• Title/Summary/Keyword: teaching of the meaning of proof

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A Study on the Historic-Genetic Principle of Mathematics Education(1) - A Historic-Genetic Approach to Teaching the Meaning of Proof (역사발생적 수학교육 원리에 대한 연구(1) - 증명의 의미 지도의 역사발생적 전개)

  • 우정호;박미애;권석일
    • School Mathematics
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    • v.5 no.4
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    • pp.401-420
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    • 2003
  • We have many problems in the teaching and learning of proof, especially in the demonstrative geometry of middle school mathematics introducing the proof for the first time. Above all, it is the serious problem that many students do not understand the meaning of proof. In this paper we intend to show that teaching the meaning of proof in terms of historic-genetic approach will be a method to improve the way of teaching proof. We investigate the development of proof which goes through three stages such as experimental, intuitional, and scientific stage as well as the development of geometry up to the completion of Euclid's Elements as Bran-ford set out, and analyze the teaching process for the purpose of looking for the way of improving the way of teaching proof through the historic-genetic approach. We conducted lessons about the angle-sum property of triangle in accordance with these three stages to the students of seventh grade. We show that the students will understand the meaning of proof meaningfully and properly through the historic-genetic approach.

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A Study on the Meaning of Proof in Mathematics Education (수학 교육에서 ‘증명의 의의’에 관한 연구)

  • 류성림
    • The Mathematical Education
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    • v.37 no.1
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    • pp.73-85
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    • 1998
  • The purpose of this study is to investigate the understanding of middle school students on the meaning of proof and to suggest a teaching method to improve their understanding based on three levels identified by Kunimune as follows: Level I to think that experimental method is enough for justifying proof, Level II to think that deductive method is necessary for justifying proof, Level III to understand the meaning of deductive system. The conclusions of this study are as follows: First, only 13% of 8th graders and 22% of 9th graders are on level II. Second, although about 50% students understand the meaning of hypothesis, conclusion, and proof, they can't understand the necessity of deductive proof. This conclusion implies that the necessity of deductive proof needs to be taught to the middle school students. One of the teaching methods on the necessity of proof is to compare the nature of experimental method and deductive proof method by providing their weak and strong points respectively.

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Student's difficulties in the teaching and learning of proof (학생들이 증명학습에서 겪는 어려움)

  • Kim, Chang-Il;Lee, Choon-Boon
    • Journal for History of Mathematics
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    • v.21 no.3
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    • pp.143-156
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    • 2008
  • In this study, we divided the teaching and learning of proof into three steps in the demonstrative geometry of the middle school mathematics. And then we surveyed the student's difficulties in the teaching and learning of proof by using of questionnaire. Results of this survey suggest that students cannot only understand the meaning of proof in the teaching and learning of proof but also they cannot deduce simple mathematical reasoning as judgement for the truth of propositions. Moreover, they cannot follow the hypothesis to a conclusion of the proposition It results from the fact that students cannot understand clearly the meaning and the role of hypotheses and conclusions of propositions. So we need to focus more on teaching students about the meaning and role of hypotheses and conclusions of propositions.

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Teaching of the Meaning of Proof Using Historic-genetic Approach - based on Pythagorean Theorem - (역사.발생적 전개를 따른 증명의 의미 지도 - 피타고라스 정리를 중심으로 -)

  • Song, Yeong-Moo;Lee, Bo-Bae
    • School Mathematics
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    • v.10 no.4
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    • pp.625-648
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    • 2008
  • We collected the data through the following process. 36 third-grade middle school students are selected, and we conducted ex-ante interviews for researching how they understand the nature of proof. Based on the results of survey, then we chose two students we took a lesson with the Branford's among the 36 samples. After sampling, historic-genetic geometry education, inspected carefully whether the Branford's method helps the students.

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A Study on the Proof Education in the Middle School Geometry - Focused on the Theory of van Hiele and Freudenthal - (중학교 기하의 증명 지도에 관한 소고 - van Hiele와 Freudenthal의 이론을 중심으로 -)

  • 나귀수
    • Journal of Educational Research in Mathematics
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    • v.8 no.1
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    • pp.291-298
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    • 1998
  • This study deals with the problem of proof education in the middle school geometry bby examining van Hiele#s geometric thought level theory and Freudenthal#s mathematization teaching theory. The implications that have been revealed by examining the theory of van Hie이 and Freudenthal are as follows. First of all, the proof education at present that follows the order of #definition-theorem-proof#should be reconsidered. This order of proof-teaching may have the danger that fix the proof education poorly and formally by imposing the ready-made mathematics as the mere record of proof on students rather than suggesting the proof as the real thought activity. Hence we should encourage students in reinventing #proving#as the means of organization and mathematization. Second, proof-learning can not start by introducing the term of proof only. We should recognize proof-learning as a gradual process which forms with understanding the meaning of proof on the basic of the various activities, such as observation of geometric figures, analysis of the properties of geometric figures and construction of the relationship among those properties. Moreover students should be given this natural ground of proof.

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Study on recognition of the dependent generality in algebraic proofs and its transition to numerical cases (대수 증명에서 종속적 일반성의 인식 및 특정수 전이에 관한 연구)

  • Kang, Jeong Gi;Chang, Hyewon
    • The Mathematical Education
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    • v.53 no.1
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    • pp.93-110
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    • 2014
  • Algebra deals with so general properties about number system that it is called as 'generalized arithmetic'. Observing students' activities in algebra classes, however, we can discover that recognition of the generality in algebraic proofs is not so easy. One of these difficulties seems to be caused by variables which play an important role in algebraic proofs. Many studies show that students have experienced some difficulties in recognizing the meaning and the role of variables in algebraic proofs. For example, the confusion between 2m+2n=2(m+n) and 2n+2n=4n means that students misunderstand independent/dependent variation of variables. This misunderstanding naturally has effects on understanding of the meaning of proofs. Furthermore, students also have a difficulty in making a transition from algebraic proof to numerical cases which have the same structure as the proof. This study investigates whether middle school students can recognize dependent generality and make a transition from proofs to numerical cases. The result shows that the participants of this study have a difficulty in both of them. Based on the result, this study also includes didactical implications for teaching the generality of algebraic proofs.

Analysis on Students' Abilities of Proof in Middle School (중학교 학생의 증명 능력 분석)

  • 서동엽
    • Journal of Educational Research in Mathematics
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    • v.9 no.1
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    • pp.183-203
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    • 1999
  • 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.

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Teaching Diverse Proofs of Means and Inequalities and Its Implications (여러 가지 평균과 부등식을 이용한 대학수학 학습)

  • Kim, Byung-Moo
    • Communications of Mathematical Education
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    • v.19 no.4 s.24
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    • pp.699-713
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    • 2005
  • In this paper, we attempted to find out the meaning of several means and inequalities, their relationships and proposed the effective ways to teach them in college mathematics classes. That is, we introduced 8 proofs of arithmetic-geometric mean equality to explain the fact that there exist diverse ways of proof. The students learned the diverseproof-methods and applied them to other theorems and projects. From this, we found out that the attempt to develop the students' logical thinking ability by encouraging them to find out diverse solutions of a problem could be a very effective education method in college mathematics classes.

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An analysis of trends in argumentation research: A focus on international mathematics education journals (논증 연구의 동향 분석: 국외의 수학교육 학술지를 중심으로)

  • Jinam Hwang;Yujin Lee
    • The Mathematical Education
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    • v.63 no.1
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    • pp.105-122
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    • 2024
  • This study analyzed the research trends of 101 articles published in prominent international mathematics education journals over 24 years from 2000, when NCTM's recommendation emphasizing argumentation was released, until September 2023. We first examined the overall trend of argumentation research and then analyzed representative research topics. We found that students were the focus of the studies. However, several studies focused on teachers. More studies were examined in secondary school than in elementary school, and many were conducted in argumentation in classroom contexts. We also found that argumentation research is becoming increasingly popular in international journals. The representative research topics included 'teaching practice,' 'argumentation structure,' 'proof,' 'student understanding,' and 'student reasoning.' Based on our findings, we could categorize three perspectives on argumentation: formal, contextual, and purposeful. This paper concludes with implications on the meaning and role of argumentation in Korean mathematics education.

How can we teach the 'definition' of definitions? (정의의 '정의'를 어떻게 가르칠 것인가?)

  • Lee, Jihyun
    • Journal of the Korean School Mathematics Society
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    • v.16 no.4
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    • pp.821-840
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    • 2013
  • Definition of geometric figure in middle school geometry seems to mere meaning of the term which could be perceived visually through its shape. However, Much research reported the low achievements of definitions of basic geometric figures. It suggested the limitation of instrumental understanding. In this research, I guided gifted middle school students to reinvent definitions of basic geometric figure by the deductive organization of its properties as Freudenthal pointed. These students understood relationally about why some geometric figure can be defined this way and how it could be defined equally via other properties. This analysis of reinventing of definitions will be a stepping stone to reflect on the pedagogical problems in teaching geometry and to search the new alternatives.

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