一個重構高中生機率概念的行動研究

王安蘭

指導教授 ： 金鈐

國立臺灣師範大學/理學院/數學系在職進修碩士班/碩士(2004年)

摘要

本研究透過行動研究法，調查個人所任教之三個社會組班級高三學生的原生機率概念及其影響，並實施試驗性直觀教學。本研究共分為兩階段，持續一個學年。第一階段為全班教學，教學策略包括利用類推比較、認知衝突和引入科學性知識來幫助學生察覺並驗證原生直觀，同時，運用直觀法則和引動後設認知。研究結果顯示，直觀迷思比例有降低的趨勢，學生在認識直觀法則之後，已經學會驗證自己的原生直觀，但也發現有部分學生因教學介入而否定直觀的意義與價值。
為了解決第一階段所遭遇的問題，在第二階段中，調整原來的教學策略，加入全班討論的活動。全班討論教學對大部分的學生有正面的影響，但是有少數學生仍受制於直觀的強制性，而感到疑惑並堅持其原生的想法，因此，進行了後半段的焦點小組晤談教學。這階段的研究結果顯示，學生對許多機率問題的答對率提高，大部分學生已能清楚表達自己所寫數學式的意義，更有些個案學生，能比較不同機率問題間的差異，同時，也增加了數學思考的信心。
經過兩階段(包含三循環)研究的觀察發現，有些學生雖然能察覺直觀的迷思，卻又無法抗拒或克服直觀迷思；有些學生於修正原先的直觀迷思之後，經過一段時間，卻又再次折返至原有的迷思，或在不熟悉的情境下，又再度使用原生直觀。這些觀察表明，直觀的特性不但會影響學生的機率概念學習，而且，原始直觀不可能消失，即使經過相當程度的教學介入，它仍會一直潛在地影響學生的機率思維。令人振奮的是，有少數學生，不但察覺而且能修正自己的原生直觀迷思，更進一步將其轉換成科學性的二階直觀；有些學生，甚至提出「修正直觀」的看法。這些學生的表現，證實了Fischbein(1987) 和Resnick(1999)“直觀是可以學習的，它是可以經由教學介入而改變”的教學猜測。
面對直觀的兩種極端面貌，為避免直觀迷思的影響和妨礙抽象思維的發展，教學時應直觀與邏輯並重，如能正確地運用直觀，將有助於學生對抽象概念的瞭解。

關鍵字

直觀 ； 直觀法則 ； 直觀教學 ； 概念改變 ； 機率迷思概念 ； 行動研究

並列摘要

Using action research methodology, three classes of students taught by the author were investigated focusing on their primitive/intuitive probabilistic conceptions and its influence. The study consists of 2 stages with 3 cycles, lasting for one academic year. In the first stage, a whole class teaching strategy was used, including analogical comparison, cognitive conflict, and introducing scientific knowledge to help the students become conscious of and test on their primitive intuitions, and then the Intuitive Rules and meta-cognitive arousal were activated. The initial results showed that students’ intuitive misconceptions were significantly reduced however for some students their primary intuitions were still active and refuted to accept the value of teaching intervention.
To resolve these problems, in the second stage, the author introduced several formats of classroom discussion and focus groups investigation. Although this exploratory teaching had a positive influence on most students, and yet few students were still interfered by the coerciveness of intuition, and felt confused holding firmly their primary thoughts. The results showed that the percentage of students’ correct answers were increased, they were also more able to explain what they wrote, and a few students could even make the underlying differences between varied probabilistic questions and also get more confidence in the process of probabilistic thinking.
Based on the results of this 2-stage/3-cycle study, for some students, whilst being able to grasp the intuitive misconceptions, they were still unable to resist or conquer the nature of mathematical intuition. After having successfully corrected the original intuitive misconceptions, they either returned to the primitive misconceptions or reverted to their primary intuition when they encountered unfamiliar questions. These evidences seem to suggest that the unique features of intuition influence not only student present learning of probabilistic concepts but also their future learning of the concepts. In other words, those primitive intuitions never completely disappear they exist a considerable period after learning it. What the most encouraging for the author is that several students were not only able to amend their primitive misconceptions, but also able to go a step further and transform this into scientific secondary intuition even modifying their views about primary and secondary intuition. This corroborates with the teaching hypotheses of “Primary intuition can be learned, and through teaching can be intervened and corrected” (Fischbein, 1987; Resnick, 1999).
In order to cope with students’ intuitive misconceptions in teaching, teachers should carefully integrate those students’ primitive probabilistic intuitions with mathematical logic.