數學臆測的思維模式

A Thinking Model of Mathematics Conjecturing

10.6173/CJSE.198806_6(2).0005

陳英娥(Ing-Er Chen)；林福來(Fou-Lai Lin)

反駁 ； 臆測 ； 數學探究 ； 思維模式 ； conjecturing ； refuting ； mathematics investigating ； thinking model

科學教育學刊

6卷2期（1988 / 06 / 01）

191 - 218

繁體中文;英文

本研究從個體思維的角度考量數學臆測，目的在了解數學臆測的思維歷程和它的特徵，並透過數學家臆測的思維模式檢視學生臆測的模式。研究方法採面談法，研究對象包括五位大一升大二的學生、五位高二學生和兩位數學家。研究工具包括數學猜測測驗和數學家面談工具。研究結果發現：學生數學臆測的思維歷程可形成一個共通的模式圖，每位學生的臆測歷程圖都是此圖的一部分，而且學生與數學家的數學臆測思維模式是相容的。此臆測思維模式是一個有方向的思維循環，思維循環有兩層，包括內循環和外循環。內循環表徵猜想的精煉過程，外循環表徵原猜想被丟棄及重構的歷程。學生的臆測思維在猜測、檢測、相信和反駁之間遞迴。從猜測問題的本質看，如果是猜測未知結果，思維起點和思維路徑比較複雜；如果是猜測命題的對錯，思維起點和思維路徑都比較簡單。本研究發展的數學臆測思維模式可用來表徵猜測問題的難度以及評量學生臆測思維的品質。

This study aimed to investigate students' thinking processes of mathematics conjecturing. Five grade eleven students, 5 undergraduate students and two mathematicians were interviewed with a special survey in the study. The survey included two parts: The Mathematics Conjecturing Test for students and The Expert's Interview Questionnaire for mathematicians. It was found that that pictorial representations of students' thinking processes of mathematics conjecturing could be constructed as a unified model. It was further found that students' conjecturing model and mathematician' conjecturing model are compatible. The conjecturing model contained two directed cycles, an inner cycle and outer cycle. The inner cycle represents the process of refining the primitive conjecture and the outer cycle represents the process of rejecting the primitive conjecture and reforming a new conjecture. The conjecturing process appears to move dynamically and recursively between four stages: guessing, checking, confirming and refuting. When students work on tasks of guessing an unknown conclusion, the model reveals that the starting point of students' thinking and their thinking paths are more complicated than the corresponding representations on tasks of judging the correctness of a proposition. This mathematics conjecturing model can be used to study the difficulty of the tasks and to evaluate the quality of individual thinking.

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