金鈐黃乃文Nai-wen Huang2019-09-052005-9-12019-09-052005http://etds.lib.ntnu.edu.tw/cgi-bin/gs32/gsweb.cgi?o=dstdcdr&s=id=%22G0069140008%22.&%22.id.&http://rportal.lib.ntnu.edu.tw:80/handle/20.500.12235/101457本研究描述一位新手國中數學教師，於在職進修期間，透過反思重新檢視自己代數相關單元的教學，而展開一段教學行動研究的歷程。作者想探討：如何透過函數觀點，幫助國中生發展算術-代數思維？並評估其對學生數學學習的影響。據此，本研究的目的包括：設計相應的單元教學活動，幫助七年級生發展算術-代數的思維；評估學生代數與函數概念及態度的學習成效；以及，透過學生對教學的回饋和教師的教學-學理反思，再次檢視和重構教學概念與信念的內涵，以提升作者的代數教學知能。 研究的結果顯示，函數觀點似乎可以扮演算術過渡至代數思維的一種學習媒介。藉由營造似真的代數問題情境，觀察和理解數量之間的關係，可以讓學生在獲得符號操作的技巧之外，同時理解基本的代數概念。在歷經挑戰(測試)、澄清(引入)、和轉變(擴展)三個階段的函數觀點，引動代數思維的教學(學習)行動中，作者一再地檢視其教學活動在認知、情意、和社會三個面向的學習成效，嘗試重新建構她自己的數學教學概念和課堂教學實務。教學重構的內涵包括：以函數觀點，發展算術-代數思維的教學策略與內容；學習起點行為、概念啟蒙、合作學習、及HLT-HTT的教學認知；以及，如何營造似真算術-代數問題解決的學習情境。根據研究的心得與省思，作者以「三階段三面向的教師-學生學習狀態脈絡圖」，表徵教(師)與學(生)的概念轉變歷程與內涵。其中，教師教學概念與實務的轉變和學生算術-代數思維認知、情意、與社會面的發展，形成一個相輔相成、動態互動的「雙學習環」，它們一同轉變也伴隨著發展。在這樣的轉變和發展的過程中，不僅學生能藉由函數觀點、合作學習、和似真學習情境，學得抽象的代數概念，教師也因而更深刻地體會到，代數的深層結構和教學概念以及國中生的算術-代數思維過程與特質。 最後，作者認為，透過函數觀點似乎可以部分克服算術思維與代數思維的學習認知差異，進而局部解決國中階段代數教與學的問題。同時，也可以達成學校數學課程與教學的潛在目標(即變數的概念)；亦即，藉由具體操作與課堂討論活動，使國中生了解符號的(變數)意義和等號的(變數)概念。另外，藉由教學功力(數學、教學、和反思)的深化，作者也提升了她的課堂代數教學的實做能力。希望，本階段性的研究方法、過程、模式、和心得，能提供給其他國中數學教師作為教學與專業發展的參照，以解決他(她)們自己的代數教學問題。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.行動研究函數觀點算術-代數思維反思教學概念教學功力Action researchFunctional aspectArithmetic-algebraic thinkingReflectionPedagogical conceptPedagogical power