Most mathematics beyond middle school requires that students operate with all real numbers, both positive and negative quantities. Later mathematics (e.g., linear and quadratic equations, coordinate graphing, absolute value equations, transformations, and matrices) and science topics (e.g., vectors, sound waves, and charged particles) all require use of negative numbers. Yet, some of the identified difficulties students have with more advanced studies, such as algebra, are due to difficulties with negative number arithmetic and notation (Vlassis, 2008).With the intent to help students overcome these difficulties, multiple instructional models are promoted in school textbooks and teacher resources. Multiple models are used although little is known about what students learn with these models. Different integer instructional models have different implications for learning mathematics, because they draw on different conceptual metaphors and students physically move their bodies in different ways to enact these models. In light of conceptual metaphor theory, the difficulties students have had with negative numbers and even those of mathematicians in history (Hefendehl-Hebeker, 1991), reveal that the collecting objects metaphor may be a cognitive obstacle to those first learning negative number arithmetic. Moreover, consistency of humans’ physical motions with targeted ideas is a factor of cognition, so the influence of students’ physical movements on their learning may be a critical, yet underexplored factor.In order to compare how these model differences influence learning, this study randomly assigned eight classes of initial learners to a specific collecting-objects (chip model) or moving-on-a-path metaphor-based model (number line model) to learn integer arithmetic with the four primary operations (addition, subtraction, multiplication, and division). This dissertation presents the main results of this pre-post-delayed posttest study to answer the questions: (a) With respect to ordering numbers and integer arithmetic, what do students demonstrate learning by enacting each model and what, if any differences in learning are found between models? (b) What meanings do students express for “-” symbols (negative signs, subtraction signs, or opposite signs) and how does the integer model used influence these meanings? Findings indicated that either instructional model did support significant learning gains for integer arithmetic and qualitative expression of basic “-” symbol meanings. These findings, moreover, did support theory that a motion-aligned model using a moving-on-a-path metaphor (walk-it-off number line model) was a better first model, because it supported initial integer learning better than a collecting objects metaphor based model both for integer arithmetic learning and opposite meanings of “-.”
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