Necessary Knowledge for Teaching Statistics: Example of the Concept of Variability (original) (raw)
Related papers
The Teaching and Learning of Statistics, 2016
This article introduces a conceptual framework for statistical knowledge for teaching (henceforth SKT), which addresses some noted gaps identified in the research literature on statistics education. It is proposed that the use and adaptation-for the case of statistics-of the model of mathematical knowledge for teaching (henceforth MKT) developed by Ball, Thames & Phelps (2008), as well as an extension of that model-and of almost all the few conceptualizations of SKT proposed to date-addressing some of its limitations, may help to gain a deeper insight into the knowledge needed to teach statistics effectively. In the present paper, the components of this new framework for SKT are elicited, identified and described through a set of tasks that examine teachers' conceptions of variability in diverse statistical contexts (as in Shaughnessy, 2007), as well as teachers' subject matter and pedagogical content knowledge in relation to the statistical ideas involved in such tasks.
International Journal of Computers for Mathematical Learning, 2007
New capabilities in TinkerPlots 2.0 supported the conceptual development of fifth-and sixth-grade students as they pursued several weeks of instruction that emphasized data modeling. The instruction highlighted links between data analysis, chance, and modeling in the context of describing and explaining the distributions of measures that result from repeatedly measuring multiple objects (i.e., the height of the school's flagpole, a teacher's head circumference, the arm-span of a peer). We describe the variety of data representations, statistics, and models that students invented and how these inscriptions were grounded both in their personal experience as measurers and in the affordances of TinkerPlots, which assisted them in quantifying what they could readily display with the computer tool. By inventing statistics, students explored the relation between qualities of distribution and methods for expressing these qualities as a quantity. Attention to different aspects of distribution resulted in the invention of different statistics. Variable invention invited attention to the qualities of ''good'' measures (statistics), thus meshing conceptual and procedural knowledge. Students used chance simulations, built into TinkerPlots, to generate models that explained variability in a sample of measurements as a composition of true value and chance error. Error was, in turn, decomposed into a variety of sources and associated magnitudes-a form of analysis of variance for children. The dynamic notations of TinkerPlots altered the conceptual landscape of modeling, placing simulation and world on more equal footing, as first suggested by Kaput (Journal of Mathematical Behavior, 17(2), 265-281, 1998). Keywords Statistics education Á Modeling Á Learning The discipline of statistics originated in problems of modeling variability (Stigler, 1986). History has not changed all that much: Professional practices of statisticians invariably include efforts to model variability (Wild and Pfannkuch 1999). It is through the contest among alternative models that statistical concepts become more widespread and stable
Exploring variation in measurement as a foundation for statistical thinking in the elementary school
International Journal of STEM Education, 2015
Background: This study was based on the premise that variation is the foundation of statistics and statistical investigations. The study followed the development of fourth-grade students' understanding of variation through participation in a sequence of two lessons based on measurement. In the first lesson all students measured the arm span of one student, revealing pathways students follow in developing understanding of variation and linear measurement (related to research question 1). In the second lesson each student's arm span was measured once, introducing a different aspect of variation for students to observe and contrast. From this second lesson, students' development of the ability to compare their representations for the two scenarios and explain differences in terms of variation was explored (research question 2). Students' documentation, in both workbook and software formats, enabled us to monitor their engagement and identify their increasing appreciation of the need to observe, represent, and contrast the variation in the data. Following the lessons, a written student assessment was used for judging retention of understanding of variation developed through the lessons and the degree of transfer of understanding to a different scenario (research question 3). Results: The results were based either on the application of the hierarchical SOLO model or on non-hierarchical clustering of responses to individual questions in the student workbooks. Students' progress throughout the lessons displayed a wide range of explanations for the estimate of a single student's arm span, general surprise at the variation in measurements, and a large variety of hand-drawn representations based on the values or frequencies of measurements. Many different representations were also created in the software for the single student measurements and for the comparison of measurements for the two scenarios. Although the students' interpretations of their plots were generally more basic than sophisticated, the results of the assessment indicated that many students had developed the ability to transfer their appreciation of variation to another context and could clearly explain the meaning of variation. Conclusions: The findings highlight the importance of an early focus on variation and distribution, with meaningful activities that motivate students to conduct and observe measurements, together with creating both hand-drawn and software representations to relate their experiences.
Development of the concept of statistical variation: An exploratory study
Mathematics Education Research Journal, 2000
An appreciation of variation is central to statistical thinking, but very little research has focused directly on students' understanding of variation. In this exploratory study, four students from each of grades 4, 6, 8, and 10 were interviewed individually on aspects of variation present in three settings. The first setting was an isolated random sampling situation, whereas the other two settings were real world sampling situations. Four levels of responding were identified and described in relation to developing concepts of variation. hnplicatioris for teaching and future research on variation are considered.
Data analysis or how high school students “read” statistics
Statistics Education and the Communication of Statistics International Association for Statistical Education Satellite Conference
In most countries, statistics are included in the mathematics curriculum and taught by mathematics teachers. This leads to students learning the elements of statistical concepts as mathematical and to more emphasis placed on being able to compute different measures (e.g. mean, median, standard deviation) rather than their meaning and use. Moreover, in Quebec, the high school curriculum favours a scattered presentation of statistical concepts: tables and simple graphical representations are seen in the first year; averages, medians and histograms in the third; position measures in the fourth and some aspects of correlation and standard deviation are seen in the fifth. Some elements of probability are seen in the second year. But “statistics requires a different kind of thinking” (Cobb & Moore, 1997). Is it possible by making students compute statistical measures to foster the development of statistical thinking and prepare to draw conclusions from different data sets - all important ...
Statistical variability: Comprehension of children in primary school
2015
This study examined the understanding of 48 Brazilians students of 2nd and 5th grades (seven and ten years old) of statistical variability of data in bar charts. A Piaget’s clinical interview was conducted involving four activities of variability: description or explanation of the variability; representation of variability; prediction results from the variability of the data; comparison between data sets. Students showed ease in recognizing endpoints, but did not make predictions based on what they had observed. The representation of the variability was shown to be an important factor in data interpretation. Make comparison between data sets was complex for most students. Therefore, it is necessary to promote interrelationship among different aspects in order to make students reflect on the data and predictions.
Where do students get lost? The concept of variation
Many college students have difficulties in understanding and making connections among the main concepts of statistics. Compounding the difficulties is the misconception of a variety of statistical concepts that students hold even before taking any statistics course. It is, thus, crucial to investigate how the understanding of statistical concepts is constructed and at which stage students start to lose making connections among various concepts. This article reports some findings from our study of investigating the path of learning statistical concepts, specifically on how students learn the concept of variation. We focus on investigating the missing connections about their understanding of variation. The framework of statistical thinking, PPDAC investigative cycle, is used as our guideline for analyzing our interview data.
Pfannkuch (1997) contends that variation is a critical issue throughout the statistical inquiry process, from posing a question to drawing conclusions. This is particularly true for K-6 teachers when they attempt to use the process of statistical investigation as a means of teaching and learning across the spectrum of the K-6 curricula. In this context statistical concepts and ideas are taught and learned in conjunction with the important content area ideas and concepts. For a K-6 teacher, this means that the investigation must not only be planned in advance, but also aimed at being responsive to students. The potential for surprise questions, unanticipated responses and unintended outcomes is high, and teachers need to "think on their feet" statistically and react immediately in ways that accomplish content objectives, as well as convey correct statistical principles and reasoning. The intellectual demands in this context are no different than in other instances where tea...
This paper examines ways in which coherent reasoning about key concepts such as variability, sampling, data, and distribution can be developed as part of statistics education. Instructional activities that could support such reasoning were developed through design research conducted with students in grades 7 and 8. Results are reported from a teaching experiment with grade 8 students that employed two instructional activities in order to learn more about their conceptual development. A "growing a sample" activity had students think about what happens to the graph when bigger samples are taken, followed by an activity requiring reasoning about shape of data. The results suggest that the instructional activities enable conceptual growth. Last, implications for teaching, assessment and research are discussed.
Introducing Students to Data Representation and Statistics
I describe the design and iterative implementation of a learning progression for supporting statistical reasoning as students construct data and model chance. From a disciplinary perspective, the learning trajectory is informed by the history of statistics, in which concepts of distribution and variation first arose as accounts of the structure inherent in the variability of measurements. Hence, students were introduced to variability as they repeatedly measured an attribute (most often, length), and then developed statistics as ways of describing "true" measure and precision. The design of the learning progression was guided by several related principles: (a) posing a series of tasks and situations that students perceived as problematic, thus creating a need for developing mathematical understanding as a means of resolving prospective impasses; (b) creating opportunities for developing representational fluency and meta-representational competence as constituents of concep...