中學生二次函數學習進程之探究

本研究旨在建立二次函數的學習進程，協助學習進程的使用者瞭解學習者對於二次函數相關概念的掌握順序，提供使用者一個參考去設計合適的教學活動，使學習者更能掌握二次函數的概念。基於此目的，本研究主要的研究問題為：「我國中學生對於二次函數的學習進程內容為何？」為了回答研究問題，本研究採用文獻分析法、質性訪談以及問卷調查法，並經歷探索期、調整期以及驗證期三個階段得到研究結果。探索期之研究的目的在於熟悉學習進程的建立方法，以及蒐集學生在學習二次函數時可能有的學習表現。此時期利用文獻分析法瞭解過去相關研究中學生有的學習表現，設計訪談稿後，藉由質性訪談，訪談了6位台北市十年級的學生以及2位台中市十一年級的學生，而後建立了初步的學習進程。根據學習進程，研究者開發評量試卷，蒐集了台中市42位十年級的學生以及38位十一年級的學生的學習表現資料，由於此試卷未做信效度的檢驗，且學生填答情形不如預期，僅作為蒐集學生的學習表現用。調整期之研究的目的在於解決探索期所遇到的困難，並調整二次函數學習進程的內容。此時期增加了對學習進程理論文獻的探討量，也利用文獻分析法瞭解更多學生在學習二次函數時可能有的學習表現，並提出一個新的初步的學習進程。驗證期之研究的目的在於驗證研究者所開發的學習進程之有效性，並嘗試提出更能描述學生在學習二次函數時的學習進程。此時期利用問卷調查法，根據調整期所提出的學習進程設計評量試卷，針對北北基八到十二年級的學生進行施測，蒐集了共604位學生的資料，由於試卷設計的緣故僅分析了399位高中學生的資料。另外，利用質性訪談的方式，邀請五位有經驗的現場教師進行排序活動，作為效化的證據之一。本研究所建立之初步的二次函數學習進程共有六個等級（等級0～等級5），內容包含二次函數的定義、作圖、係數意義、幾何變換、極值、對稱。利用蒐集到的學生資料僅能將13.67%的學生歸到對應的等級，利用教師所排序的資料，在完全一致率僅達到24.3%～48.8%，若考慮差一個等級的一致率可達68.3%～87.8%，本研究所建立之初步的學習進程仍有調整的空間。研究者根據學生資料，將等級4與等級5合併，並刪除了學生進行幾何變換時的思維的相關描述後，能將63.04%的學生歸到對應的等級。研究者並根據教師資料，將幾何變換時的思維的相關描述放回調整後的學習進程之中。由於研究者對於教師資料的處理尚未釐出頭緒，因此，以學生資料為主、教師資料為輔調整二次函數的學習進程，教師資料的使用有待研究者日後做更進一步的分析。本研究所建立之二次函數學習進程仍有很大改善的空間，但已能供使用者做一個參考，在教學上更去注意學生認知的發展。此外，本研究有別於過往的學習進程研究，利用現場教師的資料進行效化，雖然在教師資料的使用上還不夠好，但已是一個突破，可供未來建立學習進程的研究者作為一個參考。關鍵字：學習進程、二次函數。

關鍵字

學習進程；二次函數

並列摘要

The purpose of this study is to depict the learning progression for quadratic functions of secondary students in Taiwan. Through this learning progression, we hope to assist the users to understand the sequence how the learners grasp the concepts of quadratic functions, and to offer the users a reference for designing appropriate learning activities; thus, the learners can learn the concepts of quadratic functions more efficiently and better. Based on this purpose, our main research question is “what the content of the learning progression for quadratic functions of secondary students in Taiwan?” In order to answer the research question, we used literature analysis, qualitative interview and questionnaire, and went through three periods to obtain the results. First period is exploration period. The purpose of exploration period is to make us familiar with the method of depicting a learning progression, and to collect the probable learning performance when secondary students learn the concepts of quadratic functions. We used literature analysis to understand what the mathematicians and the educators expected students to learn, what content students really contacted with, and what kind of error students had through the past study. Then, we designed a draft for interview, and interviewed 8 students in total. Finally, we depicted an initial learning progression for quadratic functions. To validate the learning progression, we developed an assessment and collected the data of forty-two 16-year-old students and thirty-eight 17-year-old students. Since we didn’t test the validity and reliability of this assessment, we only collected the data in qualitative analysis rather than quantitative analysis. Second period is adjustment period. The purpose of adjustment period is to solve the challenge we faced in exploration period, and to adjust the content of the learning progression for quadratic functions. We reviewed more literature to understand the learning progression theory and to understand what kind of learning performance students had in the past research. Finally, we depicted a new initial learning progression for quadratic functions. To validate this new initial learning progression, we came to the third period, validation period. We used questionnaire and developed a new assessment based on the new initial learning progression. After testing the validity and reliability of this assessment, we collected the data of 604 students, through 14-year-old to 18-year-old, and finally analyzed the data of 399 senior high school students because of bad design of assessment for junior high school students. In addition, we used qualitative interview and invited five veterans of mathematics teachers to arrange 41 descriptions in our learning progression as one of evidence for validation. There are six levels, from 0 to 5, in our new initial learning progression for quadratic functions with the concepts including definition, graphing, the meaning of coefficients, transformations, extreme values and symmetry of quadratic functions. However, we could only depict 13.67% students in our data with this learning progression, and had 24.3 to 48.8 percent exact agreement and 68.3 to 87.8 percent adjacent agreement with five veterans of mathematics teachers. This informs us of the room of improvement of our learning progression. Based on students’ data, we combined the contents of level 4 and 5, and deleted the descriptions about the thoughts of transformations. We could finally depict 63.04% students in our data after the adjustment. And, we put the descriptions about the thoughts of transformations back into the learning progression by teachers’ data. Since we had no idea to use the teachers’ data, we mainly relied on students’ data supplemented by teachers’ data to adjust our learning progression. Keywords: learning progression, quadratic function