• School Mathematics
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• v.14 no.3
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• pp.357-375
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• 2012

This research intends to examine how 6th graders (age 12) generalize various increasing patterns. In this research, 6 problems corresponding to the ax, x+a, ax+c, ax2, and ax2+c patterns were given to 290 students. Students' generalization methods were analysed by the generalization level suggested by Radford(2006), such as arithmetic and algebraic (factual, contextual, and symbolic) generalization. As the results of the study, we identified that students revealed the most high performance in the ax pattern in the aspect of the algebraic generalization, and lower performance in the ax2, x+a, ax+c, ax2+c in order. Also we identified that students' generalization methods differed in the same increasing patterns. This imply that we need to provide students with the pattern generalization activities in various contexts.

• Journal of Educational Research in Mathematics
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• v.26 no.1
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• pp.79-101
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• 2016

Understanding the equal sign is of great significance to the development of algebraic thinking. Given this importance, this study investigated in what ways a total of 695 students from second to sixth graders understand the equal sign. The results showed that students were successful in solving standard problems whereas they were poor at items demanding high relational thinking. It was noticeable that some of the students were based on computational thinking rather than relational understanding of the equal sign. The students had a difficulty both in understanding the structure of equations and in solving equations in non-standard problem contexts. They also had incomplete understanding of the equal sign. This paper is expected to explore the understanding of the equal sign by elementary school students in multiple problem contexts and to provide implications of how to promote students' understanding of the equal sign.

• Journal of Elementary Mathematics Education in Korea
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• v.22 no.2
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• pp.115-142
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• 2018

This research analyzed domestic researches on early algebra education which are published in six major mathematics education journals in Korea. The purpose of this work is to grasp trends of early algebra education in Korea and to draw up future tasks. From 2000 to 2017, 89 papers which are related to early algebra education published in 6 journals. The 89 papers were categorized by research period, academic journals, research topics, and research subjects. As a result, the number of researches on early algebra education in Korea has increased since 2000. Although early algebra education belongs to the field of elementary mathematics education, lots of papers were published in other math education journals than in the math education journals for elementary school mathematics. Most research focused on proportional reasoning across the algebraic content area. The majority of the research subjects were students, especially upper-grade students of elementary school. Based on the results of this study, some implications for early algebra education in Korea were suggested.

• The Mathematical Education
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• v.60 no.4
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• pp.409-429
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• 2021

This study investigated instructional methods for fractional division emphasizing algebraic thinking with sixth graders. Specifically, instructional elements for fractional division emphasizing algebraic thinking were derived from literature reviews, and the fractional division instruction was reorganized on the basis of key elements. The instructional elements were as follows: (a) exploring the relationship between a dividend and a divisor; (b) generalizing and representing solution methods; and (c) justifying solution methods. The instruction was analyzed in terms of how the key elements were implemented in the classroom. This paper focused on the fractional division instruction with problem contexts to calculate the quantity of a dividend corresponding to the divisor 1. The students in the study could explore the relationship between the two quantities that make the divisor 1 with different problem contexts: partitive division, determination of a unit rate, and inverse of multiplication. They also could generalize, represent, and justify the solution methods of dividing the dividend by the numerator of the divisor and multiplying it by the denominator. However, some students who did not explore the relationship between the two quantities and used only the algorithm of fraction division had difficulties in generalizing, representing, and justifying the solution methods. This study would provide detailed and substantive understandings in implementing the fractional division instruction emphasizing algebraic thinking and help promote the follow-up studies related to the instruction of fractional operations emphasizing algebraic thinking.

• School Mathematics
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• v.10 no.1
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• pp.23-42
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• 2008

The purpose of this study was to describe characteristics of thinking in elementary gifted students' solutions to algebraic tasks. Especially, this paper was focused on the students' strategies to develop generalization while problem solving, the justifications on the generalization and metacognitive thinking emerged in stildents' problem solving process. To find these issues, a case study was conducted. The subjects of this study were four 6th graders in elementary school-they were all receiving education for the gifted in an academy for the gifted attached to a university. Major findings of this study are as follows: First, during the process of the task solving, the students varied in their use of generalization strategies and utilized more than one generalization strategy, and the students also moved from one strategy toward other strategies, trying to reach generalization. In addition, there are some differences of appling the same type of strategy between students. In a case of reaching a generalization, students were asked to justify their generalization. Students' justification types were different in level. However, there were some potential abilities that lead to higher level although students' justification level was in empirical step. Second, the students utilized their various knowledges to solve the challengeable and difficult tasks. Some knowledges helped students, on the contrary some knowledges made students struggled. Specially, metacognitive knowledges of task were noticeably. Metacognitive skills; 'monitoring', 'evaluating', 'control' were emerged at any time. These metacognitive skills played a key role in their task solving process, led to students justify their generalization, made students keep their task solving process by changing and adjusting their strategies.

• Journal of Educational Research in Mathematics
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• v.19 no.1
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• pp.81-98
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• 2009

This study analyzed the types of quantitative reasoning and the characteristics of representation in order to figure out the characteristics of quantitative reasoning of the sixth graders. Three students who used quantitative reasoning in solving problems were interviewed in depth. Results showed that the three students used two types of quantitative reasoning, that is difference reasoning and multiplicative reasoning. They used qualitatively different quantitative reasoning, which had a great impact on their problem-solving strategy. Students used symbolic, linguistic and visual representations. Particularly, they used visual representations to represent quantities and relations between quantities included in the problem situation, and to deduce a new relation between quantities. This result implies that visual representation plays a prominent role in quantitative reasoning. This paper included several implications on quantitative reasoning and quantitative approach related to early algebra education.

• Education of Primary School Mathematics
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• v.26 no.1
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• pp.45-63
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• 2023

The activity of finding rules is useful for enhancing the algebraic thinking of elementary school students. This study analyzed the pattern activities of a finding rules unit in 10 different government-authorized mathematics curricular materials for fourth graders aligned to the 2015 revised national mathematics curriculum. The analytic elements included three main activities: (a) activities of analyzing the structure of patterns, (b) activities of finding a specific term by finding a rule, and (c) activities of representing the rule. The three activities were mainly presented regarding growing numeric patterns, growing geometric patterns, and computational patterns. The activities of analyzing the structure of patterns were presented when dealing mainly with growing geometric patterns and focused on finding the number of models constituting the pattern. The activities of finding a specific term by finding a rule were evenly presented across the three patterns and the specific term tended to be close to the terms presented in the given task. The activities of representing the rule usually encouraged students to talk about or write down the rule using their own words. Based on the results of these analyses, this study provides specific implications on how to develop subsequent mathematics curricular materials regarding pattern activities to enhance elementary school students' algebraic thinking.

• School Mathematics
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• v.14 no.3
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• pp.275-296
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• 2012

This study investigated elementary school students' understanding of basic functional relationships. It analyzed the written responses from a total of 2087 students of second, fourth, and sixth graders using tests that examined their understanding of five types of functional relationships. The results of this study showed that students tended to be more successful as their grades went up with regard to all the problem types. There were statistically differences among the three grade levels. Even lower graders were quite successful in dealing with additive relation, direct proportion, and inverse proportion. However the items dealing with square relation and linear relation were difficult even to sixth graders. It was common that students were good at completing the table by looking for a pattern from the given numbers but that they had difficulties in anticipating the value of 'y' when the value of 'x' is given either as a big number or as a symbol. Given these results, this paper includes issues and implications on how to foster functional thinking ability at the elementary school.

• Education of Primary School Mathematics
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• v.23 no.1
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• pp.45-61
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• 2020

The concept of equality is given as a way of reading the equal sign without dealing it explicitly in elementary school mathematics. The meaning of the equal sign can be largely categorized as operational and relational views. However, most elementary school students understand the equal sign as an operational symbol for just writing the required answers. It is essential for them to understand a relational concept of the equal sign because algebraic thinking in middle school mathematics is based on students' understanding of a relational view of the equal sign. Recently, the relational meaning of the equal sign is emphasized in arithmetic. Hence it is necessary for elementary school students to have some activities so that they experience a relational meaning of the equal sign. In this study, we investigate the meaning of the equal sign and contexts of the equal sign in elementary school mathematics to discuss explicit ways to emphasize the concept of equality and relational views of the equal sign.

• School Mathematics
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• v.5 no.3
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• pp.343-360
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• 2003

In this paper, we deal with the teaching of algebra based on patterns and generalization. The past algebra curriculum starts with letters(variables), algebraic expressions, and equations, but these formal approaching method has many difficulties in the school algebra. Therefore we insist the new algebraic approaches should be needed. In order to develop these instructions, we firstly investigate the relationship of patterns and algebra, the relationship of generalization and algebra, the steps of generalization from patterns and levels of difficulties. Next we look into the algebra instructions based arithmetic patterns, visual patterns and functional situations. We expect that these approaches help students learn algebra when they begin school algebra.