• Title/Summary/Keyword: irrational numbers

### A textbook analysis of irrational numbers unit: focus on the view of process and object (무리수 단원에 대한 교과서 분석 연구: 과정과 대상의 관점으로)

• Oh, Kukhwan;Park, Jung Sook;Kwo, Oh Nam
• The Mathematical Education
• /
• v.56 no.2
• /
• pp.131-145
• /
• 2017
• The representation of irrational numbers has a key role in the learning of irrational numbers. However, transparent and finite representation of irrational numbers does not exist in school mathematics context. Therefore, many students have difficulties in understanding irrational numbers as an 'Object'. For this reason, this research explored how mathematics textbooks affected to students' understanding of irrational numbers in the view of process and object. Specifically we analyzed eight textbooks based on current curriculum and used framework based on previous research. In order to supplement the result derived from textbook analysis, we conducted questionnaires on 42 middle school students. The questions in the questionnaires were related to the representation and calculation of irrational numbers. As a result of this study, we found that mathematics textbooks develop contents in order of process-object, and using 'non repeating decimal', 'numbers cannot be represented as a quotient', 'numbers with the radical sign', 'number line' representation for irrational numbers. Students usually used a representation of non-repeating decimal, although, they used a representation of numbers with the radical sign when they operate irrational numbers. Consequently, we found that mathematics textbooks affect students to understand irrational numbers as a non-repeating irrational numbers, but mathematics textbooks have a limitation to conduce understanding of irrational numbers as an object.

### Preservice secondary matheamtics teachers' understanding of irrational numbers (예비 중등 교사들의 무리수에 대한 이해)

• Lee, Sunbi
• Journal of the Korean School Mathematics Society
• /
• v.16 no.3
• /
• pp.499-518
• /
• 2013
• The purpose of this study is to examine the preservice secondary mathematics teachers understanding and dimensions of knowledge about definition of irrational numbers and irrational numbers and operations. I adopted a framework consisting of formal dimensions, intuitive numbers, algorithmic dimentions suggested by Tirosh et al.(1998) by adding instrumental dimension for his study. I surveyed 65 preservice secondary mathematics teachers who are in bachelor program and post-bachelor program for teacher certificate by using a questionnaire suggested by Sirotic and Zazkis(2007). The results of this study suggest that 83.1% of the participants gave correct answers in definitions of irrational numbers. 43% of the preservice secondary teachers gave correct answers in adding with irrational numbers. Also 91% of the preservice teachers gave correct answers in multiplying irrational numbers. The preservice teachers appeared to understand irrational numbers and operations at formal dimension. More than half of the preservice teachers gave incorrect answers in adding irrational numbers and a few participants gave incorrect in multiplying irrational numbers. The preservice teachers seemed to understand irrational numbers and operations at intuitive or instrumental dimension. The results also suggest that the preservice secondary mathematics teachers have incorrect understanding about irrational numbers.

### Inducing Irrational Numbers in Junior High School (중학교에서의 무리수 지도에 관하여)

• Kim, Boo-Yoon;Chung, Young-Woo
• Journal for History of Mathematics
• /
• v.21 no.1
• /
• pp.139-156
• /
• 2008
• We investigate the inducing method of irrational numbers in junior high school, under algebraic as well as geometric point of view. Also we study the treatment of irrational numbers in the 7th national curriculum. In fact, we discover that i) incommensurability as essential factor of concept of irrational numbers is not treated, and ii) the concept of irrational numbers is not smoothly interconnected to that of rational numbers. In order to understand relationally the incommensurability, we suggest the method for inducing irrational numbers using construction in junior high school.

### A Study on Changes of the Textbooks due to the shift of Pythagorean Theorem (피타고라스 정리의 이동으로 인한 제곱근과 실수 단원의 변화에 관한 연구)

• Ku, Nayoung;Song, Eunyoung;Choi, Eunjeong;Lee, Kyeong-Hwa
• Journal of the Korean School Mathematics Society
• /
• v.23 no.3
• /
• pp.277-297
• /
• 2020
• The purpose of this study is to understand how the shift of the Pythagorean theorem influenced the representation of irrational numbers in the 3rd grade textbook of 2015 revised mathematics curriculum by textbook analysis. Specifically, the changes in the representation of irrational numbers were examined in two aspects based on the nature of irrational numbers and the teaching and learning methods of the 2015 revised mathematics curriculum. First, we analyzed the learning opportunities related to the existence of irrational numbers that were potentially provided by treating irrational numbers as geometric representations in textbooks, and confirmed that Pythagorean theorem was used. Next, we analyzed opportunities to recognize the necessity of irrational numbers provided by numerical representations of irrational numbers. This study has significance in that it confirmed the possibility and limitation of learning opportunities related to the existence and necessity of irrational numbers that were potentially provided by changes in irrational number representations in the 2015 revised textbooks.

### A Case Study on the Introducing Method of Irrational Numbers Based on the Freudenthal's Mathematising Instruction Theory (Freudenthal의 수학화 학습지도론에 따른 무리수 개념 지도 방법의 적용 사례)

• Lee, Young-Ran;Lee, Kyung-Hwa
• Journal of Educational Research in Mathematics
• /
• v.16 no.4
• /
• pp.297-312
• /
• 2006
• As research on the instruction method of the concept of irrational numbers, this thesis is theoretically based on the Freudenthal's Mathematising Instruction Theory and a conducted case study in order to find an introduction method of irrational numbers. The purpose of this research is to provide practical information about the instruction method ?f irrational numbers. For this, research questions have been chosen as follows: 1. What is the introducing method of irrational numbers based on the Freudenthal's Mathematising Instruction Theory? 2 What are the Characteristics of the teaming process shown in class using introducing instruction of irrational numbers based on the Freudenthal's Mathematising Instruction? For questions 1 and 2, we conducted literature review and case study respectively For the case study, we, as participant observers, videotaped and transcribed the course of classes, collected data such as reports of students' learning activities, information gathered through interviews, and field notes. The result was analyzed from three viewpoints such as the characteristics of problems, the application of mathematical means, and the development levels of irrational numbers concept.

### 무리수 개념의 역사적 발생과 역사발생적 원리에 따른 무리수 지도

• 장혜원
• Journal for History of Mathematics
• /
• v.16 no.4
• /
• pp.79-90
• /
• 2003
• This paper aims to consider the genesis of irrational numbers and to suggest a method for teaching the concept of irrational numbers. It is the notion of “incommensurability” in geometrical sense that makes Pythagoreans discover irrational numbers. According to the historica-genetic principle, the teaching method suggested in this paper is based on the very concept, incommensurability which the school mathematics lacks. The basic ideas are induced from Clairaut's and Arcavi's.

### Pre-Service Teachers' Understanding of the Concept and Representations of Irrational Numbers (예비교사의 무리수의 개념과 표현에 대한 이해)

• Choi, Eunah;Kang, Hyangim
• School Mathematics
• /
• v.18 no.3
• /
• pp.647-666
• /
• 2016
• This study investigates pre-service teacher's understanding of the concept and representations of irrational numbers. We classified the representations of irrational numbers into six categories; non-fraction, decimal, symbolic, geometric, point on a number line, approximation representation. The results of this study are as follows. First, pre-service teachers couldn't relate non-fractional definition and incommensurability of irrational numbers. Secondly, we observed the centralization tendency on symbolic representation and the little attention to other representations. Thirdly, pre-service teachers had more difficulty moving between symbolic representation and point on a number line representation of ${\pi}$ than that of $\sqrt{5}$ We suggested the concept of irrational numbers should be learned in relation to various representations of irrational numbers.

### On Explaining Rational Numbers for Extending the Number system to Real Numbers (실수로의 수 체계 확장을 위한 유리수의 재해석에 대하여)

• Shin, Bo-Mi
• Journal of the Korean School Mathematics Society
• /
• v.11 no.2
• /
• pp.285-298
• /
• 2008
• According to the 7th curriculum, irrational numbers should be introduced using infinite decimals in 9th grade. To do so, the relation between rational numbers and decimals should be explained in 8th grade. Preceding studies remarked that middle school students could understand the relation between rational numbers and decimals through the division appropriately. From the point of view with the arithmetic handling activity, I analyzed that the integers and terminating decimals was explained as decimals with repeating 0s or 9s. And, I reviewed the equivalent relations between irrational numbers and non-repeating decimals, rational numbers and repeating decimals. Furthermore, I suggested an alternative method of introducing irrational numbers.

### Teacher Knowledge Necessary to Analyze Student's Errors and Difficulties about the Concept of Irrational Numbers (무리수 개념에 관한 학생의 오류와 어려움 해석에 필요한 교사지식)

• Kang, Hyangim;Choi, Eunah
• School Mathematics
• /
• v.19 no.2
• /
• pp.319-343
• /
• 2017
• In this study, we hope to reveal specialized content knowledge(SCK) and its features necessary to analyze student's errors and difficulties about the concept of irrational numbers. The instruments and interview were administered to 3 in-service mathematics teachers with various education background and teaching experiments. The results of this study are as follows. First, specialized content knowledge(SCK) were characterized by the fixation to symbolic representation like roots when they analyzed the concentration and overlooking of the representations of irrational numbers. Secondly, we observed the centralization tendency on symbolic representation and the little attention to other representations as the standard of judgment about irrational numbers. Thirdly, In-service teachers were influenced by content of students' error when they analyzed the error and difficulties of students. Lately, we confirmed that the content knowledge about the viewpoint of procept and actual infinity of irrational numbers are most important during the analyzing process.

### The Meaning of the Definition of the Real Number by the Decimal Fractions (소수에 의한 실수 정의의 의미)

• Byun Hee-Hyun
• Journal for History of Mathematics
• /
• v.18 no.3
• /
• pp.55-66
• /
• 2005
• In our school mathmatics, the irrational numbers and the real numbers are defined and instructed on the basis of decimal fractions. In relation to this fact, we identified the essences of the real number and the irrational number defined by the decimal fractions through the historical analysis. It is revealed that the formation of real numbers means the numerical measurements of all magnitudes and the formation of irrational numbers means the numerical measurements of incommensurable magnitudes. Finally, we suggest instructional plan for the meaninful understanding of the real number concept.