• Title/Summary/Keyword: 공변 추론

### Covariational Reasoning of Ninth Graders in Reciprocal Peer Tutoring Process (상호또래교수과정에서 나타나는 중학교 3학년 학생들의 공변 추론)

• Gil, Seung Ho;Shin, Jaehong
• Journal of the Korean School Mathematics Society
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• v.23 no.3
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• pp.323-349
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• 2020
• In this study, we conducted eight reciprocal peer tutoring classes where each student took either role of a tutor or a tutee to study covariational reasoning in ninth graders. Students were given the opportunity to teach their peers with their covariational reasoning as tutors, and at the same time to learn covariational reasoning as tutees. A heterogeneous group was formed so that scaffolding could be provided in the teaching and learning process. A total of eight reciprocal peer tutoring worksheets were collected: four quantitative graph type questions and four questions of the qualitative graph to the group. The results of the analysis are as follows. In reciprocal peer tutoring, students who experienced a higher level of covariational reasoning than their covariational reasoning level showed an improvement in covariational reasoning levels. In addition, students enhanced the completeness of reasoning by modifying or supplementing their own covariational reasoning. Minimal teacher intervention or high-level peer mediation seems to be needed for providing feedback on problem-solving results.

### A Study of Two Pre service Teachers' Development of Covariational Reasoning (모의실험을 통한 두 예비교사의 공변추론 이해에 관한 연구)

• Shin, Jae-Hong;Lee, Joong-Kweon
• Journal of the Korean School Mathematics Society
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• v.12 no.4
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• pp.453-472
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• 2009
• This article describes the interview data with two preservice teachers where they dealt with five water-filling problems for the investigation of their covariational thinking. The study's results revealed that two students developed their covariation levels from Direction level to Instantaneous Rate with an aid of the pre-constructed GSP simulations for the problem situations. However, this study also points out that there is a missing important feature for a function notion, 'causality' in the covariation framework and suggests that future research should combine students' conception of causality with their covariational thinking for the investigation of their development of a function concept.

### A Case Study on the Students' Covariational Reasoning in the Continuous Functional Situations (함수적 상황에 대한 초등학생들의 공변추론 사례연구)

• Hur, Joonho;Park, Mangoo
• Education of Primary School Mathematics
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• v.21 no.1
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• pp.55-73
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• 2018
• The purpose of this study is to investigate the effects of cognitive activity on cognitive activities that students imagine and cope with continuously changing quantitative changes in functional tasks represented by linguistic expressions, table of value, and geometric patterns, We identified covariational reasoning levels and investigated the characteristics of students' reasoning process according to the levels of covariational reasoning in the elementary quantitative problem situations. Participants were seven 4th grade elementary students using the questionnaires. The selected students were given study materials. We observed the students' activity sheets and conducted in-depth interviews. As a result of the study, the students' covariational reasoning level for two quantities that are continuously covaried was found to be five, and different reasoning process was shown in quantitative problem situations according to students' covariational reasoning levels. In particular, students with low covariational level had difficulty in grasping the two variables and solved the problem mainly by using the table of value, while the students with the level of chunky and smooth continuous covariation were different from those who considered the flow of time variables. Based on the results of the study, we suggested that various problems related with continuous covariation should be provided and the meanings of the tasks should be analyzed by the teachers.

### Analyzing Students' Works with Quantitative and Qualitative Graphs Using Two Frameworks of Covariational Reasoning (그래프 유형에 따른 두 공변 추론 수준 이론의 적용 및 비교)

• Park, JongHee;Shin, Jaehong;Lee, Soo Jin;Ma, Minyoung
• Journal of Educational Research in Mathematics
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• v.27 no.1
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• pp.23-49
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• 2017
• This study examined two current learning models for covariational reasoning(Carlson et al.(2002), Thompson, & Carlson(2017)), applied the models to teaching two $9^{th}$ grade students, and analyzed the results according to the types of graphs(a quantitative graph or qualitative graph). Results showed that the model of Thompson and Carlson(2017) was more useful than that of Carlson et al.(2002) in figuring out the students' levels in their quantitative graphing activities. Applying Carlson et al.(2002)'s model made it possible to classify levels of the students in their qualitative graphs. The results of this study suggest that not only quantitative understanding but also qualitative understanding is important in investigating students' covariational reasoning levels. The model of Thompson and Carlson(2017) reveals more various aspects in exploring students' levels of quantitative understanding, and the model of Carlson et al.(2002) revealing more of qualitative understanding.

### Comparison of Middle School Students' Similarities Revealed in the Process of Word Problems Solving According to Covariational Reasoning (두 중학생의 공변 추론 수준에 따른 연립방정식 문장제의 해결에서 나타나는 유사성 비교)

• Ma, Minyoung
• Communications of Mathematical Education
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• v.35 no.3
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• pp.323-340
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• 2021
• The purpose of this case study is to explore the similarities revealed in the process of solving and generalizing word problems related to systems of linear equations in two variables according to covariational reasoning. As a result, student S, who reasoned with coordination of value level, had a static image of the quantities given in the situation. student D, who reasoned with smooth continuous covariation level, had a dynamic image of the quantities in the problem situation and constructed an invariant relationship between the quantities. The results of this study suggest that the activity that constructs the relationship between the quantities in solving word problems helps to strengthen the mathematical problem solving ability, and that teaching methods should be prepared to strengthen students' covariational reasoning in algebra learning.

### Gifted Middle School Students' Covariational Reasoning Emerging through the Process of Algebra Word Problem Solving (대수 문장제의 해결에서 드러나는 중등 영재 학생간의 공변 추론 수준 비교 및 분석)

• Ma, Minyoung;Shin, Jaehong
• School Mathematics
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• v.18 no.1
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• pp.43-59
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• 2016
• The purpose of this qualitative case study is to investigate differences among two gifted middle school students emerging through the process of algebra word problem solving from the covariational perspective. We collected the data from four middle school students participating in the mentorship program for gifted students of mathematics and found out differences between Junghee and Donghee in solving problems involving varying rates of change. This study focuses on their actions to solve and to generalize the problems situations involving constant and varying rates of change. The results indicate that their covariational reasoning played a significant role in their algebra word problem solving.

### Students' Problem Solving Based on their Construction of Image about Problem Contexts (문제맥락에 대한 이미지가 문제해결에 미치는 영향)

• Koo, Dae Hwa;Shin, Jaehong
• Journal of the Korean School Mathematics Society
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• v.23 no.1
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• pp.129-158
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• 2020

### How does the middle school students' covariational reasoning affect their problem solving? (연속적으로 공변하는 두 양에 대한 추론의 차이가 문제 해결에 미치는 영향)

• KIM, CHAEYEON;SHIN, JAEHONG
• The Mathematical Education
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• v.55 no.3
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• pp.251-279
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• 2016
• There are many studies on 'how' students solve mathematical problems, but few of them sufficiently explained 'why' they have to solve the problems in their own different ways. As quantitative reasoning is the basis for algebraic reasoning, to scrutinize a student's way of dealing with quantities in a problem situation is critical for understanding why the student has to solve it in such a way. From our teaching experiments with two ninth-grade students, we found that emergences of a certain level of covariational reasoning were highly consistent across different types of problems within each participating student. They conceived the given problem situations at different levels of covariation and constructed their own quantity-structures. It led them to solve the problems with the resources accessible to their structures only, and never reconciled with the other's solving strategies even after having reflection and discussion on their solutions. It indicates that their own structure of quantities constrained the whole process of problem solving and they could not discard the structures. Based on the results, we argue that teachers, in order to provide practical supports for students' problem solving, need to focus on the students' way of covariational reasoning of problem situations.

### A student's conceiving a pattern of change between two varying quantities in a quadratic functional situation and its representations: The case of Min-Seon (이차함수에서 두 변량사이의 관계 인식 및 표현의 발달 과정 분석: 민선의 경우를 중심으로)

• Lee, Dong Gun;Moon, Min Joung;Shin, Jaehong
• The Mathematical Education
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• v.54 no.4
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• pp.299-315
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• 2015
• The aim of this qualitative case study is twofold: 1) to analyze how an eleventh-grader, Min-Seon, conceive and represent a pattern of change between two varying quantities in a quadratic functional situation, and 2) further to help her form a concept of 'derivative' as a tool to express the relationship with employing a concept of 'rate of change.' The result indicates that Min-Seon was able to construct graphs of piecewise functions that take average rates of change as range of the functions, and managed to conjecture the derivative of a quadratic function, $y=x^2$. In conclusion, we argue that covariational approach could not only facilitate students' construction of an initial function concept, but also support their understanding of the concept of 'derivative.'

### Students' Recognition and Representation of the Rate of Change in the Given Range of Intervals (구간에서의 변화율에 대한 인식과 표현에 대한 연구)

• Lee, Dong Gu;Shin, Jaehon
• Journal of Educational Research in Mathematics
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• v.27 no.1
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• pp.1-22
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• 2017
• This study investigated three $10^{th}$ grade students' concept of rate of change while they perceived changing values of given functions. We have conducted a teaching experiment consisting of 6 teaching episodes on how the students understood and expressed changing values of functions on certain intervals in accordance with the concept of rate of change. The result showed that the students did use the same word of 'rate of change' in their analysis of functions, but their understanding and expression of the word varied, which turned out to have diverse perceptions with regard to average rate of change. To consider these differences as qualitatively different levels might need further research, but we expect that this research will serve as a foundational study for further research in students' learning 'differential calculus' from the perspective of rate of change.