have focused on students’ ways of reasoning about integers (exceptions include Behrend & Mohs (2006) and Wilcox (2008)). We focus on Grades 2 and 4 students’ ways of reasoning about integers because, as a field, we have little knowledge about the ways in which young children approach this mathematical topic. Further, we believe that by understanding students’ informal, preinstructional ways of reasoning, teachers, educational researchers, and curriculum designers can build on these ways so that students can develop more sophisticated reasonings about integers. 

In our research design and analyses, we take a children’s mathematical thinking perspective (Carpenter et. al, 1999; Empson & Levi, 2011). Thus, in starting with how students make sense of mathematics, we try to uncover their ways of reasoning about integers.

Methods

Participants

We are reporting on a subset of a large, cross-sectional, multi-grade-level study. In this work session, we report findings from interviews with Grades 2 and 4 students only. Participants included 40 students in each of Grades 2 and 4 from seven schools in San Diego County, selected to represent a broad cross-section of students according to ethnicity, socioeconomic status, and scores on California Standardized Tests.  None of these 2^nd- and 4^th-grade students had received formal, school instruction on integers. Our goal was to document these children’s informal and intuitive integer understandings prior to school-based instruction.

Interview Tasks

We developed tasks specific to integer reasoning for problem-solving interviews on the bases of an historical analysis of integer development and a review of the relevant literature. We posed open number sentences (i.e., problems of the form -3 + c = 6 and c + 6 = 4), varying the location of the unknown. We also posed comparison tasks (e.g., Which is larger, -7 or -8?) and problems in context. The hour-long clinical interviews were conducted during the school day.

Analysis

We began analyzing interviews with a process of open coding (Strauss & Corbin, 1998), focusing on students’ solution strategies and underlying ways of reasoning about number and operations. We used the constant-comparative method to identify emergent, distinguishing themes and features of students’ reasoning about integers. We have found that for young children who had negative numbers in their conceptual domains (that is, students who knew of negative numbers), their responses fell primarily into one of three categories of reasoning about integers—leveraging the ideas of order, magnitude, or formalisms. Though individual children did not typically use a single way of reasoning across all problems, many children were consistent in their problem-solving approaches and invoked a primary way of reasoning. Thus, we find useful considering how a particular view of number, consistently applied across problems, both supported and constrained students’ integer understanding.

We also consider the relative difficulty of problems for the students by examining patterns of reasoning (and correctness) both within and across students. Understanding which tasks are more challenging (or easier) than others and hypothesizing about what makes some tasks more challenging than others can inform sequencing of tasks for teachers and curriculum designers.

Session Organization

In this work session we seek to engage the audience in a subset of the findings from a cross-sectional study designed to document students’ conceptions of integers. Presenters will share findings about Grades 2 and 4 students’ conceptions and relative problem difficulty and will engage the audience in extended discussion about the findings and elicit their ideas about implications. The session will serve as an occasion to share with the community young children’s ways of reasoning about integers, an area that has received scant attention in the literature.

5 minutes        Session and speaker introduction

50 minutes      Presentation and audience discussion of Grades 2 and 4 students’ ways of reasoning

During this time we will show several carefully chosen video clips. After each clip, the audience will share what they notice and react to findings. As appropriate, we will facilitate discussions about students’ underlying integer conceptions and relative problem difficulty.

30 minutes      Audience discussion about implications of findings

During this time we will orchestrate conversations about implications for teaching and for further research.

5 minutes        Wrap up and audience debrief of session

Five Central Questions

1. What do our findings about Grades 2 and 4 students’ strategies suggest about the students’ conceptions of integers?
2. What was the relative level of difficulty of tasks for Grades 2 and 4 students, and what task characteristics affected the level of difficulty?
3. In what ways do findings resonate with audience members? What findings are surprising?
4. What are possible implications of the findings about young children’s informal conceptions of integers for teachers and teacher educators?
5. What are possible implications of the findings about young children’s informal conceptions of integers for researchers?

References

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