國中基測實施迄今已十年，但是各種多元管道仍以國中基測量尺分數作為分發篩選之重要參據，多元能力評量參採比重偏低，國中學生升學競爭壓力未得緩解。本研究透過數學與統計分析的工具，尋找採用學生在校成績的方法，希望能找出更好的方式來代表學生在校三年的學習現況與學習成果，以做為免試升學採計在校成績的參考與依據。 本研究主要目的是要探討如何取決各科在校成績的權重(也就是在每個科目的分數之前乘上一個加權比重係數)，以求得一個新的合成變量(由數個科目分數組成的線性組合)，並用這個新合成變量做為學生在校的“綜合學科能力表現分數”，代表學生在校三年的基本學習能力及程度。 研究方法運用主成份分析與典型相關分析的觀念，但因限制條件設定的範圍與傳統主成分分析及典型相關分析的要求不一致，因此，我們便將所用的研究方法命名為「類主成分分析」與「類典型相關分析」。 研究中，方法主要在比較「類主成分分析」、「主成分分析」、「類典型相關分析」、以及「典型相關分析」四種方法與一般學校常用的「等加權比重」算平均成績的方法之分別；了解這些不同加權機制對同一所學校內學生的學科加權平均分數之成績排名百分比結果，以及與基測排名結果的差異。 「類主成份分析」研究結果發現，各科學科成績中變異數大的科目將獲得較大的權重比例，成為主導學生加權平均成績中舉足輕重的科目。另外；運用「類典型相關分析法」所求得的典型相關係數，其結果與傳統典型相關分析法以及使用最佳數值分析軟體(GAMS)所得的典型相關係數完全相同。 本研究最重要的貢獻之一，是我們在「類典型相關分析法」中證明並推導出一個求得各科權重的公式，只要使用此公式代入簡單的MATLAB程式，其所得的權重結果與最佳化數值分析軟體(GAMS)所得的結果完全相同，但花費的計算時間及成本卻遠少於GAMS所需，是一個求權重極便捷的方法，讀者可以在本論文附錄7.5.2或政大應數系網站上下載此程式。本研究最後結論也發現，類主成份分析的變異解釋率是所有方法中較高的；與基測總分結果較相近的則是類典型相關分析所得的權重機制；而等加權方法所得的排名結果則與基測排名結果差異最小。 / The BCTEST (Basic Competence Test) for junior high school students has been implemented for ten years, however, the screenings for a variety of entrance programs are still based on the scale scores of the BCTEST with a low proportion of multi-intelligence. Hence, the competitive entrance pressure for junior high school students remains un-relieved. In view of this, via mathematics and statistics, this study is to explore an alternative approach which can not only reflect students' in-school grade, their learning situations and achievements but also represent a reference for entering senior high schools and junior colleges without entrance examinations. The purpose of this study is to determine the different weightings of five learning subjects (that is, multiply the score of each subject by a weighted coefficient) and acquire a new composite variable from the linear combinations of five learning subjects. Then, use this new composite variable as the synthetic score of students' in-school academic performance. Principal Component Analysis and Canonical Correlation Analysis are used in this study. Due to inconsistent restraints, the other two approaches we use are based on the concept of previously mentioned methodologies and denominated Principal Component Type of Analysis and Canonical Correlation Type of Analysis. In the study, we compare with the different results of Principal Component Analysis, Principal Component Type of Analysis, Canonical Correlation Analysis, Canonical Correlation Type of Analysis and identical weighted method to realize how these different weighted schemes affect the rankings of students from the same school on both their weighted in-school grade and scores of the BCTEST (Basic Competence Test). The outcomes of Principal Component Type of Analysis show that subjects with greater variance acquire larger weightings and play a dominant role in weighted in-school grade. Moreover, the correlation coefficients of Canonical Correlation Type of Analysis are completely the same as the ones of Canonical Correlation Analysis and GAMS. One of the most important contributions in this study is we have proven and derived a formula to acquire different weightings of five learning subjects by using Canonical Correlation Type of Analysis. The acquired weightings are completely the same as the ones of GAMS with less time consuming. Readers can download this program in appendix 7.5.2 or from the website of Department of Mathematical Sciences, National Chengchi University(NCCU). We have also found that, the explanation rate of variance obtained from Principal Component Type of Analysis is the highest; the weighted scheme of Canonical Correlation Type of Analysis is more similar to the scores of the BCTEST; the difference of the rankings between identical weighted method and the BCTEST is the smallest.

University of Macau / Faculty of Education

實施十二年國民基本教育，目的是為促進學生五育均衡發展，兼顧國中學習品質及日常生活表現。由於各校對成績的評分標準與評分方式皆不相同，因此如何使在校成績採計達到公平性將成為一項重要的問題。 戴岑熹(2011) 考慮了國中在校綜合學科分數與基測總分間的相關性，以決定在校各學科的權重。而本研究延伸其概念與方法，將基測各科量尺分數考慮進來，於在校綜合學科分數與基測綜合量尺分數的關聯性最密切的情況下，分析各學科權重的取決方式，希望能找出較理想的模式來代表學生在校三年的整體學習表現與成果，以做為免試升學採計在校成績的參考與依據。 本文的研究方法是運用典型相關分析的理論，但因權重的限制條件與傳統典型相關分析的要求不同，因此，便將其命名為「類典型相關分析」。在類典型相關分析中，我們證明了在校各學科分數及基測各科量尺分數的最佳權重，可先透過典型相關分析求得典型相關向量，若有必要的話，使用Rao-Ghangurad 方法加以修正，最後，再將所獲得的非負典型相關向量正規化，即可獲得所要的結果，這是一個求最佳權重向量極便捷的途徑。在實例分析方面，我們發現了一個有趣的現象，即在校學科分數與基測考科量尺分數的最佳權重向量相當接近，即名稱相同的學科與考科幾乎有相同的權重。在比較了幾個權重分配方式不同的在校綜合學科分數後，我們也發現一般學校常用的等加權模式，其表現結果也頗優異。 / The purpose of implementing the twelve-year compulsory education is to promote the balanced development of learning in students, taking into account their learning quality and normal daily performances in school. As the evaluation standard and method vary among schools, achieving fairness in calculating in-school grades has become an important issue. Dai (2011) considered the correlations between the scores of in-school academic performance and the total score of the BCTEST for junior high schools, which decided to the weightings of all learning subjects. This study extended his concept and method, and took into account the scale scores of all learning subjects. In the closest case of the weightings of all learning subjects and find out the correlations between the scores of in-school academic performance and the BCTEST, and analyse the weightings of all learning subjects. We hope the study can find a better approach that can not only reflect students’ learning situations and achievements for the three years in school but also provide a reference for the evaluation of entering senior high schools without entrance examinations. The research method in this paper employs the theory of canonical correlation analysis.However, due to that fact that weight restrictions are different from the requirements of canonical correlation analysis, it is named as the canonical correlation analysis type approach. In the canonical correlation analysis type approach, we proved that the optimal weights for school subject score and test subject score scales can be obtained by finding the canonical correlation vectors using canonical correlation analysis. Then the Rao-Ghangurad method can further be used for amending, if needed. Finally, the nonnegative canonical correlation vectors generated would be normalized to get the desired result. It is an extremely convenient way to obtain the optimal weight vector. In the case study, we found an interesting phenomenon as follows: When the optimal weight vectors for school subject score and test subject score scales were very close, subjects and tests of the same name had almost the same weight. After comparing several comprehensive school subject scores of different weight distribution, we also found that the results of the equal weighting model commonly used in schools also showed quite good results.