• School Mathematics
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• v.13 no.1
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• pp.19-36
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• 2011

This study provides middle school students with an opportunity to construct elementary functions with dynamic geometry based on the proportion between lengths of triangle to activate students' intuition in handling elementary algebraic functions and their geometric properties. In addition, this study emphasizes the process of justification about the choice of students' construction method to improve students' deductive reasoning ability. As a result of the pilot lesson study, this paper shows the characteristics of the students' construction process of elementary functions and the roles the teacher plays in the process.

• Education of Primary School Mathematics
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• v.4 no.1
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• pp.63-73
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• 2000

The purpose of this paper is to address He possibility of the teaching of 'proof' in elementary mathematics, on the assumption that proof in school mathematics should be used in the broader, psychological sense of justification rather than in the narrow sense of deductive, formal proof. 'Proof' has not been taught in elementary mathematics, traditionally. Most students have had little exposure to the ideas of proof before the geometry. However, 'Proof' cannot simply be taught in a single unit. Rather, proof must be a consistent part of students' mathematical experience in all grades. Or educators and mathematicians need to rethink the nature of mathematical proof and give appropriate consideration to the different types of proof related to the cognitive development of a notion of proof.

• School Mathematics
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• v.9 no.4
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• pp.487-506
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• 2007

The aims of this study is figure out the characteristics of justification patterns and mathematical representation which are derived from 14 mathematically gifted middle school students in the process of solving the spatial tasks on Archimedean solid. This study shows that mathematically gifted students apply different types of justification such as empirical, or deductive justification and partial or whole justification. It would be necessary to pay attention to the value of informal justification, by comparing the response of student who understood the entire transformation process and provided a reasonable explanation considering all component factors although presenting informal justification and that of student who showed formalization process based on partial analysis. Visual representation plays an valuable role in finding out the Idea of solving the problem and grasping the entire structure of the problem. We found that gifted students tried to create elaborated symbols by consolidating mathematical concepts into symbolic re-presentations and modifying them while gradually developing symbolic representations. This study on justification patterns and mathematical representation of mathematically gifted students dealing with spatial geometry tasks provided an opportunity for understanding their the characteristics of spacial geometrical thinking and expending their thinking.

• 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 Korea Game Society
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• v.13 no.6
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• pp.95-110
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• 2013

Current up-to-date courses of study put emphasis on raising creative students. However, the cramming methods of teaching mathematics in the school seems far from the creativity and the number of students who feels mathematics difficult is increasing. To overcome this situation, the government proposed 'the mathematics education using storytelling', which leads to lots of developments of mathematics using serious game in many areas. However most of the current serious games couldn't do away with the deductive framework of mathematics, which makes it impossible to achieve the purpose of raising creative students. This is because existing mathematics serious games have not deeply contemplated many aspects such as the purpose and theories of teaching and teaching mathematics. Therefore, in order to overcome the limitations of cramming methods in existing mathematics educations, this research proposes the new method of developing serious game contents for elementary geometry that is useful to improve mathematical intuition, based on RME, the theory of teaching/learning mathematics.

• Journal of the Korean School Mathematics Society
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• v.11 no.2
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• pp.299-314
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• 2008

The purposes of this study were to investigate what attitudes students have toward learning proofs and what difficulties they have in learning proofs, and to examine how the use of dynamic geometry software, the Geometer's Sketchpad, helps students' proof learning. The study involved 117 9th graders in 2 high schools. According to questionnaire data, over 50 percent of the total respondents(116) indicated negative attitudes toward learning proofs, on the other hand, only 16 percent of the total respondents indicated positive attitudes toward the learning. Memorizing and remembering many kinds of theorems, definitions, and postulates to use in proving statements was the most difficult part in learning proofs, which the largest proportion of the total respondents indicated. The study found that the use of the Geometer's Sketchpad played positive roles in developing students' understanding of proofs and stimulating students' interests in learning proofs.

• 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 for History of Mathematics
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• v.23 no.1
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• pp.41-52
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• 2010

Ancient Indian mathematical works, all composed in Sanskrit, usually consisted of a section of sturas in which a set of rules or problems were stated with great economy in verse in order to aid memorization by a student. And rules or problems of the mathematics were transmitted both orally and in manuscript form.Indian mathematicians made early contributions to the study of the decimal number system, arithmetic, equations, algebra, geometry and trigonometry. And many Indian mathematicians were appearing one after another in Ancient. This paper is a comparative study of mathematics developments in ancient India and the other ancient civilizations. We have found that the Indian mathematics is quantitative, computational and algorithmic by the principles, but the ancient Greece is axiomatic and deductive mathematics in character. Ancient India and the other ancient civilizations mathematics should be unified to give impetus to further development of mathematics education in future times.

• Journal of Educational Research in Mathematics
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• v.20 no.4
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• pp.511-528
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• 2010

Students' intuition in formal proof should be expressed as symbols according to the deductive process. The symbol will play a role of the mediation between the intuition and the formal proof. This study examined the evolution process of intuition mediated by the symbol in geometry proof. According to the results first, symbol took the great roles when students had the non-formed intuition for the proposition. The signification of symbols could explain even the proof process of the proposition with the non-expectable intuition. And when students proved it by symbols, not by figure nor words, they could evolute the conclusive intuition about the proposition.

• Education of Primary School Mathematics
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• v.14 no.1
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• pp.13-26
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• 2011

The purpose of this research is to analyze geometrical level and the justification process in the proofs of construction by mathematically gifted elementary students. Justification is one of crucial aspect in geometry learning. However, justification is considered as a difficult domain in geometry due to overemphasizing deductive justification. Therefore, researchers used construction with which the students could reveal their justification processes. We also investigated geometrical thought of the mathematically gifted students based on van Hieles's Theory. We analyzed intellectual of the justification process in geometric construction by the mathematically gifted students. 18 mathematically gifted students showed their justification processes when they were explaining their mathematical reasoning in construction. Also, students used the GSP program in some lessons and at home and tested students' geometric levels using the van Hieles's theory. However, we used pencil and paper worksheets for the analyses. The findings show that the levels of van Hieles's geometric thinking of the most gifted students were on from 2 to 3. In the process of justification, they used cut and paste strategies and also used concrete numbers and recalled the previous learning experience. Most of them did not show original ideas of justification during their proofs. We need to use a more sophisticative tasks and approaches so that we can lead gifted students to produce a more creative thinking.