Children's Concepts of Average and Representativeness.
01 Jan 1995-Journal for Research in Mathematics Education (TERC Communications, 2067 Massachusetts Avenue, Cambridge, MA 02140 ($5).)-Vol. 26, Iss: 1, pp 20-39
TL;DR: This article studied the characteristics of fourth through eighth grade students' constructions of "average" as a representative number summarizing a data set and identified five basic representativeness constructs for the statistical concept of average.
Abstract: Whenever the need arises to describe a set of data in a succinct way, the issue of mathematical representativeness arises. The goal of this research is to understand the characteristics of fourth through eighth graders' constructions of "average" as a representative number summarizing a data set. Twenty-one students were interviewed, using a series of open-ended problems that called on children to construct their own notion of representativeness. Five basic constructions of representativeness are identified and analyzed. These approaches illustrate the ways in which students are (or are not) developing useful, general definitions for the statistical concept of average. One objective of statistics is to reduce large, unmanageable, and disordered collections of information to summary representations. The need to summarize data is present even among young children. For example, in the surveys conducted by primary-grade students, we see movement from focusing on individual pieces of data ("I have one brother") to highlighting and summarizing the data in some manageable form ("Most of the class members have only one brother or sister"). As soon as there is the need to describe a set of data in a more succinct way, the notion of representativeness arises: What is typical of these data? How can we capture their range and distribution?
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TL;DR: Two studies on teaching descriptive statistics to 9th-grade students examined whether invention activities may prepare students to learn found that invention activities, when coupled with subsequent learning resources like lectures, led to strong gains in procedural skills, insight into formulas, and abilities to evaluate data from an argument.
Abstract: Activities that promote student invention can appear inefficient, because students do not generate canonical solutions, and therefore the students may perform badly on standard assessments. Two studies on teaching descriptive statistics to 9th-grade students examined whether invention activities may prepare students to learn. Study 1 found that invention activities, when coupled with subsequent learning resources like lectures, led to strong gains in procedural skills, insight into formulas, and abilities to evaluate data from an argument. Additionally, an embedded assessment experiment crossed the factors of instructional method by type of transfer test, with 1 test including resources for learning and 1 not. A "tell-and-practice" instructional condition led to the same transfer results as an invention condition when there was no learning resource, but the invention condition did better than the tell-and-practice condition when there was a learning resource. This demonstrates the value of invention activ...
671 citations
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...... do not notice relevant features—for example, if they treat probabilistic outcomes as single events rather than distributions (Konold, 1989)—then they will not understand what a statistical explanation is referring to. Furthermore, without an understanding of the relevant features, students may transfer in vague intuitions that do not have sufficient precision to motivate the form or function of a particular statistical formalization ( Mokros ......
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...Furthermore, without an understanding of the relevant features, students may transfer in vague intuitions that do not have sufficient precision to motivate the form or function of a particular statistical formalization (Mokros & Russell, 1995)....
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01 Jan 2004
TL;DR: This chapter discusses the development of Instructional Design for Supporting the Development of Students' Statistical Reasoning and research on Statistical Literacy, Reasoning, and Thinking.
Abstract: Statistical Literacy, Reasoning, and Thinking: Goals, Definitions, and Challenges.- Towards an Understanding of Statistical Thinking.- Statistical Literacy.- A Comparison of Mathematical and Statistical Reasoning.- Models of Development in Statistical Reasoning.- Reasoning about Data Analysis.- Learning to Reason About Distribution.- Conceptualizing an Average as a Stable Feature of a Noisy Process.- Reasoning About Variation.- Reasoning about Covariation.- Students' Reasoning about the Normal Distribution.- Developing Reasoning about Samples.- Reasoning about Sampling Distribitions.- Primary Teachers' Statistical Reasoning about Data.- Secondary Teachers' Statistical Reasoning in Comparing Two Groups.- Principles of Instructional Design for Supporting the Development of Students' Statistical Reasoning.- Research on Statistical Literacy, Reasoning, and Thinking: Issues, Challenges, and Implications.
477 citations
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...Moore (1990) has argued that in statistics, the context motivates procedures; data should be viewed as numbers with a context, and hence the context is the source of meaning and basis for interpretation of obtained results....
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TL;DR: The authors provide an overview of current research on teaching and learning statistics, summarizing studies that have been conducted by researchers from different disciplines and focused on students at all levels, and suggest what can be learned from the results of each of these questions.
Abstract: Summary
This paper provides an overview of current research on teaching and learning statistics, summarizing studies that have been conducted by researchers from different disciplines and focused on students at all levels. The review is organized by general research questions addressed, and suggests what can be learned from the results of each of these questions. The implications of the research are described in terms of eight principles for learning statistics from Garfield (1995) which are revisited in the light of results from current studies.
418 citations
TL;DR: In this article, a case study analyzes ways in which an experienced physics teacher uses questioning to guide student thinking during a benchmark discussion about measurement, and analyzes reflective tosses in terms of the immediate action plans they instantiated, the emergent goals they served, and the underlying beliefs they embodied during an episode that involved the public refinemen.
Abstract: This case study analyzes ways in which an experienced physics teacher uses questioning to guide student thinking during a benchmark discussion about measurement. Interactional issues involve ways of speaking: Why the teacher decided to ask what he did, when he did, of whom, in what way, and for what purpose. Conceptual issues involves ways of thinking: How students seemed to understand measurement concepts such as calculating an average value. We define a particular kind of question, a reflective toss, that the teacher uses to try to give students responsibility for thinking. A reflective toss sequence typically consists of a student statement, teacher question, and additional student statements. This unit of analysis directs attention to ways in which a teacher question influences student thinking. We analyze reflective tosses in terms of the immediate action plans they instantiated, the emergent goals they served, anal underlying beliefs they embodied during an episode that involved the public refinemen...
334 citations
Journal Article•
TL;DR: For the past several years, Ball and Chazan as discussed by the authors have been developing and studying teaching practices through their own efforts to teach school mathematics, and using their teaching as a site for research into, and as a source for formulating a critique of, what it takes to teach in the ways reformers promote.
Abstract: For the past several years, we have been developing and studying teaching practices through our own efforts to teach school mathematics. Ball's work has been at the elementary level, in third grade, and Chazan's at the secondary level, grade ten and above, in Algebra I. In our teaching, we have been attempting, among other things, to create opportunities for classroom discussions of the kinds envisioned in the US National Council for Teachers of Mathematics Standards (NCTM, 1989, 1991). At the same time, we have been exploring the complexities of such practice. By using our teaching as a site for research into, and as a source for formulating a critique of, what it takes to teach in the ways reformers promote, we have access to a particular 'insider' sense of the teacher's purposes and reasoning, beyond that which a researcher might have. [1] This article originated with frustration at current math education discourse about the teacher's role in discussion-
246 citations
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...really is, but as it might be reimagined This kind of reimagining is characteristic of mathematics (O'Cormm, 1998) It is this hypothetical, or 'abstract', quality of the arithmetic mean which causes much difficulty for students (Mokros and Russell, 1995) I had two reasons fOr wanting to fOcus the conversation on the meaning of the average....
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References
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TL;DR: In this paper, the authors report data from interviews in which students attempted to solve problems involving the appropriate weighting and combining of means into an overall mean, and find that a surprisingly large proportion of them do not understand the concept of the weighted mean.
Abstract: In statistics, and in everyday life as well, the arithmetic mean is a frequently used average. The present study reports data from interviews in which students attempted to solve problems involving the appropriate weighting and combining of means into an overall mean. While mathematically unsophisticated college students can easily compute the mean of a group of numbers, our results indicate that a surprisingly large proportion of them do not understand the concept of the weighted mean. When asked to calculate the overall mean, most subjects answered with the simple, or unweighted, mean of the two means given in the problem, even though these two means were from different-sized groups of scores. For many subjects, computing the simple mean was not merely the easiest or most obvious way to initially attack the problem; it was the only method they had available. Most did not seem to consider why the simple mean might or might not be the correct response, nor did they have any feeling for what their results represented. For many students, dealing with the mean is a computational rather than a conceptual act. Knowledge of the mean seems to begin and end with an impoverished computational formula. The pedagogical message is clear: Learning a computational formula is a poor substitute for gaining an understanding of the basic underlying concept.
178 citations
"Children's Concepts of Average and ..." refers background or methods in this paper
...Weighted Means Problem The Elevator Problem (adapted from Pollatsek et al., 1981) was intended to reveal children's understanding of a more complex weighted averaging problem....
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...…1988); (b) sixth-grade students are generally unable to use the mean to compare two different-sized sets of data (Gal, Rothschild, & Wagner, 1989, 1990); and (c) even college students have difficulty working with familiar averaging problems that involve weighted means (Pollatsek et al., 1981)....
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TL;DR: This article investigated a model used by non-mathematically oriented students in solving problems in descriptive statistics and found that most students were able to acquire the appropriate schema of statistical concepts by engaging in diagnostic activities embedded within a feedback-corrective procedure.
Abstract: The present study investigated a model used by non-mathematically oriented students in solving problems in descriptive statistics. Analyses show that college students mistakenly assume that a set of means together with simple mean computation constitutes a mathematical group satisfying the four axioms of closure, associativity, identity, and inverse. This set of misconceptions is so deeply ingrained in a students' underlying knowledge base that mere exposure to a more advanced course in statistics is not sufficient to overcome those misconceptions. However, results of an experiment indicated that most students were able to acquire the appropriate schema of statistical concepts by engaging in diagnostic activities embedded within a feedback-corrective procedure.
129 citations
"Children's Concepts of Average and ..." refers background in this paper
...Other research identifies several characteristics of the mean and then examines children's and adults' understanding of these characteristics (Goodchild, 1988; Mevarech, 1983; Strauss & Bichler, 1988)....
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...Even college students, after being given a balance beam analogy for the mean, "mistakenly believe that the mean of a group does not depend on what the elements of the group are" (Mevarech, 1983)....
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TL;DR: This paper conducted a study to determine the development of children's understanding of seven properties of the arithmetic mean and assess the effects of the material used in the testing (continuous, discontinuous) and the medium of presentation (story, concrete, and numerical).
Abstract: The study was conducted to (a) determine the development of children's understanding of seven properties of the arithmetic mean and (b) assess the effects of the material used in the testing (continuous, discontinuous) and the medium of presentation (story, concrete, and numerical). Twenty children were selected at each of the ages 8, 10, 12, and 14 years. Different developmental courses of the children's reasoning were found on some tasks measuring the properties of the average. No significant effects were found for the materials used or the medium of presentation. The findings are discussed in terms of their importance for developmental psychology and educational practice. The Lord must have loved the average man because he made more of them than the others.
124 citations
"Children's Concepts of Average and ..." refers background in this paper
...These notions, while including some of the properties of the mean articulated by Strauss and Bichler (1988) and Goodchild (1988), more explicitly connect different averages to the data they represent....
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...…the subject, researchers have found that (a) fourth- through eighth-grade students have a difficult time understanding the properties of the mean (Strauss & Bichler, 1988); (b) sixth-grade students are generally unable to use the mean to compare two different-sized sets of data (Gal, Rothschild,…...
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...Other research identifies several characteristics of the mean and then examines children's and adults' understanding of these characteristics (Goodchild, 1988; Mevarech, 1983; Strauss & Bichler, 1988)....
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36 citations