The error analysis of geometry proofs which require high-school students deal with auxiliary lines

• Main Authors: Chen, Tz-Yen, 陳姿妍
• Other Authors: Lin Fou-Lai
• Format: Others
• Language: zh-TW
• Published: 1996
• Online Access: http://ndltd.ncl.edu.tw/handle/05898522019959111551

碩士 === 國立臺灣師範大學 === 數學系 === 84 === The research aimed at analyzing the common errors of geometry proofs which require high-school students deal with auxiliary lines. The research is based on the investigation tests. We find three common error styles from the tests'' data of 100 1-grade junior college students and 91 3-grade junior highschool students. 1.Students impose extra property for the construction of auxiliary lines. 2.Students do not use auxiliary lines and introduce other auxiliary ele ments imposed extra property. 3.Students do not introduce auxiliary elements and use cycle auguments. The error style that the research probed into impose extra propertyon auxiliary elements and cyele auguments. The research investigated the cognition and related factors of students with the two performances by testsand interviews and investigated the affection performances of students using auxiliary lines conducting geometry proofs. We investigated one by one 31 students identified with the actions. We analyzed the data and got the results by the theory of ploya''s "How to solveit" ,Mason et al''s "Thinking Mathematically" ,Information Processing system and intuition cognition. A.The background for students using auxiliary lines to do geometry arguements : 1.Eucliad geometry knowledge is not structured connection for many high school students. 2.The auxiliaryl ines outside the diagrams is more difficult for high school students. 3.Some students perform monitor ability. 4.The influences of affection weaken the warning of monitior. B.The reasons for students to impose extra properties on auxiliauy elements. 1.Students have the intention of solving problems with the information in the diagram. 2.What students learned is a releated chunk without logical sequences. 3.With the influence of diagrams and theorematic chunks,students rely on intuition cognition and use "extra property" strategy to write proofs. 4.Students are likely to use "extra property" strategy because of the jump or ignorance of their thinking process. 5.Students offen use "extra property" strategy when they can''t introduce suitable auxiliauy elements. C.The resons for students to apply cycle arguements. 1.What students learn are some geometry properties chunks. There are no logical sequences between the properties. 2.Students''intuition cognition supports the legality of the applied properties. 3.Students can''t introduce suitable auxiliary elements. The reseach came to some conclusions by the above analysis. 1.Intuition cognition direts the problem solving strategy when they can''t write suitable proofs. 2.Because of the correctness of "extra properties" and "cycle arguements" in the theorem chunks,students are likely to be influences by affation and their monitors can''t detect the errors easily. 3.Students are likely to apply "cycle arguments" when they can not introduce suitable auxiliary elements. The result of the research confims "Many students cited the theorem to be proved in their proofs"、"The fact that many students had difficulty with embedded figures and anxiliary lines exemplifies the need to teach students how ,why and whem they can transform a diagram in a proof", which senk <1985>indicated.