• Journal of the Korean School Mathematics Society
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• v.16 no.1
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• pp.31-61
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• 2013

The purpose of this study is to confirm constructivists' assumption that when a little low level learners are taken in learner-centered instruction based on a constructivism they can also construct knowledge by themselves. To achieve this purpose, the researchers compare the effects of learner-centered instruction based on the constructivism and teacher-centered instruction based on the objective epistemology where second graders learn multiplication facts through the each treatment on learners' reasoning ability and achievement. Some conclusions are drawn from results as follows. First, learner-centered instruction based on a constructivism has significant effect on learners' reasoning ability. Second, learner-centered instruction has slightly positive effect on learners' deductive reasoning ability. Third, learner-centered instruction has more an positive influence on understanding concepts and principles of not-presented mathematical knowledge than teacher-centered instruction when implementing it with a little low level learners.

• Journal of Educational Research in Mathematics
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• v.23 no.4
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• pp.585-604
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• 2013

This study is based on the problem that most middle school students cannot recognize the generality of geometrical theorems even after having proved them. By considering this problem from the point of view of empirical verification, the particularity of geometrical representations, and the role of geometrical variables, we suggest that some experiences in dynamic geometry environment (DGE) can help students to recognize the generality of geometrical theorems. That is, this study aims to observe students' cognitive changes related to their recognition of the generality and to provide some educational implications by making students experience some geometrical explorations in DGE. To do so, we selected three middle school students who couldn't recognize the generality of geometrical theorems although they completed their own proofs for the theorems. We provided them exploratory activities in DGE, and observed and analyzed their cognitive changes. Based on this analysis, we discussed the effects of DGE on studensts' recognition of the generality of geometrical theorems.

• Journal of Elementary Mathematics Education in Korea
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• v.15 no.3
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• pp.619-640
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• 2011

The study aims the introducing the items for the assessment of mathematical thinking including mathematical reasoning, problem solving, and communication and the analyzing on the responses of the 5th grade pupils. We categorized the area of mathematical reasoning into deductive reasoning, inductive reasoning, and analogy; problem solving into external problem solving and internal one; and communication into speaking, reading, writing, and listening. And we proposed the examples of our items for each area and the 5th grade pupils' responses. When we assess on pupil's mathematical reasoning, we need to develop very appropriate items needing the very ability of each kind of mathematical reasoning. When pupils solve items requesting communication, the impact of the form of each communication seem to be smaller than that of the mathematical situation or sturucture of the item. We suggested that we need to continue the studies on mathematical assessment and on the constitution and utilization of cognitive areas, and we also need to in-service teacher education on the development of mathematical assessments, based on this study.

• School Mathematics
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• v.9 no.3
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• pp.353-373
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• 2007

The purpose of this study was to provide instructional suggestions by investigating the spatial sense and spatial reasoning ability of 6th grade students. The questionnaire consisted of 20 questions, 10 for spatial visualization and 10 for spatial orientation. The number of subjects for the survey was 145. The processes through which the students solved the problems were the basis for the assessment of their spatial reasoning. The result of the survey is as follows: First, students performed better in spatial visualization than in spatial orientation. With regard to spatial visualization, they were better in transformation than in rotation. With regard to spatial orientation, students performed better in orientation sense and structure cognitive ability than in situational sense. Second, the students that weren't excellent in spatial visualization tended to answer the familiar figures without using mental images. The students who lacked spatial orientation experienced difficulties finding figures observed from the sides. Third, students had high frequency rate on the cognition and use of transformation, the development and application of visualization methods and the use of analysis and synthesis. However they had a lower rate on a systematic approach and deductive reasoning. Further detailed investigation into how students use spatial reasoning, and apply it to actual teaching practice as a device for advancing their geometric thinking is necessary.

• Journal of Educational Research in Mathematics
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• v.19 no.4
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• pp.479-491
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• 2009

This paper researches the possibility of introducing Descartes' theorem to mathematically gifted students. Not only is Descartes' theorem logically equivalent to Euler's theorem but is hierarchically connected with Gauss-Bonnet theorem which is the core concept on differential geometry. It is possible to teach mathematically gifted students Descartes' theorem by generalizing mathematical property in solid geometry through analogical reasoning, that is, so in a polyhedrons the sum of the deficient angles is $720^\circ$ as in an polygon the sum of the exterior angles is $360^\circ$. This study introduces an alternative method of instruction that we enable mathematically gifted students to reinvent Descartes' theorem through analogical reasoning instead of deductive reasoning.

• Journal of the Korean Chemical Society
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• v.68 no.1
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• pp.40-53
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• 2024

The study investigates the number and proportion of questions in each area by examining Chemistry I questions from the College Scholastic Ability Test from 2019 to 2022. The analysis was conducted using a three-dimensional framework that included key concepts in chemistry, behavioral domains in chemistry, and behavioral domains in mathematics. The results indicated that Chemistry I questions on the College Scholastic Ability Test had a relatively even distribution of questions across core individual topics, but highly difficult questions were predominantly biased toward stoichiometry. In terms of the behavioral domains in chemistry, there was a remarkably low proportion of questions related to problem recognition and hypothesis establishment, as well as designing research and implementing research. Conversely, highly difficult questions were more inclined towards drawing conclusions and evaluations. Regarding behavioral domains in mathematics, there was a limited number of questions addressing heuristic reasoning and deductive reasoning. On the other hand, high-difficulty questions favored internal problem-solving ability. Additionally, certain key concepts in chemistry and behavioral domains in chemistry exhibited a strong correlation with specific behavioral domains in mathematics. This characteristic was particularly evident in questions that encompassed higher-dimensional behavioral domains in mathematics, which students tend to find challenging.

• The Mathematical Education
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• v.24 no.2
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• pp.105-116
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• 1986

For any thought and knowledge, its growth and development has close relation with the society where it is developed and grow. As Feuerbach says, the birth of spirit needs an existence of two human beings, i. e. the social background, as well as the birth of body does. But, at the educational viewpoint, the spread and the growth of such a thought or knowledge that influence favorably the development of a society must be also considered. We would discuss the goal and the function of mathematics education in relation with the prosperity of a technological civilization. But, the goal and the function are not unrelated with the spiritual culture which is basis of the technological civilization. Most societies of today can be called open democratic societies or societies which are at least standing such. The concept of rationality in such societies is a methodological principle which completes the democratic society. At the same time, it is asserted as an educational value concept which explains comprehensively the standpoint and the attitude of one who is educated in such a society. Especially, we can considered the cultivation of a mathematical thinking or a logical thinking in the goal of mathematics education as a concept which is included in such an educational value concept. The use of the concept of rationality depends on various viewpoints and criterions. We can analyze the concept of rationality at two aspects, one is the aspect of human behavior and the other is that of human belief or knowledge. Generally speaking, the rationality in human behavior means a problem solving power or a reasoning power as an instrument, i. e. the human economical cast of mind. But, the conceptual condition like this cannot include value concept. On the other hand, the rationality in human knowledge is related with the problem of rationality in human belief. For any statement which represents a certain sort of knowledge, its universal validity cannot be assured. The statements of value judgment which represent the philosophical knowledge cannot but relate to the argument on the rationality in human belief, because their finality do not easily turn out to be true or false. The positive statements in science also relate to the argument on the rationality in human belief, because there are no necessary relations between the proposition which states the all-pervasive rule and the proposition which is induced from the results of observation. Especially, the logical statement in logic or mathematics resolves itself into a question of the rationality in human belief after all, because all the logical proposition have their logical propriety in a certain deductive system which must start from some axioms, and the selection and construction of an axiomatic system cannot but depend on the belief of a man himself. Thus, we can conclude that a question of the rationality in knowledge or belief is a question of the rationality both in the content of belief or knowledge and in the process where one holds his own belief. And the rationality of both the content and the process is namely an deal form of a human ability and attitude in one's rational behavior. Considering the advancement of mathematical knowledge, we can say that mathematics is a good example which reflects such a human rationality, i. e. the human ability and attitude. By this property of mathematics itself, mathematics is deeply rooted as a good. subject which as needed in moulding the ability and attitude of a rational person who contributes to the development of the open democratic society he belongs to. But, it is needed to analyze the practicing and pursuing the rationality especially in mathematics education. Mathematics teacher must aim the rationality of process where the mathematical belief is maintained. In fact, there is no problem in the rationality of content as long the mathematics teacher does not draw mathematical conclusions without bases. But, in the mathematical activities he presents in his class, mathematics teacher must be able to show hem together with what even his own belief on the efficiency and propriety of mathematical activites can be altered and advanced by a new thinking or new experiences.

• Journal of Elementary Mathematics Education in Korea
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• v.15 no.2
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• pp.247-282
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• 2011

In the elementary school mathematics textbooks of the 7th national curriculum, just simple construction education is provided by having students draw a circle and triangle with compasses and drawing vertical and parallel lines with a set square. The purpose of this study was to examine the mathematical thinking of sixth-grade elementary school students in the construction process in a bid to give some suggestions on elementary construction guidance. As a result of teaching the sixth graders in gifted and nongifted classes about the equal division of line segments and evaluating their mathematical thinking, the following conclusion was reached, and there are some suggestions about that education: First, the sixth graders in the gifted classes were excellent enough to do mathematical thinking such as analogical thinking, deductive thinking, developmental thinking, generalizing thinking and symbolizing thinking when they learned to divide line segments equally and were given proper advice from their teacher. Second, the students who solved the problems without any advice or hint from the teacher didn't necessarily do lots of mathematical thinking. Third, tough construction such as the equal division of line segments was elusive for the students in the nongifted class, but it's possible for them to learn how to draw a perpendicular at midpoint, quadrangle or rhombus and extend a line by using compasses, which are more enriched construction that what's required by the current curriculum. Fourth, the students in the gifted and nongifted classes schematized the problems and symbolized the components and problem-solving process of the problems when they received process of the proble. Since they the urally got to use signs to explain their construction process, construction education could provide a good opportunity for sixth-grade students to make use of signs.

• Education of Primary School Mathematics
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• v.13 no.2
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• pp.85-97
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• 2010

The purpose of this study is to develop a teaching program in consideration of the geometrical thinking levels of students to make a contribution to teaching figures effectively. To do this, we checked the geometrical thinking levels of fourth-graders, developed a teaching program based on van Hiele's theory, and investigated its effect on their geometrical thinking levels. The teaching program based on van Hiele's theory put emphasis on group member interaction and specific activities through offering various geometrical experiences. It contributed to actualizing activity-centered, student-oriented, inquiry-oriented and inductive instruction instead of sticking to expository, teacher-led and deductive instruction. And it consequently served to improving their geometrical thinking levels, even though some students didn't show any improvement and one student was rather degraded in that regard - but in the former case they made partial progress though there was little marked improvement, and in the latter case she needs to be considered in relation to her affective aspects above all. The findings of the study suggest that individual variances in thinking level should be recognized by teachers. Students who are at a lower level should be given easier tasks, and more challenging tasks should be assigned to those who are at an intermediate level in order for them to have a positive self-concept about mathematics learning and ultimately to foster their thinking levels.

• Journal of Educational Research in Mathematics
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• v.21 no.4
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• pp.327-344
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• 2011

This study is the first step for us toward improving high school students' capability of statistical inferences, such as obtaining and interpreting the confidence interval on the population mean that is currently learned in high school. We suggest 5 underlying concepts of 'discretion of contingency and inevitability', 'discretion of induction and deduction', 'likelihood principle', 'variability of a statistic' and 'statistical model', those are necessary to appreciate statistical inferences as a reliable arguing tools in spite of its occasional erroneous conclusions. We assume those 5 concepts above are to be gradually developing in their school periods and Korean mathematics textbooks of grades 1-12 were analyzed. Followings were found. For the right choice of solving methodology of the given problem, no elementary textbook but a few high school textbooks describe its difference between the contingent circumstance and the inevitable one. Formal definitions of population and sample are not introduced until high school grades, so that the developments of critical thoughts on the reliability of inductive reasoning could not be observed. On the contrary of it, strong emphasis lies on the calculation stuff of the sample data without any inference on the population prospective based upon the sample. Instead of the representative properties of a random sample, more emphasis lies on how to get a random sample. As a result of it, the fact that 'the random variability of the value of a statistic which is calculated from the sample ought to be inherited from the randomness of the sample' could neither be noticed nor be explained as well. No comparative descriptions on the statistical inferences against the mathematical(deductive) reasoning were found. Few explanations on the likelihood principle and its probabilistic applications in accordance with students' cognitive developmental growth were found. It was hard to find the explanation of a random variability of statistics and on the existence of its sampling distribution. It is worthwhile to explain it because, nevertheless obtaining the sampling distribution of a particular statistic, like a sample mean, is a very difficult job, mere noticing its existence may cause a drastic change of understanding in a statistical inference.