• Communications of Mathematical Education
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• v.23 no.3
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• pp.707-734
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• 2009

The purpose of the study was to investigate how children understand addition and subtraction of fractions and how their understanding influences the solutions of fractional word problems. Twenty students from 4th to 6th grades were involved in the study. Children's understanding of operations with fractions was categorized into "joining", "combine" and "computational procedures (of fraction addition)" for additions, "taking away", "comparison" and "computational procedures (of fraction subtraction)" for subtractions. Most children understood additions as combining two distinct sets and subtractions as removing a subset from a given set. In addition, whether fractions had common denominators or not did not affect how they interpret operations with fractions. Some children understood the meanings for addition and subtraction of fractions as computational procedures of each operation without associating these operations with the particular situations (e.g. joining, taking away). More children understood addition and subtraction of fractions as a computational procedure when two fractions had different denominators. In case of addition, children's semantic structure of fractional addition did not influence how they solve the word problems. Furthermore, we could not find any common features among children with the same understanding of fractional addition while solving the fractional word problems. In case of subtraction, on the other hand, most children revealed a tendency to solve the word problems based on their semantic structure of the fractional subtraction. Children with the same understanding of fractional subtraction showed some commonalities while solving word problems in comparison to solving word problems involving addition of fractions. Particularly, some children who understood the meaning for addition and subtraction of fractions as computational procedures of each operation could not successfully solve the word problems with fractions compared to other children.

• 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.

• Education of Primary School Mathematics
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• v.26 no.3
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• pp.125-140
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• 2023

In elementary school mathematics, the addition and subtraction of fractions are difficult for students to understand but very important concepts. This study aims to examine the teaching methods and visual aids utilized in the context of common denominator fraction addition and subtraction. The analysis focuses on evaluating the lesson flow and the utilization of visual representations in one national textbook and ten certified textbooks aligned with the current 2015 revised curriculum. The results show that each textbook is composed of chapter sequences and topics that reflect the curriculum faithfully, with each textbook considering its own order and content. Additionally, each textbook uses a different variety and number of visual representations, presumably intended to aid in learning the operations of fractions through the consistency or diversity of the visual representations. Identifying the characteristics of each textbook can lead to more effective instruction in fraction operations.

• Journal of Elementary Mathematics Education in Korea
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• v.18 no.2
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• pp.237-255
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• 2014

The purpose of this study is to investigate elementary school students' understanding the concept and operations of decimal fraction. The survey research was performed for this study. This survey was done by selecting 156 students. Questionnaire were made in five areas with reference to the 2007 revised mathematics curriculum. Five areas were the concept of decimal fraction, the addition, the subtraction, the multiplication and the division of decimal fraction. The results of such analysis are as follow: The analyzed result of understanding about concepts and operation of decimal fraction showed a high rate of correct answer, more than 85%. Students thought that multiplication and division of decimal fraction is more difficult than addition, subtraction, concept of decimal fraction. As the learning about concepts and operation of decimal fraction progress, the learning gap is bigger. Effort to reduce the learning deficits are needed in the lower grades. Mathematics is the study of the hierarchical. Learning deficits in low-level interfere with the learning in next-level. Therefore systematic supplementary guidance for a natural number and decimal fraction in low-level is needed. And understanding concepts and principles of calculations should be taught first.

• Journal of the Korean School Mathematics Society
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• v.17 no.3
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• pp.409-435
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• 2014

This study examines relations between the 5th graders' fraction operation skills and error types on constructed-response items. As results, first, the participants have lower fraction operation skills on 'multiplication of fraction' than 'addition and subtraction of fraction'. Second, the participants have different error types depend on their constructed-response items. Most of error types which group with high ability made was 'leap of solving process', both groups error type with medium ability as well as low ability is 'misunderstanding of questions'. Third, the operation skills on 'addition and subtraction of fraction' have an influence on their operation skills on 'multiplication of fraction', and error types of 'understanding of questions' and 'understanding of solving process' have the most effects on the influence.

• Education of Primary School Mathematics
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• v.25 no.3
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• pp.213-232
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• 2022

With the importance of number line in learning fractions, this study investigated how 4th grade students understand fractions and its operations in number line. The questionnaire consisted 22 items which were related to representing fractions, comparing the size of fractions, and operating addition and subtraction of fractions. Both structured number line and sub-structured number line were presented in the items. As results of the study, the overall success rates were not high and even some items showed higher incorrect answer rates than the success rates. Also, the students showed a difficulty in solving non-structured number line tasks. It was also noticeable that students showed several types of incorrect answers, which means that students had incomplete understanding of both fractions and number line. This paper is expected to shed light on elementary students' understanding of fractions and number line and to provide implications of how to deal with number line in teaching and learning fractions in the elementary school.

• The Mathematical Education
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• v.40 no.2
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• pp.195-215
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• 2001

The purpose of this paper is to present a specific set of home teaching methods in hopes to prevent slow learner of the elementary mathematics. This paper deals with the number and operations, one of five topics in the elementary mathematics A survey of two hundred elementary school teachers was made to see the teacher's opinions of the role of home studying and to concretize the contents of the research topics. There were asked which is the most essential contents for the concrete loaming and which is the most difficult monad that might cause slow leaner. And those were found to be; counting, and arithmetic operations(addition and subtraction) of one or two-digit numbers and multiplication and their concepts representations and operations(addition and subtraction) of fractions. The home teaching methods are based on the situated learning about problem solving in real life situations and on the active teaming which induces children's participation in the process of teaching and learning. Those activities in teaching each contents are designed to deal with real objects and situations. Most teaching methods are presented in the order of school curriculum. To teach the concepts of numbers and the place value, useful activities using manipulative materials (Base ten blocks, Unifix, etc.) or real objects are also proposed. Natural number's operations such as addition, subtraction and multiplication are subdivided into small steps depending upon current curriculum, then for understanding of operational meaning and generalization, games and activities related to the calculation of changes are suggested. For fractions, this paper suggest 10 learning steps, say equivalent partition, fractional pattern, fractional size, relationship between the mixed fractions and the improper fraction, identifying fractions on the number line, 1 as a unit, discrete view point of fractions, comparison of fractional sizes, addition and subtraction, quantitative concepts. This research basically centers on the informal activities of kids under the real-life situation because such experiences are believed to be useful to prevent slow learner. All activities and learnings in this paper assume children's active participation and we believe that such active and informal learning would be more effective for learning transfer and generalization.

• School Mathematics
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• v.18 no.3
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• pp.625-645
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• 2016

This study clarified the big ideas related to teaching addition and subtraction of fractions with different denominators based on quantitative reasoning with unit and recursive partitioning. An analysis of this study urged us to re-consider the content related to the addition and subtraction of fraction. As such, this study analyzed textbooks and teachers' manuals developed from the fourth national mathematics curriculum to the most recent 2009 curriculum. In addition and subtraction of fractions with different denominators, it must be emphasized the followings: three-levels unit structure, fixed whole unit, necessity of common measure and recursive partitioning. An analysis of this study showed that textbooks and teachers' manuals dealt with the fact of maintaining a fixed whole unit only as being implicit. The textbooks described the reason why we need to create a common denominator in connection with the addition of similar fractions. The textbooks displayed a common denominator numerically rather than using a recursive partitioning method. Given this, it is difficult for students to connect the models and algorithms. Building on these results, this study is expected to suggest specific implications which may be taken into account in developing new instructional materials in process.

• Journal of Elementary Mathematics Education in Korea
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• v.14 no.2
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• pp.241-262
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• 2010

This study was carried out to identify the cognitive obstacles while using addition and subtraction with fractions, and to analyze the sources of cognitive obstacles. For this purpose, the following research questions were established : 1. What errors do elementary students make while performing the operations with fractions, and what cognitive obstacles do they have? 2. What sources cause the cognitive obstacles to occur? The results obtained in this study were as follows : First, the student's cognitive obstacles were classified as those operating with same denominators, different denominators, and both. Some common cognitive obstacles that occurred when operating with same denominators and with different denominators were: the students would use division instead of addition and subtraction to solve their problems, when adding fractions, the students would make a natural number as their answer, the students incorporated different solving methods when working with improper fractions, as well as, making errors when reducing fractions. Cognitive obstacles in operating with same denominators were: adding the natural number to the numerator, subtracting the small number from the big number without carrying over, and making errors when doing so. Cognitive obstacles while operating with different denominators were their understanding of how to work with the denominators and numerators, and they made errors when reducing fractions to common denominators. Second, the factors that affected these cognitive obstacles were classified as epistemological factors, psychological factors, and didactical factors. The epistemological factors that affected the cognitive obstacles when using addition and subtraction with fractions were focused on hasty generalizations, intuition, linguistic representation, portions. The psychological factors that affected the cognitive obstacles were focused on instrumental understanding, notion image, obsession with operation of natural numbers, and constraint satisfaction.

• Journal of the Korean School Mathematics Society
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• v.25 no.4
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• pp.307-332
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• 2022

This action research was carried out to help students learn fractions computation by making and using semi-concrete fraction manipulatives that can be used continuously in math classes. For this purpose, the researcher and students made semi-concrete fraction manipulatives and learned how to use these through reviewing the previously learned fraction contents over 4 class sessions. Afterward, through the 14 classes (7 classes for learning to reduce fractions and to a common denominator, 7 classes for adding and subtracting fractions with different denominators) in which the principle inquiry learning model was applied, students actively engaged in learning activities with fraction manipulatives and explored the principles underneath the manipulations of fraction manipulatives. Students could represent various fractions using fraction manipulatives and solve fraction computation problems using them. The achievement evaluation after class found that the students could connect the semi-concrete fraction manipulatives with fraction representation and symbolic formulas. Moreover, the students showed interest and confidence in mathematics through the classes using fraction manipulatives.