一位體制外教師三年級數學臆測任務設計及實踐之行動研究

本研究是描述研究者在體制外的數學補習班場域，如何選定與設計數學臆測任務，如何將臆測任務融入國小三年級數學課堂教學的行動研究歷程，說明研究者所遭遇到的問題及所因應的解決策略，進而從中了解教學問題及學生的學習情形，增進自己的數學教學知識及數學教學專業知能。研究者以自己任教補習班的三年級班級為研究場域，臆測任務選定，都是依循前一次臆測教學實踐後的省思，修正調整，依序為「周長與陎積」、「數字謎題」、「雞兔問題」。分別進行教材分析及設計數學臆測任務，並以數學臆測教學模式的五個階段（造例組織觀察、發現關係並提出猜想、效化猜想、猜想一般化、證明一般化）來融入數學教學活動以達到教學目標。本研究結果發現，在體制外選定的臆測任務，若與體制內相關的教學單元，必頇進行相關教材與體制外學生學習脈絡的分析；臆測任務的設計，造例是一個非常重要的起點，需掌握學生具備的前置經驗，設計適當的造例內容，以符合臆測任務的教學目標；臆測任務實踐時，小組共同合作的成員需做適當的調配，以達最佳的互動學習；個人造例、小組彙整單頇確認其正確性與完整性，教師需適時的引導學生，觀察彙整單上下左右的變化關係，教師需全班確認猜想時的共同數學語言，並鼓勵學生大膽的提出想法；檢驗猜想需依組內的例子說明清楚，反駁也頇提出具體的例子，養成有憑有據的民主素養；效化猜想時學生是學習的真正主導者，教師只需引導學生學習如何驗證、補充、修正、分類、合併猜想，以提升學生運用精確的數學語言；教師需釐清效化階段的重點是擴大驗證猜想的範圍，以進行猜想一般化，不頇證明猜想。本研究學生只經歷三次的臆測教學，故本研究並未徹底執行證明一般化階段。數學臆測教學研究的建議：日後若能增加臆測教學的次數，可探討猜想一般化與證明一般化學生的學習歷程。可探討學生若在低年級即開始經歷數學臆測任務的啟發，延續至高年級時其數學知識學習發展的情形為何。關鍵字：體制外教育、數學臆測、數學臆測任務設計、數學臆測任務實踐

關鍵字

體制外教育；數學臆測；數學臆測任務設計；數學臆測任務實踐

並列摘要

This action research describes the process of how the researcher decided and designed the mathematical conjecturing task and how to implement the conjecturing task in 3rd-grade math classes in a mathematics learning institution and demonstrates the problems encountered by and the corresponding solving strategies come up with by the researcher, thereby, realizing the issues of teaching and students’ learning status and enhancing mathematics teaching knowledge and expertise. The researcher takes her own 3rd grade classes as research field, the selection of conjecturing task was based on the reflection and adjustment from the previous teaching practice, “Perimeter and Area”, “Digit Puzzle” and “Chicken and Rabbit Cages”. Analyzing teaching materials, designing mathematics conjecturing task and the five stages in the framework of mathematics conjecturing teaching mode (product - organization - observing example, discovering relations and proposing hypotheses, verifying the hypotheses, generalizing the hypotheses and generalizing the proofs) are included in the math teaching activities to complete the teaching objectives. The results showed that “choosing conjecturing task”, it is necessary to pre-analyze the related teaching materials and the way of students’ learning if the task is relevant to the subject taught at school. “Designing conjecturing task”, exampling is an important starting point, it needs to take students’ prior knowledge into account, design the proper example to meet the objectives of conjecturing task. “Implementing conjecturing task”, the team members should be properly deployed to achieve the best interactive learning. “Individual exampling and team worksheet”, correctness and completeness need to be confirmed. Teachers have to guide students at the right time, to observe the changes of their worksheet, make sure the common mathematical terms are used when verifying the hypotheses and encourage students to put forward ideas boldly. “Verifying hypotheses”, hypotheses should be clarified in accordance with the team examples. It is also necessary to put forward concrete examples to refute. “Validating hypotheses”, students take the leading role of learning process, teachers only need to guide students to verify, supplement, revise, classify and combine the hypotheses to help students use accurate mathematical language. Further research is needed to explore the process of students’ learning in “generalizing the hypotheses” and “generalizing the proofs” if the number of conjecturing teaching is increased and how students develop mathematics concept in higher grades if they start to experience the mathematical conjecturing task in lower grades. Keywords: learning institution, mathematical conjecturing, mathematical conjecturing design, mathematical conjecturing implementi