本研究主旨在因應國中數學領域潛能優異及不同能力學生去編製工具及執行相關研究。其目的有三：第一、發展「知覺數學差異化教學環境量表」、「數學知識信念量表」與「數學解題自我調整學習量表」。第二、探究不同數學能力學生在「知覺數學差異化教學環境量表」、「數學知識信念」與「數學解題自我調整學習」的差異。第三、探究數學潛能優異學生與非數學潛能優異學生「知覺數學差異化教學環境」和「數學知識信念」對「數學解題自我調整學習」的預測力。
為達成研究目的，本研究共分為二個部分。第一個部分為量表發展（目的一），根據文獻及採用統計方法編製目前國內缺乏「知覺數學差異化教學環境量表」、「數學知識信念量表」與「數學解題自我調整學習量表」等三項量表工具。經研究者蒐集高雄市國中二年級學生資料進行預試（n=258）與正式量表（n=523）的發展，建立量表的信度與效度。
第二個部分為探索性的相關研究分析（目的二及目的三），根據第一部分完成之工具及523份樣本資料作進一步的分析，其中數學潛能優異學生有103人；非數學潛能優異學生有420人（分四等級之人數為150人、103人、69人及98人）。利用單因子多變量分析針對5種能力學生在3個變項進行差異比較（目的二）。至於預測力（目的三），研究採用逐步多元迴歸分析針對數學潛能優異學生與非數學潛能優異學生進行「知覺數學差異化教學環境」和「數學知識信念」對「數學解題自我調整學習」的預測力分析。

根據以上三個目的，本研究主要發現如下：
一、本研究編製完成「知覺數學差異化教學環境量表」、「數學知識信念量表」與「數學解題自我調整學習量表」三份量表工具。信度與效度分析顯示，這些工具具有研究與實務參考價值。
二、數學潛能優異學生與其他不同數學能力學生在「知覺數學差異化教學環境」有顯著差異。此結果顯示，數學教師可能對數學潛能優異學生較常採用的差異化教學策略包括「教材內容自主彈性」、「高層次思考」與「正向支持」；也可能是數學潛能優異學生對教師採用上述策略的知覺比較強烈。
另外，數學潛能優異學生與其他不同數學能力學生在「在數學知識信念」與「數學解題自我調整學習」有顯著差異。數學潛能優異學生的數學知識信念趨近精熟數學知識信念，其在數學知識的確定性、數學知識的單一性、數學知識的來源與數學學習的辯證性皆趨近精熟數學知識信念。同時，數學潛能優異學生也具有較強的數學解題自我調整學習行為表現，尤其在認知調整與動機情感調整方面。
三、無論在數學潛能優異學生群體或非數學潛能優異學生群體，學生知覺數學差異化教學環境的程度和本身數學知識信念的精熟程度，皆可有效預測數學解題自我調整學習行為表現。
最後，研究者根據研究發現提出建議，以提供教師教學以及未來研究作為參考。

The aim of this study is to consider middle school students who are promising in mathematics and other different mathematical abilities students in the development of instruments and administration of related studies. The objectives are three. First, to develop instruments: Perception of Differentiated Mathematics Instruction Environment Scale (PDMIES); Mathematics Epistemological Belief Scale (MEBS), and Self-regulated Learning in Mathematical Problem Solving Scale (SRL-MPS-S). Second, to explore differences of PDMIES, MEBS and SRL-MPS-S among promising students and other different mathematical abilities students. Third, to investigate the scores of students’ PDMIES and MEBS on the prediction of students’ SRL-MPS-S.
To fulfill the above objectives there are two parts. The first part is to develop instruments (Objective 1). The investigators used literature and statistical methods and develop 3 needy instruments not yet found locally: PDMIES, MEBS and SRL-MPS-S. Data collection included grade 8 middle school students in Kaohsiung (n=258 for pilot; n=523 for main study) in attaining validity and reliability during the development.
The second part is an exploratory study on related studies (Objective 2 & Objective 3), using the results of part one and subsequent analyses of 523 students from main study. The number of promising students was 103 and other students counting to 420 (the number of students in four ability levels are150, 103, 69, and 98 respectively). Statistical methods used in part two were one-way MANOVA (Objective 2) and stepwise regression (Objective 3).
Based on the above three objectives the main results of this research were as follow:
1. There were adequate validity and reliability in the three scales for practitioners and for research: Perception of Differentiated Mathematics Instruction Environment Scale (PDMIES); Mathematics Epistemological Belief Scale (MEBS), and Self-regulated Learning in Mathematical Problem Solving Scale (SRL-MPS-S).
2. Promising students and other different mathematical abilities students had significant differences in PDMIES. This results indicated that teachers can used different strategies in teaching promising students, including the four subscales in PDMIES: material elasticity and autonomy, higher-order thinking, and positive support.
In addition, promising students and other different mathematical abilities students had significant differences in MEBS and SRL-MPS-S. This results indicated that teachers can used different strategies in teaching promising students, including the four subscales in PDMIES. The mathematics epistemological belief of mathematical promising students approaching to sophisticated belief, including certainty of knowledge, simplicity of knowledge, sources of knowledge, and justification for knowing. Also, mathematical promising students had stronger behavior in self-regulated learning in mathematical problem solving, especially in the regulation of cognition and the regulation of motivation/ affect.

3. For both promising students and other different mathematical abilities students, their scores in PDMIES and MEBS had significant influence on SRL-MPS-S .
Based on these findings, implications for practice and future researches were discussed.

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