初探多元解題教學情境下高中生的數學創造力與創造歷程

Abstract

本研究旨在探討在多元解題教學的情境之下高中生在幾何證明題、幾何問題與代數問題的數學創造力表現，以及學生在面對多元解題任務時的創造歷程。為探討此研究問題，研究者以提升數學創造力的五項原則作為課程架構，搭配多元解題任務作為教學工具，發展出多元解題教學法。研究樣本為某公立高中二年級理組資優班學生，共29位學生完整參與研究。學生數學創造力的評分架構為參照Leikin (2013) 的評分方式，透過數學思維形式的流暢性、變通性與原創性來評估學生的數學創造力表現。學生創造歷程的分析則是參照格式塔模型以及大衛創造模型建立多元解題情境創造模型剖析學生在嘗試解題時的思考過程。研究結果發現：(1) 學生在經過多元解題教學法的介入後，依照前測結果將全班學生分為多解組與少解組，發現在證明題與幾何問題中，有較高比例的少解組學生後測的流暢性分數提升，而在代數問題中，少解組學生的流暢性分數則是以不變的比例為最高，顯示多元解題教學法對於證明題與幾何問題有提升流暢性的效果，對於代數問題的效果則較不明顯；(2) 學生在經過多元解題教學法的介入後，三種題型的變通性分數皆提升，顯示多元解題教學法能夠有效幫助學生寫出差異程度更大的解法；(3) 學生在經過多元解題教學法的介入後，證明題的原創性分數有所提升，而幾何問題與代數問題的原創性分數發生下降，同時發現解法種類的得分情形出現『去極端化』的現象，顯示鮮少有學生能產生特異且稀有的解法；(4) 學生創造歷程的分析結果顯示獲得高創造力得分解法的思考歷程並沒有特定的模式，但是有較高比例的解法未包含準備期，顯示對於複雜度較高的問題，若解題時先設定好一個主要概念來切入問題，接著再針對後續搭配的概念進行發散性思考，亦能達到提升創造力得分的效果。This study aims to explore senior secondary students’ mathematical creativity and their creative process through multiple-solution problem solving in geometric proofs, geometry, and algebra. To achieve this aim, the present research developed an instructional module based on the Five Overarching Principles to Maximize Creativity and included a series of multiple solution tasks . A total of 29 eleventh graders from a public senior high school in Kaohsiung participated in the study. The study used the framework established by Leikin (2013) to evaluate the students’ mathematical creativity including fluency, flexibility, and originality. Regarding creative process, the study modified the Gestalt model and David’s creative process model to develop the Creation Model for Multiple-Solution Problem Solving to analyze the students’ thinking process when solving mathematical problems with multiple solutions.Compared to the pre-test, the results show that: (1) the students’ fluency in mathematical creativity was facilitated in geometric proof and geometric problems, but not in algebra problems; (2) the students’ flexibility was enhanced in all of the types of mathematical problems, which indicated that multiple solution tasks promoted students to generate distinct solutions; (3) While the students’ originality was improved in geometric proof problems, the scores decreased in geometric and algebra problems. It was also found that few students were able to generate rare solutions; (4) The analysis of the students’ creative processes of higher creativity scores showed no specific thinking patterns. However, a relatively high proportion of the creative processes with higher creativity scores did not start from the preparation stage. It seems that the preparation may not be necessary for complex problems. It is likely that the initial selection of a key concept and the use of diverse thinking played an important role in advancing mathematical creativity.