發展問題知識－以數學能力低學童對乘除法算式為例

The Knowledge of Developing Problems: An Example of the Students with Lowly Mathematical Abilities in the Equations of Multiplication and Division

馬秀蘭(Hsiu-Lan Ma)

《高雄師大學報：自然科學與科技類》 23期 (2007/12) Pp. 99-123

https://doi.org/10.7060/KNUJST.200712.0099

摘要

本文旨在探討學童對乘除法算式發展問題歷程中涉及的知識，作者修正Mayer(1992)之解題知識成「發展乘除問題知識」，它指出學生若要成功發展問題必須具有策略性（檢視列式）、基模（情境、結構類型）、語意（事物表示）和語文（正確中文表之）四種知識；研究對象為兩位國小六年級數學能力較低之學童。結果發現在發展問題時，學生之認知結構可透露出此四種知識，且學童涉及的知識及內涵會受算式數字型式（如乘數是小數，非整數之被除數大於或小於除數）影響。其中1.乘法算式：為配合乘數是整數或非整數，有人用離散量或連續量事物，以「單位量（數）×單位數（量）」比例因子結構；有人保握住單位數為整數之累加模式，總用離散量事物，以「單位量×單位數」比例因子結構。有人在乘數是純小數之多步驟算式因不知何者是單位量（數），及在乘數是小數之單步驟算式因不識表示物相關資料，而發展不正確問題。2.除法算式：為配合非整數之被除數大或小於除數，學童大多以離散量或連續量事物，以「總數量÷單位數」之等分除結構。有人在被除數小於除數之單步驟算式因錯誤檢視列式或不識表示物相關資料、在小數被除數大於除數算式因錯誤檢視列式及不識表示物、或在有餘數算式因不完整檢視列式，而發展不正確問題。上述結果可作教師瞭解學童數學概念與文字表徵之參考。

關鍵字

知識 ； 乘除法 ； 發展問題擬題 ； 算式 ； 數學能力

並列摘要

The purpose of this paper was to explore the knowledge the students developed problems in the equations of multiplication and division. The knowledge of ”developing multiplication and division problems” was revised from Mayer's (1992) knowledge of solving problems by the author of this paper. It denoted that the students need to have four kinds of knowledge, strategic, schematic, semantic, and linguistic knowledge, when developing proper problems. The subjects were the two sixth graders with lowly mathematical abilities. The findings were the following.
The students' cognition structure can reveal the knowledge above. Their knowledge and the content of knowledge would be influenced by different numbers (multiplier is decimal, non-integer dividend is more or less than divisor). 1. Multiplicative equation: Some students applied the structure of rate factor with discrete or continued objects to adjust integer or non-integer multipliers. Among multiple-step equations with multiplier between 0 and 1 and single-step equations with decimal multiplien, some students failed because they did not know which one was the number in each subset or the number of sets that can be made nor did they understand the related information of representative objects. 2. divisive equation: Most students applied the structure of partition division with discrete or continued objects to adjust non-integer dividend being more or less than divisor. Some students failed because the following reasons. Among single-step equations with dividend being less than divisor, equations with the decimal dividend being more than divisor, and equations with remainder, some students did not correctly view the equation or understand the related information of representative objects, correctly view the equation or understand the representative objects, or completely view the equation respectively. These findings might be a reference to understand students' mathematical concept and word representation for teachers.

並列關鍵字

knowledge ； multiplication and division ； developing problems posing problems ； equation ； mathematical ability

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