本研究以「通分」、「顛倒相乘」及「整合通分與顛倒相乘」等三種不同類型的範例教學，探討學生於分數除法之概念性知識與程序知識的學習成效，尤其關注低先備知識學生的學習。前導研究以115名五年級學生和108名六年級學生分別預試先備知識試題及概念性知識評量試題，結果顯示大部分的試題呈現難度較低，但鑑別度優良的情況，進一步進行信效度分析（效度指標為學生100學年度第二學期數學期中考成績），亦顯示試題內部一致性良好，也與數學成就有適當的相關，故本研究僅修訂少部份試題，其他試題在正式實驗則予以沿用。正式實驗的研究參與者為台北市與桃園縣各一所國小、各三個五年級班級的學生，有效樣本為148 名學生。受試者以班級為單位分派至三種不同的範例教學。教學實驗前施測先備知識；隨後進行二節（整合組）或三節（另外兩組）的範例教學，課堂讓學生兩兩異質分組，透過討論範例與各自練習解題進行學習；教學後施以分數除法的程序性知識及概念性知識試卷。結果顯示學生透過範例教學學習分數除法的概念性知識及程序性知識，其正確率達八成與九成，達到一般教學所欲達成的教學效果，然由於三組學生的表現都很好，透過共變數分析控制先備知識的影響後三種範例組別未達顯著差異。低先備知識的學生則有組別效果，以整合組學習概念性知識的成效優於顛倒相乘組，且整合或顛倒相乘組學習程序性知識的效果優於通分組，顯示對於低先備知識學生而言，整合組的範例不僅最有效率，也最有效益。本文最後對課堂中的學生行為進行描述，也討論研究限制與教學上的建議。

In this study, three different prototypes of “common denominator”, “reciprocal multiplication” , and “integration” as the worked-out examples were applied to explore the learning effect on conceptual knowledge and procedural knowledge of fraction division, especially that on the students with low prior knowledge. The pilot study pre-examined 115 5th graders and 108 6th graders via the examination of prior knowledge and conceptual knowledge. The pilot study result shows lower difficulties in accordance with better discrimination in most of the items. The analysis of internal consistency reliability fitted the goodness; criterion-related validity was qualified for the significant correlation with the mathematics-achievement scores. Therefore, the study only revised minor part of the items. Other items were expected to be adopted in the formal examination. The valid samples in the formal examination were 148, including 5th graders sampling from elementary schools in Taipei and Taoyuan. The participants divided by classes as a unit were designed to three conditions with different worked-out-example teaching instruction. The examination of prior knowledge was proceeded before the instruction. The worked-out-example instruction was then proceeded in the second (integrated group) or third (the other groups). The students were divided into heterogeneous group and they could learn via discussing the examples as well as solving the questions. The examination of fraction division with procedural knowledge and conceptual knowledge were provided after the worked-out examples. The results exhibited the anticipated learning effects on the students with 80% and 90% correct rate via learning conceptual knowledge and procedural knowledge of fraction division respectively. The difference between three groups was not significant according to ANCOVA analysis since the performance of three groups were fine simultaneously. The effect resulting from different groups occurred in the students with low prior knowledge. The effects of learning conceptual knowledge in integrated group were better than those in the reciprocal multiplication group. The effects of learning procedural knowledge in integrate group or reciprocal multiplication group were better than those in the common denominator group. In conclusion, the prototype of worked-out examples in integrated group demonstrated the best efficiency and benefit for the students with low prior knowledge. Additionally, the acts of the students in class, the restriction of the research, and the advice of instruction were all described and discussed in this article.

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