Journal Article•
Exploring Students' Conceptions of the Standard Deviation
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.
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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TL;DR: A broad overview of research conducted in ensemble perception can be found in this paper, where the authors discuss how principles of ensemble encoding have been applied to the research in data visualization, and showcase the barriers graphs can pose to learning statistical concepts, using histograms as a specific example.
Abstract: In the age of big data, we are constantly inventing new data visualizations to consolidate massive amounts of numerical information into smaller and more digestible visual formats. These data visualizations use various visual features to convey quantitative information, such as spatial position in scatter plots, color saturation in heat maps, and area in dot maps. These data visualizations are typically composed of ensembles, or groups of related objects, that together convey information about a data set. Ensemble perception, or one’s ability to perceive summary statistics from an ensemble, such as the mean, has been used as a foundation for understanding and explaining the effectiveness of certain data visualizations. However, research in data visualization has revealed some perceptual biases and conceptual difficulties people face when trying to utilize the information in these graphs. In this tutorial review, we will provide a broad overview of research conducted in ensemble perception, discuss how principles of ensemble encoding have been applied to the research in data visualization, and showcase the barriers graphs can pose to learning statistical concepts, using histograms as a specific example. The goal of this tutorial review is to highlight possible connections between three areas of research—ensemble perception, data visualization, and statistics education—and to encourage research in the practical applications of ensemble perception in solving real-world problems in statistics education.
7 citations
TL;DR: The authors investigated the conceptual understanding of measures of spread among community college students in an introductory statistics course and found that students' ability to organize concepts of spread in a way that is meaningful to them individually was significant.
Abstract: This study investigated the conceptual understanding of measures of spread among community college students in an introductory statistics course. The course is centered around deemphasizing computational skills and focused, rather, on development of conceptual understanding. Open-ended questions were developed to explore and assess students' conceptual understanding of measures of spread. A detailed analysis of the students' responses is presented to reveal the range of students' conceptions of the measures of spread. The analysis of a wide variety of responses provides evidence of the students' ability to organize concepts of spread in a way that is meaningful to them individually. Some common student misconceptions revealed by this study should be examined closely and taken into consideration to promote students' development of understanding of spread.
7 citations
Cites methods from "Exploring Students' Conceptions of ..."
...The item Q5 was adapted from Delmas & Liu (2005) and modified by taking out the labels indicating the location and value of the mean and the value of the standard deviation....
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Posted Content•
TL;DR: This paper used think-aloud interviews to explore student reasoning and iteratively draft questions to evaluate student learning and understand how students think, in which students answer questions while narrating their thinking aloud.
Abstract: As undergraduate statistics education rapidly changes to incorporate new topics and skills, statistics educators need tools to evaluate student learning and understand how students think. Think-aloud interviews, in which students answer questions while narrating their thinking aloud, are a valuable education research tool for detecting misconceptions and developing robust assessments.
While think-aloud interviews have been widely used for education research in other fields, in statistics education they have been primarily used to vet concept inventory questions for confusing wording or poor design. Here, we argue for their much more comprehensive use to explore student reasoning and iteratively draft questions. To motivate the use of think-aloud interviews, we describe two case studies on correlation, causation, and sampling distributions. In these, think-alouds revealed unexpected student reasoning and suggested new ways to assess difficult statistical topics. These case studies illustrate the usefulness of think-aloud interviews for studying student thinking, and we argue for their wider use in statistics education research.
7 citations
01 Jan 2006
TL;DR: For example, this article found that many students had difficulty integrating their knowledge of variance and heritability, and could not apply their knowledge to the solution of practical problems in the field of quantitative genetics.
Abstract: Genetics has been identified as a subject area which many students find difficult to comprehend. The researcher, who is also a lecturer at the University of KwaZulu-Natal, had noted over a number of years that students find the field of quantitative genetics particularly challenging. The aim of this investigation was two-fold. Firstly, during the diagnostic phase of the investigation, to obtain empirical evidence on the nature of difficulties and alternative conceptions that may be experienced by some students in the context of quantitative genetics. Secondly, to develop, implement and assess an intervention during the remediation phase of the study which could address the identified difficulties and alternative conceptions. The research was conducted from a human constructivist perspective using an action research approach. A mixed-method, pragmatic paradigm was employed. The study was conducted at the University of KwaZulu-Natal over four years and involved third-year students studying introductory modules in quantitative genetics. Empirical evidence of students' conceptual frameworks, student difficulties and alternative conceptions was obtained during the diagnostic phase using five research instruments. These included: free-response probes, multiple-choice diagnostic tests, student-generated concept maps, a word association study and student interviews. Data were collected, at the start and completion of the modules, to ascertain the status of students' prior knowledge (prior knowledge concepts), and what they had learnt during the teaching of the module (quantitative genetics concepts). Student-generated concept maps and student interviews were used to determine whether students were able to integrate their knowledge and link key concepts of quantitative genetics. This initial analysis indicated that many students had difficulty integrating their knowledge of variance and heritability, and could not apply their knowledge of quantitative genetics to the solution of practical problems. Multiple-choice diagnostic tests and interviews with selected students were used to gather data on student difficulties and alternative conceptions. The results suggested that students held five primary difficulties or alternative conceptions with respect to prior knowledge concepts: (1) confusion between the terms variation and variance; (2) inappropriate association of heterozygosity with variation in a population; (3) inappropriate association of variation with change; (4) inappropriate association of equilibrium with inbred populations and with values of zero and one; and, (5) difficulty relating descriptive statistics to graphs of a normal distribution. Furthermore, three major difficulties were detected with respect to students understanding of quantitative genetics concepts: (1) students frequently confused individual and population measures such as breeding value and heritability; (2) students confused the terms heritability and inheritance; and, (3) students were not able to link descriptive statistics such as variance and heritability to
7 citations
Cites background or methods or result from "Exploring Students' Conceptions of ..."
...Furthermore, it has been postulated that some difficulties may be due, in part to the instructional neglect of the core concept of variation (Meletiou 2000) and the lack of understanding of graphical distributions (DelMas and Liu 2005; Makar and Confrey 2005)....
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...2002) and to co-ordinate and link foundational concepts (DelMas and Liu 2005)....
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...r--Stu-der1ts-understandin-gOfmeasures-oi-------- -Tri--orderfor-stu-denis-to-have-some-------------- GraphiCaTcfistrTbutlons--sh-o-uicfbe-us-ed--to-------------I variation, such as the standard deviation, meaningful understanding of the standard provide a visual structure with which students i are determined only by the ability to deviation they must perceive that a are able to conceive the aggregate features of I calculate the value using a particular histogram depicts one variable and the data sets (Bakker 2004; Pfannkuch 2005) such I procedure (DelMas and Liu 2005) or as a frequency of its possible values, as measures of centre and variation (Ben-Zvi I characteristic shape ill a distribution understand the concept of a mean as well 2004)....
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...In support of the results obtained in this investigation, DelMas and Liu (2005) point ....
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...as the idea of values deviating from the "hills" or "bumps" can be used by students in i mean (DelMas and Liu 2005)....
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TL;DR: In this article, the authors designed measurement activities on the basis of authentic professional practices in which linear regression is used, to study considerations of variability by students in Grade 12 (aged 17-18).
Abstract: Measurement activities were designed in this study on the basis of authentic professional practices in which linear regression is used, to study considerations of variability by students in Grade 12 (aged 17-18). The question addressed in this article is: In what ways do secondary students consider variability within these measurement activities? Analysis of students' reasoning during these activities in one classroom (N = 13) suggests that students considered variability in four ways: noticing and acknowledging variability, measuring variability, explaining variability, and using investigative strategies to handle variability. We conclude that the measurement tasks based on authentic professional practices helped students to reason with relevant aspects of variability. Finally, we discuss curricular and research implications.
6 citations
Cites background from "Exploring Students' Conceptions of ..."
...Graphical representation of the data can help students to see the variability around the regression line and develop their understanding of variability (delMas & Liu, 2005)....
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References
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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.
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.
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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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.
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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