Statistics as unbiased estimators: exploring the teaching of standard deviation

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This manuscript presents findings from a study about the knowledge for and planned teaching of standard deviation. We investigate how understanding variance as an unbiased (inferential) estimator – not just a descriptive statistic for the variation (spread) in data – is related to teachers’ instruction regarding standard deviation, particularly around the issue of division by n-1. In this regard, the study contributes to our understanding about how knowledge of mathematics beyond the current instructional level, what we refer to as nonlocal mathematics, becomes important for teaching. The findings indicate that acquired knowledge of nonlocal mathematics can play a role in altering teachers’ planned instructional approaches in terms of student activity and cognitive demand in their instruction.

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... Formula untuk perhitungan root mean square error (RMSE) dapat ditunjukkan pada persamaan (1), (Li et al., 2015): Selain perhitungan root mean square error (RMSE), perkiraan nilai parameter standar deviasi juga penting dalam menentukan standar akurasi dari pekerjaan yang telah dilakukan dengan definisi sebagai nilai statistik dalam menentukan persebaran data dalam sampel, serta pendekatan titik data individu ke rata-rata nilai sampel (Hidayat et al., 2019). Standar deviasi dapat dilakukan perhitungan dengan menggunakan persamaan seperti dibawah ini (Wasserman et al., 2017): ...
... This paper is motivated by our teaching statistical inferences about the population standard deviation σ for undergraduate students (cf. [1], [4], [5]). The common rendition is to use the sample-variance s 2 to estimate σ 2 by the fact that the mathematical expectation E s 2 = σ 2 , thereby s to estimate σ, an approach that is so universally adopted that few studies exclusively on s have been found by our extensive literature research (see [4]). ...
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While teacher content knowledge is crucially important to the improvement of teaching and learning, attention to its development and study has been uneven. Historically, researchers have focused on many aspects of teaching, but more often than not scant attention has been given to how teachers need to understand the subjects they teach. Further, when researchers, educators and policy makers have turned attention to teacher subject matter knowledge the assumption has often been that advanced study in the subject is what matters. Debates have focused on how much preparation teachers need in the content strands rather than on what type of content they need to learn. In the mid-1980s, a major breakthrough initiated a new wave of interest in the conceptualization of teacher content knowledge. In his 1985 AERA presidential address, Lee Shulman identified a special domain of teacher knowledge, which he referred to as pedagogical content knowledge. He distinguished between content as it is studied and learned in disciplinary settings and the "special amalgam of content and pedagogy" needed for teaching the subject. These ideas had a major impact on the research community, immediately focusing attention on the foundational importance of content knowledge in teaching and on pedagogical content knowledge in particular. This paper provides a brief overview of research on content knowledge and pedagogical content knowledge, describes how we have approached the problem, and reports on our efforts to define the domain of mathematical knowledge for teaching and to refine its sub- domains.
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This study (N = 48) examined the relationship between preservice secondary teachers’ subject-matter expertise in mathematics and their judgments of students’ algebra problem-solving difficulty. As predicted by the “expert blind spot” hypothesis, participants with more advanced mathematics education, regardless of their program affiliation or teaching plans, were more likely to view symbolic reasoning and mastery of equations as a necessary prerequisite for word equations and story problem solving. This view is in contrast with students’ actual performance patterns. An examination across several subject areas, including mathematics, science, and language arts, suggests a common pattern. This article considers how teachers’ developmental views may influence classroom practice and professional development, and calls into question policies that seek to streamline the licensure process of new teachers on the basis of their subject-matter expertise.
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This paper draws on videotapes of mathematics lessons prepared and conducted by pre-service elementary teachers towards the end of their initial training at one university. The aim was to locate ways in which they drew on their knowledge of mathematics and mathematics pedagogy in their teaching. A grounded approach to data analysis led to the identification of a ‘knowledge quartet’, with four broad dimensions, or ‘units’, through which mathematics-related knowledge of these beginning teachers could be observed in practice. We term the four units: foundation, transformation, connection and contingency. This paper describes how each of these units is characterised and analyses one of the videotaped lessons, showing how each dimension of the quartet can be identified in the lesson. We claim that the quartet can be used as a framework for lesson observation and for mathematics teaching development.
The notion of practice-based models for mathematical knowledge for teaching has played a pivotal role in the conception of teacher knowledge. In this work, teachers' knowledge of mathematics that is outside the scope of what is being taught (nonlocal mathematics) is considered more explicitly. Drawing on a cognitive model for the development of mathematical knowledge for teaching, this paper explores the implications for the underlying theory being applied to (nonlocal) knowledge outside the scope what is being taught as being influential for the teaching of (local) mathematics. The work provides a unique perspective on the role of advanced mathematics with respect to secondary teaching. Considerations for mathematics teacher education are discussed.
This paper explores the potential for aspects of abstract algebra to be influential for the teaching of school algebra (and early algebra). Using national standards for analysis, four primary areas common in school mathematics – and their progression across elementary, middle, and secondary mathematics – where teaching may be transformed by teachers' knowledge of abstract algebra are developed. In each of the four content areas (arithmetic properties, inverses, structure of sets, and solving equations), descriptions and examples of the transformational influence on teaching these topics are used to depict and support ways that study of more advanced mathematics can influence teachers' practice. Implications for the mathematical preparation and professional development of teachers are considered.
The aim of this chapter is to summarise the major studies related to teachers’ understanding of variability in both data analysis and chance contexts. Since there is a relation between this research and previous studies dealing with students, some results on students’ reasoning on variation are also described. At the end some recommendations for teaching and research are presented.
This paper discusses the thought processes involved in statistical problem solving in the broad sense from problem formulation to conclusions. It draws on the literature and in-depth interviews with statistics students and practising statisticians aimed at uncovering their statistical reasoning processes. From these interviews, a four-dimensional framework has been identified for statistical thinking in empirical enquiry. It includes an investigative cycle, an interrogative cycle, types of thinking and dispositions. We have begun to characterise these processes through models that can be used as a basis for thinking tools or frameworks for the enhancement of problem-solving. Tools of this form would complement the mathematical models used in analysis and address areas of the process of statistical investigation that the mathematical models do not, particularly areas requiring the synthesis of problem-contextual and statistical understanding. The central element of published definitions of statistical thinking is "variation". We further discuss the role of variation in the statistical conception of real-world problems, including the search for causes.
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