- 作者： 林福來; 鄭英豪
- 作者服務機構： 國立台灣師範大學數學系; 私立中國工商專科學校共同科
- 中文摘要：
反證法是利用對偶命題等價的原理進行論證。調查我國高二學生對反證法的了解，發現有約55%的學生瞭解反證法的證明格式(否定結論，推得矛盾)，然而只有不到5%的學生瞭解反證法的論證原理。這一方面顯示在數學教室中學生只學到一種演繹的格式，而非完整的論理原理；另一方面也顯示學生在生活中利用對偶命題進行論證的經驗並未與邏輯原理相連結。
思考生活中的命題時，往往因語言使用的習慣與個人經驗的累積，而產生與邏輯規則不同甚至相違背的情況。本文分析了生活命題的思維特質，詮釋四種在生活命題中之所以產生異於邏輯規則的思維成因。進一步提出克服此種思維限制以發展邏輯思維的教學假設 — 「將生活情境中命題的真偽關係符號化，利用符號規則推理再轉譯回生活命題，可增進學生對反證法論證原理的暸解」，並據以發展教案進行探究性教學。
探究性教學時，著重觀察當下學生的學習反應，尤其注重學生能否發現規律?能否符號化?能否將符號規律應用於其他命題等瞭解的歷程上。這些瞭解歷程的分析，主要根據教師於教學中之觀察記錄來進行。最後佐以教學後一個月的延後測，對照具代表性的高二學生對反證法的瞭解狀況以評估學習效果。
教學活動共約三堂課(150分鐘)，教學對象為五專商科一年級學生47人與五專工科一年級學生23人。
教學過程的觀察記錄顯示在教師的導引與學生的小組討論下，學生確可發現對偶命題等價並符號化，而在求等價命題的問題上，將原命題符號化再輔以符號規則是成功的關鍵步驟，教學假設獲得支持。
延後測結果顯示，實驗班學生有19%的學生能辨識對偶命題等價，相對於參照組已學過反證法的高二學生之4.5%高出許多。 - 英文摘要：
The argument principle of the method of proof by contradiction is base on equivalent relation of dual propositions. A study on 11th grade students’ understanding of proof by contradiction revealed that there are 55% of them understand its format of proof, ie. Negating the result of a proposition first, then inducing a contradiction; however, there are only less than 5% of them understand its argument principle. This result not only revealed the learning situations in mathematics classroom of senior high school, but also indicated that students’ experiences of argument in everyday life are not linked to their logical reasoning.
Language, thinking and logical reasoning are mutually interacted and correlated. Due to different interpretations of everyday language, argument in everyday life sometimes might be against logical rule. This paper analyzed the features of thinking propositions in everyday life and described four different types of process which cause the inconsistency between everyday life argument and logical argument. Based on this analysis, we then made a teaching assumption to get rid of such kind of thinking barrier, ie. “Symbolizing the propositions in everyday life and finding dual proposition according to symbolic rule will enhance students’ understanding of the argument principle of proof by contradiction. “Two explorative teaching were carried out by two teachers in two classes with 47 and 23 of 10th grade students in two vocational junior colleges respectively. Each class lasted three lessons, 150 minutes. Those explorative teachers are also researchers of the project. They were observing and recording their students’ learning process during their teaching. Particularly, they focused on how students’ capability of discovering rule, symbolizing a given proposition and applying the symbolized rule. One month later, the students in explorative teaching classes were assessed with an understanding test paper of proof by contradiction.
Results showed that the teaching assumption is supported by this study. Guided discovery and peer discussion are effective instruction strategies for symbolizing a proposition and finding symbolic rule of equivalence relation of dual propositions. The result of understanding test paper on proof by contradiction showed that 19% of 10th grade students in those explorative teaching classes are able to recognize the equivalency relation of dual propositions. However, only 4.5% of 11th graders are able to recognize this relationship. - 中文關鍵字： 反證法;生活論證; 探究性教學; 對偶命題
- 英文關鍵字： Proof by contradiction; Dual propositions; Argument in everyday life; Explorative teaching
