Senior Middle School Students' Understanding About Proof By Contradiction

The proof by contradiction is one of the most important methods of proving propositions in mathematics. Being used properly, it can reduce the complexity to simplicity, difficulty to easiness, impossibility to possibility, thus playing a distinctive role which the direct proof cannot play. Because the logical thinking involved is strong and the precision of mathematical language is high, the proof by contradiction also plays an important role in the training of students' abilities of rigorous logical thinking and reading comprehension and establishing of correct conceptions of mathematics. Moreover, this method is widely used in college mathematics. However, the proof by contradiction is difficult both to teach and to learn. How to effectively improve the teaching of this method is an important task facing middle school mathematics teachers.This paper reports an empirical study on senior middle school students' understanding about the proof by contradiction. Through a test conducted to 268 senior middle school students from Shanghai and Zhejiang and interviews with some of the subjects from Shanghai whose answers are typical, the following results are revealed:(1) The students' ability to negate statements is of low levels, which directly affect their understanding and application of the proof by contradiction;(2) The scope understood by the students within which the proof by contradiction is applicable is narrow, and the logical basis and the types of contradictions are beyond their knowledge.(3) The students' awareness to apply the proof by contradiction is weak, and their operation level is low.(4) With the grade, students' ability to negate statements increases, but the difference is not evident.(5) There exists difference in levels of understanding of the proof by contradiction between subjects from Shanghai and Zhejiang, the latter being superior...