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This study describes the story of a novice mathematics teacher's action research on re-examining her classroom teaching of junior high school algebra, during master of mathematics education courses. The major research question for her is: How to develop arithmetic-algebraic thinking in terms of the functional aspect? Moreover, the study assesses its effects via that way. Thus, there are three aims. First is to design teaching activities in order to develop 7th graders' arithmetic-algebraic thinking; second, to assess teaching effects concerning the students' algebraic/functional concepts and attitudes toward learning; at last, to examine the possibility of re-conceptualizing the author's pedagogical/mathematical powers of teaching school algebra in terms of students' feedbacks and self-reflection. The results of this study showed that the functional aspect could be acted as a kind of learning media in transition of arithmetic to algebraic thinking for those 7th graders. By being embedded in an experientially real problem situation, observing and understanding quantitative relationships, the students were able to understand the fundamental algebraic concepts while obtaining the relevant symbol skills. Through challenging (test), clarifying (intervention), and changing (extension) phases, the author examined constantly the teaching effects in three orientations related to cognitive, affective, and social-interactive aspects, trying to re-construct her own mathematical/pedagogical powers and instruction practice. The re-construction consisted of developing teaching tactics/content of arithmetic-algebraic thinking; understanding student pre-requisites, generic concepts, cooperative learning approaches, Martin Simon's ideas of HLT-HTT, and learning how to build the experientially real arithmetic-algebraic learning situation. Based on the results, the author proposed “a 3-phases-3-aspcts model of teacher-student learning”, which included processes and content of changing concepts of teaching (teacher) and learning (students). Among the above, the transition of the teacher's mathematical/pedagogical concepts/practice and the development of the students' arithmetic-algebraic thinking were represented as a dynamic/interactive “doubled cycle of learning” informed each other. In this dynamically interactive process, not only students learned cooperatively the algebraic concepts with the functional aspect in the more or less real situation, the author also realized more deeply the underlying mathematical structures of algebra, relevant pedagogical concepts, and her students' algebraic thought processes. It seemed that the functional aspect could partly overcome the cognitive gaps of learning between arithmetic and algebraic thinking, and thus resolved partially the problems of teaching and learning algebra in the junior high level. It could also accomplish the implicit goal of school algebra, i.e. the concept of variables, through concrete operation with and classroom discussion on the meaning of symbols and equivalences in terms of variables concept. In addition, the author's abilities of algebraic teaching were improved in the process of re-conceptualizing or re-constructing mathematical/pedagogical/reflective powers. Hopefully, this phased research approaches and results can contribute references about teaching of algebra and the professional development to other mathematics teachers of junior high school, in resolving his or her problems of classroom mathematics teaching.
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