Analysis on Students' Abilities of Proof in Middle School

중학교 학생의 증명 능력 분석

In this study, we analysed the constituents of proof and examined into the reasons why the students have trouble in learning the proof, and proposed directions for improving the teaming and teaching of proof. Through the reviews of the related literatures and the analyses of textbooks, the constituents of proof in the level of middle grades in our country are divided into two major categories 'Constituents related to the construction of reasoning' and 'Constituents related to the meaning of proof. 'The former includes the inference rules(simplification, conjunction, modus ponens, and hypothetical syllogism), symbolization, distinguishing between definition and property, use of the appropriate diagrams, application of the basic principles, variety and completeness in checking, reading and using the basic components of geometric figures to prove, translating symbols into literary compositions, disproof using counter example, and proof of equations. The latter includes the inferences, implication, separation of assumption and conclusion, distinguishing implication from equivalence, a theorem has no exceptions, necessity for proof of obvious propositions, and generality of proof. The results from three types of examinations; analysis of the textbooks, interview, writing test, are summarized as following. The hypothetical syllogism that builds the main structure of proofs is not taught in middle grades explicitly, so students have more difficulty in understanding other types of syllogisms than the AAA type of categorical syllogisms. Most of students do not distinguish definition from property well, so they find difficulty in symbolizing, separating assumption from conclusion, or use of the appropriate diagrams. The basic symbols and principles are taught in the first year of the middle school and students use them in proving theorems after about one year. That could be a cause that the students do not allow the exact names of the principles and can not apply correct principles. Textbooks do not describe clearly about counter example, but they contain some problems to solve only by using counter examples. Students have thought that one counter example is sufficient to disprove a false proposition, but in fact, they do not prefer to use it. Textbooks contain some problems to prove equations, A=B. Proving those equations, however, students do not perceive that writing equation A=B, the conclusion of the proof, in the first line and deforming the both sides of it are incorrect. Furthermore, students prefer it to developing A to B. Most of constituents related to the meaning of proof are mentioned very simply or never in textbooks, so many students do not know them. Especially, they accept the result of experiments or measurements as proof and prefer them to logical proof stated in textbooks.