• 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.

• Journal of Educational Research in Mathematics
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• v.17 no.2
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• pp.179-199
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• 2007

The purpose of this study was to discover differences between mathematically gifted students (MGS) and non-gifted students (NGS) when making probability judgments. For this purpose, the following research questions were selected: 1. How do MGS differ from NGS when making probability judgments(answer correctness, answer confidence)? 2. When tackling probability problems, what effect do differences in probability judgment factors have? To solve these research questions, this study employed a survey and interview type investigation. A probability test program was developed to investigate the first research question, and the second research question was addressed by interviews regarding the Program. Analysis of collected data revealed the following results. First, both MGS and NGS justified their answers using six probability judgment factors: mathematical knowledge, use of logical reasoning, experience, phenomenon of chance, intuition, and problem understanding ability. Second, MGS produced more correct answers than NGS, and MGS also had higher confidence that answers were right. Third, in case of MGS, mathematical knowledge and logical reasoning usage were the main factors of probability judgment, but the main factors for NGS were use of logical reasoning, phenomenon of chance and intuition. From findings the following conclusions were obtained. First, MGS employ different factors from NGS when making probability judgments. This suggests that MGS may be more intellectual than NGS, because MGS could easily adopt probability subject matter, something not learnt until later in school, into their mathematical schemata. Second, probability learning could be taught earlier than the current elementary curriculum requires. Lastly, NGS need reassurance from educators that they can understand and accumulate mathematical reasoning.

• Journal of Elementary Mathematics Education in Korea
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• v.18 no.1
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• pp.1-17
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• 2014

Instructions for the commutative property of multiplication at elementary schools tend to be based on checking the equality between the quantities of 'a times b 'and b' times a, ' for example, $3{\times}4=12$ and $4{\times}3=12$. This article critically examined the approaches to teach the commutative property of multiplication from Kant's perspective of mathematical knowledge. According to Kant, mathematical knowledge is a priori. Yet, the numeric exploration by checking the equality between the amounts of 'a groups of b' and 'b groups of a' does not reflect the nature of apriority of mathematical knowledge. I suggest we teach the commutative property of multiplication in a way that it helps reveal the operational schema that is necessarily and generally involved in the transformation from the structure of 'a times b' to the structure of 'b times a.' Distributive reasoning is the mental operation that enables children to perform the structural transformation for the commutative property of multiplication by distributing a unit of one quantity across the other quantity. For example, 3 times 4 is transformed into 4 times 3 by distributing each unit of the quantity 3, which results in $3{\times}4=(1+1+1){\times}4=(1{\times}4)+(1{\times}4)+(1{\times}4)+(1{\times}4)=4+4+4=4{\times}3$. It is argued that the distributive reasoning is also critical in learning the subsequent mathematics concepts, such as (a whole number)${\times}10$ or 100 and fraction concept and fraction multiplication.

• School Mathematics
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• v.18 no.3
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• pp.625-645
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• 2016

This study clarified the big ideas related to teaching addition and subtraction of fractions with different denominators based on quantitative reasoning with unit and recursive partitioning. An analysis of this study urged us to re-consider the content related to the addition and subtraction of fraction. As such, this study analyzed textbooks and teachers' manuals developed from the fourth national mathematics curriculum to the most recent 2009 curriculum. In addition and subtraction of fractions with different denominators, it must be emphasized the followings: three-levels unit structure, fixed whole unit, necessity of common measure and recursive partitioning. An analysis of this study showed that textbooks and teachers' manuals dealt with the fact of maintaining a fixed whole unit only as being implicit. The textbooks described the reason why we need to create a common denominator in connection with the addition of similar fractions. The textbooks displayed a common denominator numerically rather than using a recursive partitioning method. Given this, it is difficult for students to connect the models and algorithms. Building on these results, this study is expected to suggest specific implications which may be taken into account in developing new instructional materials in process.

• Journal for History of Mathematics
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• v.24 no.3
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• pp.109-143
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• 2011

Augustus De Morgan became to be deeply interested in the education of pure mathematics since he came to teach in UCL because of the specific nature of natural philosophy lectures, the academical knowledge and reasoning powers of the students, and the negative attitudes of London society on mathematics. During his long tenure, he really tried his best to make his students understand the important concepts and the principles of pure mathematics, and logically explain the processes of inducing and proving the laws of pure mathematics. When he could not stay as a mere researcher, he had to concern himself with and pay attention to the problems of educating students. And then his teaching style was constructed in a specific way by the various attitudes about mathematics, the boundary relationship between the adjacent academical branches, and the social and systematic nature of UCL.

• Journal of Educational Research in Mathematics
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• v.23 no.3
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• pp.335-351
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• 2013

This research assumes that abduction is so important as much as all the creative plausible reasoning to be based upon. We expect it to be deeply appreciated and be taught positively in school mathematics. We are noticing that every problem solving process must contain some steps of abduction and thus, we believe that those who are afraid of abduction cannot solve any newly faced problem. Upon these thoughts, we are looking into the middle school mathematics textbooks to see that how strongly various abductions are emphasized to solve problems in it. We modified types of abduction those were suggested by Eco(1983) or by Bettina Pedemonte, David Reid (2011) and investigated those books to see if, we may regard, various types of abduction be intended to be used to solve their problems. As a result of it, we found that more than 92% of the problems were not supposed to use creative abduction necessarily to solve it. And we interpret this as most authors of the textbooks have emphasis more on the capturing and understanding of basic knowledge of school mathematics rather than the creative reasoning through them. And we believe this need innovation, otherwise strong debates are necessary among the professionals of it.

• School Mathematics
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• v.13 no.1
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• pp.1-17
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• 2011

This study investigated the Chinese national university entrance examination (Gaokao) in mathematics administered in 2009 and 2010 to draw out some implications on the College Scholastic Ability Test (CSAT) in mathematics of Korea. To evaluate the attainments of basic mathematical skills and multilateral abilities required for further studies in university, the Gaokao mathematics is set in two forms(Art/Science), based on the Chinese national mathematics curriculum. The types of items in the Gaokao mathematics are multiple-choice, single-answer, and write-out-answer. The mathematical abilities that the Gaokao mathematics evaluates are mathematical reasoning, operation, geometrical imagination, application, and creativity. As a result, some implications on the Korean CSAT are drawn out in terms of the level of difficulty, the types of items, the arrangements, and the scores of items.

• Journal of Educational Research in Mathematics
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• v.27 no.4
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• pp.747-764
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• 2017

Problems in mathematics textbooks are very important because there is a high reliance on textbooks in elementary school mathematics classes and there is a strong belief that mathematics is to find the solution to problems. Considering this importance, we analyzed problems in elementary mathematics textbooks quantitatively and qualitatively. Concretely, problems of addition and subtraction with carry on in the range of two digit numbers in the mathematics textbooks from the 1st to the 2015 national revised curriculum were analyzed. As a result, the problems in each textbook were found to reveal important features of the textbook reflecting changes in curriculum and educational background. And the problem of textbooks has changed in the direction of enhancing students' reasoning, communication, and problem solving ability. Based on these results, we suggested several implications for dealing with problems in elementary mathematics textbooks.

• Communications of Mathematical Education
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• v.23 no.1
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• pp.73-83
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• 2009

Mathematics Olympiad aims to identify and encourage students who have superior ability in mathematics, to enhance students' understanding in mathematics while stimulating interest and challenge, to increase learning motivation through self-reflection, and to speed up the development of mathematical talent. Participating mathematical competition, students are going to solve a variety of types of mathematical problems and will be able to enlarge their understanding in mathematics and foster mathematical thinking and creative problem solving ability with logic and reasoning. In addition, parents could have an opportunity valuable information on their children's mathematical talents and guidance of them. Although there should be presenting diversified mathematical problems in competitions, the real situations is that resent most mathematics Olympiads present mathematical problems which narrowly focus on types of solving problems. In order to diversifying types of problems in mathematics Olympiads and making mathematics popular, this study will discuss a Olympiad for problem solving ability, a Olympiad for exploring mathematics, a Olympiad for task solving ability, and a mathematics fair, etc.

• The Mathematical Education
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• v.61 no.2
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• pp.339-357
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• 2022

This study explores pre-service secondary mathematics teachers (PSTs)' noticing competency. 17 PSTs participated in this study as a part of the mathematics teaching method class. Individual PST's essays regarding the question 'what effective mathematics teaching would be?' that they discussed and wrote at the beginning of the course were collected as the first data. PSTs' written analysis of an expert teacher's teaching video, colleague PSTs' demo-teaching video, and own demo-teaching video were also collected and analyzed. Findings showed that most PSTs' noticing level improved as the class progressed and showed a pattern of focusing on each key aspect in terms of the Teaching for Robust Understanding of Mathematics (TRU Math) framework, but their reasoning strategies were somewhat varied. This suggests that the TRU Math framework can support PSTs to improve the competency of 'what to attend' among the noticing components. In addition, the instructional reasoning strategies imply that PSTs' noticing reasoning strategy was mostly related to their interpretation of noticing components, which should be also emphasized in the teacher education program.