Difference of the scores of multiplechoice and descriptive problem and students’ perceptions of the difference  Focused on high school geometry course 
황재우 Hwang Jaewoo , 부덕훈 Boo Deok Hoon
57(3) 197213, 2018
황재우 Hwang Jaewoo , 부덕훈 Boo Deok Hoon
DOI: JANT Vol.57(No.3) 197213, 2018
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 multiplechoice or shortanswer 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 multiplechoice 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 multiplechoice ones and it is convenient for the students to describe the solving process. Last, multiplechoice problems have to be carefully reviewed that the possibility of students’ choosing incorrect answer with a small mistake is minimal.

The Reliability and Validity of Online Peer Assessment on Proofs in a Number Theory Course
오예린 Oh Yaerin , 권오남 Kwon Oh Nam , 박주용 Park Jooyong
57(3) 215229, 2018
오예린 Oh Yaerin , 권오남 Kwon Oh Nam , 박주용 Park Jooyong
DOI: JANT Vol.57(No.3) 215229, 2018
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 interrater 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.

An analysis on the secondary students’ conceptualization level of the formula of quadratic equation based on Sfard's reification theory
장현석 Chang Hyun Suk , 이봉주 Lee Bongju
57(3) 231246, 2018
장현석 Chang Hyun Suk , 이봉주 Lee Bongju
DOI: JANT Vol.57(No.3) 231246, 2018
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.

A study on the mathematical disposition and communication level in process of applying mathematical journal writing to the 3^{rd} graders in a mathematics classroom
양현수 Yang Hyeonsu , 김민경 Kim Min Kyeong
57(3) 247270, 2018
양현수 Yang Hyeonsu , 김민경 Kim Min Kyeong
DOI: JANT Vol.57(No.3) 247270, 2018
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 pretest and posttest 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.

Analysis of Collaborative Utterances among Elementary Students in ProblemSolving Process
이보람 Lee Boram , 박만구 Park Mangoo
57(3) 271287, 2018
이보람 Lee Boram , 박만구 Park Mangoo
DOI: JANT Vol.57(No.3) 271287, 2018
This is a case study that defined collaborative utterances and analyzed how they appear in the problemsolving process when 5thgrade 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 problemsolving process right after it was provided and could solve problems even though they had some mathematics errors. In interchangetype collaborative utterances, each student’s participation varies with their individual achievement. In delivertype collaborative utterances, students who solved problems by themselves participated dominantly. The conclusions of this paper are as follows. First, interchangetype collaborative utterances fostered students’ active participation and accelerated students’ arguments. Second, interchangetype collaborative utterances positively influenced the problemsolving 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.

Reconstruction and application of reforming textbook problems for mathematical modeling process
박선영 Park Sunyoung , 한선영 Han Sunyoung
57(3) 289309, 2018
박선영 Park Sunyoung , 한선영 Han Sunyoung
DOI: JANT Vol.57(No.3) 289309, 2018
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 realworld focused on the Operating and Interpreting stage of the mathematical modeling process, while no realworld 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.

A study on expression of students in the process of constructing average concept as mathematical knowledge
이동근 Lee Dong Gun
57(3) 311328, 2018
이동근 Lee Dong Gun
DOI: JANT Vol.57(No.3) 311328, 2018
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.
