Describing pre-service teachers' developing understanding of elementary number of theory topics
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Although elementary number theory topics are closely linked to foundational topics in number and operations and are prevalent in elementary and middle grades mathematics curricula, little is currently known about how students and teachers make sense of them. This study investigated pre-service elementary teachers' developing understanding of elementary number theory topics, including factors, divisibility, greatest common factor, and least common multiple. Fifty-nine participants in a college mathematics course for pre-service elementary teachers participated in a three-week unit of instruction on number theory. All participants completed the Number Theory Knowledge Test (NTKT) before and after instruction. Additionally, individual clinical interviews were conducted with six participants before and after instruction. Each interview was recorded and transcribed. In order to describe how participants' understanding of number theory developed during instruction, analysis of the interview data was done using Dubinsky's (1991) APOS theory. Results showed that the most notable changes were in participants' attention to prime factorization, their notions of factor, and their abilities to coordinate multiple processes. Prior to instruction, participants' understanding was characterized by a need to convert prime factored numbers into decimal form, the notion of factor as a number that is visible within the prime factorization of another number, and an inability to coordinate multiple processes. Following instruction, participants attended to a number's prime factorization in order to solve problems, identified factors as combinations of prime numbers within the prime factorization of another number, and coordinated multiple processes. Statistical analysis of the NTKT scores supported the interview data, showing that participants significantly improved in their performance on questions requiring both procedural and conceptual knowledge. Contrary to prior research, this study provides evidence that pre-service elementary teachers can develop a deep and connected understanding of number theory topics. Future research should continue to investigate pre-service teachers' understanding of number theory by examining their work during periods of classroom instruction. Additionally, mathematical tasks that attend to prime factorization as a foundational tool should be investigated for their effects on pre-service teachers' understanding of number theory.
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