| 摘要: | 臺灣的現階段機率課程由九年級才開始，相較於他國時程明顯晚了許多，此現象即為本研究發展之緣由：在八年級設計全新的機率課程，並實驗其教學成效。在數學教育研究的方法論上，引進法國新興教學理論：教學工法（Didactic Engineering, DE）以設計機率課程和教材，達到理論與實務之實踐。故本研究將兼具三面向：一、DE的實踐：課程發展四階段的工法；二、樹狀圖：以圖象表徵處理機率運算；三、數學素養：培養機率思維與解決問題。
機率實驗課程採用研究團隊之自編教材，將樹狀圖設計為一致性的機率學習工具，引導八年級學生發展主觀機率、頻率機率、古典機率的概念，並達到處理餘事件的思維層次，也發展了解決生活實際問題之素養。並以DE從事教學設計與課堂實踐的規準。雖然事前準備工作繁雜，但它能避免研究困境，以及提高學生的課堂參與率。
研究發現，未經過機率教學的學生，已具有部分機率值的範圍、主觀機率、頻率機率、古典機率的自發性概念，猜測學生可以從生活經驗中習得機率概念，可見機率與生活經驗之密切相關性。而經過研究團隊的機率教學後，發現八年級學生在學後測驗的成績上有成長，表示學生需要一套有系統的機率教學，且少數學生甚至能延伸至獨立性與乘法原理的思維層次，也有少數學生隱約表露出條件機率的概念。
另一方面，在學生的文本中，本研究亦認出學生的機率與樹狀圖迷思：（1）理想值的概念，（2）樣本空間分群，（3）樹狀圖的分類，（4）不對稱事件的機率值，（5）加法原理與乘法原理。不過，在機率課程結束後六個月進行的延後測驗，發現學生成績相較於機率學後測驗，呈現小幅進步，表示本研究的機率教學，能使學生成功地習得機率課程的教學目標，成為其素養。
綜上所述，本研究建議機率主題可提前至八年級學習，以生活經驗作為內容主軸，並以樹狀圖為解題之技術工具。對於九年級的機率課程，建議以樹狀圖之圖形特徵導入互斥和事件、獨立性以及乘法原理的概念。而DE確實提供一套從事數學教育研究的方法論，特別是針對創新的實驗性課程。本研究也歸納出幾項在使用DE上的難處，將提供給未來以DE從事教學設計的教育同仁參考。;Currently, the probability curriculum of Taiwan has only been officially implemented since the ninth grade. Compared with other countries, our curriculum is obviously very late. This phenomenon is the motivation for this study, and it is necessary to redesign the new probability curriculum of the eighth grade and practice it. On the methodology of mathematics education research, this study uses the French teaching theory Didactic Engineering (DE) to design the probability curriculum and materials to achieve the theory and practice. The study will realize the following ideologies. (1) Practice of DE: Four stages of curriculum development. (2) Tree diagram: Using Graph representation to deal with the probability problems. (3) Mathematics literacy: Cultivate probabilistic concepts and problem solving.
The curriculum of this study used the teaching materials designed by our team. The core feature of our method is the role of tree diagram as a consistent probability learning tool for developing concepts of subjective probability, frequentist probability and classical probability. Students reach the level of dealing with the complementary events, and developing a literacy to solve practical problems in life. Also, using DE as a set of methodology for this study, it provides the discipline of instructional design and classroom practice. Although it takes a lot of time to prepare for the materials, it can prevent the research failure and develop the students′ participation in the curriculum.
The study found that students who were not taught the probability curriculum, have already contained the spontaneous concept of range of probability values, subjective probability, frequentist probability, and classical probability. The researcher speculated that students might learn the concept of probability from daily experience. After our probability curriculum, it was found that the eighth-grade students had a growth in the post-study test, indicating that students need a systematic teaching of probability, and a few students can even extend to the concept of independence and multiplication, and few students vaguely revealed the concept of conditional probability.
On the other hand, in the students’ writings, this study identifies students’ myths of probability and tree diagram. (1) The concept of ideal value. (2) The sort of sample space. (3) Classification of tree diagram. (4) Probability value of asymmetric events (5) Adding principle and multiplication principle. However, six months later of the probability curriculum, the postponement test was conducted. Compared to the probationary post-test, the students′ grades showed a slight improvement indicating that the probability curriculum of the study was quite successful.
In summary, this study suggests that the probability subject can be moved ahead to the eighth-grade, with daily experience as the main content of probability curriculum, and tree diagram as the technical tool for solving the problem. As for the ninth-grade probability curriculum, it may include the concepts of exclusion-or events, independence, and multiplication principle. This study also shows that DE really provides a methodology for mathematics education, especially for the innovation of experimental courses. This study also summarized several difficulties in using DE. It will provide as a reference for educational colleagues who are going to engage in teaching design in the future. |
