Randall E. Groth

Preservice Elementary Teachers' Conceptual and Procedural Knowledge of Mean, Median, and Mode.

This article describes aspects of the statistical content knowledge of 46 preservice elementary school teachers. The preservice teachers responded to a written item designed to assess their knowledge of mean, median, and mode. The data produced in response to the written item were examined in light of the Structure of the Observed Learning Outcome (SOLO) taxonomy (Biggs & Collis, 1982, 1991) and Mas (1999) conception of Profound Understanding of Fundamental Mathematics (PUFM). The article describes 4 levels of thinking in regard to comparing and contrasting mean, median, and mode. Several different categories of written definitions for each measure of central tendency are also described. Connections to previous statistical thinking literature are discussed, implications for teacher education are given, and directions for further research are suggested.

Characterizing Key Developmental Understandings and Pedagogically Powerful Ideas within a Statistical Knowledge for Teaching Framework.

A hypothetical framework to characterize statistical knowledge for teaching (SKT) is described. Empirical grounding for the framework is provided by artifacts from an undergraduate course for prospective teachers that concentrated on the development of SKT. The theoretical notion of “key developmental understanding” (KDU) is used to identify landmarks in the development of SKT subject matter knowledge. Sample KDUs are given for the subject matter knowledge categories of common content knowledge, specialized content knowledge, and horizon knowledge. The theoretical notion of “pedagogically powerful idea” is used to describe how KDUs must be transformed to become useful in teaching. Examples of pedagogically powerful ideas for the pedagogical content knowledge categories of knowledge of content and teaching and curriculum knowledge are provided. Knowledge of content and students is hypothesized as a basis for the development of pedagogically powerful ideas.

High School Students' Levels of Thinking in Regard to Statistical Study Design

The study describes levels of thinking in regard to the design of statistical studies. Clinical interviews were conducted with 15 students who were enrolled in high school or were recent high school graduates, and who represented a range of mathematical backgrounds. During the clinical interview sessions students were asked how they would go about designing studies to answer several different quantifiable questions. Several levels of sophistication were identified in their responses, and are discussed in terms of the Biggs and Collis (1982, 1991) cognitive model.

Understanding Teachers' Resistance to the Curricular Inclusion of Alternative Algorithms

This study focuses on a group of practitioners from a school district that adopted reform-oriented curriculum materials but later rejected them, partially due to the inclusion of alternative algorithms in the materials. Metaphors implicit in a conversation among the group were analysed to illuminate their perspectives on instructional issues surrounding alternative algorithms. Several possible sources of resistance to folding alternative algorithms into instruction were found, including the ideas that: successful learning does not involve struggling with mathematics, the teacher’s role in the classroom is primarily to present information, and that mathematics learning progresses according to a fixed sequence of levels.

An exploration of two online approaches to mathematics teacher education

The purpose of the present study was to examine the nature of the discourse generated by two different online approaches to mathematics teacher professional development. Thirty mathematics teachers participated in online activities involving analysis and discourse about artifacts of teaching practice. Half were randomly assigned to a group that analyzed and discussed students work samples, and the other half to a group focused upon descriptions of classroom teaching episodes. Teachers formed threads of conversation on asynchronous discussion boards as they considered the different aspects of each artifact. Within the threads, different orientations toward the reform agenda in mathematics education were shown. Some messages were strongly rooted in the reform paradigm, others in the traditional paradigm, and many others contained elements characteristic of each paradigm. The distribution and characteristics of the messages reflected teachers’ frequent attempts to try to reconcile the largely incompatible paradigms undergirding reform-oriented and traditional approaches to mathematics instruction. This paper describes the nature of the discourses occurring within each group of teachers with the aim of providing empirical grounds to inform the actions of teacher educators and researchers designing online learning environments that attempt to bring teachers more fully into the reform-oriented discourse about teaching mathematics. Two Online Approaches 3 Teachers shape and are shaped by conversations with other professionals (Rust & Orland, 2001). Not all of these professional discourses, however, are necessarily productive. Putnam and Borko (2000) noted, “patterns of classroom teaching and learning have historically been resistant to fundamental change, in part because schools have served as powerful discourse communities that enculturate participants (students, teachers, administrators) into traditional school activities and ways of thinking” (p. 8). Mathematics education researchers have called for changes to “traditional school activities and ways of thinking” for several decades (Brownell, 1954; Hiebert & Carpenter, 1992). However, traditional practices in mathematics have remained entrenched and resistant to change (Jacobs, Hiebert, Givvin, Hollingsworth, Garnier, & Wearne, 2006). This phenomenon suggests that many teachers have been enculturated into professional discourses that value traditional rather than reform-oriented mathematics instruction. Ross, McDougall, and Hogaboam-Gray (2002) provided a helpful conceptualization of how traditional and reform-oriented ideas in mathematics education differ from one another. They identified ten dimensions distinguishing traditional approaches to teaching mathematics from reform-oriented ones: (i) Traditional programs tend to focus heavily on number and operation to the exclusion of multiple strands of mathematics; (ii) Reform-oriented discourse emphasizes that all students have the ability to do mathematics, whereas traditional discourse does not; (iii) The main task in traditional classrooms is to apply learned procedures to decontextualized problems, whereas reform-oriented curricula emphasize learning from contextualized, open-ended tasks amenable to multiple solutions; (iv) Teaching in traditional classrooms rests on a “transmission” theory of learning, whereas reform-oriented classrooms emphasize students’ talk as a means of knowledge construction; (v) The teacher in a reform-oriented classroom is a co-learner of mathematics and not the sole knowledge expert; (vi) Manipulatives, calculators, computers, and Two Online Approaches 4 other such tools are featured in reform-oriented classrooms but not traditional ones; (vii) Student-to-student interaction is encouraged in reform classrooms but seen as an off-task distraction by traditional teachers; (viii) Continual formative assessments shape reform-oriented instruction, whereas assessment in traditional instruction relies solely on summative assessments; (ix) Reform-oriented teachers conceive of mathematics as a dynamic rather than static discipline; (x) Building students’ mathematical self-confidence is an explicit goal in reform-oriented instruction but not in traditional. Several types of professional development programs have been designed for the purpose of bringing teachers into the discourse surrounding reform-oriented mathematics instruction. In reviewing several different programs, Kilpatrick, Swafford, and Findell (2001) argued that the focus should be upon teachers’ construction of mathematical knowledge, knowledge of students, and knowledge of classroom practice. They identified three approaches with the potential to achieve these goals: Some of these programs begin with mathematical ideas from the school curriculum and ask teachers to analyze those ideas from the learners’ perspective. Other programs use students’ mathematical thinking as a springboard to motivate teachers’ learning of mathematics. Still others begin with teaching practice and move toward a consideration of mathematics and students’ thinking (p. 385). This paper is an exploration of the latter two approaches in an online environment. Whereas the first approach begins with the analysis of mathematics problems, the latter two begin with the analysis of practice-related artifacts like student work samples and written cases of teaching. Two Online Approaches 5 Shifting Mathematics Teachers’ Discourse through Professional Development Discussion of artifacts of practice like student work samples and teaching cases can help shift teachers’ discourse toward reform-oriented ideas by helping them form empiricallygrounded general conclusions and theories about teaching through the consideration of specific examples (Lampert & Ball, 1998). From the perspective that language mediates knowledge construction (Bakhtin, 1981; Wertsch, 1997), discourse among teachers about such artifacts is a key mechanism through which the formation of such general conclusions takes place. In particular, as teachers discuss practice with one another, they can challenge one another’s interpretations of curriculum-related materials and prompt one another to extend local knowledge to the theoretical level (Manouchehri, 2002). Artifacts of practice facilitate such discourse by providing a common referent that allows for various interpretations to emerge during conversations (Newman, Griffin, & Cole, 1989). In the present study, the two types of practice-based artifacts of practice utilized as catalysts for discourse were student work samples and teaching cases. Discourse Catalyst 1: Student Work Samples Borasi and Fonzi (2002) described mathematics teacher education that focuses on student work samples in the following terms: Teachers analyze students’ thinking as revealed in students’ written assignments, thinkaloud problem-solving tasks, class discussions, and clinical interviews...teachers learn to observe various types of student mathematical activity and to interpret what they observe, with the ultimate goal of enhancing their students’ learning opportunities (p. 53). Various programs have incorporated elements of this approach, including Cognitively Guided Instruction (CGI) (Carpenter, Fennema, Peterson, Chiang, & Loef, 1989), Quantitative Two Online Approaches 6 Understanding: Amplifying Student Achievement and Reasoning (QUASAR) (Silver & Stein, 1996), and Integrating Mathematical Assessment (IMA) (Gearhardt, Saxe, & Stipek, 1995). Analyzing students’ work samples can motivate discussions of mathematics because the artifacts analyzed contain mathematics problems themselves. Understanding the problems presented in the artifacts is a prerequisite to understanding students’ thinking about them (Kilpatrick, Swafford, & Findell, 2001). For example, the CGI approach typically begins with an analysis of mathematics problems and problem types before proceeding to an examination of students’ thinking about them (Carpenter, Fennema, Franke, & Levi, 1999). Analyzing samples of students’ work can also motivate discourse about students. In particular, teachers have the opportunity to observe that students often have thought patterns that diverge from their own and from the structure of the discipline of mathematics (Even & Tirosh, 2002). Finally, although instructional practices are not the initial focus of professional development that deals with the analysis of students’ work, shifts in teachers’ pedagogical practices can occur. Teachers participating in CGI professional development, for example, adapted their practices to include teaching via problem-solving and student-to-student discourse (Franke, Fennema, & Carpenter, 1998; Lubinski & Jaberg, 1998). Discourse Catalyst 2: Teaching Cases The case approach to teacher education is one in which, “teachers analyze and discuss ‘cases’ that are written narratives or video excerpts of events that are used as catalysts for raising and discussing important issues regarding school mathematics reform” (Borasi & Fonzi, 2002, p. 67). While excerpts of students’ work may be included in a case, the primary focus is upon showing the social interactions between teachers and students. Programs incorporating this Two Online Approaches 7 approach include the Mathematics Case Methods Project (Barnett, 1991; Barnett, Goldstein, & Jackson, 1994) and Teaching for the Big Ideas (Schifter, Bastable, & Russell, 1997). Cases designed for mathematics teacher education are generally constructed around descriptions of the teaching and learning of important mathematics problems. This allows for discussion of the complexity of concepts in the school mathematics curriculum as a group reads about teachers’ and students’ different approaches to the problems (Barnett, 1998). Through this process, teachers can identify the limits of their own mathematical content knowledge (Davenport & Sassi, 1995) and begin to acknowledge the need to continue their mathemat

Research on Statistics Teachers’ Cognitive and Affective Characteristics

Research about statistics teachers faces a unique challenge. It is not sufficient to account only for teachers’ cognition and affect in regard to the subject matter of statistics. We also need to understand the personal characteristics teachers have related to developing the statistics-related cognitive and affective traits of students. Toward this end, researchers have supplemented studies of teachers’ subject matter knowledge with studies of their pedagogical content knowledge, technological pedagogical statistical knowledge (TPSK), beliefs, and attitudes relevant to teaching statistics. We describe existing models and empirical research concerning each of these characteristics. Written assessments, interview techniques, and observation methods for assessing teachers’ development of the characteristics are described as well. Strengths and limitations of existing models and assessments are discussed. We conclude by summarizing statistics teacher education research in the specific areas of data, uncertainty, and statistical inference. We close with recommendations about how statistics teachers’ cognitive and affective characteristics may be developed by learning from teaching practice, immersion in statistical content, and use of technological environments. Opportunities and directions for future research appear throughout the chapter. Some specific research needs include progressive development of improved models for statistics teachers’ cognition and affect along with robust qualitative and quantitative assessment tools.

Prospective Teachers' Procedural and Conceptual Knowledge of Mean Absolute Deviation

Abstract The recommendation to study statistical variation has become prevalent in recent curriculum documents. At the same time, research on teachers’ knowledge of variation is in its beginning stages. This study investigated prospective teachers’ knowledge in regard to a specific measure of statistical variation that is new to many curriculum documents: the mean absolute deviation (MAD). Seventy-six prospective teachers participated in the study. Participants exhibited various procedural and conceptual characteristics in their thinking about the MAD. The majority were able to successfully select and carry out a procedure for computing the MAD. However, some had difficulty dealing with procedures for absolute deviations, and others confused the procedure for the MAD with the procedure for a different descriptive statistic. Conceptually, participants offered a variety of interpretations of the MAD, with some demonstrating deep understanding of the measure and others demonstrating shallower understanding or misconceptions. Those who demonstrated the strongest conceptual knowledge of the MAD also exhibited sound procedural understanding, suggesting that the two types of knowledge are intertwined in the process of fully understanding the measure.

Unpacking Implicit Disagreements Among Early Childhood Standards for Statistics and Probability

Numerous recent curriculum documents around the world recommend that children begin to develop understanding of probability and statistics during early childhood and primary school. Although there is widespread agreement that such learning should occur, standards documents are not uniform in their specific recommendations. In particular, there are implicit disagreements about the roles of student-posed statistical questions, probability language, and variability in children’s learning. Unpacking these implicit disagreements is in the interest of teachers, researchers, and curriculum developers because it can stimulate thought and debate about the proper emphasis for the concepts in standards documents. This chapter will help define the space for such thought and debate by summarizing how some key concepts are addressed differently in various early learning standards for probability and statistics. Defensible interpretations of the research literature are considered. Strategies teachers and curriculum developers can use to cope with situations in which standards documents conflict with desirable learning goals for children are also described. Boundary objects, which allow related communities of practice to operate jointly in absence of consensus, are discussed as a means for advancing teaching and research despite the existence of disagreement. Suggestions for working toward a greater degree of consensus across early childhood standards for statistics and probability are also offered.

A framework for characterizing students’ cognitive processes related to informal best fit lines

ABSTRACT Informal best fit lines frequently appear in school curricula. Previous research collectively illustrates that the adjective informal does not translate to cognitive simplicity. Using existing literature, we create a hypothetical framework of cognitive processes associated with studying informal best fit lines. We refine the framework using data from a cycle of design-based research about building students’ understanding of covariation. The refined framework includes student thinking processes for signifying observations as data, signifying data with scatterplots, perceiving aggregates in scatterplots, perceiving trends in aggregates, signifying trends with straight lines, and using straight lines as estimation tools. We explain how students’ perceptions of aggregates can proceed from the inside-out as well as from the outside-in. We also demonstrate how the amounts of variation encountered at different points in time and the extent to which students perceive straight lines to be abbreviations of linear covariation are important considerations for teaching and research.