Abstract
The purposes of this study were to (1) investigate non-accelerated middle schools students' understanding of pre-algebraic concepts, in particular, addition and subtraction of integers and the notion of equality; (2) describe and analyze how non-accelerated middle school students interact with The Geometer's Sketchpad (GSP) applets depicting pre-algebraic concepts in order to discover how these applets can elucidate or mask these concepts and hence facilitate or hinder understanding; (3) to chronicle how students' conceptions of these pre-algebraic concepts change during and after instructional use of the GSP applets; (4) to receive feedback from students to help develop the applets in order to improve the value of such applets in the algebra classroom. The participants in this study were eight middle school students taking part in the after-school math-tutoring program. All participants had had some algebra instruction and needed remediation in addition and subtraction of integers and solving equations. These eight students consisted of five females, three males, three sixth graders, three seventh graders, and two eighth graders. Each participant met with me for at least two sessions, totaling of at least two hours and fifteen minutes. The study included three self-contained modules consisting of two sections: an interview section without technology and an instructional section with technology. The interview section was audio recorded, and the instructional section was video and audio recoded. During the interview section, participants were asked to complete a variety of tasks based on literature findings of pre-algebraic and algebraic concepts in order to elicit participant conceptions. I used the task-based interview section of each module to assess the students' knowledge of the module's topic. If during this section of the module I saw that the student could not solve the task or articulate conceptual reasoning for his answer, I used that task to segue into the accompanying instructional section for that module. Thus, I met the student where he was in terms of understanding and guided him with technology and questions to answer and understand the mathematical idea. However, if the student could solve and satisfactorily explain the correct reasoning behind the answer in the task-based interview section, I moved to the next question in the task-based interview section of the same module. If a student had difficulty with one question in the task-based interview section, he could have not fully finished this section, for he would have been segued into the accompanying instructional section of the module. If a student could correctly solve and reason through every task-based interview question in one module, I continued to the next module's task-based interview section. After data collection, data will be transcribed and reviewed for coding an analysis. Results will be presented in case studies and a cross case analysis. Each case study will illustrate (1) a participant's initial understanding of operations on integers, the notion of equality, and solving equations, (2) the interactions with the dynamic technology applets GSP, and (3) the conceptual changes regarding the pre-algebraic and algebraic concepts. The findings of this study will have implications for mathematics teachers as they prepare students' for learning algebra. These findings will also inform mathematics researchers on the conceptions of these topics for sixth and seventh-grade students and possible future research avenues in algebra.
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