Efforts to develop Pre-K-12 data science curricula have accelerated in recent years. Data science curriculum documents generally specify desired student learning outcomes, but they do not always specify what teachers need to know and do to support students in attaining the outcomes. This article addresses the issue by offering a process that can be used to set and refine goals for data science teacher education. The process draws upon research and theory about mathematical knowledge for teaching, statistical knowledge for teaching, and technological pedagogical statistical knowledge. These types of knowledge are defined and exemplified in the context of Pre-K-12 data science. Examples of teaching actions that require the different types of knowledge to support students' learning are given to illustrate how meaningful goals for data science teacher education can be set. An agenda for future research and development is also proposed. The proposed agenda includes generating curriculum-specific data science teacher education goals, identifying and prioritizing teacher education strategies that have the greatest impact on students' learning, and continuously refining and improving theory and practice in data science teacher education using empirical data.

Pedagogical content knowledge (PCK) traditionally has been used in research concerning prospective and practicing classroom teachers. This essay argues that PCK is also relevant to other professions including those advancing quantitative reasoning (QR). To illustrate, the case of PCK for teaching QR is considered. Those in fields such as public health, journalism, meteorology, and government increasingly find themselves responsible for helping the public understand an ever-growing amount of quantitative information that has a bearing on societal well-being. Several examples illustrate how such professionals’ responsibilities require knowing prevalent QR patterns in society, strategies for fostering sound reasoning, and the general nature of school curricula. Professional organizations in education, colleges of education, and educational researchers would benefit from expanding PCK research to encompass multiple professions. These expanded research and development efforts would simultaneously address urgent societal needs.

"Lesson study" has garnered considerable attention from educational researchers and practitioners as a promising method for improving instruction. At its core, lesson study reflects a collaborative inquiry process, grounded in cycles of planning, enacting, and reflecting, that promotes teacher learning through active engagement in lesson refinement. In this article, we leverage lesson study as an exemplar case to foreground an examination and application of theory-based evaluation. We overview a novel evaluation framework contextualized to this multidimensional instructional program and draw out the contributions of theory-based evaluation to the study of program effects. To ground the description of this framework, we summarize representative findings from our ongoing evaluation of a lesson study-based instructional program and discuss processes for selecting, using, and managing evidence sources as well as determining causal strands supporting program outcomes. We conclude by discussing lessons learned and implications for theory-based evaluation of complex programs.

Learning to interpret data in context is an important educational outcome. To assess students' attainment of this outcome, it is necessary to examine the interplay between their contextual and statistical reasoning. We describe a research method designed to do so. The method draws upon Toulmin's (1958, 2003) model of argumentation for the first stage of qualitative data analysis and the Structure of the Observed Learning Outcome (SOLO) (Biggs & Collis, 1991) model for the second stage. Toulmin analyses help identify the justifications and expressions of uncertainty students provide in their interpretive arguments. Subsequent analyses based on the multi-modal conceptualization of SOLO help characterize the quality of student arguments relative to one another. Existing literature and an empirical example are drawn upon to explain how the Toulmin and SOLO models can be used in tandem to analyze students' interpretations of contextualized data. We also explain how pairing Toulmin and SOLO can address theoretical and practical limitations that arise when using just one of the two models on its own.

In this report, we analyze students’ learning of compound probability by describing connections they generated during individual interviews and group lessons. Several of their connections were compatible with the development of expertise, such as recognizing the need to determine sample spaces across a variety of situations and noting structural similarities among tasks, even when their task solutions were incomplete from a normative standpoint. Students reasoned about dimensions of context, variation, mathematical structure, sample space, and probability quantification. We describe the extent to which they coordinated these dimensions. We also describe teaching moves, such as posing idealized situations and shifting to structurally similar tasks, which prompted students to attend to multiple relevant task dimensions.

The Journal of Statistics and Data Science Education (JSDSE) has a history of initiatives to expand its readership and impact. In this article, we reflect on JSDSE’s relevance to K-12 education and how it might increase its influence in this area. We characterize JSDSE’s K-12 impact partially in terms of its ability to foster statistical knowledge for teaching (SKT). We also introduce a new construct, statistical knowledge for the transformation of teaching (SKT 2), to more fully capture the K-12 contributions JSDSE can make. Our analysis draws upon the perspectives of an experienced high school teacher (the first author) and a statistics education researcher (the second author). The first author of the article surveyed recent JSDSE issues to identify and reflect upon articles with implications for his teaching practice. The second author framed the first author’s article reflections in terms of their connections to SKT and SKT2. Drawing upon our collaborative analysis, we propose strategies to help JSDSE more fully realize its potential to contribute to SKT and SKT2. We explain how acting on the proposed strategies may help JSDSE readers, authors, and editors bridge the persistent historical gap between formal scholarship and practice in K-12 education.

This study considers the evolving influence of variation and expectation on the development of school students’ appreciation of distribution as displayed in their construction of graphical representations of data sets. Three interview protocols are employed, presenting different contexts within which 109 students, ranging in age from 6 to 15 years, could display and interpret their understanding. Responses are analyzed within a hierarchical cognitive framework. It is hypothesized from the analysis that, contrary to the order in which expectation and variation are introduced in the school curriculum, the natural tendency for students is to acknowledge variation first and then expectation. First published May 2009 at Statistics Education Research Journal Archives

As lesson study becomes more prevalent, there is a need to continuously develop theoretical and methodological infrastructure to support and refine its use. In this article, we present a critical methodological analysis of the challenges and benefits of using Toulmin's argumentation model in mathematics education to assess the debriefing phase of lesson study. During debriefing sessions, teachers offer arguments about how to improve teaching that are grounded in observations of students' learning. Toulmin's model provides a means to analyze the structure of such arguments. Using an empirical example, we illustrate challenges of using the model, such as determining appropriate grain sizes for data and claims, evaluating qualifiers, recognizing multiple categories of backing, identifying implicit warrants, and deciding between the individual or the group as a unit of analysis. We also discuss benefits such as being able to systematically compare mathematics teachers' pedagogical arguments against one another, assess attainment of debriefing session goals, and characterize group discursive dynamics. Despite the challenges of using the Toulmin model, we conclude that it provides a useful framework for systematic analysis of lesson study debriefing sessions. The present article can help researchers anticipate and address challenges of conducting Toulmin-based qualitative analyses of debriefing session discourse.

Statistics is an important component of the knowledge base for health care professionals. In this essay, it is argued that statistical knowledge for teaching (SKT) should be considered an important component of their preparation as well. Health care professionals often must help others understand the statistical basis for recommendations they make. A COVID-19 press briefing is used to illustrate the need for SKT when making high-stakes recommendations related to public health. It is conjectured that efforts to educate the public during the press briefing would have been enhanced if the presenters had deeper knowledge of the general public’s common statistical thinking patterns, the typical statistics curriculum experienced by members of their audience, and contemporary tools for teaching statistics. The importance of such knowledge to support smaller-scale individual interactions is also discussed. A call for SKT-centered partnerships between educational researchers and medical researchers is made; such partnerships could be mutually beneficial to the development of both fields and to society at large.

Normative discourse about probability requires shared meanings for disciplinary vocabulary. Previous research indicates that students’ meanings for probability vocabulary often differ from those of mathematicians, creating a need to attend to developing students’ use of language. Current standards documents conflict in their recommendations about how this should occur. In the present study, we conducted microgenetic research to examine the vocabulary use of four students before, during, and after lessons from a cycle of design-based research attending to probability vocabulary. In characterizing students’ normative and nonnormative uses of language, we draw implications for the design of curriculum, standards, and further research. Specifically, we illustrate the importance of attending to incrementality, multidimensionality, polysemy, interrelatedness, and heterogeneity to foster students’ probability vocabulary development.

Undergraduate research is increasingly prevalent in many fields of study, but it is not yet widespread in mathematics education. We argue that expanding undergraduate research opportunities in mathematics education would be beneficial to the field. Such opportunities can be impactful as either extracurricular or course-embedded experiences. To help readers envision directions for undergraduate research experiences in mathematics education with prospective teachers, we describe a model built on a design-based research paradigm. The model engages pairs of prospective teachers in working with faculty mentors to design instructional sequences and test the extent to which they support children’s learning. Undergraduates learn about the nature of systematic mathematics education research and how careful analyses of classroom data can guide practice. Mentors gain opportunities to pursue their personal research interests while guiding undergraduate pairs. We explain how implementing the core cycle of the model, whether on a small or large scale, can help teachers make instructional decisions that are based on rich, qualitative classroom data.

For many years, lesson study has been woven into the professional lives of Japanese teachers. Recently, it has become more widespread in the U.S. Core elements of Japanese Lesson Study include collaborative lesson design, empirical trials of lessons, collective analysis of empirical trials, and subsequent lesson re-design. We describe how these core elements were implemented in a teacher education project that brought together prospective and practicing secondary-level science and mathematics teachers in interdisciplinary groups facilitated by university faculty members. The four groups described in this report designed and analyzed lessons about attending to scale factor in the context of microscopic images, constructing a geologic timeline, examining states of matter, and building a scale model of the solar system. The Japanese Lesson Study process helped these groups reflect on the nature of the content being taught, pedagogical concerns, and content-specific pedagogy. We found that discursive moves such as connecting separate strands of conversation and identifying mistakes helped facilitate group reflection. A growing body of research suggests that Japanese Lesson Study can be implemented in a wide array of contexts; we encourage others to include it among clinical experiences for prospective teachers because of its potential to give more undergraduates access to the most skilled mentor teachers and to counteract feelings of professional isolation that cause many to leave teaching. We offer our own experiences facilitating lesson study and dealing with challenges along the way to help others start to implement the model in their own professional settings.

The role of probability in curricula for children has fluctuated greatly over the past several decades. Recently, some countries have removed probability from their preschool and primary curricula, and others have retained it. One reason for such lack of agreement is that theory about early probability learning is still relatively new and under development. The purpose of this report is to sketch a tentative theoretical structure with the potential to anchor curricular decisions and inform further research on early probability learning. Toward that end, we begin by reviewing existing literature. We attend, in particular, to the probabilistic thinking tendencies exhibited by children whom researchers consider to be in the earliest stages of learning the subject. We then use these tendencies to posit two different cycles for early probabilistic thinking. One of these cycles is compatible with disciplinary norms and supports normative thinking; thinking tendencies in this normative compatible cycle include attending to the position of objects in a container, forming images of random generators, attending to the operation of random generators, and thinking about past experiences playing games of chance. Thinking tendencies in the other posited cycle lead to belief systems that conflict with normative disciplinary practice; these belief systems include elements such as myths, superstitions, animism, and determinism. We illustrate the two cycles using empirical data from design-based research. We then reflect on how the two cycles comprising our structure of early probabilistic thinking (SEPT) framework can provide a basis for further curricular and theoretical work.

The Common Core State Standards for Mathematics have a widespread impact on children’s statistical learning opportunities. The Grade 6 standards are particularly ambitious in the goals they set. In this critique, experiences helping children work toward the Grade 6 Common Core statistics expectations are used in conjunction with previous research to identify ways in which the Grades 4–6 standards might be supplemented or revised to help maximize learning. It is suggested that opportunities for children to perceive datasets as aggregates and to draw reasonable conclusions about statistical data by attending to context should be purposefully introduced in Grades 4–5. Currently, the Common Core does not have explicit learning standards for these activities in fourth and fifth grade. It is also suggested that teachers help students question their natural tendencies to focus extensively on the mode when summarizing data. The current standards do not specifically mention the mode. Revising or supplementing the Common Core in the suggested ways holds potential to make the Grade 6 statistical learning standards more attainable for children and to help teachers better anticipate the statistical thinking tendencies that are likely to emerge during classroom discourse.

SummaryWe describe how we used puppets as tools to draw 9 to 10‐year‐old students into conversations about probability. Puppets supported classroom discourse by putting forth probabilistic arguments for critique, introducing extreme and unusual examples of concepts, and introducing an element of surprise.

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.

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 forworking toward a greater degree of consensus across early childhood standards for statistics and probability are also offered.

Qualitative classroom data from video recordings and students' written work can play important roles in improving mathematics instruction. In order to take full advantage of these data sources, it is helpful to have a strong analytic lens to orient one's reflections on the data. One promising analytic lens is the National Research Council's five stands of mathematical proficiency framework. It prompts teachers to examine the extent to which their students have attained conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive dispositions. This article describes how a pair of prospective teachers used the five strands to analyze and reflect upon qualitative classroom data from a series of lessons they taught. The insights gained during the process can help others anticipate dynamics involved in using the five strands framework during practitioner research and reflection on classroom data.