| Summary: | 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.
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