• The Mathematical Education
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• v.60 no.1
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• pp.61-76
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• 2021

The purpose of this study is to analyze the errors and difficulties which pre-service secondary teachers shows during the task modification in consideration of the cognitive demand and to provide significant implications to the pre-service teacher education program related to the modification of the mathematical tasks. In the pursuit of this purpose, tasks were selected from perpendicular bisector units and 24 pre-service teachers were asked to modify the tasks to higher and lower level tasks. After the modification activities, opportunities for reflection and modification were provided. The findings from analysis are as follows. Pre-service teachers had a difficulty to distinguish between PNC tasks and PWC tasks. Also, We identified the interference phenomena that pre-service teachers depended on the apparent elements of the task. Pre-service teachers showed a tendency to overlook the learning objectives and learning hierarchy during the task modification, and to focus on some types of task modification. However, pre-service teachers were able to have meaningful learning opportunities and extend the category of tools to technology including Geogebra through self-reflection and correction activities on task modification. The above results were summed up and we presented the implications to the task modification program in the pre-service secondary teacher education.

• The Mathematical Education
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• v.60 no.1
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• pp.21-39
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• 2021

The aim of this study was to deduce implications of the growth of mathematics teachers' specialty for effective instruction about the formulae of exponentiation with rational exponents by analyzing teachers' understanding of the definition of rational exponent. In order to accomplish the aim, this study ascertained the nature of the definition of rational exponent through examining previous literature and established specific research questions with reference to the results of the examination. A questionnaire regarding the nature of the definition was developed in order to solve the questions and was taken to 50 in-service high school teachers. By analysing the data from the written responses by the teachers, this study delineated four characteristics of the teachers' understanding with regard to the definition of rational exponent. Firstly, the teachers did not explicitly use the condition of the bases with rational exponents while proving f'(x)=rxr-1. Secondly, few teachers explained the reason why the bases with rational exponents must be positive. Thirdly, there were some teachers who misunderstood the formulae of exponentiation with rational exponents. Lastly, the majority of teachers thought that $(-8)^{\frac{1}{3}}$ equals to -2. Additionally, several issues were discussed related to teacher education for enhancing teachers' knowledge about the definition, features of effective instruction on the formulae of exponentiation and improvement points to explanation methods about the definition and formulae on the current high school textbooks.

• The Mathematical Education
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• v.60 no.2
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• pp.133-157
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• 2021

Ill-structured problems have drawn attention in that they can enhance problem-solving skills, which are essential in future societies. The purpose of this study is to analyze and evaluate students' spatial reasoning(Intrinsic-Static, Intrinsic-Dynamic, Extrinsic-Static, and Extrinsic-Dynamic reasoning) and problem solving abilities(understanding problems and exploring strategies, executing plans and reflecting, collaborative problem-solving, mathematical modeling) that appear in ill-structured problem-solving. To solve the research questions, two ill-structured problems based on the geometry domain were created and 11 lessons were given. The results are as follows. First, spatial reasoning ability of sixth-graders was mainly distributed at the mid-upper level. Students solved the extrinsic reasoning activities more easily than the intrinsic reasoning activities. Also, more analytical and higher level of spatial reasoning are shown when students applied functions of other mathematical domains, such as computation and measurement. This shows that geometric learning with high connectivity is valuable. Second, the 'problem-solving ability' was mainly distributed at the median level. A number of errors were found in the strategy exploration and the reflection processes. Also, students exchanged there opinion well, but the decision making was not. There were differences in participation and quality of interaction depending on the face-to-face and web-based environment. Furthermore, mathematical modeling element was generally performed successfully.

• The Mathematical Education
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• v.60 no.2
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• pp.159-172
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• 2021

The purpose of this study is to compare and analyze the effects of physical manipulative and exploratory geometry software on the spatial sense for 5^th grade elementary school students in learning nets. For this purpose, ton experimental group used Geofix, an operational learning tool, and the experimental group used Cabri 3D, an exploratory geometry software to learn the nets of solids. The comparison group was learned by worksheet only without any manipulative or software. Spatial sense tests were conducted before and after to determine the level, and eye tracking were used to analyze the strategies of students in solving nets problems. As a result, it was confirmed that the using Geofix group was the most effective for the spatial sense, and Cabri 3D could also be a good tool for learning the nets of solids. In addition, after learning the nets of solids, the analytical strategy, which was the most effective strategy for students' solving strategies, increased. In the process of solving spatial tasks such as the spatial sense tasks, eye tracking technology become a very useful tool for exploring students' strategies, so it is expected that objective and useful data will be collected through more active use in the future.

• The Mathematical Education
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• v.60 no.2
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• pp.191-207
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• 2021

Recently, the number of multicultural families has significantly increased in Korea, and this trend creates a need to understand how successfully students from multicultural families achieve their mathematics learning. To understand and predict the changes in mathematics learning achievement of these students over time, we conducted in this study a latent growth mixture model analysis. The study findings show that the majority (92%) of the students from multicultural families experience a decrease in their mathematics achievement over time as their grade level goes up. It was found, in particular, that female students are likely to have lower initial achievements and rapid decline over time more than male students and that the decline over time was more severe for female students than their male counterparts. The findings of this study convey several implications on the how to support the students from multicultural families. First, the result of this study was different from the outcomes of previous studies that presented the income of the household and the education level of the students' parents as major factors that determine the academic achievement of the students from multicultural families. Furthermore, the study indicates the need for more research to identify variables related to the mathematical achievements of the students from multicultural families and the need to use these research findings to develop public support plans for the students from multicultural families.

• The Mathematical Education
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• v.60 no.2
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• pp.209-228
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• 2021

The purpose of this study is developed the Item Feature Analysis (IFA) frameworks for curriculum-based assessments, focusing on Math competency and representation in secondary schools and implemented the IFA in National Assessment of Educational Achievement. To conduct the study, previous studies were analyzed, and feasibility studies were conducted twice. As a result of the study, we structured the IFA framework based on the 2015 revised mathematics curriculum in Korea and developed a method to analyze the characteristics of the math items. The results of structuring the framework for math included two categories: math competency in the content aspects, and representation type in the formal aspects. Specifically, 12 features of math competency and 8 features of representation type were identified, and an item feature analysis framework composed of these features was developed. The math competency was developed based on the subject competency of 2015 national curriculum. Math assessments in high schools, which have been changed to the competency-based assessments, had more frequency of the feature of math competency compared to middle schools. In this study, implemented the IFA in National Assessment of Educational Achievement and explored the way of ensuring the validity. These have been proved as critical applications for ensuring the validity of curriculum-based student assessment as well as building a tool for assessment.

• The Mathematical Education
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• v.60 no.2
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• pp.229-247
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• 2021

Students' mathematical thinking is represented via various forms of outcomes, such as written response and verbal expression, and teachers could infer and respond to their mathematical thinking by using them. This study analyzed 39 elementary teachers' competency to notice students' problem-solving strategies containing mathematical errors in fraction addition and subtraction with uncommon denominators problems. Participants were provided three types of students' problem-solving strategies with regard to fraction addition and subtraction problems and asked to identify and interpret students' mathematical understanding and errors represented in their artifacts. Moreover, participants were asked to design additional questions and problems to correct students' mathematical errors. The findings revealed that first, teachers' noticing competency was the highest on identifying, followed by interpreting and responding. Second, responding could be categorized according to the teachers' intentions and the types of problem, and it tended to focus on certain types of responding. For example, in giving questions responding type, checking the hypothesized error took the largest proportion, followed by checking the student's prior knowledge. Moreover, in posing problems responding type, posing problems related to student's prior knowledge with simple computation took the largest proportion. Based on these findings, we suggested implications for the teacher noticing research on students' artifacts.

• The Mathematical Education
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• v.60 no.1
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• pp.111-131
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• 2021

This study aims to investigate how secondary mathematics teachers understand and intend to use textbooks for their mathematics instruction. For this purpose, we developed a set of survey items in order to unpack what the teachers understand the mathematical tasks suggested in the textbooks in terms of the levels of cognitive demand and how they intended to use the tasks in the textbooks for their teaching. Twenty-five teachers participated in the survey. The data from the survey were analyzed. The findings from the data analysis suggested as follows: a) the teachers seemed to closely follow textbooks without attempting to modify the tasks, even when the teachers consider it is necessary to modify textbook tasks to high-level tasks, b) the teachers seemed to be unstable in regards that they admitted themselves very competent for modifying tasks for developing students' mathematical thinking but, at the same time, they were uncomfortable with transforming tasks into cognitively demanding tasks that promote students' mathematical understanding, and c) the teachers appeared to consider textbooks as significant criteria in conducting tests including midterm and final exam. In conclusion, the teachers seemed to intend to follow closely the contents and sequence of mathematics textbooks in their mathematics classrooms.

• The Mathematical Education
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• v.60 no.1
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• pp.41-59
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• 2021

The purpose of this study is to synthesize a comprehensive and general conclusion about the effects of mathematics classes using peer tutoring on the cognitive (mathematics learning achievement) and affective domains. For this purpose, a total of 61 individual studies were meta-analyzed in this study to calculate the effect size, measuring the strength of the relationship between mathematics classes using peer tutoring and either the cognitive or affective domain. As a result of this study, it was confirmed that mathematics classes using peer tutoring generally have a medium effect size in both cognitive and affective domains. Also, it was found that level of school, type of student, learning location, class time, tutor education or prior training are significant variables that affect the impact of mathematics classes using peer tutoring on the cognitive and affective domains. These results suggest specific ideas on how to design and operate peer tutoring in school mathematics classes on the basis of different variables.

• Communications of Mathematical Education
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• v.34 no.4
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• pp.485-506
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• 2020

The purpose of this study is to analyze of the effect in Mathematics Teachers beliefs on their students beliefs by Latent Class Regression Model (LCRM). For this analysis, the study used the findings and surveys of Kang, Hong (2020) who developed a belief profile by analyzing the mathematical beliefs of 60 high school teachers and 1,850 second-year high school students learning from them through the Latent Class Analysis (LCA). As a result It was observed that 'Nature of Mathematics', 'Mathematic Teaching' and 'Mathematical Ability' of mathematics teachers beliefs influence the mathematical beliefs of students. The teacher's belief of 'Nature of Mathematics' statistically significant effects on students' beliefs in 'School Mathematics', 'Problem Solving', 'Mathematics Learning'. The teacher's belief of 'Teaching Mathematics', 'Mathematical Ability' statistically significant effects on students' beliefs in 'School Mathematics', 'Problem Solving', 'Self-Concept'. The results of this study can give a preview of the phenomenon in which teacher's mathematical beliefs are reproduced into student's mathematical beliefs. In addition, the results of observation of this study can be used to the contents that can achieve the purpose of reorientation for mathematics teachers.