Preservice and novice in-service mathematics teachers' understanding of concepts related to number theory
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Learning about numbers is a big part of learning mathematics. Basic concepts of number theory give students tools to understand the structure of the number system. Starting from relatively few and easy to understand ideas students may arrive at valid number theory conjectures using different strategies. Number theory is a promising area of mathematics for developing and assessing conceptual understanding. Teaching for understanding is often stated as the most important goal of mathematics education. In order to teach mathematics meaningfully, teachers themselves should have deep conceptual knowledge. That is why information about teachers' understanding is crucial for the development of productive teacher education. Investigation of preservice and novice in-service high school teachers' understanding of concepts related to introductory number theory has received little attention. My research is about preservice and novice in-service teachers' understanding of number theory ideas. To what extent is their understanding relational? How well can they explain concepts related to number theory? How fluent are they with algebraic proofs? Participants in this research were six young mathematics teachers who received their education and started their teaching careers in Western New York. Task-based interviews constructed as sequences of non-routine problems requiring the use of different types of representations and the development of connections between the concepts were used to obtain information about teachers' understanding of ideas related to number theory. The results of this study suggest that more attention should be paid to the development of conceptual understanding of mathematics ideas in general and basic number theory in particular within teacher education programs.