• Journal of Educational Research in Mathematics
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• v.21 no.2
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• pp.201-220
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• 2011

This study investigates levels of thinking of elementary and middle school students doing their tasks of explaining and dealing with variability. According to results, on the task of explaining variability in the measurement settings five levels of thinking were identified: a lack of understanding of explanation of the causes, an insufficient understanding of the causes, an offer of physical causes, consideration of unexplained causes as the source of variability, and consideration of unexplained causes as quasi-chance variability. Also, in the chance settings five levels of thinking were identified: a lack of understanding of explanation of the causes, an insufficient understanding of the causes, an offer of physical causes, recognition of chance variability, and consideration of causes of distribution. On the task of dealing with variability in both the measurement and chance settings five levels of thinking were identified: a lack of understanding of dealing with variability, no physical control and improper statistical control, no physical control and proper statistical control, physical control and improper statistical control, and physical control and proper statistical control.

• School Mathematics
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• v.14 no.4
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• pp.495-516
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• 2012

This study distinguished thinking related to statistical variability into six components - the noticing of variability, the explanation of variability, the control of variability, the modeling of variability, the understanding of samples, and the understanding of sampling distribution and investigated the relationships among the thinking components. This study found that this distinction of thinking components related to statistical variability is reasonable. The results showed that each correlation coefficient of the modeling of variability, the understanding of samples, and the understanding of sampling distribution with regard to the noticing of variability, the explanation of variability, and the control of variability is similar. Based on this results, new variable, the understanding of sampling, has been drawn. The results also showed that while the noticing of variability and the control of variability influence the understanding of sampling, the explanation of variability does not influence it.

• Journal of Educational Research in Mathematics
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• v.20 no.4
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• pp.493-509
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• 2010

Researchers have suggested that educators have to focus their attention on variability and reasoning about variation as means of developing students' statistical thinking in school mathematics. This paper investigated knowledge for the teaching of variability and reasoning about variation; what are sources of variability, how to cope with variability, what are types of variability, how to recognize variability, and the relationship between statistical problem solving and variability. The results involve: discussion on the sources of variability and how to cope with variability promotes students' awareness of different types of variability and students' motivation in the following steps in the statistical activity; emphasis on reasoning about variation in teaching representation of data accords with objectives of statistics education; reexamination of curriculum for statistics education is needed, which has a content-oriented arrangement.

• Journal of Educational Research in Mathematics
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• v.20 no.3
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• pp.339-356
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• 2010

It is important for children to develop statistical reasoning as they think through data. In particular, it is imperative to provide children instructional situations in which they are encouraged to consider variability in data because the ability to reason about variability is fundamental to the development of statistical reasoning. Many researchers argue that even highperforming mathematics students show low levels of statistical reasoning; interventions attending to pedagogical concerns about child ren's statistical reasoning are, thus, necessary. The purpose of this study was to investigate 15 gifted elementary students' various ways of understanding important statistical concepts, with particular attention given to 3 students' reasoning about data that emerged as they engaged in the process of generating and graphing data. Analysis revealed that in recognizing variability in a context involving data, mathematically gifted students did not show any difference from previous results with general students. The authors suggest that our current statistics education may not help elementary students understand variability in their development of statistical reasoning.

• Journal of Educational Research in Mathematics
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• v.16 no.3
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• pp.221-249
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• 2006

The object of this study is to understand the characteristics of mathematical knowledge that elementary 5th graders have regarding the statistical variation concept and the changes after taking lessons. This study includes a pretest to examine the characteristics of mathematical knowledge that elementary 5th graders have regarding the statistical variation concept. And It was followed by a lesson on statistical variation concept to be able to correct error which was revealed by the inspection, and to improve good points. It turned out that after five lessons on the statistical variation concept, the insufficient aspects were properly improved, and as for the points they already understand, they came to understand better than before. They came to consider the statical variation concept instead of the frequency, preponderance, average, stable traits for the optimum value. Also, through the lesson on drawing tables and graphs, they came to better understand them, analyzing correctly the exercises in which tables and graphs were combined. When comparing data sets whose general distributions and extents were similar, students came up with the right answers in a stable way by considering averages combining statistical variation too. Since they tended to interpret a situation with their own subjective views adding conditions, teachers need to examine the proper situation and conditions prior to the lessons on the statistical variation concept.

• School Mathematics
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• v.9 no.1
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• pp.29-44
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• 2007

The aim of statistics education is to enhance statistical thinking. Variability is the key components of statistical thinking. The research has been reviewed preceding research about variability of data. Proceeding from what has been considered above, this research developed learning materials that investigated the concept of variability as it relates to Freudenthal's context by having students sort a particular context. The research is executed the case study evidently aimed at Junior High School 3rd Grade Student's Understanding of Variability. The study of variability in data can be an important start to reach a testing of statistical hypothesis; students reduce data and draw graphs by relating probability distribution to relative frequency and normal distribution. Thus, this study offers basic materials into developing both contents and methods of education need to consider with this sense of purpose held by students to achieve this goal.

• Journal of Elementary Mathematics Education in Korea
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• v.22 no.3
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• pp.221-240
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• 2018

In this study, we focus on the classifying and representing data in the elementary mathematics curriculum for 1st and 2nd grades which have been rarely addressed in the previous studies. We analyze the significance of classifying and representing sata in terms of statistical problem solving and variability as the core of statistical literacy. As a result, the classifying and representing data are important for students to recognize the variability which is ubiquitous in the data and to construct distribution of data, respectively. They are reflected in the 2015 revised mathematics curriculum as the statistical literacy for addressing data. We suggest some implications to teach the classifying and representing data as the practice of statistical literacy education in their statistics classes for 1st and 2nd grades.

• Journal of the Korean School Mathematics Society
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• v.21 no.2
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• pp.183-206
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• 2018

It arises that teachers' professional competence should be considered not only with a cognitive perspective but also with a situative perspective. In this study, we considered mathematics teacher noticing as situational professional competencies of a mathematics teacher, and explored how mathematics teachers noticing on children's development of reasoning about variability in a video club has changed with the situative perspective. Findings illustrate that the 'interpreting' component among the three components of noticing-attending, interpreting, and deciding how to respond-was critically decisive for the change of the participant teachers' noticing. We also discussed how the video club intervention(the framework of children's development of reasoning about variability) can support the development of teacher noticing as a professional competence. This study has implications on the design of a video club to improve mathematics teacher noticing.

• Education of Primary School Mathematics
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• v.21 no.2
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• pp.131-150
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• 2018

The structure of a mathematics task shapes the aspects of learning of those who solve the task. This study explores the process of understandings on the statistical variability of primary school students. Students were given two problems with different degrees of structuring - a well-structured problem (WSP) and an ill-structured problem (ISP) - and discussed in a group to solve each task. The highest level of development achieved in both cases appeared to be similar. However, when given the ISP, students dynamically proposed ideas and justified the conclusion based on their hypothesis. Furthermore, all students actively participated in solving the ISP until the end whereas some students were marginalized while solving the WSP. This discrepancy results from the difference in the degrees of task structuring.

• Journal of Gifted/Talented Education
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• v.23 no.3
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• pp.387-406
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• 2013

This study compared levels of mathematically talented students' statistical thinking with those of non-talented students in the noticing of statistical variability. t tests were conducted to test for statistically significant differences between mathematically gifted students and non-gifted students. Results for the t-test shows that there is no difference between the TE students' and NE students' noticing of variability in the measurement settings. Meanwhile, the t-test results also show that there is a difference between the TM students' and NM students' noticing of variability in the both measurement and chance settings. Table of frequencies of each level, however, shows that levels of mathematically gifted students' thinking were not distributed at the high levels but were overlapped with those of non-gifted students. These results are thought-provoking results in statistics instruction for mathematically talented students.