Exploring Students' Conceptions of the Standard Deviation

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Exploring Students' Conceptions of the Standard Deviation

Journal Article•

Exploring Students' Conceptions of the Standard Deviation

Robert delMas, Yan Liu¹•Institutions (1)

01 May 2005-Statistics Education Research Journal (International Association for Statistical Education (IASE))-Vol. 4, Iss: 1, pp 55-82

TL;DR: This article investigated introductory statistics students' conceptual understanding of the standard deviation and found that students moved from simple, one-dimensional understandings of standard deviation that did not consider variation about the mean to more mean-centered conceptualizations that coordinated the effects of frequency (density) and deviation from the mean.

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Abstract: SUMMARY This study investigated introductory statistics students’ conceptual understanding of the standard deviation. A computer environment was designed to promote students’ ability to coordinate characteristics of variation of values about the mean with the size of the standard deviation as a measure of that variation. Twelve students participated in an interview divided into two primary phases, an exploration phase where students rearranged histogram bars to produce the largest and smallest standard deviation, and a testing phase where students compared the sizes of the standard deviation of two distributions. Analysis of data revealed conceptions and strategies that students used to construct their arrangements and make comparisons. In general, students moved from simple, one-dimensional understandings of the standard deviation that did not consider variation about the mean to more mean-centered conceptualizations that coordinated the effects of frequency (density) and deviation from the mean. Discussions of the results and implications for instruction and further research are presented.

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Journal Article•DOI•

Lexical ambiguity: making a case against spread

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Jennifer J. Kaplan¹, Neal Rogness², Diane G. Fisher³•Institutions (3)

Michigan State University¹, Grand Valley State University², University of Louisiana at Lafayette³

01 Jun 2012-Teaching Statistics

TL;DR: The authors argue for increasing the use of the word variability as a replacement for spread in introductory statistics courses, and argue that spread is not the best word for describing the statistical idea of dispersion or variability in statistics courses.

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Abstract: We argue for decreasing the use of the word spread when describing the statistical idea of dispersion or variability in introductory statistics courses. In addition, we argue for increasing the use of the word variability as a replacement for spread.

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20 citations

Cites background from "Exploring Students' Conceptions of ..."

...55) as reported by delMas and Liu (2005). Such students would claim that figure 1, a histogram with a smooth top, but large range, would have less spread (or variability) than figure 2, a ‘bumpy’ histogram with small range. Researchers delMas and Liu (2005) also noticed that students believed that more variability or spread was present in a distribution when the bars of a histogram were equally spaced across the entire x-axis (as seen Lexical ambiguity: making a case against spread 57...
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...55) as reported by delMas and Liu (2005). Such students would claim that figure 1, a histogram with a smooth top, but large range, would have less spread (or variability) than figure 2, a ‘bumpy’ histogram with small range....
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Journal Article•DOI•

Robust Understanding of Statistical Variation.

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Susan A. Peters¹•Institutions (1)

University of Louisville¹

31 May 2011-Statistics Education Research Journal

TL;DR: The authors present a framework that captures the complexity of reasoning about variation in ways that are indicative of robust understanding and describes reasoning as a blend of design, data-centric, and modeling perspectives.

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Abstract: This paper presents a framework that captures the complexity of reasoning about variation in ways that are indicative of robust understanding and describes reasoning as a blend of design, data-centric, and modeling perspectives. Robust understanding is indicated by integrated reasoning about variation within each perspective and across perspectives for four elements: variational disposition, variability in data for contextual variables, variability in relationships among data and variables, and effects of sample size on variability. This holistic image of robust understanding of variation arises from existing expository and empirical literature, and additional empirical study.

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20 citations

Book•

Developing an understanding of variation: AP statistics teachers' perceptions and recollections of critical moments.

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Susan A. Peters

01 Jan 2011

18 citations

Cites background from "Exploring Students' Conceptions of ..."

...…spread relative to a center coupled with thoughtful consideration of formal measures of center and spread to reason about data are marks of sophisticated statistical thinking (delMas & Liu, 2005; Groth, 2005) and seen as necessary for deep understandings of variation (Garfield & Ben-Zvi, 2005)....
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...…dynamic conception of distribution that coordinates changes to the relative density of 22 values about the mean with their deviation from the mean (delMas & Liu, 2005), then it would appear that an understanding of mean as a mathematical point of balance is needed to reason about estimating a…...
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...…standard deviations for multiple pairs of distributions by attempting to create rules to generalize patterns of histogram bars to make comparisons (delMas & Liu, 2005). delMas and Liu noted that very few students employed a conceptual approach to coordinate the location of the mean, estimated by…...
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...…deviation, that individual may develop a dynamic conception that coordinates changes to the relative density of values about the mean with their deviation from the mean, a conception suggested by some researchers as necessary for understanding standard deviation (delMas & Liu, 2005, p. 56)....
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Journal Article•DOI•

Statistical Analysis When the Data is an Image: Eliciting Student Thinking About Sampling and Variability

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Margaret A Hjalmarson¹, Tamara J. Moore², Robert delMas³•Institutions (3)

Purdue University¹, University of Minnesota², George Mason University³

31 May 2011-Statistics Education Research Journal

TL;DR: In this paper, a first-year undergraduate engineering activity was held where teams of students were asked to develop a procedure for quantifying the roughness of a surface at the nanoscale.

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Abstract: Results of analysis of responses to a first-year undergraduate engineering activity are presented. Teams of students were asked to develop a procedure for quantifying the roughness of a surface at the nanoscale, which is typical of problems in Materials Engineering where qualities of a material need to be quantified. Thirty-five teams were selected from a large engineering course for analysis of their responses. The results indicate that engagement in the task naturally led teams to design a sampling plan, use or design measures of center and variability, and integrate those measures into a model to solve the stated problem. Team responses revealed misunderstandings that students have about measures of center and variability. Implications for instruction and future research are discussed. First published May 2011 at Statistics Education Research Journal: Archives

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17 citations

Cites background from "Exploring Students' Conceptions of ..."

...DelMas and Liu (2005) provided evidence that experience with a computer environment designed to promote students’ ability to coordinate characteristics of variation of values about the mean moved from simple, one-dimensional understandings of the standard deviation toward more mean-centered…...
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...Items and tasks from research studies on students' understanding of variability and the standard deviation (e.g., Chance et al., 2004; delMas & Liu, 2005; C Reading & Shaughnessy, 2004; Shaughnessy et al., 1999) and of sampling methods (Watson & Kelly, 2005, 2006) could be administered prior to and…...
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Journal Article•DOI•

Statistical reasoning ability, self-efficacy, and value beliefs in a reform based university statistics course

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Aboma Olani, Rink Hoekstra, Evert Harskamp, van der Margaretha Werf

19 Nov 2017-Electronic Journal of Research in Educational Psychology

16 citations

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Journal Article•DOI•

Accommodation of a scientific conception: Toward a theory of conceptual change

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George J. Posner¹, Kenneth A. Strike¹, Peter W. Hewson¹, William A. Gertzog¹•Institutions (1)

Cornell University¹

01 Apr 1982-Science Education

TL;DR: In this paper, a general model of conceptual change is proposed, which is largely derived from current philosophy of science, but which they believe can illuminate * This model is partly based on a paper entitled "Learning Special Relativity: A Study of Intellectual Problems Faced by College Students,” presented at the International Conference Celebrating the 100th Anniversary of Albert Einstein, November 8-10, 1979 at Hofstra University.

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Abstract: It has become a commonplace belief that learning is the result of the interaction between what the student is taught and his current ideas or concepts.’ This is by no means a new view of learning. Its roots can be traced back to early Gestalt psychologists. However, Piaget’s (1929, 1930) early studies of children’s explanations of natural phenomena and his more recent studies of causality (Piaget, 1974) have perhaps had the greatest impact on the study of the interpretive frameworks students bring to learning situations. This research has led to the widespread study of students’ scientific misconceptions.2 From these studies and, particularly, from recent work by researchers such as Viennot ( 1979) and Driver (1 973), we have developed a more detailed understanding of some of these misconceptions and, more importantly, why they are so “highly robust” and typically outlive teaching which contradicts them (Viennot, 1979, p. 205). But identifying misconceptions or, more broadly speaking, “alternative frameworks” (Driver & Easley, 1978), and understanding some reasons for their persistence, falls short of developing a reasonable view of how a student’s current ideas interact with new, incompatible ideas. Although Piaget (1974) developed one such theory, there appears to be a need for work which focuses “more on the actual content of the pupil’s ideas and less on the supposed underlying logical structures” (Driver & Easley, 1978, p. 76). Several research studies have been performed (Nussbaum, 1979; Nussbaum & Novak, 1976; Driver, 1973; Erickson, 1979) which have investigated “the substance of the actual beliefs and concepts held by children” (Erickson, 1979, p. 221). However, there has been no well-articulated theory explaining or describing the substantive dimensions of the process by which people’s central, organizing concepts change from one set of concepts to another set, incompatible with the first. We believe that a major source of hypotheses concerning this issue is contemporary philosophy of science, since a central question of recent philosophy of science is how concepts change under the impact of new ideas or new information. In this article we first sketch a general model of conceptual change which is largely derived from current philosophy of science, but which we believe can illuminate * This article is partly based on a paper entitled “Learning Special Relativity: A Study of Intellectual Problems Faced by College Students,” presented at the International Conference Celebrating the 100th Anniversary of Albert Einstein, November 8-10, 1979 at Hofstra University.

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5,052 citations

"Exploring Students' Conceptions of ..." refers background in this paper

...Reading and Shaughnessy (2004) present evidence of different levels of sophistication in elementary and secondary students’ reasoning about sample variation....
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Journal Article•DOI•

The Role of Anomalous Data in Knowledge Acquisition: A Theoretical Framework and Implications for Science Instruction

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Clark A. Chinn, William F. Brewer¹•Institutions (1)

University of Illinois at Urbana–Champaign¹

01 Mar 1993-Review of Educational Research

TL;DR: In this article, a detailed analysis of the ways in which scientists and science students respond to anomalous data is presented, giving special attention to the factors that make theory change more likely.

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Abstract: Understanding how science students respond to anomalous data is essential to understanding knowledge acquisition in science classrooms. This article presents a detailed analysis of the ways in which scientists and science students respond to such data. We postulate that there are seven distinct forms of response to anomalous data, only one of which is to accept the data and change theories. The other six responses involve discounting the data in various ways in order to protect the preinstructional theory. We analyze the factors that influence which of these seven forms of response a scientist or student will choose, giving special attention to the factors that make theory change more likely. Finally, we discuss the implications of our framework for science instruction.

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1,434 citations

Annual Meeting of the American Educational Research

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James H. McMillan

01 Jan 2001

793 citations

Book•

On the Shoulders of Giants

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Stephen W. Hawking, Johannes Kepler, Galileo Galilei, Nicolaus Copernicus

01 Jan 2002