- 學位論文
資優數學之高次聯立方程式的解題策略
The strategies of solving high-order simultaneous equations in gifted mathematics
摘要
本研究旨在探討國高中生在資優數學之高次聯立方程式的解題策略之方法論的實際範例，做分類統整之探究，並就解題過程中做解題分析、使用的解題方法論進而理出解題策略。就試題是以國高中資優數學之「聯立方程式」為範圍，從「二元一次聯立方程式」追溯，進而最主要以「二元高次聯立方程式」為主，並涉略至「多元高次聯立方程式」的資優數學題目，共113道試題進行篩選分類之統整。然而在「多元高次聯立方程式」的資優數學題目中，尚有更多變化之試題，甚至可再深究「聯立分式方程式」、「聯立無理方程式」、「聯立指數與對數函數方程式」與「聯立三角函數方程式」的資優數學試題。而本研究的探討對象為研究者本身曾任教的數理資優班學生，在解題表現上是否會因歷屆程度的不同而有所不同呈現，值得將來探究；或者將數理資優生與一般數學能力較佳的學生做比較分析，看看兩者在解題表現上有何異同，亦值得將來探究。 解題策略方面，教師在解題教學時，可以根據題目的特性，先運用學生偏好的解題策略，明確地幫助學生解題，提高學生對解題的興趣，等學生學會之後，就能靈活思考，將來面對各種問題時，就會使用更多樣的解題策略，進而成功解題。更進一步，可引導學生讀題時注意題目中的關鍵字句，以利組織相關訊息統整分析，並訓練學生解題時的自我監控能力，進而成功解題。同時多給學生一些時間去獨立思考解決問題，不求練習很多的數學問題，而是讓數理資優生從解題過程中，累積解題經驗，增強解題能力，產生學習遷移，才能應用到新的問題情境中，解題重質不重量，老師也可以與學生教學相長，豐富教學經驗及能力，進而回饋到一般班級的課室教學上，提升整體的教學成效。
並列摘要
This study was designed to investigate students in junior high school and high school for their practical examples of problem solving strategies with the methodology of simultaneous equations and do classification Integration of the inquiry, and do problem-solving analysis of the problem solving process, and then sort out the problem-solving methodology. Questions based on gifted junior high school and high school mathematics so called "Simultaneous equations" ranging from "Of simultaneous equations" to the "Binary high-order simultaneous equations" based, including slightly "Multiple high-order simultaneous equations" gifted math problems, a total of 113 questions were sorting out from the entire system. However, in the "Multiple high-order simultaneous equations" in gifted students, there are more changes for questions, or even get to the bottom again, i.e., gifted mathematics questions like "Fractional simultaneous equations", "Unreasonable simultaneous equations", "Exponential and logarithmic simultaneous equations" and "Trigonometric simultaneous equations". The object of this study was to explore the researchers themselves if they had taught mathematics classes for gifted students in problem-solving performance depending on whether previous levels presented, it is worth exploring; or doing a comparative analysis for mathematical gifted students and general math skills preferably students. While looking at both similarities and differences in performance on problem-solving, it is also worth exploring in the future. Speaking of problem-solving strategies, teaching teacher in problem-solving is based on characteristics of the subject, which is the first preference of the students for problem-solving strategies, specifically to help students solve problems, improve students' interest in problem solving. Therefore, after the students learn, they can think flexibly, and then may face the future along with the various problems. They will use a greater variety of problem solving strategies, and then successfully solve these problems. Furthermore, it can guide students to pay attention to the subject in questions with relevant information to facilitate the organization Integration analysis and problem-solving ability, and then lead to the success of problem-solving. At the same time, giving the students some time to think independently to solve problems, instead of practicing a lot of math problems. With the focus on mathematical gifted students by the problem solving process, the cumulative problem solving experience will enhance capability for problem-solving. The learning habit will transfer to appliance to the new problem situation in solving quality instead of quantity. Both the teachers and students can learn teaching experience and ability, and then back to the classroom teaching general class, to improve the overall effectiveness of teaching.