A Case Study Of Senior High School Students Using Proof By Contradiction To Solve Mathematical Problems

Proof by contradiction is an important mathematical proof method,but there is little attention to proof by contradiction in middle school mathematics education.This study considers the actual situation of high school students in our country using proof by contradiction to solve mathematical problems.This study will explore the process of two groups of students —— excellent students and ordinary students using proof by contradiction to solve mathematical problems.From the perspective of three aspects,such as the plot,proof by contradiction obstacle and metacognitive monitoring experienced in the process of problem solving,the research problems of this study will be determined.(1)What are the characteristics of the plots in the process of solving problems by contradiction between excellent students and ordinary students in high school mathematics?(2)What obstacles will arise when excellent students and ordinary students of high school mathematics solve problems by proof by contradiction?(3)What are the characteristics of metacognitive monitoring for excellent students and ordinary students in high school mathematics when they solve problems by contradiction?This study mainly adopts the method of thinking aloud to detect the problem solving process of four subjects,while assisting the method of observation and interview.After literature reading,the analysis framework of the process of proving by contradiction,the analysis framework of proving obstacles and the analysis framework of metacognitive monitoring is determined,and the case study is carried out.The results show that:(1)The proportion of analysis episodes of the two excellent students was less than that of the ordinary students,the proportion of planning episodes of the excellent students was higher than that of the ordinary students,and the proportion of implementation episodes of the four subjects was the highest.(2)The total number of the obstacle of proof by contradiction is significantly less than that of two ordinary students.In terms of operational disorders,each subject had problems in error correction,such as falsely claiming contradiction and failing to find contradiction.In the sense of using proof by contradiction,the excellent students were better than the ordinary students.In the aspect of the principle type of obstacle,the study of excellent 2A once confused the contradiction with the positive.In terms of non-cognitive factors,ordinary students1 B once showed rejection to the emotion of contradiction method;The excellent students 1A showed an emotional disorder of the mind-set when solving problem 1.(3)In terms of metacognitive monitoring,the four subjects showed different characteristics.The frequency of metacognitive monitoring behavior of two ordinary students was in the middle level,and the frequency of metacognitive monitoring behavior of two excellent students was at the highest and the lowest level respectively.According to the above research conclusions,the author puts forward some suggestions on teaching by contradiction,teaching materials and curriculum arrangement,and students’ learning by contradiction.