황재우 Hwang Jae-woo , 부덕훈 Boo Deok Hoon

DOI: JANT Vol.57(No.3) 197-213, 2018

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Descriptive problems can be used to grow student’s ability of thinking logically and creatively, because it shows if the students had a reasonable way of thinking. Rate of descriptive problems is increasing in middle and high school exams. However, students in middle and high schools are generally used to answering multiple-choice or short-answer questions rather than describing the solving process. The purpose of this paper is to gain a theoretic ground to increase the rate of descriptive problems. In this study, students were to solve some multiple-choice problems, and after a few weeks, to solve the problems of same contents in the form of descriptive problems which requires the students to write the solving process. The difference of the scores were measured for each problems to each students, and students were asked what they think the reason for rise or fall of the score is. The result is as follows: First, average scores of 7 of 8 problems used in this study had fallen when it was in descriptive form, and for 5 of them in the rate of 11.2%~16.8%. Second, the main reason of falling is that the students have actual troubles of describing the solving process. Third, in the case of rising, the main reason was that partial scores were given in the descriptive problems. Last, there seems a possibility gender difference in the reason of falling. From these results, followings are suggested to advance the learning, teaching and evaluation in mathematics education: First, it has to be emphasized enough to describe the solving process when solving a problem. Second, increasing the rate of descriptive problems can be supported as a way to advance the evaluation. Third, descriptive problems have to be easier to solve than multiple-choice ones and it is convenient for the students to describe the solving process. Last, multiple-choice problems have to be carefully reviewed that the possibility of students’ choosing incorrect answer with a small mistake is minimal.

오예린 Oh Yaerin , 권오남 Kwon Oh Nam , 박주용 Park Jooyong

DOI: JANT Vol.57(No.3) 215-229, 2018

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Despite the importance of learning to do mathematical proofs, researchers have reported that not only secondary school students but also undergraduate students have difficulties in learning proofs. In this study, we introduced a new toll for learning proofs and explored the reliability and the validity of peer assessment on proofs. In the < Number Theory > course of a university in Seoul, students were given weekly proof assignments prior to class. After solving the proofs, each student had to assess other students’ proofs. The inter-rater reliabilities of weekly peer assessment was higher than .9 over 90 percent of the observed cases. To examine the validity of peer assessment, we check whether students’ assessments were similar to expert assessment. Analysis showed that the equivalence has been quite high throughout the semester and the validity was low in the middle of the semester but rose by the end of the semester. Based on these results, we believe instructors can consider the application of peer assessment on proving tasks as a tool to help students learn.

장현석 Chang Hyun Suk , 이봉주 Lee Bongju

DOI: JANT Vol.57(No.3) 231-246, 2018

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In this paper, we applied Sfard's reification theory to analyze the secondary students’ level of conceptualization with regard to the formula of quadratic equation. Through the generation and development of mathematical concepts from a historical perspective, Sfard classified the formulation process into three stages of interiorization, condensation, and reification, and proposed levels of formulation. Based on this theory, we constructed a test tool reflecting the reversibility of the nature of manipulation of Piaget’s theory as a criterion of content judgement in order to grasp students’ conceptualization level of the formula of quadratic equation. By applying this tool, we analyzed the conceptualization level of the formula of quadratic equation of the 9^th and 10^th graders. The main results are as follows. First, approximately 45% of 9^th graders can not memorize the formula of quadratic equation, or even if they memorize, they do not have the ability of accurate calculation to apply for it. Second, high school curriculum requires for students to use the formula of the quadratic equation, but about 60% of 10^th graders have not reached at the level of reification that they can use the formula of quadratic equation. Third, as a result of imaginarily correcting the error of the previous concept, there was a change in the levels of 9^th graders, and there was no change in 10^th graders.

양현수 Yang Hyeonsu , 김민경 Kim Min Kyeong

DOI: JANT Vol.57(No.3) 247-270, 2018

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The purpose of this study is to investigate the mathematical disposition and mathematical communication level of elementary school students in the process of applying mathematical journal writing activities. For this study, 21 third grade students in elementary school were observed when they participated in mathematical journal writing activities while studying number and operation area. According to the Mathematical disposition pre-test and post-test results, mathematical confidence, mathematical flexibility, mathematical will, and mathematical reflection increased and it was statistically proved. Expression and explanation level of the mathematical communication writing area also increased as the mathematical journal writing activity continued. Thus, mathematical journal writing activities can help to enhance the core competencies of the 2015 revised mathematics curriculum while make students 'to develop and transform mathematical expressions' and ‘to express oneself'. Also, it provides implications of including active writing activities such as mathematical journal writing activities into mathematics classroom. Furthermore, the change in mathematical communication level according to mathematical disposition level was not statistically significant. Therefore, when providing active writing activities including mathematical journal writing activities into classroom, it is necessary to understand students’ individual characteristics and to encourage communication to be active rather than giving feedback based on one’s mathematical disposition level.

이보람 Lee Boram , 박만구 Park Mangoo

DOI: JANT Vol.57(No.3) 271-287, 2018

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This is a case study that defined collaborative utterances and analyzed how they appear in the problem-solving process when 5th-grade students solved problems in groups. As a result, collaborative utterances consist of an interchange type and a deliver type and the interchange type is comprised of two process: the verification process and the modification process. Also, in groups where interchange type collaborative utterances were generated actively and students could reach an agreement easily, students applied the teacher’s help to their problem-solving process right after it was provided and could solve problems even though they had some mathematics errors. In interchange-type collaborative utterances, each student’s participation varies with their individual achievement. In deliver-type collaborative utterances, students who solved problems by themselves participated dominantly. The conclusions of this paper are as follows. First, interchange-type collaborative utterances fostered students’ active participation and accelerated students’ arguments. Second, interchange-type collaborative utterances positively influenced the problem-solving process and it is necessary to provide problems that consider students’ achievement in each group. Third, groups should be comprised of students whose individual achievements are similar because students’ participation in collaborative utterances varies with their achievement.

박선영 Park Sunyoung , 한선영 Han Sunyoung

DOI: JANT Vol.57(No.3) 289-309, 2018

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There has been a gradually increasing focus on adopting mathematical modeling techniques into school curricula and classrooms as a method to promote students' mathematical problem solving abilities. However, this approach is not commonly realized in today's classrooms due to the difficulty in developing appropriate mathematical modeling problems. This research focuses on developing reformulation strategies for those problems with regard to mathematical modeling. As the result of analyzing existing textbooks across three grade levels, the majority of problems related to the real-world focused on the Operating and Interpreting stage of the mathematical modeling process, while no real-world problem dealt with the Identifying variables stage. These results imply that the textbook problems cannot provide students with any chance to decide which variables are relevant and most important to know in the problem situation. Following from these results, reformulation strategies and reformulated problem examples were developed that would include the Identifying variables stage. These reformulated problem examples were then applied to a 7th grade classroom as a case study. From this case study, it is shown that: (1) the reformulated problems that included authentic events and questions would encourage students to better engage in understanding the situation and solving the problem, (2) the reformulated problems that included the Identifying variables stage would better foster the students’ understanding of the situation and their ability to solve the problem, and (3) the reformulated problems that included the mathematical modeling process could be applied to lessons where new mathematical concepts are introduced, and the cooperative learning environment is required. This research can contribute to school classroom’s incorporation of the mathematical modeling process with specific reformulating strategies and examples.

이동근 Lee Dong Gun

DOI: JANT Vol.57(No.3) 311-328, 2018

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In school mathematics, the concept of an average is not a concept that is limited to a unit of statistics. In particular, high school students will learn about arithmetic mean and geometric mean in the process of learning absolute inequality. In calculus learning, the concept of average is involved when learning the concept of average speed. The arithmetic mean is the same as the procedure used when students mean the test scores. However, the procedure for obtaining the geometric mean differs from the procedure for the arithmetic mean. In addition, if the arithmetic mean and the geometric mean are the discrete quantity, then the mean rate of change or the average speed is different in that it considers continuous quantities. The average concept that students learn in school mathematics differs in the quantitative nature of procedures and objects. Nevertheless, it is not uncommon to find out how students construct various mathematical concepts into mathematical knowledge. This study focuses on this point and conducted the interviews of the students(three) in the second grade of high school. And the expression of students in the process of average concept formation in arithmetic mean, geometric mean, average speed. This study can be meaningful because it suggests practical examples to students about the assertion that various scholars should experience various properties possessed by the average. It is also meaningful that students are able to think about how to construct the mean conceptual properties inherent in terms such as geometric mean and mean speed in arithmetic mean concept through interview data.