臺灣博碩士論文加值系統

中文摘要
許多科學概念建立在數學的比例概念上。中學生需要學習許多物理化學概念，例如溫度，速度，加速度，密度……等等，即使在日常生活、工作職場所也可能會運用比例概念。因此，有必要在國小階段，了解兒童比例概念的發展，以及教材對比例概念學習的影響。研究者在高雄縣市有效取樣135名小學六年級兒童，以面談和問卷的方式進行三階段研究分析。在比例、比值、比例尺三種的型式下，分析四個向度，包含中文用語、現實感、等值概念、直觀。
研究結果顯示：在中文用語方面，兒童對於「倍」字大多持有「擴大、大於一」的意涵，因此部分兒童會拒絕「0.7倍」的描述，儘管兒童已經在教材中學過前述用語，也歷經相關的考試。兒童在現實生活的量感方面，普遍表現不佳，特別是對重量；雖然兒童對於長度距離的掌握較佳，然而，由於數學課室裡遠離現實的啟蒙例，例如以大尺度的比例尺為教學案例，本研究發現，那樣的設計不利於中低成就兒童的現實感發展。在等值概念方面，兒童多能掌握具體數量比，低成就兒童在十倍數擴分的表現較好，其次是約分，最差的是「約分後再擴分」，此與現行教材的安排略有差異；對中、高成就兒童而言，基於熟練的運算技能，因此約分、擴分正確表現率相當一致；但是在大數字的表現則不佳，可能來自機械式理解，因此不具解題信心。在直觀方面，兒童學得後天直觀，認為「%」代表小於100份，因此認為150%不合理，顯然脫離真實生活中所使用的意義表徵。
此外，本研究印證「相對性思考」為比例運思基礎，樣本中超過85%的六年級兒童具備該能力。而根據研究發現，兒童在處理比例式、比值，往往要求給「單位」，特別是低成就兒童，有時甚至是中等成就者。比例式以非整數比方式出現(1/300：1)，兒童作答的正確率會陡降，因此，研究者認為不適合以這種表達方式做為引入比值的教學，此與國內教科書的編寫主張不同；又比例式中的數值型態轉換仍是兒童學習障礙，例如：小數、分數的轉換。低成就兒童易受學習情境影響；超過70%的兒童會接受「70%元」，主要來自日常商業折扣的情境。

ABSTRACT
Partial science conception is built upon mathematics. Pupils in senior and junior high school would further learn temperature, velocity, acceleration and density, etc. Therefore it would be better for children to understand the mathematical concept of proportion and ratio in the primary school. In fact, talking about application, the ratio, proportion and percentage are worthy to be studied because of their usefulness in everyday and many occupations as well.
In this study, the objects of this investigate are 135 sixth grade children of Kaohsiung city and Kaohsiung county. By means of interviews and questionnaires, the researcher studies the ratio and proportion conception of them. There are four divisions to be the investigational framework; Chinese character, equivalent, reality, and intuition are included.
As the finding in the division of Chinese character, ‘倍’(it is somehow similar to ‘times’ in English), which always means bigger than ‘1’ in children’s mind. Therefore partial children would not accept the representation of ‘0.7倍’ illustrated in the mathematics textbooks before. Even they had learned it and passed examines in the mathematics classroom.
In sense of reality, children can sense ‘distance’ better than ‘weight’. And the sense developments of the middle level and low level children are deeply impressed by the generic sample. As investigated, the proportion of large scale, such as shown in maps, could not provide the advantages to the development of children’s reality sense.
In the equivalent, all children’s performance is good in the concrete context. Meanwhile, in the performance in purifying the proportion, there are discriminate differences among three levels of them. For the low level, to enlarge by 10 times easier than to purify, and to enlarge after purifying is the most difficult for them. The middle level and high level have skillful operation to solve the problem. But, in generally, big numbers would disturb the performance of both levels. The researcher suggests that children might learn the ratio and proportion conception with instrumental understanding but not by proportion reasoning. Chinese ancient mathematics, such as concrete operation strategies, might be further studied in detail for helping children deal with big numbers to overcome the basic fear.
In intuition division, children get the secondary intuition from the context of mathematics classroom that ‘%’ means less than 100 pieces. Therefore 150%, shown in daily life, is out of reason in children’s opinions. In the proportion problem, more than the half of objects, would eager to ask for units to make sure their reality sense. It often happens to the low level and sometimes to the middle level.

CONTENTS PAGE
LIST OF TABLES
LIST OF FIGURES
ABSTRACT- CHINESE
ABSTRACT- ENGLISH
CHAPTER 1.INTRODUCTION
1.1The Background and Importance of this Study
1.2The Purpose of this Study
CHAPTER 2.LITERATURE REVIEW
2.1 Children’s Mathematics
2.1.1 Children and Mathematics
2.1.2 The Mathematics Curriculum
2.1.3 The Curriculum Development
2.1.4 Problem Solving and Case-Base Reasoning
2.1.5 Ancient Mathematics and Children’s Learning
2.2 Children’s Mathematics Learning
2.2.1 The Piagetian Theory
2.2.2 The Intuition Theory
2.2.3 The Vygotskyan theory
2.3 The Researches of Ratio, Proportion Learning
CHAPTER 3. RESEARCH METHODS
3.1 Subjects and Research Design
3.2 The procedure and Analytical Approaches
3.2.1 The Exploratory Stage
3.2.2 The Pre-test Stage
3.2.2.1 Item Difficulty and Item Discrimination
3.2.2.2. Reliability
3.2.2.3 Validity
3.2.3 The Data Collection Stage
CHAPTER 4.RESULT AND DISCUSSION
4.1 Performance of Overall
4.2 Performances of different levels
4.3 Performance of different divisions
REFERENCES
APPENDICES
Appendix A.1 The questionnaire of Pre-Test
Appendix A.2 The Factor Analysis of Pre-Test
Appendix B.1 The formal questionnaire
Appendix B.2 The Factor Analysis of formal questionnaire
LIST OF TABLES
Table 1. Two—way specification table
Table 2. The corrections between Mathematics and Science teachers
Table 3. Test-retest reliability
Table 4. Internal consistency
Table 5. External consistency
LIST OF FIGURES
Fig. 1 Item Difficulty and Item Discrimination
Fig. 2 All performance
Fig. 3.1 Average performance
Fig. 3.2 Top performance
Fig. 3.3 Middle performance
Fig. 3.4 Bottom performance
Fig. 4.1 The Chinese Characters — Ratio performance
Fig. 4.2 Equivalence — Proportion performance
Fig. 4.3 Reality — Proportion Performance
Fig. 4.4 Reality— Scale Performance
Fig. 4.5 Reality — Ratio Performance
Fig. 4.6 Intuition — Proportion Performance
Fig. 4.7 Intuition — Ratio Performance

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