建立中學生絕對值與絕對值方程式學習進程之研究

Abstract

本研究的目的主要為探討學生在學習絕對值的學習進程，並初步發展絕對值學習進程評量試題，進而協助教師了解中學生在學習絕對值概念的理解情況，做為後續教學調整之依據，希望研究的結果能夠提供教師作為補救教學或改進教學策略的依據，增進教學成效，並作為未來教學及研究的參考。 本研究主要分為兩部分，第一部分為建立學習進程，採文獻分析及調查研究法，經由對絕對值相關概念作文獻探討，與指導教授及研究小組初步討論，並輔之以訪談學生及教師，初步擬定學生絕對值的學習進程。 研究者係以過去對於絕對值的相關研究文獻為基礎，同時為兼顧過去的研究結果與現階段學生的實際表現情形，研究者針對七年級、八年級與九年級各2位學生，一共有6位國中生，以及台北市高中一年級5位學生，總共11位學生進行訪談，以瞭解不同年級學生在絕對值相關概念上的表現，並協助發展初步的學習進程架構。 擬定初步學習進程的內容後，第二部分為驗證學習進程，採調查研究法，在擬定初步學習絕對值學習進程的內容後，研究者發展絕對值概念試題工具，以檢驗研究者所初步發展之學習進程。在評量施測階段，研究者蒐集151位學生做為研究對象進行施測，以檢驗研究者所初步發展之學習進程與學生的學習表現是否相符。 本研究所初步開發的學習進程評量，在絕對值概念的部分，能夠將118人進行學習進程的歸類，無法歸類的人數共33人，可歸類的人數百分比為78.1%，無法歸類的人數百分比為21.9%，研究者認為本評量工具在測量學生學習進程的功能上有一定程度的可行性，並且在進行調整與修正後，將能夠解釋學生在學習進程中表現。 學生在絕對值概念上的表現上， (一)在絕對值定義理解的部分: 國中生在掌握含絕對值的加減運算以及兩數互為相反數，則其絕對值相等的能力是具備的，在本研究之絕對值學習進程的第1層符合學生實際能力表現。(二)絕對值幾何意義的部分: 學生大部分能夠用絕對值式子表示兩點間的距離，並且說明兩點間的距離關係。(三)在絕對值代數運算的部分: 大部分學生在解絕對值方程式時能夠正確作答，九年級學生有約半數的學生能夠解含兩個絕對值的方程式，而部分學生透絕對值距離含意進行解題，展現更高能力層次的作答表現。(四)在絕對值代數與幾何之間的表徵轉換: 部分學生在以代入數字的方式進行作答，而少部分在試卷較高分的學生會以距離方式判斷題目。(五) 含絕對值方程式圖形概念部分: 七年級學生對絕對值方程式圖形概念較不掌握，八、九年學生相對較能夠以絕對值概念判斷圖形。繪製絕對值方程式圖形的部分，多數九年級學生能夠帶點正確畫出圖形，少部分學生會忽略負數的部分而畫出一條直線的圖形。 根據上述的研究發現，本研究建議，未來的研究者可以針對在概念試題工具上已開放性的試題進行探討，針對學生在絕對值的學習過程進行更多元的探討，在評量工具的設計與使用方面，建議使用更多的評量試題以及多元的判準原則來判斷學生的所屬階層。 本研究因應於12年國教可能將國中絕對值內容的改變(刪減部分內容)，學習進程在第1至2層基本上並無太大差異，而未來研究上建議將本研究之學習進程第3至6層再行延伸(例如:加入第7層)，並且搜集高中學生的樣本作驗證，即可更符合12年國教絕對值的學習進程。故此論文雖主要為中學生的學習進程，但可作為日後12國教絕對值學習進程的藍圖。The main purpose of this study is to investigate students’ learning progression of absolute value and preliminarily develop learning progression assessment items to further assist teachers in understanding high school students’ concepts of it, which will become a basis for following teaching. The results of the study expect to provide a basis for remedial teaching or improvements of teaching strategies to enhance teaching effectiveness and to serve as a reference for teaching and research in the future. This study is divided into two parts. The first part is establishing a learning progression in which survey research and documentary analysis methods are used. Through literature review of absolute value related concepts, preliminary discussions with principal advisor and research groups, and interviews with students and teachers as supplements, students’ learning progressions of absolute value are drafted. Based on past relevant research literatures of absolute value and giving consideration to both the previous research and the actual performance of students at the moment, this study focuses on 7th grade, 8th grade and 9th grade junior high school students, two from each grade and 6 in total, as well as 5 senior high school students in Taipei. A total of 11 students were interviewed in order to understand the performance of students at different grades in absolute value related concepts and to help develop rudimentary framework of their learning progression. The second part is to verify the learning progression conducted by survey research. The authors developed assessment item tools to test the attributive level of students. During the testing, 151 students were examined as research subjects to verify if the learning progression developed preliminarily by the authors corresponds to students’ performance. In the learning progression assessment for the concepts of absolute value, 118 students were classified while 33 students weren’t, corresponding to 78.1% and 21.9% in percentage. The authors deemed this assessment tool feasible to some extent in measuring students’ learning progression. After making adjustments and corrections, it can be used to explain students’ performance in the learning progression. In terms of students’ learning performance in absolute value: (1) Understanding the definition: junior high school students were able to master the addition and subtraction of absolute value as well as the notion of two opposite numbers having the same absolute value. The first level of this learning progression was therefore in line with students' actual performance. (2) Geometric meanings of absolute value: most students were capable of using absolute value equations to represent the distance between two points and explained the relation between them. (3) Algebraic computation of absolute value:most students answered correctly when solving absolute value equations. Half of the 9th-graders were able to solve two equations containing absolute values, while some students solved the problems by applying the distance meanings of absolute value, showing higher level ability in answering. (4) Representational conversion of absolute values between algebra and geometry: part of the students answered by substituting numbers, while few of them scoring high used distance to consider the question. (5) Absolute value equations containing graphic concepts: seventh-graders were weak in these concepts. 8th and 9th graders were relatively more capable of differentiating the graphs by using the concept of absolute value. As for graph drawing of the equations, most of the 9th graders could draw the graph almost correctly, but some students ignored the negative numbers and drew a graph with a line. According to the above findings, the study suggests future researchers focus on open-ended questions of conceptual item tools and probe into students’ learning process in more diverse aspects. As for the design and application of assessment tools, it suggests more assessment items and judging criteria be used to determine the levels students belong. Considering the possible changes in the content of absolute value in 12-year compulsory education (part of the content may be eliminated), there’s basically not so much difference between the 1-2 levels. It suggests that future research extend 3-6 levels, (add level 7, for example) and collect samples of high school students for verification to make it more in line with the learning progression of absolute value in 12-year compulsory education. Although this study mainly centers on the learning progression of high school students, it could serve as a blueprint for the learning progression of absolute value of 12-year compulsory education in the future.