Keywords

1 Introduction

The previous chapters in this handbook have dealt with the nature of statistics and students’ knowledge of the discipline. We now turn our attention to important mediators between students and the discipline: statistics teachers (primary, secondary, and tertiary). Teachers can be considered the third vertex in a didactic triangle, with students and content as the other two vertices (Goodchild & Sriraman, 2012). The mediating role of the teacher has motivated researchers to study teachers’ cognitive and affective characteristics. Certainly, not all teacher and student interactions can be fully explained by these characteristics alone, since contextual constraints such as supervisor expectations , institutional policies, and instructional resources available are influential (Forgasz & Leder, 2008; Sullivan & Mousley, 2001). Diversity among students and equity concerns also come into play (Cobb, 1999). Nonetheless, research suggests that understanding teachers’ individual characteristics is an essential part of studying teachers’ impact on students’ learning. For example, student achievement in statistics has been found to be positively associated with teachers’ knowledge (Callingham, Carmichael, & Watson, 2016).

In this chapter, we focus specifically on research findings related to the cognitive and affective characteristics of statistics teachers. Precise definitions of “cognition ” and “affect” are elusive in the literature, but we conceive of cognitive characteristics as being related to the knowledge and reasoning processes needed for teaching statistics and affective characteristics as being related to dispositions, emotions, attitudes, and beliefs about teaching statistics (McLeod, 1992). In many cases, it is difficult to separate cognition from affect. Beliefs, for example, though often discussed in connection with affect, are widely acknowledged to have cognitive components as well (Philipp, 2007). Hence, in this chapter, our primary goal is not to separate characteristics neatly into “cognitive” and “affective” bins, but rather to identify characteristics that may help shed light on the nature of teachers’ mediating role between students and statistics.

We begin the chapter with descriptions of salient theoretical constructs related to statistics teachers’ cognition (Sect. 10.2) and affect (Sect. 10.3). We then turn to methods for assessing attainment of these constructs (Sect. 10.4). Finally, we summarize findings from research in regard to the constructs (Sect. 10.5). In doing so, we seek to portray the current state of the art and identify fruitful directions for further research (Sects. 10.6 and 10.7).

2 Constructs for Describing Teachers’ Cognitive Characteristics

Researchers employ various theoretical models to study cognition related to teaching statistics. These models generally acknowledge that knowing statistics is a necessary, but not sufficient, condition for teaching it. This resonates with Shulman’s (1987) influential assertion that teachers need pedagogical content knowledge , which is a “special amalgam of content and pedagogy that is uniquely the province of teachers” (p. 8). Building on Shulman’s work, the Learning Mathematics for Teaching (LMT) Project conceptualized content knowledge for teaching as consisting of both subject matter knowledge and pedagogical content knowledge (Ball, Thames, & Phelps, 2008; Hill, Ball, & Schilling, 2008). Several studies of statistical knowledge for teaching (SKT) have profitably used adaptations of the LMT model (e.g., Burgess, 2011; González, 2014; Groth, 2013; Leavy, 2015; Noll, 2011; Wassong & Biehler, 2010). Hence, we describe possible elements of subject matter knowledge (Sect. 10.2.1) and pedagogical content knowledge (Sect. 10.2.2) related to the LMT model next.

Although the LMT model appears frequently in statistics teacher education research, it would be inaccurate to portray it as the only model employed. We will also describe work that challenges the field to continue to think critically about the precise nature of the elements of SKT , their relationships with one another, and how they develop (Sect. 10.2.3). We close with a discussion of technological pedagogical content knowledge and how it relates to other research on teachers’ cognitive characteristics (Sect. 10.2.4).

2.1 Subject Matter Knowledge

Subject matter knowledge can be conceptualized as having three sub-domains: common content knowledge, specialized content knowledge , and horizon knowledge (Ball et al., 2008; Hill et al., 2008).

Hill et al. (2008) described common content knowledge as “knowledge that is used in the work of teaching in ways in common with how it is used in many other professions or occupations that also use mathematics” (p. 377). At the university level, prospective teachers often study aspects of common statistical knowledge alongside those preparing for other professions. For example, knowing how to compute and interpret descriptive statistics such as mean, median, and interquartile range is valuable both to teachers and to other professionals (Groth, 2007).

Specialized content knowledge can be described as “the mathematical knowledge that allows teachers to engage in particular teaching tasks, including how to accurately represent mathematical ideas, provide mathematical explanations for common rules and procedures, and examine and understand unusual solution methods to problems” (Hill et al., 2008, p. 378). Specialized knowledge of statistics might involve knowing how to represent the mean as a typical value, a fair share, and a signal amid noise (Wassong & Biehler, 2010). It might also entail the ability to analyze students’ statistically naïve interpretations of data (Burgess, 2011). Similarly, appraising novel student-invented graphical representations may be done by drawing upon specialized knowledge (Groth, 2013).

Ball and Bass (2009) spoke of horizon knowledge as that which allows teachers to see connections between content studied at a particular grade level and major disciplinary structures, ideas, practices , and sensibilities. Consider the case of standard deviation. Seventh-grade US teachers using the Common Core State Standards (National Governors Association for Best Practices & Council of Chief State School Officers, 2010) would not teach this idea directly to their students, but they would be responsible for teaching the related idea of mean absolute deviation (MAD). Teachers might conceive of the MAD in at least two different ways: (1) as an algorithm for students to compute and master or (2) as a precursor for the study of standard deviation (Groth, 2014). Teachers holding the latter view would seem more likely to select tasks and ask questions that lead toward the concept of standard deviation that is on the statistical horizon. Teachers holding the former view may not be able to imbue their instruction with this perspective, and they may reduce the study of the MAD to rote learning of a procedure . Many questions about horizon knowledge remain for exploration by researchers, such as: (1) What specific aspects does horizon knowledge entail? (2) How is horizon knowledge best developed? (3) What can teachers with well-developed horizon knowledge do for students that others cannot?

Research involving teachers frequently focuses on their subject matter knowledge. Specific findings regarding the nature of different elements of teachers’ subject matter knowledge appear in Sect. 10.5.

2.2 Pedagogical Content Knowledge

Hill et al. (2008) hypothesized that pedagogical content knowledge has three sub-domains: knowledge of content and students, knowledge of content and teaching, and curriculum knowledge.

The first sub-domain, knowledge of content and students, pertains to teachers’ knowledge of students’ thinking patterns and problem-solving strategies (Hill et al., 2008). The importance of this type of knowledge is well established in the literature. Teachers participating in professional development about how students think about content tend to be more effective in facilitating students’ learning (Franke, Kazemi, & Battey, 2007). Statistics education researchers have hypothesized that knowledge of content and students for statistics may consist of elements such as understanding students’ difficulty learning the mean conceptually (Wassong & Biehler, 2010), comprehending student difficulties sorting data (Burgess, 2011), and knowing differences between how students tend to read dot plots and box plots (Groth, 2013). Comprehensively conceptualizing the nature of knowledge of content and students for statistics and its impact on student learning constitute important tasks for future research.

Knowledge of content and teaching is a combination of knowing about teaching and knowing about subject matter. It can help teachers with tasks such as choosing models and examples that bring out important aspects of content (Ball et al., 2008). It appears that knowledge of content and students contributes to knowledge of content and teaching. Consider, for example, a teacher who knows that transitioning from dot plot displays to box plots and histograms is a difficult task for students. Such a teacher is in position to select tasks to help students gradually transition from one display to the next (Groth, 2013).

Ball and Bass (2009) spoke of curriculum knowledge as including knowledge of educational goals, standards, and grade levels where specific concepts appear. This type of knowledge may help teachers appropriately sequence the introduction of statistical ideas in a curriculum (Godino, Ortiz, Roa, & Wilhelmi, 2011). However, there is considerable variability in how teachers interpret curriculum materials. When given a curriculum, some implement it with a high amount of fidelity to the curriculum authors’ intentions, and others do not (Tarr et al., 2008). Sometimes this degrades the quality of instruction, but other times may help improve it (Brown, Pitvorec, Ditto, & Kelso, 2009). Hence, carefully examining statistics teachers’ curriculum knowledge has the potential to help explain underlying reasons for instructional dynamics observed in the classroom.

Although pedagogical content knowledge appears as a separate category from content knowledge in the LMT framework , in practice , it is difficult, and often not advisable, to separate the two. Hence, in the summary of research appearing in Sect. 10.5, we report findings about teachers’ pedagogical content knowledge, for the most part, alongside subject matter knowledge findings.

2.3 Continuing the Work of Precisely Defining SKT Elements, Their Relationships to One Another, and Their Development

Given the preceding description of the elements of SKT , some may gain the impression that it is a static trait rather than one that evolves and changes continuously within classroom contexts. LMT-based models are sometimes perceived in this manner (Venkat & Adler, 2014). Some theoretical work serves to cast SKT in a more dynamic light.

Working from the LMT framework and empirical data, Groth (2013) theorized about processes involved in individuals’ transformation of statistics subject matter knowledge into forms that are useful for teaching. Central to the analysis is the idea that teachers’ key developmental understandings of subject matter knowledge (Simon, 2006) are, alone, not sufficient for teaching. Teachers who have key developmental understandings must also learn to view subject matter knowledge from students’ perspectives in order to create pedagogically powerful ideas (Silverman & Thompson, 2008). In terms of the LMT framework , this suggests that knowledge of content and students is a precursor to developing knowledge of content and teaching. That is, teachers should understand students’ learning needs in order to design and select teaching methods suitable for addressing them. The potential link between knowledge of content and students and knowledge of content and teaching deserves more research attention, as it is difficult to conceive of a teacher with robust knowledge of content and teaching but underdeveloped knowledge of content and students. Investigating whether one sort of knowledge is usually prerequisite to another could help effectively sequence learning experiences for teachers.

Studies in the traditions of design research (Bakker & van Eerde, 2015) and didactic engineering (Artigue, 2015) may help further reveal dynamic processes involved in SKT development. Such studies involve iterative cycles of research and development of instructional sequences in classroom settings. As the cycles occur, data yield contextually rich information about teachers’ knowledge and its enactment in practice . Working from the perspective of teachers’ knowledge situated in an institutional context, Godino et al. (2011) proposed facets of professional knowledge for teaching statistics that differ from those in the LMT model. One such facet was that teachers need knowledge of “students’ attitudes, emotions, and motivations regarding the content” (p. 279). This type of knowledge is similar to knowledge of content and students in its focus on student characteristics, but different in that it deals with the importance of knowing children’s affect in regard to statistics (and not just children’s statistical cognition ). It would be profitable for researchers to take advantage of different conceptualizations of the nature of SKT as starting points for comparing and contrasting viewpoints. Done systematically, such theoretical comparisons could lead to the incremental development of progressively more sophisticated models of SKT and how it develops.

In any model of SKT that is constructed, it is important for researchers to acknowledge that mathematics and statistics are distinct disciplines. Mathematics and statistics differ in their “origins, subject matter, foundational questions, and standards” (Moore, 1988, p. 3). Therefore, it is reasonable to assume that the knowledge needed for teaching statistics is not precisely equivalent to the knowledge needed for teaching mathematics (Groth, 2007). Hence, as theoretical work on the conceptualization of SKT continues, researchers must be careful to distinguish, as necessary, between professional knowledge needed for teaching mathematics and that needed for teaching statistics.

2.4 Technological Pedagogical Content Knowledge

Shulman’s (1987) notion of pedagogical content knowledge is the basis for another related, yet somewhat distinct, body of research on teachers’ knowledge. As digital technologies became more prevalent in classrooms, it was apparent that teachers needed technological pedagogical content knowledge (TPCK) to effectively use them for instruction (Koehler & Mishra, 2008). TPCK is a complex interaction among knowledge of content, pedagogy, and technology. Some theoretical work to conceptualize TPCK for statistics appears in this section, and some work to help develop teachers’ TPCK appears in Sect. 10.6.3.

Lee and Hollebrands (2011) offered a framework to operationalize TPCK for statistics. Their framework posits statistical knowledge as the basis for technological statistical knowledge (TSK). TSK is a blend of technology knowledge and statistics knowledge. TSK includes knowledge of technologies that are both amplifiers and reorganizers (Ben-Zvi, 2000; Lee & Hollebrands, 2008). Amplifiers help automate processes that could be done by hand, such as computing least-squares regression lines (Lee & Hollebrands, 2008). Reorganizers “extend what teachers may be able to do without technology to help students reorganize and change their statistical conceptions” (Lee & Hollebrands, 2008, p. 329). For instance, TinkerPlots (Konold & Miller, 2011) affords the opportunity to generate and link multiple graphical representations . Using TinkerPlots to produce suitable representations for data is another activity that engages TSK (Lee et al., 2014).

TSK must ultimately merge with pedagogical knowledge if teachers are to develop technological pedagogical statistical knowledge (TPSK). An example of a task requiring TPSK was discussed by Wilson, Lee, and Hollebrands (2011). Teachers used pedagogical, statistical, and technological knowledge in analyzing a video case of children working with TinkerPlots . To analyze the case, teachers attended to how students thought about statistical tasks, how they used TinkerPlots in solving them, how the technology assisted or hindered students’ learning, and the strengths and weaknesses of the task given to students. Statistical knowledge, or even TSK, would not be sufficient for analyzing these elements of the case.

At present, the literature provides a more comprehensive portrait of teachers’ TSK than it does TPSK , though investigation of both types of knowledge is in its beginning stages. Also requiring research attention are the potential links among statistical knowledge, TSK, and TPSK. The Lee and Hollebrands (2011) framework provides a starting point for such investigations, but Lee et al. (2014) acknowledge that empirical work remains to be done to test the conjecture that teachers’ TSK impacts their TPSK and teaching practices .

3 Constructs for Describing Teachers’ Affective Characteristics

In the affective domain, beliefs and attitudes of statistics teachers have received research attention. However, the terms “beliefs” and “attitudes” are not used uniformly across studies. Philipp (2007) encountered the same dilemma in writing about beliefs and attitudes related to mathematics. To address the problem, he offered general descriptions that capture much of what authors often mean when using the two terms:

  • Attitudes: “manners of acting, feeling, or thinking that show one’s disposition or opinion. Attitudes change more slowly than emotions, but they change more quickly than beliefs” (p. 259).

  • Beliefs: “Psychologically held understandings, premises, or propositions about the world that are thought to be true. Beliefs are more cognitive, are felt less intensely, and are harder to change than attitudes” (p. 259).

These characterizations provide starting points for our discussion of statistics teachers’ beliefs (Sect. 10.3.1) and attitudes (Sect. 10.3.2).

3.1 Beliefs

Research describes several types of beliefs connected to teaching statistics. These include beliefs about the relationship between mathematics and statistics, goals and strategies for statistics instruction, and self-efficacy to teach statistics.

3.1.1 Beliefs About the Relationship Between Mathematics and Statistics

Statistics is often taught as part of a mathematics curriculum or in a mathematics department. This arrangement can support the belief that statistics is a branch of mathematics rather than a discipline in its own right (Burrill & Biehler, 2011). Rossman, Chance, and Medina (2006) argued that this is not a useful belief for teachers to hold, as it may lead to lack of instructional emphasis on the nature and role of context, measurement, data collection, and uncertainty in statistics. Similar concerns have been expressed by others (Gattuso, 2008; Scheaffer, 2006). Empirical data lend support to the validity of such concerns. Begg and Edwards (1999) found that teachers tended to acknowledge the cross-curricular nature of statistics yet still generally taught it as a unit of mathematics. Yang (2014) suggested that teacher beliefs about the differences between statistics and mathematics may be influenced by national curricula and assessments and that it would be worthwhile to explore the influence of these factors.

3.1.2 Beliefs About Goals and Strategies for Statistics Instruction

Eichler (2007) provided an empirically grounded framework for characterizing teachers’ beliefs about the goals of statistics instruction. The framework included four categories of beliefs: traditionalist, application preparer, everyday life preparer, and structuralist. Traditionalists focus on the study of probability and algorithms in the abstract and not on applications. Application preparers value teaching students the interplay between theory and applications, focusing on the use of algorithms to solve real-world problems. Everyday life preparers take the focus on applications a step further, believing that the study of statistics should be driven by applications rather than theory. Structuralists focus heavily on probability theory, mathematical structure, and algorithms. Structuralists differ from traditionalists in that they believe in using applications as the basis for instruction. Structuralists differ from other groups in that their primary goal is to help students abstract mathematical structure from the applications rather than apply mathematical principles to make sense of situations students encounter outside of school.

Aspects of Eichler’s framework resonate with other researchers’ findings in regard to teachers’ beliefs about strategies and goals of statistics instruction. Sedlmeier and Wassner (2008) found that teachers believed it to be valuable to relate statistics content to daily issues (similar to everyday life preparers), but did not believe in placing as much emphasis on student data gathering or student interests. Pierce and Chick (2011) found that some teachers believe in teaching procedures first and then using applications merely to try to make the procedures more interesting. Such a strategy may reflect application preparer and/or structuralist tendencies. Comprehensively mapping the relationships between observed teaching strategies and specific beliefs about the goals of statistics instruction is an interesting empirical task for which some infrastructure currently exists, and it awaits additional research attention.

3.1.3 Self-Efficacy Beliefs About Teaching Statistics

Harrell-Williams, Sorto, Pierce, Lesser, and Murphy (2014) argued that it is important to measure self-efficacy to teach statistics. Teacher self-efficacy can be defined as a teacher’s belief that he or she has the ability to bring about student learning (Ashton, 1985). Harrell-Williams et al. synthesized existing research to conclude that self-efficacy influences teachers’ choices of instructional techniques and students’ learning. They argued that it is particularly important to consider self-efficacy in regard to teaching statistical investigations . Such a domain-specific portrait of teacher self-efficacy is potentially more informative to teacher educators than more generic assessments.

3.2 Attitudes

There is a voluminous body of research on individuals’ attitudes toward statistics (Nolan, Beran, & Hecker, 2012), but literature about teachers’ attitudes toward statistics is more sparse (Estrada, Batanero, & Lancaster, 2011). Available research suggests that teachers tend to value statistics as a subject but find it difficult to enjoy, teach, and learn (Estrada, Batanero, Fortuny, & Díaz, 2005; Martins, Nascimento, & Estrada, 2012). Teachers’ attitudes toward statistics are potentially important because they are hypothesized to relate to their persistence in gaining statistical knowledge (Estrada et al., 2005) and willingness to teach the subject (Leavy, Hannigan, & Fitzmaurice, 2013). Teachers’ attitudes toward statistics are hypothesized to influence their knowledge of statistics, their teaching practices , and their students’ attitudes (Martins et al., 2012). Several attitude-related hypotheses, however, await strong empirical support. In studies of the impact of attitudes on teachers’ content knowledge , for example, researchers have found moderate to low correlations (Hannigan, Gill, & Leavy, 2013; Nasser, 2004). Negative attitudes toward statistics appear to be clearly detrimental (Onwuegbuzie, 2000), but there seems be a limit on the extent to which positive attitudes relate to increased knowledge (Hannigan et al., 2013).

It appears that the field has not yet made a strong distinction between “teachers’ attitudes toward statistics” and “attitudes toward teaching statistics.” Although assessment items about attitudes toward teaching statistics have been included in some research studies (Martins et al., 2012; Pierce & Chick, 2011), many studies of teachers’ attitudes have used instruments intended to measure the attitudes of the general population (Estrada et al., 2011; Hannigan et al., 2013). This might explain why empirical evidence about the impact of attitudes is elusive. If, for example, the field were to systematically conceptualize and investigate teachers’ attitudes toward pedagogical elements such as statistics curriculum, children’s statistical learning, and technology for teaching statistics, might we better understand the impact of teachers’ attitudes on statistics teaching and learning?

4 Methods for Assessing Statistics Teachers’ Cognition and Affect

Assessments of cognition and affect related to teaching statistics come in a variety of forms, spanning the spectrum of written assessments, interviews, and observations . Many studies make use of more than one type of assessment and may involve more than one aspect of teachers’ cognition and affect. Below, a representative sample of assessments is discussed. Due to the scope of the chapter, we focus on assessments specifically designed for teachers rather than general standardized scales of cognition and affect that are sometimes used as part of research with teachers.

4.1 Written Assessments

Written assessments are often the most practical way to gather information from large groups of teachers. One such assessment, the Diagnostic Teacher Assessment of Mathematics and Science, includes a separate scale of multiple-choice and open-ended items for statistics (Saderholm, Ronau, Brown, & Collins, 2010). The LMT project also designed a scale of multiple-choice items specific to teaching statistics (G. Phelps, personal communication, June 11, 2010). An international comparison of teacher education , the Teacher Education and Development Study in Mathematics, included some items on pedagogical content knowledge for statistics among items pertaining to mathematics, though algebra, geometry, and number were more heavily assessed (Blömeke & Delaney, 2012).

Some studies of teachers’ affect in regard to statistics and statistics teaching have used collections of questions from larger scales intended for a broad population . Estrada and Batanero (2008), for example, used a subset of items from the Survey of Attitudes Toward Statistics (Schau, Stevens, Dauphine, & del Vecchio, 1995) that had previously yielded lower scores for teachers. More recently, Harrell-Williams et al. (2014) designed a scale to assess teachers’ self-efficacy to teach statistics. It measures teachers’ feelings of preparedness to teach content from the Guidelines for Assessment and Instruction in Statistics Education ( GAISE) report for grades pre-K–12 (Franklin et al., 2007). This sort of assessment, which is specifically designed and psychometrically tested to measure affect in regard to teaching statistics, is relatively rare.

Some written assessments deal with both cognition and affect related to teaching statistics. Watson (2001) designed a survey to assess teachers’ pedagogical content knowledge , self-efficacy to teach statistics, and beliefs about the value of statistics. Watson, Callingham, and Donne (2008) built on this survey to devise a 12-item scale of pedagogical content knowledge. The Statistics Teaching Inventory (Zieffler, Park, Garfield, del Mas, & Bjornsdottir, 2012) contains questions about statistics teachers’ teaching practices , course characteristics, assessment practice, teaching beliefs , and assessment beliefs. González (2014) designed a written assessment of teachers’ subject matter knowledge , pedagogical content knowledge , and beliefs and conceptions of variability. Instruments that assess aspects of both cognition and affect have the potential to help researchers understand complex relationships among teacher characteristics such as knowledge, beliefs, attitudes, goals, and teaching practices .

4.2 Interviews

Clinical interviews allow a high degree of interactivity between the researcher and study participants. They are more time intensive than written assessments. Interviews come in a variety of forms. They may be driven by a formal protocol, such as the StatSmart teacher interview protocol (Watson & Nathan, 2010), which probes the nature of teachers’ subject matter knowledge and pedagogical content knowledge . More often, however, interview tasks and questions are designed to meet the specific objectives of a research study. For example, Noll (2011) interviewed graduate assistants to assess their statistical content knowledge of sampling. Participants were asked about written items they had completed and were given some new tasks to solve. Similarly, Browning, Goss, and Smith (2014) conducted interviews with preservice teachers to gain better understanding of the thinking they employed while solving written statistical tasks. Other studies incorporating interviews have probed subjects such as teachers’ classroom practices (Casey, 2010), beliefs about the nature of statistics (Leavy et al., 2013), and perceptions of professional development sessions (Peters, Watkins, & Bennett, 2014).

4.3 Observations

Written assessments and interviews provide proxies of teachers’ classroom practices and quality of instruction, but observation techniques allow researchers to see these firsthand. Using structured observations, researchers can infer the nature of teachers’ SKT and knowledge of statistical investigations . For instance, Burgess (2011) illustrated how an SKT framework can guide such observations of teachers’ practice . Casey (2010) described an observation-based process for assessing knowledge for teaching statistical association . Pfannkuch (2006) used observations to describe a teacher’s knowledge of comparing distributions with box plots. Jacobbe and Horton (2010) used observations of teachers’ lessons to detect their misconceptions related to data displays.

Another strategy is to observe teachers’ interactions during professional development sessions. Although these are not always firsthand observations of teaching, they still provide data that may not be easily obtained through interviews or written tasks. Wilson et al. (2011) studied video of teacher education sessions to better understand teachers’ pedagogical content knowledge in connection with the use of dynamic statistics software. Peters et al. (2014) observed how teachers’ learning of measures of center developed as they interacted with one another. Leavy (2010, 2015) observed prospective teachers involved in lesson study. Lesson study engages a group of teachers in planning a lesson, carrying it out, observing students’ reactions, and then debriefing on the lesson’s effectiveness. Observations of these activities provide a window into teachers’ thinking about planning and analyzing lessons. In general, teachers’ discourse with one another during professional development can help explain the nature and origin of the knowledge and beliefs that guide their instruction.

Observations of teachers’ lessons and professional development sessions typically yield a variety of artifacts. These may include written lesson plans (Garfield & Ben-Zvi, 2008), statistical tasks a teacher assigns to a class (Burgess, 2011), field notes (Casey, 2010), and teachers’ responses to professional development tasks (Wilson et al., 2011). Artifacts of this nature can be synthesized with other data to help researchers develop portraits of teachers’ cognitive and affective characteristics related to teaching statistics.

5 Research on Teachers’ Statistical Knowledge

The research models, constructs, and techniques described up to this point in the chapter have been used in studies spanning various statistical content areas. In this section, we summarize findings from two broad, interrelated bodies of literature about teachers’ knowledge related to data (Sect. 10.5.1) and uncertainty and statistical inference (Sect. 10.5.2).

5.1 Research on Data

We present a brief overview of research on teachers’ subject matter knowledge and pedagogical content knowledge in regard to data displays, distribution and variability, associations between variables, and statistical literacy, reasoning, and thinking.

5.1.1 Data Displays

Studies of preservice or practicing teachers’ graph reading and interpretation skills show a tendency to underestimate the complexities of learning and teaching graphical representations (Leavy, 2010). Teachers tend to express confidence in their understandings of graphical representations and to feel better equipped to teach this topic compared to other statistical ideas (González & Pinto, 2008; Watson, 2001). However, despite their positive attitudes and confidence toward teaching graphs, many educators have limited subject matter knowledge of graphical representations (González & Pinto, 2008; Jacobbe & Horton, 2010; Pierce & Chick, 2013; Sorto, 2004). They sometimes confuse histograms with bar diagrams (Bruno & Espinel, 2009), fail to integrate graphical knowledge with problem context (Burgess, 2002), and have trouble with graph selection and understandings of data type (Leavy, 2010). Monteiro and Ainley (2007) introduced the idea of “critical sense” as a key skill in the analysis and interpretation of graphical artifacts. They investigated critical sense in graphing among prospective primary school teachers from Brazil and England. They found that many preservice teachers did not have adequate mathematical knowledge to read graphs from the daily press. However, the majority displayed an ability to think critically and justify their ideas by combining statistical knowledge with personal experience and contextual knowledge.

A small number of studies have examined teachers’ pedagogical content knowledge of graphs. González and Pinto (2008) concluded that teachers need more knowledge of the process of learning statistical graphs and the difficulties that students might have with them. Espinel, Bruno, and Plasencia (2008) observed lack of coherence between prospective primary teachers’ graph building and their evaluation of tasks carried out by fictitious future students. Heaton and Mickelson (2002) observed that graph construction often became the endpoint of statistical investigation for preservice elementary teachers, who focused on the technical aspects of graph construction rather than on engaging children in reasoning with the data. However, some studies indicate that using dynamic data exploration tools (e.g., Finzer, 2002; Konold & Miller, 2011) can help teachers portray graph production as a means for understanding data rather than an end in itself (Meletiou-Mavrotheris, Mavrotheris, & Paparistodemou, 2011).

5.1.2 Distribution and Variability

Much of the research investigating teachers’ reasoning with distributions has focused on their understanding of measures of central tendency, measures of variability , distributional thinking, and procedural aspects of statistics. Pedagogical content knowledge has also been explored.

Studies examining teachers’ conceptions of measures of center have focused on the arithmetic mean (e.g., Batanero, Godino, & Navas, 1997; Gfeller, Niess, & Lederman, 1999); the mean, median, and mode (e.g., Groth & Bergner, 2006; Jacobbe, 2012); and the general concept of average (e.g., Begg & Edwards, 1999; Estrada, Batanero, & Fortuny, 2004; Leavy & O’Loughlin, 2006). Evidence from such studies illustrates that attaining deep understanding of these statistical concepts is nontrivial. Like students, many teachers struggle to view measures of central tendency as representative (or “typical”) values. Although teachers can readily calculate the mean, they tend not to use it to compare groups (Canada, 2004; Hammerman & Rubin, 2004; Leavy & O’Loughlin, 2006; Makar, 2004; Makar & Confrey, 2002, 2004; Peters, 2009). Like students, teachers may rely upon procedural algorithms and need conceptual understanding (Gfeller et al., 1999; Leavy & O’Loughlin, 2006; Peters et al., 2014; Sorto, 2004).

As with measures of center, teachers’ understandings of standard deviation and other formal measures of variation tend to be procedural (Leavy, 2006; Makar & Confrey, 2005; Sorto, 2004). Research indicates difficulties with the concept of variability for teachers of various grade levels (Mooney, Duni, van Meenen, & Langrall, 2014; Vermette, 2013) and similar misunderstandings to those seen in students (e.g., perceiving the normality shape of a distribution as an indication of low variability). Teachers often hold competing beliefs about random variation when the setting of a problem changes (Canada, 2004; Makar, 2004). Kuntze (2014) found that some secondary teachers did not consider learning about statistical variation to be an important instructional goal, though others did recognize the importance of teaching the concept.

Measures of center and measures of spread are inseparable. Conceptual understanding of standard deviation, for example, requires “a dynamic conception of distribution that coordinates changes to the relative density of values about the mean with their deviation from the mean” (Peters, 2009, p. 21). Teachers often have difficulty coordinating understandings of central tendency and dispersion (Dabos, 2014; Lee & Lee, 2011). Many teachers tend to focus either only on the center of the distribution, or on its range, or on small clusters or individual points, rather than integrating different aspects of data distributions (Canada, 2008; Makar & Confrey, 2005; Mooney et al., 2014). When beginning to reason about distributions, teachers can be encouraged to use informal terminology to describe spread and distribution, such as “clump,” “bump,” “bulk of this data,” “scattered,” and “bunched” (Canada, 2004; Makar, 2004; Makar & Confrey, 2005). Since children use similar language (e.g., Bakker & Gravemeijer, 2004), Makar and Confrey (2005) suggested recognizing and valuing this informal “variation talk” as a way to encourage intuitive statistical sensemaking .

Explorations with dynamic data software can also help improve teachers’ distributional reasoning and pedagogical content knowledge (Canada, 2004; Hammerman & Rubin, 2004; Leavy, 2006; Lee & Lee, 2011; Makar, 2004; Meletiou-Mavrotheris, Paparistodemou, & Stylianou, 2009; Peters et al., 2014). For example, Meletiou-Mavrotheris and Serradó (2012) reported on EarlyStatistics, an intercultural professional development course in which teachers took part in authentic educational activities. The activities gave them the opportunity to reflect on the “big ideas ” of statistics and their applications and to explore ways of improving statistics instruction through the adoption of a coherent technology-rich curriculum based on the statistical problem-solving process. Findings from the study indicated that EarlyStatistics met its objectives, improving teachers’ knowledge of key statistical ideas including distributions. Data obtained from follow-up teaching interventions in some of the participants’ classrooms suggested positive gains in student learning and attitudes toward statistics (Meletiou-Mavrotheris et al., 2011).

5.1.3 Associations Between Variables

Some researchers have designed instruction to help teachers confront their potential misunderstandings of association and those of their students. Batanero, Estepa, and Godino (1997) examined whether a computer-based teaching experiment would improve preservice teachers’ understanding of association. They found improvement in covariational strategies and reduction in deterministic concepts of association. However, they also found that most teachers retained the belief that a strong association between two variables is adequate for drawing conclusions about cause and effect. Casey (2010) observed three experienced secondary teachers as they taught statistical association and interviewed them immediately following each observation . The research showed that to meet the demands of teaching, the teachers needed substantial knowledge of the concepts’ underlying statistical association. For example, they needed to know not only how to compute the value of a correlation coefficient but also why it was computed as it was and the implications of the computation. Teachers also need to understand the nature of informal lines of best fit and criteria for placing them on graphs. Casey and Wasserman (2015) found that teachers hold a variety of conceptions of informal lines of best fit and how they should be placed. Despite the different conceptions, teachers positioned informal best fit lines in approximately the same place the least-squares regression line would appear in a scatterplot.

Along with subject matter knowledge of association , teachers need pedagogical content knowledge . Casey (2014) synthesized the results of three research studies centered on the teaching and learning of linear regression to describe the knowledge needed by teachers regarding learners’ conceptions of linear regression. The synthesis illustrated that the knowledge needed to teach linear regression differs in significant ways from the knowledge needed to teach linear functions. Quintas, Ferreira, and Oliveira (2014) compared and contrasted the pedagogical content knowledge of two experienced secondary mathematics teachers as they taught bivariate data. The teachers had difficulty helping students reason about bivariate relationships. It was challenging for them to teach aspects of structure and strength, model fitting, and the role of the linear regression model in predicting events. Both teachers exhibited some of the common misunderstandings and errors with regard to bivariate relationships identified in the literature (e.g., Engel & Sedlmeier, 2011). Such findings indicate a need to design professional development that strengthens teachers’ content knowledge and pedagogical content knowledge for teaching association in tandem.

5.1.4 Statistical Literacy, Reasoning, and Thinking

The development of students’ statistical literacy has become an overarching goal of statistics education internationally. This broadening of the curriculum to encompass statistical literacy, reasoning, and thinking has put considerable demands on teachers (Hannigan et al., 2013). In particular, they must design lessons with engaging contexts (Chick & Pierce, 2008), focus on conceptual understanding (Watson, 2001), and pose critical questions (Reston, Jala, & Edullantes, 2006).

Research sheds light upon factors that influence the design and implementation of instruction that fosters students’ statistical literacy, reasoning, and thinking. Burgess (2011) found that the students of a teacher with well-developed SKT were able to progress further with statistical investigations than students of a teacher whose knowledge was less developed. Callingham and Burgess (2014) conjectured that the national curriculum under which teachers operate may influence their approach to teaching statistics, since Australian teachers in their study tended to focus more on procedural aspects of instruction than did their counterparts from New Zealand. Makar and Fielding-Wells (2011) found that challenges in teaching statistical inquiry may stem from difficulties coping with the uncertainties of inquiry, managing classroom logistics, and developing the requisite content knowledge . Mickelson and Heaton (2004) found that the ability to translate content knowledge into effective teaching practices is complex and urged researchers to team with classroom teachers in order to help design meaningful experiences for students. Indeed, during the past decade, several researchers have been experimenting with new innovative models of preservice and in-service teacher training that are focused on inquiry-based instruction and on statistical problem-solving (e.g., Garfield & Ben-Zvi, 2008; Groth, Bergner, Burgess, Austin, & Holdai, 2016; Makar & Fielding-Wells, 2011; Meletiou-Mavrotheris & Serradó, 2012; Serradó, Meletiou-Mavrotheris, & Paparistodemou, 2014).

5.2 Research on Uncertainty and Statistical Inference

Uncertainty and statistical inference are challenging ideas for teachers, just as they are for the general population . Researchers have documented teachers’ understanding of theoretical probability (Batanero & Díaz, 2012; Watson, 2001), empirical probability (Dollard, 2011; Groth, 2010; Theis & Savard, 2010), informal inference (Canada, 2008; Pfannkuch, 2006), samples and sampling distributions (Green, 2010; Green & Blankenship, 2014; Groth & Bergner, 2005; Maxara & Biehler, 2010; Noll, 2011), and formal inference (Liu & Thompson, 2009; Thompson & Liu, 2005).

Sound reasoning about uncertainty and inference require a departure from deterministic modes of thinking. Such modes of thinking influence teaching practices . Serradó, Azcárate, and Cardeñoso (2006), for example, found that deterministic beliefs about the nature of statistics prevented some teachers from embracing curricular goals related to probability and uncertainty. Liu and Thompson (2009) found that the majority of the high school teachers in their study tended to think deterministically. This made it difficult for the teachers to understand and portray hypothesis testing as a tool for drawing inferences.

Research suggests that enhancing teachers’ subject matter knowledge about uncertainty and statistical inference must be given high priority. Building teachers’ knowledge of pedagogical structures and tools by itself is not sufficient. Lee and Mojica (2008), for example, found that although a group of teachers involved their students in authentic statistical inquiry that included use of simulation tools, they missed the chance to develop students’ understanding of the frequentist approach to probability because of limited subject matter knowledge. Deep understanding of probability is also needed for identifying student errors and implementing effective teaching practices (Maher & Muir, 2014; Paparistodemou, Potari, & Pitta, 2006). Such understanding can be developed through well-designed professional development. For example, Theis and Savard (2010) helped high school teachers design and implement a technology-based instructional intervention. They found that the use of simulation software within the intervention allowed teachers to adopt more inquiry-oriented strategies and begin to incorporate frequentist probability.

Although having subject matter knowledge is necessary for effective teaching of uncertainty, it is not sufficient. Leavy (2010) worked with a group of prospective teachers who demonstrated relatively strong subject matter knowledge about informal inference. However, they had trouble using this knowledge to develop pedagogical contexts for advancing children’s learning. In particular, they had difficulty choosing sufficiently complex data, creating engaging contexts, handling unexpected student responses, and scaffolding children’s thought processes. In other studies, gaps in pedagogical content knowledge have been framed as contributing factors to teachers’ failure to emphasize important probability concepts when writing lesson plans (Chick & Pierce, 2008) and designing productive learning environments for students (Groth, 2010). Accordingly, researchers have begun to develop techniques capable of assessing both subject matter knowledge and pedagogical content knowledge (Meletiou-Mavrotheris, Kleanthous, & Paparistodemou, 2014) and to monitor and adjust their professional development efforts to ensure that they facilitate teachers’ development of both of these aspects of SKT (Lee & Hollebrands, 2008; Serradó et al., 2014).

6 Teacher Education Frontiers

As illustrated in this chapter and in other reports on statistics teacher education (Franklin et al., 2015), developing cognitive and affective characteristics related to teaching statistics is a complex process deserving concentrated research attention. Such research attention is particularly important in light of the move away from traditional methods of teaching statistics (Batanero & Díaz, 2012). In order to better fulfill teachers’ needs in a reform-oriented context, alternative approaches to teacher education have become prevalent. Some alternative approaches are situated within: the context of teachers’ classroom practice , deep exploration of statistical content, and technological environments. To conclude the chapter, we summarize some of the work done using these approaches. We do so to encourage others to continue to develop and extend the approaches. We also foreshadow Chap. 12, which describes approaches to professional development in greater detail.

6.1 Learning from Teaching Practice

In practice-based approaches, teachers use real classrooms as sites for investigation. Lesson study is one example. Leavy (2010, 2015) used lesson study to develop prospective teachers’ knowledge and ability to teach informal inferential reasoning and data handling. Roback, Chance, Legler, and Moore (2006) used lesson study to refine their own approaches to teaching inference. Other practice-based approaches involve researchers collaborating with teachers to design, implement, and analyze instruction. For example, Noll and Shaughnessy (2012) reported on a project in which teachers teamed with five university researchers to design and co-teach lessons to investigate secondary students’ conceptions of variability. They found the project to be mutually beneficial; teachers and researchers learned from one another during collaboration . Groth et al. (2016) collaborated with prospective teachers to design instruction suitable for meeting students’ learning needs in regard to measures of center and involved the prospective teachers in the process of disseminating the results (Groth, Kent, & Hitch, 2015). Under such approaches, the line between teachers and researchers is intentionally blurred in order to engage teachers in some of the same types of systematic classroom inquiry that are characteristic of formal research.

6.2 Immersion in Statistical Content

Examples of approaches that immerse teachers in deep exploration of statistics content can be found at the primary, secondary, and tertiary levels. Reston (2012) explored in-service elementary teachers’ conceptions of probability, finding that problem-based learning, inquiry, and statistical investigations promoted stronger conceptual understandings of probability and enhanced pedagogical skills. Makar and Confrey (2002) immersed secondary teachers in focused investigations about student data and studied their statistical reasoning when comparing two groups. They concluded that such an immersion model can help improve teachers’ conceptual understanding of inference, their instructional practices , and their disposition toward inquiry. Bargagliotti et al. (2014) developed materials capable of promoting secondary teachers’ deep immersion in the study of variability and regression . At the tertiary level, Green and Blankenship (2014) designed a course to develop teaching assistants (TAs) as statistics educators. The course focused on how TAs can foster critical thinking and enhance learning in their classrooms. The TAs left the course with improved conceptual understanding of sampling distributions and strategies for teaching and assessing students.

6.3 Technological Environments

Numerous studies have investigated the use of dynamic statistical packages (Finzer, 2002; Konold & Miller, 2011) to develop teachers’ knowledge for teaching concepts such as sampling distributions, the central limit theorem, confidence intervals , and hypothesis testing (e.g., Garfield & Everson, 2009; Maxara & Biehler, 2010; Meletiou-Mavrotheris et al., 2014). Such studies indicate that teachers’ experimentation with statistical ideas through investigations of authentic and computer-simulated data can help them develop informal inferential reasoning, construct more sophisticated understandings of the logic of inferential statistics, and improve their repertoire of teaching strategies related to statistical inference . Those interested in exploring the potential of dynamic statistics software for supporting teachers’ learning can take advantage of resources such as Lee and Hollebrands’ (2008) teacher education curriculum that incorporates the software and Madden’s (2011) framework describing the characteristics of statistically, contextually, and/or technologically provocative tasks.

Along with dynamic statistics software environments, there are many other technological frontiers to continue to explore for teacher education . These include online communities, mobile devices, and the use of big data in relation to assessment and instruction. Environments and tools of this nature help break traditional boundaries of time, location, and extent of teacher learning (Koehler & Mishra, 2008). As relatively new, emerging technologies, much of the story of their impact on statistics teachers’ learning remains to be written.

7 Conclusion

Research on cognition and affect related to teaching statistics is a relatively new endeavor. Several opportunities for future research are identified in this chapter. Work remains to be done to more clearly define the elements of SKT , their relationships among one another, and the mechanisms through which they develop. As this work is carried out, it will be important to reconcile and refine different SKT models through systematic academic discourse among researchers working from diverse inquiry paradigms. We also need better understanding of interactions among affect, cognition, and teaching practices . We know that teachers’ goals, beliefs, and attitudes influence teaching practices to an extent. The precise nature of the types of goals, beliefs, and attitudes that are most relevant and their degree of impact need further investigation. Additionally, we need better understanding of teachers’ knowledge of the impact of social and environmental factors on students’ achievement and interest in statistics and their levels of preparedness to help diverse populations of students learn statistics.

Qualitative and quantitative approaches each have roles to play as research on cognition and affect for teaching statistics continues to develop. Some psychometrically and theoretically sound quantitative instruments specific to teaching statistics exist, but many studies still have to rely on instruments developed for the general population. Qualitative research can help define the salient cognitive and affective characteristics to be assessed and can provide vivid portraits of how such characteristics may develop under different circumstances. As this work occurs, we can gain progressively better understanding of teachers’ mediating role between statistical content and students.