• The Mathematical Education
• /
• v.54 no.1
• /
• pp.83-98
• /
• 2015

By analysing the examination papers for geometry, we classified the errors occured in the proofs for geometric problems into 5 main types - logical invalidity, lack of inferential ability or knowledge, ambiguity on communication, incorrect description, and misunderstanding the question - and each types were classified into 2 or 5 subtypes. Based on the types of errors, answers of each problem was analysed in detail. The errors were classified, causes were described, and teaching plans to prevent the error were suggested case by case. To improve the students' ability to express the proof of geometric problems, followings are needed on school education. First, proof learning should be customized for each types of errors in school mathematics. Second, logical thinking process must be emphasized in the class of mathematics. Third, to prevent and correct the errors found in the proofs for geometric problems, further research on the types of such errors are needed.

• Journal of the Korean School Mathematics Society
• /
• v.9 no.4
• /
• pp.521-538
• /
• 2006

The purpose of this study is to examine the effect of teaching mathematical proofs that made use of the 'proof assisted cards' at the second year of middle school and to investigate students' ability to geometric proofs as well as changes of mathematical attitudes toward geometric proofs. The subjects are seven students at the 2nd year of D Middle School in Daegu who made use of the 'proof assisted cards' during five class periods. The researcher interviewed the students to investigate learning questions made by students as well as the 'proof assisted cards' before and after use. The findings are as follows: first, the students made change of geometric proof ability by proof activity with the 'proof assisted cards' and second, the students made significant change of mathematical attitudes toward geometric proofs by proof activity using the cards.

• Journal for History of Mathematics
• /
• v.17 no.3
• /
• pp.93-108
• /
• 2004

In this article we study on various proofs of the Steiner-Lehmus theorem(any triangle that has two equal angle bisectors is isosceles). We suggest 6 geometric proofs and 3 algebraic proofs of the theorem in detail, analyze these proofs, extract related theorems, proof ideas.

• Research in Mathematical Education
• /
• v.19 no.4
• /
• pp.229-245
• /
• 2015

In this study, I investigated how pre-service teachers (PSTs) proved three geometric problems by using Geometer's SketchPad (GSP) software. Based on observations in class and results from a test of geometric reasoning, eight PSTs were sorted into four of the five van Hiele levels of geometric reasoning, which were then used to predict the PSTs' levels of reasoning on three tasks involving proofs using GSP. Findings suggested that the ways the PSTs justified their geometric reasoning across the three questions demonstrated their different uses of GSP depending on their van Hiele levels. These findings also led to the insight that the notion of "proof" had somewhat different meanings for students at different van Hiele levels of thought. Implications for the effective integration of technology into pre-service teacher education programs are discussed.

• Journal of the Korean Mathematical Society
• /
• v.36 no.6
• /
• pp.1169-1180
• /
• 1999

Two-point comparison theorems between hyperbolic and euclidean geometry for convex regions in the complex plane are known([5], [6]). We give new geometric proofs of sharp two-point comparison theorems for convex regions.

• Korean Journal of Mathematics
• /
• v.14 no.2
• /
• pp.281-289
• /
• 2006

We introduce the resistant length and examine its properties. We also consider the geometric applications of resistant length to the boundary behavior of analytic functions, conformal mappings and derive the theorem in connection with the cluster sets, purely geometric problems. The method of resistant length leads a simple proofs of theorems. So it shows us the usefulness of the method of resistant length.

• Journal of applied mathematics & informatics
• /
• v.27 no.5_6
• /
• pp.1473-1479
• /
• 2009

We introduce the resistant length and examine its properties. We also consider the geometric applications of resistant length to the boundary behavior of analytic functions, conformal mappings and derive the theorem in connection with the fundamental sequences, purely geometric problems. The method of resistant length leads a simple proofs of theorems. So it shows us the usefulness of the method of resistant length.

• Journal of applied mathematics & informatics
• /
• v.18 no.1_2
• /
• pp.603-611
• /
• 2005

We present some geometric applications of extremal length. The method of extremal length leads a simple proofs of theorems. And we consider the applications of extremal length to the boundary behavior of analytic functions and derive theorems in connection with the conformal mappings. It shows us the usefulness of the method of extremal length.

• Journal of Educational Research in Mathematics
• /
• v.11 no.1
• /
• pp.193-206
• /
• 2001

This study was designed to develop investigation- and exploration- activities on Euclidean geometry with an exploratory type software, Geometer's Sketchpad, and to gain insights into the geometric reasoning abilities of college students working with the software. A package of Euclidean geometric activities with GSP was developed and four college students worked on the several activities with GSP and their geometric reasoning process were analyzed. Results indicated that GSP helped students solve problems in the several ways: to make conjectures and discover theorems by providing precise construction and measurement; to discover their proofs by providing the visual images and its manipulation; and to make students easily apply "what-if"strategies and expand and deepen their activities. Students' geometric reasoning was highly depended on analytic methods and their abilities with synthetic methods were shown very limited.

• School Mathematics
• /
• v.11 no.1
• /
• pp.55-70
• /
• 2009

The purpose of this study is to implement Lakatos method in the teaching and learning of geometry for middle school students. In his landmark book , Lakatos suggested the following instructional approach: an initial conjecture was produced, attempts were made to prove the conjecture, the proofs were repeatedly refuted by counterexamples, and finally more improved conjectures and refined proofs were suggested. In the study, students were selected from the high achieving students who participated in the special mathematics and science program offered by the city council of Seoul. The students were given a contradictory geometric proposition, and expected to find the cause of the fallacy. The students successfully identified the fallacy following the Lakatos method. In this process they also set up a primitive conjecture and this conjecture was justified by the proof and refutation method. Some implications were drawn from the result of the study.