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This dissertation reports the results of a teaching experiment, which explored student thinking about integer addition and subtraction. Through the lens of commognitive theory (Sfard, 2008), interpreting negative integers as secondary intuitions (Fischbein, 1987), and employing teaching experiment methodology (Steffe & Thompson, 2000), this study was a first step in developing more robust descriptions of students’ conceptual models for integer addition and subtraction. I investigated: (a) the conceptual models that students exhibited, (b) the various ways that students utilized conceptual models while learning about the addition and subtraction of integers, and (c), the ways that students’ conceptions evolved over the course of a teaching experiment. This study of students’ conceptual models led to the modification and refinement of the CMIAS descriptions (Wessman-Enzinger & Mooney, 2014, 2015).

Three Grade 5 students were selected based on results of a pilot study and students’ responses to a written assessment. Data for this study was used from the Open Number Sentence and Context Individual Sessions of the 12-week teaching experiment.

All of the Individual Sessions were videotaped and transcribed. The transcripts paired with all of the drawings produced by the students for stories for open number sentences generated and open number sentences solved during all Individual Sessions constituted the unit of analysis. A constant comparative method (Merriam, 1998) was used to modify the previous descriptions and develop new descriptions of the CMIAS (Wessman-Enzinger & Mooney, 2014). A description of the changes in mathematical discourse (i.e., word use, visual mediators, narratives, routines) was used to highlight the learning of integer addition and subtraction across the Individual Sessions.

There are seven CMIAS described in this study: Bookkeeping, Counterbalance, Translation, Relativity, Proceduralization, Analogy, and Algebraic Reasoning. These CMIAS were used differently in Individual Context Sessions and Individual Open Number Sentence Sessions. The three Grade 5 students also prominently utilized certain CMIAS over others. How the students used these CMIAS changed over time, these changes are considered to be descriptions of learning about integer addition and subtraction.

The results presented in this study extend the literature on student thinking about integer addition and subtraction by (a) describing student thinking within both contextual and symbolic problem types; (b) extending and modifying the previous descriptions of the CMIAS; and (c) providing a developmental perspective that includes learning over an extended period of time.