• School Mathematics
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• v.8 no.3
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• pp.291-300
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• 2006

Proofs in school mathematics are regarded as the procedures to examine a proposition's truth or falsehood. However, they are not based on an axiomatic system in general. This implies the possible existence of vicious circles in proofs in school mathematics. The concept of proof can be more completely acquired when accompanied with concept of circular reasoning and necessity of axiomatic system. Therefore, it is necessary to discuss on the axiomatic system in school mathematics curriculum. The vicious circle can be found in computing an area of a circle by using definite integral in differentiation/integration part of high school textbooks. This paper will first illustrate this in detail and then offer several comments on the axiomatic methods related to the dissolution of that circular reasoning. To further the discussion, Archimedes' derivation for the area of a circle will be considered next. Finally, several arguments on circular reasoning, local organization, and axiomatic system in school curriculum will be presented at the end of the paper.

• Research in Mathematical Education
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• v.18 no.3
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• pp.171-185
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• 2014

Mathematically deductive reasoning skill is one of the major learning objectives stated in senior secondary curriculum (CDC & HKEAA, 2007, page 15). Ironically, student performance during routine assessments on geometric reasoning, such as proving geometric propositions and justifying geometric properties, is far below teacher expectations. One might argue that this is caused by teachers' lack of relevant subject content knowledge. However, recent research findings have revealed that teachers' knowledge of teaching (e.g., Ball et al., 2009) and their deductive reasoning skills also play a crucial role in student learning. Prior to a comprehensive investigation on teacher competency, we use a case study to investigate teachers' knowledge competency on how to teach their students to mathematically argue that, for example, two triangles are congruent. Deductive reasoning skill is essential to geometry. The initial findings indicate that both subject and pedagogical content knowledge are essential for effectively teaching this challenging topic. We conclude our study by suggesting a method that teachers can use to further improve their teaching effectiveness.

• Journal of Elementary Mathematics Education in Korea
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• v.22 no.4
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• pp.497-516
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• 2018

The purpose of this study is to compare the process of mathematical modeling in mathematical modeling class according to group organization, and to investigate whether it shows improvement in mathematical reasoning ability. A total of 24 classes with 3 mathematical modeling activities were designed to investigate the research problem. The result of this study showed that the heterogeneous groups performed better than the homogeneous groups in terms of both the performance ability of mathematical modeling and mathematical reasoning ability. This study implies that, with respect to group design for applying mathematical modeling in teaching mathematics, heterogeneous group design would be more efficient than homogeneous group design.

• Journal of Educational Research in Mathematics
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• v.8 no.1
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• pp.351-364
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• 1998

This thesis analyzes the nature of proof in the perspective on the philosophy of mathematics. such as absolutism, quasi-empiricism and social constructivism. And this thesis searches for the improvement of teaching proof in the light of the result of those analyses of the nature of proof. Though the analyses of the nature of proof in the perspective on the philosophy of mathematics, it is revealed that proof is a dynamic reasoning process unifying the way of analytical thought and the way of synthetical thought, and plays remarkably important roles such as justification, discovery and conviction. Hence we should teach proof as a dynamic reasoning process unifying the way of analytic thought and the way of synthetic thought, avoiding the mistake of dealing with proof as a unilaterally synthetic method. At the same time, we should make students have the needs of proof in a natural way by providing them with the contexts of both justification and discovery simultaneously. Finally, we should introduce the aspect of proof that can be represented as conviction, understanding, explanation and communication to school mathematics.

• Proceedings of the Korean Society of Precision Engineering Conference
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• 1996.04a
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• pp.531-535
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• 1996

Computer aided conceptual solution of engineering problems can be effectively implemented by qualitative reasoning based on a physical model. Qualitative reasoning needs modeling paradigm which provides intellignet control of modeling assumptions and robust inferences without quantitative information about the system. We developed reasoning method using new algebra of qualitative mathematics. The method is applied to a conceptual design scheme of anadaptive control system of cutting process. The method identifies differences between proportional and proportional-integral control scheme of cutting process. It is shown that unfeasible investment could be prevented in the early conceptual stage by the qualitative reasoning procedures proposed in this paper.

• Journal of Educational Research in Mathematics
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• v.20 no.4
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• pp.493-509
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• 2010

Researchers have suggested that educators have to focus their attention on variability and reasoning about variation as means of developing students' statistical thinking in school mathematics. This paper investigated knowledge for the teaching of variability and reasoning about variation; what are sources of variability, how to cope with variability, what are types of variability, how to recognize variability, and the relationship between statistical problem solving and variability. The results involve: discussion on the sources of variability and how to cope with variability promotes students' awareness of different types of variability and students' motivation in the following steps in the statistical activity; emphasis on reasoning about variation in teaching representation of data accords with objectives of statistics education; reexamination of curriculum for statistics education is needed, which has a content-oriented arrangement.

• Education of Primary School Mathematics
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• v.24 no.1
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• pp.43-69
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• 2021

The purpose of this study is to confirm the influences of learner-centered instruction on learners' achievement and reason ability. In order to accomplish them, the fraction unit and the polygonal unit in the fourth grade were implemented with teaching methods and materials suitable for learner-centered mathematics instruction. Some conclusions could be drawn from the results as follows: First, learner-centered mathematics instruction has a more positive effect on learning of learned knowledge and generating unlearned knowledge in the experimental period than teacher-centered instructions. Second, learner-centered instruction makes an influence of low learning ability on getting achievement positively. Third, as the experimental treatment is repeated, learner-centered instruction has a positive effect on students' reasoning ability. The reasoning ability of students showed a difference in the comparison between the experimental group and the comparative group, and within the experimental group, there was a positive effect of the extension of the positive reasoning ability. Fourth, it can be estimated that the development of students' reasoning ability interchangeably affected their generation test results.

• Journal of the Korean School Mathematics Society
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• v.13 no.2
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• pp.263-287
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• 2010

The purposes of the study were to investigate how children define a triangle, their reasoning types in identifying examples and non-examples of a triangle, and the relationship between their reasoning types and geometrical levels. Twenty-nine students consisted of 3th to 6th grades were involved in the study. Using the van Hiele levels of geometrical thought, children's reasoning types for identifying a figure as a triangle or non-triangle were categorized into visual reasoning, reasoning based on the figure's attributes and formal reasoning. The figure's attributes were further divided into critical and non-critical attributes. Most children identified a figure as a triangle or non-triangle based on critical attributes of the figure(e.g. closed figure, three, vertices, straight sides etc.) Some children identified a figure based on non-critical attributes of the figure(e.g. the length of the sides, the measurement of the angles, or the orientation of the figure). Particularly, some children who had lower levels of geometry identified a figure using visual reasoning, taking in the whole shape without considering that the shape is made up of separate components.

• School Mathematics
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• v.5 no.4
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• pp.421-440
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• 2003

It is difficult for the learner to understand completely the ratio concept which forms a basis of proportional reasoning. And proportional reasoning is, on the one hand, the capstone of children's elementary school arithmetic and, the other hand, it is the cornerstone of all that is to follow. But school mathematics has centered on the teachings of algorithm without dealing with its essence and meaning. The purpose of this study is to analyze the essence of ratio concept from multidimensional viewpoint. In addition, this study will show the direction for improvement of ratio concept. For this purpose, I tried to analyze the historical development of ratio concept. Most mathematicians today consider ratio as fraction and, in effect, identify ratios with what mathematicians called the denominations of ratios. But Euclid did not. In line with Euclid's theory, ratio should not have been represented in the same way as fraction, and proportion should not have been represented as equation, but in line with the other's theory they might be. The two theories of ratios were running alongside each other, but the differences between them were not always clearly stated. Ratio can be interpreted as a function of an ordered pair of numbers or magnitude values. A ratio is a numerical expression of how much there is of one quantity in relation to another quantity. So ratio can be interpreted as a binary vector which differentiates between the absolute aspect of a vector -its size- and the comparative aspect-its slope. Analysis on ratio concept shows that its basic structure implies 'proportionality' and it is formalized through transmission from the understanding of the invariance of internal ratio to the understanding of constancy of external ratio. In the study, a fittingness(or comparison) and a covariation were examined as the intuitive origins of proportion and proportional reasoning. These form the basis of the protoquantitative knowledge. The development of sequences of proportional reasoning was examined. The first attempts at quantifying the relationships are usually additive reasoning. Additive reasoning appears as a precursor to proportional reasoning. Preproportions are followed by logical proportions which refer to the understanding of the logical relationships between the four terms of a proportion. Even though developmental psychologists often speak of proportional reasoning as though it were a global ability, other psychologists insist that the evolution of proportional reasoning is characterized by a gradual increase in local competence.

• The Mathematical Education
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• v.51 no.2
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• pp.161-171
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• 2012

Generally mathematics is regarded as a subtle subject to grasp their true meaning. And teacher's personal conceptions of mathematics influence greatly on the teaching and learning of mathematics. More over often teachers confess their difficulties in explaining the true nature of mathematics. In this paper, applying the theory of epistemology, we tried to search factors that must be counted important when trying to understand the true nature of mathematics. As results, we identified five characteristics of mathematical knowledge such as logical reasoning, abstractive concept, mathematical representation, systematical structure, and axiomatic validation. Next, we tried to investigate math education major students' conception of mathematics using these items. To proceed this research we asked 51 students from three Universities to answer their opinion on 'What do you think is mathematics?'. Analysing their answers in the light of the above five items, we got the following facts. 1. Only 38% of the students regarded mathematics as one of the five items, which can be considered to reveal students' low concern about the basic nature of mathematics. 2. The status of students' responses to the question were greatly different among the three Universities. This shows that mathematics professors need to lead students to have concern about the true nature of mathematics.