• The Mathematical Education
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• v.46 no.2 s.117
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• pp.141-153
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• 2007

The purpose of the study is to analyze children's proportional reasoning and problem solving in proportional problems with/without iconic representations. Proportional problems include 3 tasks such as (a) without any picture, (b) with simple picture, and (c) with/without iconic representation. As a result, children didn't show any significant differences in two tasks such as (a) and (b). However, children showed better proportional reasoning with iconic representation. In addition, 'build-up expression' strategy was used mostly in solving problems and 'additive strategy' was shown as an error which students didn't make an appropriate proportional relation expression and they made a wrong additive strategy.

• School Mathematics
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• v.6 no.3
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• pp.283-299
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• 2004

It is a very important objective of mathematical education to lead students to apply mathematics to the problem situations and to solve the problems. Assuming that mathematical modeling is appropriate for such mathematical education objectives, we must emphasize mathematical modeling learning. In this research, we focused what mathematical concepts are learned and what reasoning are applied and used through mathematical modeling. In the process of mathematical modeling, the students used several types of reasoning; deduction, induction and abduction. Although we cannot generalize a fact by a single case study, deduction has been used to confirm whether their model is correct to the real situation and to find solutions by leading mathematical conclusion and induction to experimentally verify whether their model is correct. And abduction has been used to abstract a mathematical model from a real model, to provide interpretation to existing a practical ground for mathematical results, and elicit new mathematical model by modifying a present model.

• Research in Mathematical Education
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• v.16 no.1
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• pp.15-29
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• 2012

One of the important aims in mathematics education is to enhance mathematical thinking for students. And students posing questions is a vital process in mathematical thinking as it is part of the reasoning and communication of their learning. This paper investigates how students develop their mathematical thinking through working on tasks in fractions and posing their own questions after successfully solved the problems. The teaching was conducted in primary five classes and the results showed that students' reasoning is related to their analogy with what previously learned. Also, posing their problems after solving the problem not only helps students to understand the structure of the problem, it also helps students to explore on different routes in solving the problem and extend their learning content.

• Journal of Elementary Mathematics Education in Korea
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• v.15 no.3
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• pp.641-654
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• 2011

In order to grow students' rational and creative problem-solving ability which is one of the primary goals in mathematics education. students' proper understanding of mathematical concepts, principles, and rules must be backed up as its foundational basis. For the relevant teaching strategies. National Mathematics Curriculum advises that students should be allowed to discover and justify the concepts, principles, and rules by themselves not only through the concrete hands-on activities but also through inquiry-based activities based on the learning topics experienced from the diverse phenomena in their surroundings. Hereby, this paper, firstly, looks into both the meaning and the inductive reasoning process of mathematical principles and rules, secondly, suggest "learning through discovery teaching method" for the proper teaching of the mathematical principles and rules recommended by the National Curriculum, and, thirdly, examines the possible discovery-led teaching strategies using inductive methods with the related matters to be attended to.

• Journal of Educational Research in Mathematics
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• v.24 no.1
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• pp.1-27
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• 2014

In the present study, we analyze the four participating 9th grade students' mathematical reasoning processes in their dragging activities while solving open-ended geometry problems in terms of abduction, induction and deduction. The results of the analysis are as follows. First, the students utilized 'abduction' to adopt their hypotheses, 'induction' to generalize them by examining various cases and 'deduction' to provide warrants for the hypotheses. Secondly, in the abduction process, 'wandering dragging' and 'guided dragging' seemed to help the students formulate their hypotheses, and in the induction process, 'dragging test' was mainly used to confirm the hypotheses. Despite of the emerging mathematical reasoning via their dragging activities, several difficulties were identified in their solving processes such as misunderstanding shapes as fixed figures, not easily recognizing the concept of dependency or path, not smoothly proceeding from probabilistic reasoning to deduction, and trapping into circular logic.

• Research in Mathematical Education
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• v.14 no.3
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• pp.195-209
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• 2010

Utilizing a structure of operations known as Dissection-Motion-Operations (DMO), a set of mathematics propositions or area-formulas in school mathematics will be introduced through shape-to-shape transforms. The underlying theme for DMO is problem-solving through visual reasoning and proving manipulatively or electronically vs. rote learning and memorization. Visual reasoning is the focus here where two operations that constitute DMO are utilized. One operation is known as Dissection (or Decomposition) operation that operates on a given region in 2D or 3D and dissects it into a number of subregions. The second operation is known as Motion (or Composition) operation applied on the resultant sub-regions to form a distinct area (or volume)-equivalent region. In 2D for example, DMO can transform a given polygon into a variety of new and distinct polygons each of which is area-equivalent to the original polygon (cf [Rahim, M. H. & Sawada, D. (1986). Revitalizing school geometry through Dissection-Motion Operations. Sch. Sci. Math. 86(3), 235-246] and [Rahim, M. H. & Sawada, D. (1990). The duality of qualitative and quantitative knowing in school geometry, International Journal of Mathematical Education in Science and Technology 21(2), 303-308]).

• Journal of Educational Research in Mathematics
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• v.27 no.1
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• pp.51-67
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• 2017

Analogical reasoning is a mathematically useful way of thinking. By analogy reasoning, students can improve problem solving, inductive reasoning, heuristic methods and creativity. The purpose of this study is to analyze the analogical reasoning of preservice mathematics teachers while constructing quadratic curves defined by eccentricity. To do this, we produced tasks and 28 preservice mathematics teachers solved. The result findings are as follows. First, students could not solve a target problem because of the absence of the mathematical knowledge of the base problem. Second, although student could solve a base problem, students could not solve a target problem because of the absence of the mathematical knowledge of the target problem which corresponded the mathematical knowledge of the base problem. Third, the various solutions of the base problem helped the students solve the target problem. Fourth, students used an algebraic method to construct a quadratic curve. Fifth, the analysis method and potential similarity helped the students solve the target problem.

• Journal of Elementary Mathematics Education in Korea
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• v.21 no.3
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• pp.461-483
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• 2017

The ability to think mathematically and to reason inductively are basics of logical reasoning and the most important skill which students need to acquire through their Math curriculum in elementary school. For these reasons, we need to conduct an analysis in their procedure in inductive reasoning and find difficulties thereof. Therefore, through this study, I found parts which covered inductive reasoning in their Math curriculum and analyzed the abilities and characteristics of students in solving a problem through inductive reasoning.

• East Asian mathematical journal
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• v.31 no.2
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• pp.145-165
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• 2015

The purpose of this study is to suggest how to go about teaching and learning secondary school algebra by analyzing problem-solving ability and problem-solving process through algebraic reasoning. In doing this, 393 students' data were thoroughly analyzed after setting up the exam questions and analytic standards. As with the test conducted with technical school students, the students scored low achievement in the algebraic reasoning test and even worse the majority tried to answer the questions by substituting arbitrary numbers. The students with high problem-solving abilities tended to utilize conceptual strategies as well as procedural strategies, whereas those with low problem-solving abilities were more keen on utilizing procedural strategies. All the subject groups mentioned above frequently utilized equations in solving the questions, and when that utilization failed they were left with the unanswered questions. When solving algebraic reasoning questions, students need to be guided to utilize both strategies based on the questions.

• Journal of the Korean School Mathematics Society
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• v.20 no.4
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• pp.405-422
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• 2017

Nim games were divided into three stages : one file, two files and three files game, and inquiry activities were conducted for middle school mathematically gifted students. In the first stage, students easily found a winning strategy through deductive reasoning. In the second stage, students found a winning strategy with deductive reasoning or inductive reasoning, but found an error in inductive reasoning. In the third stage, no students found a winning strategy with deductive reasoning and errors were found in the induction reasoning process. It is found that the tendency to unconditionally generalize the pattern that is formed in the finite number of cases is the cause of the error. As a result of visually presenting the binary boxes to students, students were able to easily identify the pattern of victory and defeat, recognize the winning strategy through game activities, and some students could reach a stage of justifying the winning strategy.