建立數學論證判讀認知機制之個案研究

證明文本的確認涉及閱讀，對不能理解的部份，可能需要構建輔助證明，又與數學解題相關，故將證明文本的確認稱為「論證判讀」，本研究目的便在探討數學論證判讀之認知機制。先依據文獻探討，提煉出論證判讀的理論模式為分析架構，繼而檢視專家、生手的判讀歷程，建立論證判讀的機制。研究對象爲三位高二學生、兩位數學碩士研究生、一位數學教授，以放聲思考法輔以晤談，對判讀原案進行分析，主要發現如下：專家、生手的差異在於專家能對文本的證法加以評估，據此擬定局部修正，生手則依據表面結構是否熟悉來判斷文本的證明有效性；判讀認知機制可分為「文本有明確證明架構」與「文本無明確證明架構」二種型態，主要由「讀題－分析－相信」及「分析－檢驗－反駁－修正」兩種內迴圈所形成。

關鍵字

數學證明；數學論證；判讀

並列摘要

Confirmation of the proof text involves reading. As to the part which cannot he understood, it possibly needs to construct an assistance proof Confirmation is also related to mathematical problem solving. Therefore confirmation of the proof text is called ”the validation of proof”. The purpose of this research is to discuss the cognitive mechanism of mathematics argumentation validating. First of all, according to literature discussion, the researcher puts forward the theoretical model of mathematics argumentation validating process as the basis of analysis. Then the researcher checks the mathematics argumentation validating processes of novices and experts, and sets up cognitive mechanism of mathematics argumentation validating from the practice. The subjects are three second-grade senior high students, two mathematics masters, and one mathematics professor. This research analyzes the validating protocol by thinking aloud and interviewing. The main findings and results are as follows: The difference between the experts and the novices is that the experts can evaluate the proving method of the proof text, and then based on the evaluation draft local correction. The novices, on the other hand, judge the validity of the proof based on whether the surface structure is familiar to them or not. Cognitive mechanism of mathematics argumentation validating can be divided into two conditions, which are ”the text with obvious proof framework” and ”the text without obvious proof framework”, and is mainly composed of two kinds of inside loops: reading-analysis-believing and analysis-verification-refutation-modification.