摘 要

本研究針對中學生處理有輔助線需求的幾何問題證明時較常見的錯誤類型
進行錯誤分析 ,根據自行發展的調查工具所做研究，由100位專一生及91
位國三生的卷面資料 ,發現三種常見的錯誤類型。 1.對輔助線的作圖
" 附加條件 "。 2.不使用輔助線 ,而引介其他含 " 附加條件 "
的輔助元素。 3.不引介輔助元素 ,使用 " 循環論證 " 。
本研究探究的錯誤類型為對輔助元素附加條件進行幾何證明與循環論證。
利用筆測及面談法調查有這兩種論證的學生之認知相關因素,並以問卷調
查學生使用輔助線進行幾何證明的情意表現。 總共一對一面談31位
經鑑定有此論證行為的學生。利用 Ploya 的 " 怎樣解題 " ,Mason et
al.的 " 數學思考 " , 訊息處理系統及直觀認知等理論基礎 ,分析資料,
得到下列結果。 一.學生使用輔助線政證明幾何問題的背景 1.歐氏
幾何知識對許多中學生而言不是結構性的連接。 2.圖形外的輔助線對
中學生而言較困難。 3.確實有些學生表現出監控能力。 4.情意的
催化減弱監控的警訊。 二.學生對輔助元素" 附加條件" 的原因 1.學
生有針對問題附圖解題的傾向。 2.學生所學的是一個相關組集,並沒
有邏輯序列。 3.學生在圖形及命題組集的影響下,僅依賴直觀認知,使
用" 附加條件" 的策略進行論 證。 4.思考過程的跳躍或忽略易
使學生產生如同" 附加條件" 策略的答題表現。 5.學生無法引介適切
的輔助元素時, 常引用" 附加條件" 的策略。 三.學生循環論證的原因
1.學生學得的是一些幾何性質的組集,性質之間沒有邏輯序列。 2.直
覺支持所引用性質為合法。 3.學生無法引介適切的輔助元素。 由
前述分析,本研究得到幾點結論: 1.學生無法產生適當的證明程序時,
直觀認知主導解題策略。 2." 附加條件" 及" 循環論證" 在命題組集
內的相容性,使學生易受情意影響,其監控 系統難以偵錯。 3.學
生無法引介適合的輔助元素時,容易陷入" 循環論證 "。
本研究結果與Senk (1985)所提的「許多學生在證明當中, 引用了要證明
的定理。」及「學生使用輔助線的困難,說明了教學上應注意如何、為什
麼及何時可以轉換證明中的圖 形。」相印證。

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.
The research aimed at analyzing the common errors of