• Journal of Educational Research in Mathematics
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• v.14 no.2
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• pp.171-185
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• 2004

Students are exposed to a concept of tangent from a specific context of the relation between a circle and straight lines at the 7th grade. This initial experience might cause epistemological obstacles regarding learning concepts of tangent to additional curves. The paper provides a method of how to introduce a series of concepts of tangent in order to lead students to revise and improve the concept of tangent which they have. As students have chance to reflect and revise a series of concepts of tangent step by step, they realize the facts that the properties such as 'meeting the curve at one point' and 'touching but not cutting the curve' may be regarded as the proper definition of tangent in some limited contexts but are not essential in more general contexts. And finally students can grasp and appreciate that concept of tangent as the limit of secants and the relation between tangent and derivative.

• School Mathematics
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• v.17 no.4
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• pp.613-632
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• 2015

In this study, we hope to reveal teacher knowledge necessary to address student errors and difficulties about ratio and rate. The instruments and interview were administered to 3 in-service primary teachers with various education background and teaching experiments. The results of this study are as follows. Specialized content knowledge(SCK) consists of profound knowledge about ratio and rate beyond multiplicative comparison of two quantities and professional knowledge about the definitions of textbook. Knowledge of content and students(KCS) is the ability to recognize students' understanding the concept and the representation about ratio and rate. Knowledge of content and teaching(KCT) is made up of knowledge about various context and visual models for understanding ratio and rate.

• Journal of Educational Research in Mathematics
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• v.27 no.4
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• pp.727-745
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• 2017

This study investigated the Aristotle's continuity and the historical development of continuity of function to explore the differences between the concepts of mathematics and students' thinking about continuity of functions. Aristotle, who sought the essence of continuity, characterized continuity as an 'indivisible unit as a whole.' Before the nineteenth century, mathematicians considered the continuity of functions based on space, and after the arithmetization of nineteenth century modern ${\epsilon}-{\delta}$ definition appeared. Some scholars thought the process was revolutionary. Students tended to think of the continuity of functions similar to that of Aristotle and mathematicians before the arithmetization, and it is inappropriate to regard students' conceptions simply as errors. This study on the continuity of functions examined that some conceptions which have been perceived as misconceptions of students could be viewed as paradigmatic thoughts rather than as errors.

• School Mathematics
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• v.14 no.3
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• pp.331-355
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• 2012

All the method used in teaching the ellipse was to have students draw the points which have the same sum of distances from the two points so that they can confirm the shapes of the ellipse before showing them the definition of ellipse. In this process, students would not get an opportunity to think or make the definition of ellipse for themselves. This deductive way can hinder students from having clear understanding of why such definition was made. This paper introduces a method of defining the ellipse based on the similarity between a circle and an ellipse, leading into the equation. This method is possible by introducing Analytic Geometry taught in current school mathematics and Transformation Geometry. By doing so, this paper will discuss a fundamental understanding about the ellipse and the feature of the ellipse expandable by intuition. Furthermore this paper will also show various advantages which can be given by defining the ellipse in Transformation Geometry.

• School Mathematics
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• v.12 no.4
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• pp.455-471
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• 2010

Understanding of randomness is essential for learning and teaching of probability and statistics. Understanding of randomness prompts to understand natural and social phenomena from the point of view of mathematics, and plays a role of base in understanding of judgments based on rational interpretation on these phenomena. This study examined whether pre-service teachers recognize this, and they understand randomness included in various contexts. According to results, they did not have a understanding of randomness in the context related to measuring, while they grasped randomness in simple and joint events. This implies that they lack the understanding of variability which is essential in the context of measuring. This study, therefore, suggests that the settings of measuring should be introduced into probability and statistics education, especially that data from measuring should be analyzed focusing on the variability in the data set.

• Journal of Elementary Mathematics Education in Korea
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• v.17 no.3
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• pp.457-481
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• 2013

Since the importance of teacher knowledge in teaching mathematics has been emphasized, there have been many studies exploring the nature or characteristics of such knowledge. However, there has been lack of research on the tools of investigating teacher knowledge. Given this background, this study explored teachers' knowledge of fraction lessons using classroom video analysis. The analyses of this study showed that knowledge of teaching methods was activated better than that of student thinking or mathematical content. Knowledge of fraction operation was activated better than that of fraction concept. The degree by which teacher knowledge was activated depended on the characteristics of the video clips used in the study. This paper raised some issues about teachers' knowledge of fraction lessons and suggested classroom video analysis as an alternative tool to measure teacher knowledge in the Korean context.

• Journal of Elementary Mathematics Education in Korea
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• v.1 no.1
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• pp.53-65
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• 1997

There are many mistakes when we estimate probability of an event, for example, we often omit some likelihoods (of an event), sometimes give too large or too small possibility for a particular case, cannot relate current cases with which were concerned before, apply at another cases as soon as discuss about it insufficiently, etc. If we go into a history of probability and statistics, we shall ascertain that many scientists and mathmaticians made essentially same mistakes with us. In the paper, we will consider the theorization of probability and statistics as a process of modification of mistakes which were made during one's estimating possibility of an event. On that point of view, we shall look at historical background of probability and statistics.

• Membrane Journal
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• v.25 no.4
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• pp.367-372
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• 2015

In pressure-retarded osmosis (PRO) and forward osmosis (FO) processes, solvent (permeate) flux depends on which surface the draw solution faces. There are two operation modes. PRO mode indicates that the active layer faces the draw solution, and FO mode means that the porous substrate fronts the draw stream. It is often observed that the PRO mode produces higher flux than that of FO under the same operating conditions. The current work uses the method of proof by contradiction, and mathematically proves the intrinsic flux inequality between the two modes.

• Research in Mathematical Education
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• v.4 no.1
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• pp.33-43
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• 2000

The purpose of this paper was to introduce sociocultural theory which is a different epistemological perspective from constructivism and to understand the sociocultural theory in a systemic way by providing four specific criteria for a sociocultural theory from the analysis of Vygotsky's ideas. The four criteria are the followings: first, the origin of learning is not at the individual level, but at the social. Second, Learning takes place in a sociocultural framework through ZPD and there exists the stage of pseudo concept before it gets to a true concept. Third, a clear focus on action, especially mediated action, and the concept of psychological tools should be discussed in the boundary of a sociocultural theory. Fourth, actors in a learning process are not an individual child alone. In consequence, the role of adults, particularly teachers, are significant in a child's learning, and this fact provides a great potential for the active role of teachers in the students' learning of mathematics from the sociocultural perspective.

• Journal of Educational Research in Mathematics
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• v.17 no.4
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• pp.453-475
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• 2007

School algebra starts with introducing algebraic expressions which have been one of the cognitive obstacles to the students in the transfer from arithmetic to algebra. In the recent studies on the teaching school algebra, algebraic thinking is getting much more attention together with algebraic expressions. In this paper, we examined the processes of the transfer from arithmetic to algebra and ways for teaching early algebra through algebraic thinking factors. Issues about algebraic thinking have continued since 1980's. But the theoretic foundations for algebraic thinking have not been founded in the previous studies. In this paper, we analyzed the algebraic thinking in school algebra from historico-genetic, epistemological, and symbolic-linguistic points of view, and identified algebraic thinking factors, i.e. the principle of permanence of formal laws, the concept of variable, quantitative reasoning, algebraic interpretation - constructing algebraic expressions, trans formational reasoning - changing algebraic expressions, operational senses - operating algebraic expressions, substitution, etc. We also identified these algebraic thinking factors through analyzing mathematics textbooks of elementary and middle school, and showed the middle school students' low achievement relating to these factors through the algebraic thinking ability test. Based upon these analyses, we argued that the readiness for algebra learning should be made through the processes including algebraic thinking factors in the elementary school and that the transfer from arithmetic to algebra should be accomplished naturally through the pre-algebra course. And we searched for alternative ways to improve algebra curriculums, emphasizing algebraic thinking factors. In summary, we identified the problems of school algebra relating to the transfer from arithmetic to algebra with the problem of teaching algebraic thinking and analyzed the algebraic thinking factors of school algebra, and searched for alternative ways for improving the transfer from arithmetic to algebra and the teaching of early algebra.