透過建模與多重表徵教學探討高二學生的建模能力與概念改變－以空間概念為例

Abstract

本研究旨在透過科學建模與科學概念改變理論的觀點來探討學生在學習數學上的空間概念時的建模能力與概念改變。在本研究中，研究者發展了一套為期六週、以建模與多重表徵為設計基礎的數學課程，並在一所普通高中內實際進行教學。研究者將三個班級（共125位高二學生）透過方便性取樣分為建模與表徵組(MR)、表徵組(R)與控制組(C)等三組。其中建模與表徵組（簡稱建表組）與表徵組由研究者進行教學，而控制組則由該校一位經驗豐富的數學教師進行教學。

在本研究中，研究者修改了張志康和邱美虹（2009）所提出的「建模能力分析指標(Modeling Ability Analytic Index, MAAI)」，發展了一套多面向的概念試題，在每單元結束後各施測一次，以作為評量與分析的依據。該試題共有五個單元，每個單元均包含許多開放性問題；學生必須完整地進行答題與解釋，才能獲得高分。在後測結束後，每位學生均接受為時20至30分鐘訪談，以協助研究者確認對學生所持之心智模式的歸類是合理的。此外，研究者尚挑選了27位持有不同心智模式與建模能力的標的學生進行為時1小時的深度訪談；此深度訪談旨在深入探討這些學生的心智模式與建模能力，以作為典型個案的呈現。本研究所有的研究工具均經過信效度檢驗，並達到合理標準。

經過資料編碼與分析，本研究顯示出以下結果：首先，三組學生在接受過建模與多重表徵教學後，在概念理解上均有顯著的進步，其中又以建表組的進步幅度最大，表徵組的學生次之，而控制組的學生進步幅度最小。在這當中，建表組與表徵組學生的進步幅度也顯著地較控制組來得大。其次，學生所持有的空間概念心智模式是多元且多樣的。本研究根據先前文獻與試測結果，將學生的心智模式分為科學模式、斜角模式、有界模式、符號模式、實體模式、二維延伸模式、以及其他等七類。在教學前，持有二維延伸模式與有界模式的學生是最多的，隨著教學活動的進展，二維延伸模式的學生開始減少，取而代之的是實體模式與符號模式。有界模式雖然一直都有學生持有，但卻不是比例最高的。在教學的尾聲以及教學完畢後，持有科學模式與符號模式的學生則是最多的。這樣的結果符合了研究者的教學目標，也符合了先前的研究發現。接下來，在學生所使用的建模歷程上，建表組的學生所使用的建模歷程顯著地較表徵組與控制組來得高。學生在六個建模歷程當中，以模型選擇、模型建立與模型應用的表現較為理想。建表組的學生在模型效化、模型調度與模型重建三個歷程上，所使用的次數與表徵組及控制組的學生有顯著的差異。接下來，在學生的建模能力方面，建表組學生的平均建模能力（2.81分）顯著地比表徵組（2.03分）與控制組（1.98分）來得高。56%的建表組學生在教學結束後具備了進階的建模能力，然而三組學生在答題層次上並未展現出顯著差異。接下來，在學生所採用的表徵形式與種類上，學生最常使用的表徵種類為數學表徵，其次是言語表徵與具體表徵。學生較少使用視覺與動作表徵。造成這樣的差異的主要來源，乃是由於建表組的學生在進階建模能力上的得分顯著地比其他兩組來得好。最後，根據回饋單、情意問卷與課堂錄影等資料顯示，學生對這樣的課程持有相當正向的態度，且在課程中有著良好的師生互動。根據本研究的結果，研究者建議教師設計建模與多重表徵課程，並以外顯化的方式實施，以提升學生的建模能力並促進概念改變。

This empirical study aims to investigate high school students' modeling ability and conceptual change about the concept of three-dimensional space in mathematics. In this study, the researcher developed a 6-week high school mathematics curriculum and put it into teaching via the use of modeling and multiple representation approaches. According to the research design, three different classes with 125 11th grade students in total were assigned into three different groups correspondingly, named as the Modeling and Multiple Representation Group (MR), the Multiple Representation Group (R), and the Control Group (C). During the 6-week teaching schedule, the two experimental groups were taught by the researcher, while the control group was taught by an experienced mathematics teacher in the same school.

In order to improve the accuracy on investigating and measuring students' modeling ability and conceptual change, the researcher modified the "Modeling Ability Analytic Index (MAAI; Chang& Chiu, 2009)", and then developed and conducted a set of tests after each unit. Each test contained several open-ended problems, in which students' full explanations were required to score high and solve the problem progressively and completely. After the posttest, each student was interviewed to clarify the researcher's classification toward students' mental models. Furthermore, the researcher selected 20 target students with different mental models and different modeling abilities, interviewed individually for an hour to reveal their modeling and conceptual change paths as typical cases. All the research tools were examined with proper validity.

After data coding and analysis, the results showed: first, on the perspective of conceptual understanding, all the three groups of students scored significantly higher in the posttest and delayed test than in the pretest. Also, the improvement of students' conceptual understanding in both two experimental groups (MR& R) were significantly better than the students in the control group. Second, students' understanding of three-dimensional space was classified into seven different mental models, named as scientific model (SC), inclined angle model (CL), boundary model (BD), symbolic model (SY), concrete model (CN), two-dimensional extension model (EX), and others (OT). Most students held EX and CN mental models in the first two units, whereas many of them turned to held the SY model in the middle of the curriculum. After instruction, 47.2% students held the scientific model, but there was no significant difference among the three groups at this stage. Third, students showed different types of modeling processes in three groups. Students in the MR group conducted model validation, model deployment, and model reconstruction more frequently than those in the R and C groups. Fourth, students in the MR group scored significantly higher in modeling ability tests; 56% of them were able to retrieve advanced modeling ability. However, there was no significant difference in basic and advanced problem-response ability among the three groups. Fifth, students in MR& R groups used more concrete, verbal, visual and gestural representations than those in the control group. Finally, students in MR and R groups showed positive attitudes toward the curriculum. It is suggested that teachers should develop modeling and multiple representation curriculum and implement in their teaching.