數學學習與理解之內涵 ─ 從心理學觀點分析

A Psychological Analysis of Mathematical Learning and Understanding

本文根據新近的認知心理學觀點，從五方面分析數學學習與理解的內涵：（一）從知識表徵的觀點而言，數學理解即是獲得有力的表徵系統並具有多重表徵的彈性，以有效地進行數學思考、與他人溝通；（二）從知識結構的觀點言，數學理解需要學習者建構適切的認知結構。數學教學的目標乃在於協助學生建立具有高度結構的知識；（三）從知識網路的觀點而言，數學理解本質上乃是相關知識密切聯結而形成網路的狀態。數學教學上需以協助學生建立概念性與程序性知識之關聯及正式與非正式數學知識之關聯為重點；（四）從知識建構的觀點而言，數學理解乃是學習者從數學活動中主動建構而得，學習者需要運用自身已有的知識在具體的數學解題活動中逐歩發展數學能力；（五）從情境認知的觀點分析，數學理解乃是個人能與所處情境有效地交互作用的狀態。換言之，此乃是個人從實際情境中獲得認知工具以解決問題的能力。不同觀點反映了學者們對於數學學習的不同著重點，亦反映了數學學習的多面性。教學者應從多重角度思考數學學習的內涵，以引導學生更有效地學習。

This study examines the nature and meaning of mathematical learning and understanding from five different perspectives. From the perspective of knowledge representation， understanding mathematics requires learners to develop proper representations， build relationships between external representations of mathematical ideas and internal cognitive representations， and acquire the ability to move flexibly among various representations of mathematical ideas. From the view of knowledge structures， understanding mathematics means establishing knowledge structures with good correspondence， connectedness， and integration. Understanding mathematics， form the view of knowledge networks， basically entails connections among various kinds of knowledge. From the perspective of learning as a constructive process， understanding mathematics requires learners to actively structure and invent their knowledge through interaction with physical and social environment. Finally， from the perspective of situated cognition， mathematical understanding is inextricably tied to the situations in which mathematical knowledge was acquired and can only be achieved in the context of authentic activity. This analysis revealed the multi─facet nature of mathematical understanding.