New Standards Require Teaching More Statistics

Abstract

Mathematics teacher education programs often need to respond to changing expectations and standards for K-12 curriculum and accreditation. New standards for high school mathematics in the United States include a strong emphasis in statistics. This article reports results from a mixed methods cross-institutional study examining the preparedness of preservice secondary mathematics teachers to teach statistics and identifying factors and experiences that influence their preparedness. Our results suggest that the cohort of teachers entering secondary mathematics classrooms in 2015-2016 were not well prepared to teach statistics. Specific suggestions are given for how teacher education programs must rise to the challenge of preparing their graduates to teach statistics.

Keywords

mathematics education preservice teacher education teacher education preparation mixed methods

Secondary teacher education preparation programs commonly struggle to meet the demands of state and national standards, accreditation, and licensure, as well as update programs based on new research in teacher education. For example, many programs have been modifying course content to better align with recent adoptions of national standards in English, mathematics, and science. In mathematics, most states have either adopted the Common Core State Standards for Mathematics (CCSSM; National Governors Association Center for Best Practice & Council of Chief State School Officers, 2010) or modified their previously adopted standards to align with CCSSM (Academic Benchmarks, 2015). Therefore, teacher education programs must adapt to meet the challenge of preparing preservice secondary (Grades 6-12) mathematics teachers (PSMTs) to teach all aspects of mathematics included in CCSSM.

With an ever increasing world where statistical literacy is imperative for most careers, daily decision making, and informed citizenry (Franklin, 2013), the inclusion of statistics and probability in secondary (Grades 6-12) curricula has been long advocated for by the National Council of Teachers of Mathematics (NCTM; 1989, 2000). In 2007, the Guidelines for Assessment and Instruction in Statistics Education (GAISE; Franklin et al., 2007) was written to provide a strong foundation to support implementation of the NCTM standards at the PreK-12 level. “The Pre-K-12 GAISE Framework, the document endorsed and published by the American Statistical Association (ASA; which heavily influenced the Statistics standards in CCSS), presents the Statistics curriculum for grades Pre-K-12 as a cohesive and coherent curriculum strand” (Franklin, 2013, p. 6).

In an effort to assist teacher education programs, the recent Mathematics Education of Teachers II (Conference Board of the Mathematical Sciences, 2012) and Statistical Education of Teachers (SET; Franklin et al., 2015) recommended additional courses to develop statistical and pedagogical knowledge in secondary mathematics teacher education programs. In addition, the last decade has included recommendations to assist college faculty to reform statistics courses, particularly introductory statistics courses which may be taken by many secondary preservice teachers (e.g., Cobb, 2015; Garfield et al., 2007).

In 2016, Utts, the President of the ASA, reached out to mathematics teacher educators to express the urgency for teacher preparation to play an important role in the growing demand of a statistically literate society by implementing the recommendations presented in the SET report. However, without knowing how PSMTs’ current collegiate experiences are impacting their content knowledge and confidence to teach statistics, it is difficult for mathematics teacher educators to advocate for changes to the secondary mathematics teacher preparation program. Therefore, the purpose of this study is to examine PSMTs’ content knowledge of statistics and confidence in their ability to teach statistics, and describe experiences that appear to influence PSMTs’ preparedness to teach statistics. Gaining such insight can assist teacher education programs in making strategic changes to meet the demands of the changing standards in mathematics that intends to promote a more conceptual approach to statistics content that was markedly absent prior to CCSSM (Franklin, 2013).

Statistical Knowledge for Teaching

In typical secondary mathematics teacher preparation programs, PSMTs should have opportunities to develop statistical knowledge for teaching as learners in statistics courses, as well as courses focused on pedagogy. Regardless of where these opportunities arise, it is critical that PSMTs engage in experiences that can develop their statistical knowledge for teaching. Building off of the work of Hill, Ball, and Schilling (2008), Groth (2013) posits that statistical knowledge for teaching consists of two domains: subject matter knowledge and pedagogical content knowledge. Within subject matter knowledge, there are three types of knowledge teachers must develop: common content knowledge, specialized content knowledge, and horizon knowledge. Common content knowledge refers to knowledge of statistical topics and processes taught in high school. For this study, we are particularly focused on the statistics topics and processes outlined in the Pre-K-12 GAISE report (Franklin et al., 2007). These topics range in sophistication across three developmental levels labeled A, B, and C (typically taught in elementary, middle, and high school) and emphasize four phases of a statistical investigation: formulate questions, collect data, analyze data, and interpret results. Therefore, PSMTs need to understand key differences between mathematics and statistics and develop robust knowledge of topics such as measures of center, variability, distributions, bivariate data analysis, sampling distributions, and inference techniques.

Developing common statistical knowledge is not adequate to be prepared to teach statistics. Teacher educators need to develop PSMTs’ pedagogical statistical knowledge (e.g., Franklin et al., 2015; Groth, 2013), including the use of technological tools (e.g., Lee & Hollebrands, 2011), and attend to noncognitive aspects of teaching (e.g., beliefs, confidence) that can impact the PSMTs’ future instructional approaches (e.g., Ball, Thames, & Phelps, 2008). We take pedagogical statistical knowledge to include a teacher’s knowledge of potential student difficulties with statistics, strategies to support student’s learning, strategies to engage students in a statistical investigative cycle, including use of technology tools, and knowledge of statistics curricula (e.g., Groth, 2013). This requires teachers to be comfortable with various approaches, representations, and results that students may use during a statistical investigation, and understanding how to use technological tools to solve statistical problems and to teach statistical topics through investigations (Lee & Hollebrands, 2011). In addition, we take a holistic approach to preparation to teach statistics that encompasses attention to how and why PSMTs may believe statistics is difficult and how this may impact their perceptions about their ability to teach statistics content (e.g., Fitzmaurice, Leavy, & Hannigan, 2014; Harrell-Williams, Sorto, Pierce, Lesser, & Murphy, 2015).

Teaching Efficacy

Given our holistic approach to understanding PSMTs’ preparation to teach statistics, we focus on PSMTs’ self-efficacy for teaching high school students the statistical standards in CCSSM. Bandura (1986) defines self-efficacy as “people’s judgments of their capabilities to organize and execute courses of action required to attain designated types of performance” (p. 391). A teacher has two types of self-efficacy for each content area they teach: self-efficacy to know the content themselves, and self-efficacy to teach the topic to students, known as teaching efficacy.

To understand the roots of teachers’ efficacy, we focus on two sources suggested by Bandura (1997): mastery and vicarious experiences. Bandura describes mastery experiences as prior experiences in performing a task that are perceived to be successful. Palmer (2011) further elaborated two forms of mastery experiences: classroom teaching experiences and cognitive mastery. Palmer describes cognitive mastery as a teacher’s perceived success in understanding content and pedagogy necessary to teach a topic and vicarious experiences as those where an individual observes another person perform the behavior successfully.

Most research on teaching efficacy has been conducted with inservice teachers, and points to the positive impact of mastery experiences, as well as developing teachers’ cognitive mastery of content and pedagogy (Thrasher, Starling, Lovett, Doerr, & Lee, 2015; Palmer, 2011). While limited research of the impact of sources of preservice teachers’ efficacy has been conducted, one study of preservice elementary teachers found that mastery experiences, specifically cognitive mastery, had the greatest impact on preservice teachers’ teaching efficacy for mathematics and science (Newton, Leonard, Evans, & Eastburn, 2012).

Other research suggests three kinds of mastery experiences that potentially influence PSMTs’ teaching efficacy: university experiences, teaching experience, and world experiences (e.g., Conner, Edenfield, Gleason, & Ersoz, 2011; Rubie-Davies, Flint, & McDonald, 2012). In the context of our study, university experiences include cognitive mastery of content and pedagogy gained through university statistics content courses and mathematics methods courses; teaching experience includes prior experience teaching (or tutoring) mathematics or statistics at the K-16 level; and world experiences refer to workplace or everyday experiences with statistics.

PSMTs have both statistics self-efficacy and statistics teaching efficacy. Statistics self-efficacy is a “teacher’s belief in his/her ability to do statistics” and statistics teaching efficacy is “a teacher’s belief in his/her ability to teach statistics to bring about student learning” (Lovett, 2016, p. 83). In our study, we focus on the latter, but as our results will show, statistics self-efficacy and knowledge will come into play. Only a few studies have been conducted on statistics teaching efficacy of preservice or inservice teachers. For example, Fitzmaurice et al. (2014) examined the statistics teaching efficacy of preservice teachers through interviews and found that they were reluctant to teach statistics during their student teaching, but that those who did teach statistics later reported an increase in efficacy.

Method

This article is part of a mixed methods study on preparedness of PSMTs to teach statistics (Lovett, 2016). The study utilizes an explanatory design, first quantitatively examining PSMTs’ statistical knowledge and statistics teaching efficacy, and then qualitatively seeking factors and experiences that influence PSMTs’ confidence through analysis of open-ended responses and interviews. Thus, three research questions are investigated:

Research Question 1: What is PSMTs’ understanding of statistics content and statistics teaching efficacy for content they are expected to teach?

Research Question 2: How does PSMTs’ confidence to teach statistics compare with other areas of high school mathematics?

Research Question 3: From the perspectives of PSMTs, what factors and experiences influence their preparedness to teach statistics?

Participants

This study focuses on PSMTs prepared through university-based teacher preparation programs in the United States. As a complete list of all institutions that prepare PSMTs was unavailable, we decided to purposefully sample PSMTs from institutions whose faculty have participated in professional development efforts focused on statistics education from 2003-2013. Our assumption was that faculty from these institutions may have been motivated to include more of a focus on statistical content, even before CCSSM emerged in 2010. Thus, PSMTs from these institutions represent a critical case (Patton, 2002) as, perhaps, they have had more opportunities in their coursework than may be represented at other institutions.

To situate the context of the cases, the institutions chosen had faculty that engaged in professional development through either the National Science Foundation (NSF)–funded Preparing to Teach Mathematics With Technology (PTMT) project (ptmt.fi.ncsu.edu) or the ASA-funded Math/Stat Teacher Education: Assessment, Methods, and Strategies (TEAMS) program to increase the emphasis of statistics education at that institution. The PTMT project engaged faculty in summer institutes and pre-conference workshops from 2009 to 2013 to learn how to implement dynamic statistics tools, such as TinkerPlots (Konold & Miller, 2011) and Fathom (Finzer, 2007), and faculty tended to use the text materials (Lee, Hollebrands, & Wilson, 2010) produced by the project in both methods and statistics content courses for teachers at their home institutions. The TEAMS conference brought together faculty for three days in fall 2003 and focused on improving strategies for integrating conceptual approaches to statistics in both content and pedagogy courses. Our study is neither an evaluation of these projects nor an evaluation of individual institutional efforts. Rather, these institutions provide a context for a critical case of PSMTs that we hypothesize may have had university course experiences that attended to the importance of statistics in the secondary curricula.

Faculty from 57 institutions participated in either (or both) the PTMT and TEAMS programs. All 57 institutions were contacted, and 18 agreed to participate in this study. The majority of institutions (61.1%) had an enrollment profile of high undergraduate, and the majority of participants (83%) attended institutions with a basic Carnegie Classification™ of research universities (very high), research universities (high), or master’s college and university with a larger program.

PSMTs were recruited by their mathematics teaching methods instructor during the final weeks of their last mathematics education course prior to student teaching. All participants were undergraduate juniors and seniors, or graduate students earning initial licensure during fall 2014 or spring 2015 semesters. The number of PSMTs participating from each institution ranged from two to 31, with a mean of 12 per institution. Fourteen institutions had 100% participation of PSMTs, with the remaining four institutions having between one and four students who did not participate. This resulted in a total of 236 PSMTs participating in some aspect of the study. The majority of PSMTs were female (70.3%), and almost all (95.3%) reported having had at least one or two statistics courses. All data were blinded so that participants and universities remained anonymous, and that collectively they were considered as a single case.

Data Collection

Statistical content assessment

To examine PSMTs’ statistical knowledge, the Levels of Conceptual Understanding of Statistics (LOCUS) assessment instrument (Jacobbe, Case, Whitaker, & Foti, 2014) was administered online (locus.statisticseducation.org). This instrument was developed using an evidence-centered design approach that began with content domain analysis using the GAISE framework, the statistics content in the CCSSM, and learning trajectories from research on students’ learning in statistics (Haberstroh et al., 2015). Participants took the Intermediate/Advanced Statistical Literacy version, designed to assess statistics content taught in Grades 10 to 12. This 30-item version has been validated and is reliable with students in Grades 6 to 12 to assess statistical concepts present in the CCSSM, across Levels B and C of the GAISE framework and four phases of an investigative cycle (T. Jacobbe, personal communication, June 7, 2016)¹; although this instrument is not intended as a high stakes assessment of knowledge, it does represent the statistics content PSMTs are expected to teach their students in the near future. Thus, teachers are expected to score fairly high on the assessment. For each participant, the LOCUS assessment produces an overall score (percent correct), as well as five subscores for Level B, Level C, Formulating Questions, Collecting Data, Analyzing Data, and Interpreting Results.

Statistics teaching efficacy instrument

To examine PSMTs’ statistics teaching efficacy, the Self-Efficacy to Teach Statistics (SETS; Harrell-Williams, Sorto, Pierce, Lesser, & Murphy, 2014a) instrument was administered. The SETS instrument is aligned with the GAISE framework and reflects statistics content PSMTs are expected to teach.

The instrument contains 44 items on a 6-point Likert-type scale, from not at all confident (1) to completely confident (6), and six open-response items. An earlier version of this instrument with 26 items aligned with Levels A and B of GAISE was validated to measure K-8 preservice teachers’ self-efficacy for teaching statistics (Harrell-Williams, Sorto, Pierce, Lesser, & Murphy, 2014b). The high school version contains the 26 items aligned to GAISE Levels A and B, and an additional 18 items validated and aligned to GAISE Level C (Harrell-Williams & Pierce, 2015; Harrell-Williams et al., 2014a).² In addition to an overall score, the instrument provides subscale scores that correspond to Levels A (11 items), B (15 items), and C (18 items). For all Likert-type items, the stem of the question was, “Rate your confidence in teaching high school students the skills necessary to complete successfully the task” (Harrell-Williams et al., 2015, p. 3).

For the open-ended portion, in each GAISE level, PSMTs were asked to identify an item which they felt least confident to teach and an item which they felt most confident to teach and to explain their reasoning. Thus, there were up to six open-ended responses possible for each PSMT. There is some concern about using self-reported data due to a tendency of participants to present favorable images of themselves (Ross, 1989); however, there is little motivation to misreport as confidentiality was preserved (Baldwin, 2000).

Based on preliminary analysis from data in fall 2014 (n = 154), in spring 2015, participants (n = 81) were asked an additional question to compare their preparedness across five areas of high school mathematics (algebra, geometry, pre-calculus/advanced algebra, calculus, and statistics) and to rank these in order of how well they feel prepared to teach them from most to least. This ranking was collected to help situate PSMTs’ statistics teaching efficacy in relationship to other areas of high school mathematics.

Interviews

Semistructured interviews of 25 PSMTs were conducted via video-conferencing to further understand how experiences and factors affect their preparedness to teach statistics. These 25 volunteers were from the spring 2015 participants, and represented nine of the 18 institutions. PSMTs were asked to expand on their open-ended SETS responses; their experiences in statistics courses and mathematics education methods courses; previous tutoring, field, and teaching experiences; and their comfort level with statistical technologies. Transcripts of interviews and field notes were used to write a narrative summary of interviewee’s experiences (Merriam, 1998).

Data Analysis

Analysis occurred in two phases. Phase one was a quantitative analysis of data from the LOCUS and SETS. For the LOCUS, we examined the time taken to complete the assessment and decided to eliminate those participants taking less than 10 min, based on recommendations from the test authors (T. Jacobbe, personal communication, August 11, 2015). Thus, there were 217 useable LOCUS scores from the sample of 236. Descriptive statistics were computed and paired-samples t tests were used to test for differences of PSMTs’ statistical knowledge between GAISE Levels B and C. The rationale for comparing PSMTs’ subscores from GAISE Levels B and C is that Level C generally represents new content that PSMTs are expected to know with the adoption of CCSSM.

For the SETS survey, we began with accounting for missing data, as every PSMT did not respond to all 44 Likert-type items. The percentage of missing values ranged from 0% to as high as 5.1% for some items, and only 91.1% of the 236 PSMTs in the sample would have been available for analysis under listwise deletion. Data were primarily missing due to nonresponse, and this was addressed by using multiple imputation under the assumption that missing values are missing at random (Allison, 2002). We made this assumption because for 61% of the cases, there were no patterns of specific items that were skipped. For the other 39%, participants skipped a cluster of questions that could be due to scrolling issues on their device while taking the survey. Even though there did not seem to be a pattern in skipped items relating to difficulty of the items, there is always a chance this assumption is incorrect.

After using imputation to replace missing values, an overall score and three subscales were calculated for each PSMT as the sum of his or her Likert-type scores on the items divided by the total number of items, resulting in scores that corresponded to the 6-point Likert-type scale. Analyses on each imputed data set were pooled according to Rubin’s (1987) rules and pooled values were compared with the original data. The results were similar; thus, imputed results will be presented. As assumptions of normality and sphericity were not validated, a repeated-measures ANOVA was used to test for significant differences in PSMTs’ statistics teaching efficacy between the three GAISE levels.

Phase two, qualitative analysis, began by coding the open-ended responses on the SETS survey using a priori codes from the three major components of the framework: experiences, statistical knowledge, and pedagogical statistical knowledge. Then, we open coded (Corbin & Strauss, 2008) within data tagged for each component to identify themes. Iteratively, themes were collapsed into codes (e.g., procedural knowledge, appropriate teaching strategies, recent experiences). We coded all open-ended responses and randomly chose 15 PSMTs’ responses for comparison and found little disagreements, which were then resolved through discussion. We then recoded all responses based on the discussion. Analysis was then focused on coding 25 interviewee summaries using the resulting codebook from the analysis of open responses. Open coding was also used to capture themes that appeared in interviews, but had not appeared in open-ended responses (e.g., traditional statistics courses, knowledge of student misconceptions, use of dynamic statistical software).

Coded data from surveys and interviews were then separated into two categories: responses describing confidence to teach statistics and responses describing a lack of confidence to teach statistics. Responses from each category were examined to identify a list of factors that influence PSMTs’ confidence (or lack thereof) for each category.

Results

Statistical Knowledge

Due to the limited scope of this article, we briefly present findings about PSMTs’ statistical knowledge to help situate results in their confidence to teach (Table 1). The PSMTs had a mean statistical content score of 69% (SD = 14.06). PSMTs scored significantly higher (t = 5.772, p < .001) on Level B items (M = 70.85, SD = 17.69) than Level C items (M = 64.87, SD = 14.16), indicating their statistical knowledge is weaker as topics increase in statistical sophistication. For the overall scores and subscores, there are at least some PSMTs who scored between 90% and 100% correct, indicating they likely have strong common statistical knowledge of topics they will soon be responsible to teach. However, there is a concern as only one quarter of PSMTs scored overall above 77%, and a quarter scored below 57%. As the LOCUS was designed to assess students’ statistical knowledge in high school, this suggests that overall, PSMTs did not demonstrate a strong understanding of high school statistics content. Examining items by phases in the statistical investigative cycle, PSMTs scored highest on Formulating Questions and lower as the cycle progresses. A repeated-measures ANOVA determined that mean scores differed significantly between the four phases of the cycle, F(3, 648) = 64.73, p < .001. Post hoc tests using a Bonferroni correction revealed that PSMTs scored significantly lower on questions as the cycle progressed (p < .001). However, there was only a slight difference between mean scores for Analyze Data and Interpret Results (p = .32).

Table 1.

PSMTs’ Percent Correct on LOCUS Instrument.

	M	SD
Overall score	68.61	14.06
GAISE levels
Level B	70.85	17.69
Level C	64.87	14.16
Phases of statistical investigative cycle
Formulating Questions	80.37	21.51
Collect Data	70.40	19.70
Analyze Data	63.34	22.22
Interpret Results	60.48	16.25

Note. PSMT = preservice secondary mathematics teacher; LOCUS = Levels of Conceptual Understanding of Statistics; GAISE = Guidelines for Assessment and Instruction in Statistics Education.

An examination of items identified several strengths and weakness of PSMTs’ statistical knowledge. One important strength is that PSMTs are proficient at identifying an appropriate measure of center for a given context, a topic heavily emphasized in school mathematics. This strength suggests PSMTs should be well-equipped to assist their future students to develop stronger conceptions of measures of center beyond the standard algorithms. PSMTs’ weaknesses involve issues with understanding variability, sampling distributions, p values, and confidence intervals.

Statistics Teaching Efficacy

In the aggregate, PSMTs’ confidence to teach statistics was typically between somewhat confident (4) and confident (5), with the rating scale from 1 (not at all confident) to 6 (completely confident; see Table 2). There is a similar trend as with the statistical knowledge scores; PSMTs have the highest confidence in the least sophisticated topics to teach (Level A items). A repeated-measures ANOVA indicate mean confidence scores differed significantly between the three levels, F(2, 470) = 66.54, p < .001, with post hoc tests using a Bonferroni correction revealing PSMTs’ confidence was significantly lower as statistical sophistication of items increased (p < .001).

Table 2.

PSMTs’ Confidence Scores on SETS Instrument.

	M	SD
Overall score	4.06	0.80
GAISE levels
Level A	4.51	0.80
Level B	4.08	0.85
Level C	3.76	0.91

Note. PSMT = preservice secondary mathematics teacher; SETS = Self-Efficacy to Teach Statistics; GAISE = Guidelines for Assessment and Instruction in Statistics Education.

Examining scores by item identified specific topics that PSMTs’ felt least confident and most confident to teach. Within Level A, PSMTs were most confident to teach creating graphical representations recognizing that statistical results may differ from group to group (an early conception related to sampling variability). They were least confident to teach selecting appropriate graphical displays and generalizing results from a small to large group. For Level B, PSMTs were most confident to teach creating histograms and computing the interquartile range, and least confident to teach interpreting measures of association. The highest confidence within Level C was for teaching characteristics of a normal distribution and identifying slope and y-intercept in a regression equation. PSMTs were least confident to teach conditional and marginal frequencies using two-way tables, and formal ideas of inference such as margin of error and testing for statistical significance using simulations.

Confidence Teaching High School Mathematics Topics

To contextualize PSMTs’ statistics teaching efficacy, we examined the ranked order of 81 PSMTs’ confidence (1 is highest confidence) to teach five areas of high school mathematics: algebra, geometry, algebra 2/pre-calculus, statistics, and calculus. Figure 1 shows that 74% of PSMTs were most confident to teach algebra, while 63% of PSMTs were least confident to teach statistics. These results indicate that although PSMTs seem confident to teach statistics as measured by SETS, their confidence in teaching statistics is lower ranked than other high school topics.

Figure 1.

PSMTs’ ranking of confidence to teach high school topics from most (1) to least (5).

Influences on Statistics Teaching Efficacy

While our quantitative results identified trends in PSMTs’ statistical knowledge and statistics teaching efficacy, our analysis of the open-ended SETS responses and interview summaries sought to provide insight as to what experiences and factors may be contributing to PSMTs’ perceived preparedness to teach statistics. Four major categories emerged in our analysis: role of statistics knowledge, role of pedagogical knowledge, impact of using technology, and view of statistics. These categories were not equally present in the open-ended SETS responses. Thus, we first present data on the frequency of occurrence of each categorical factor to give a sense of prevalence and importance, from the perspective of the PSMTs, on the impact on their confidence to teach statistics. Then, we use specific details from survey responses and interviews to elaborate on PSMTs’ confidence or lack of confidence for each of these categories in order of prevalence.

Occurrence of factors in open-ended responses

Because every PSMT did not answer each open-ended response, there were 657 responses for items that PSMTs felt least confident to teach, and 659 responses for items they felt most confident to teach. The four major categories give insight to the most dominant factors that we identified that PSMTs attribute to influencing their confidence to teach statistics, but the prevalence of the categories differed greatly (see Table 3). PSMTs’ perception of their statistical knowledge seemed to have the same large impact in both their least and most confident to teach responses. However, there is a large difference in the other categories. The factors of pedagogical statistical knowledge, the impact of using technology, and view of statistics appeared more frequently in PSMTs’ responses regarding topics they were most confident in teaching than in responses regarding topics they were least confident in teaching. Also, as PSMTs in this study felt most confident to teach algebra (Figure 1), it makes sense that PSMTs were confident in teaching items they viewed as teaching a procedure or teaching algebra.

Table 3.

Occurrence of Factors in Open-Ended Responses.

	Least confident to teachn (%)	Most confident to teachn (%)
Role of statistical knowledge	585 (89.0)	588 (89.0)
Role of pedagogical knowledge	127 (19.3)	230 (34.9)
Impact of using technology	26 (4.0)	58 (8.8)
View of statistics	26 (4.0)	99 (15.1)

Role of statistical knowledge

Of the responses regarding items PSMTs felt least confident to teach, 89% of the responses were coded as lack of statistics knowledge. Two themes emerged in responses that were coded as lack of statistics knowledge: declarative statements about their lack of knowledge and the need to review a topic. In more than half of these responses, PSMTs made short statements that did not provide much information on experiences that related to their statistical knowledge (e.g., “I’m not sure that I know what a two-way table is”). However, many PSMTs expanded on their responses and discussed previous experiences related to their statistical knowledge that had an impact on their confidence. In these responses, PSMTs often noted they needed to review or “brush up” as they had not recently had experiences with the topic. This theme often occurred when PSMTs were discussing their lack of confidence to teach graphical representations, measures of association, two-way tables, and items involving formal inference, such as developing a margin of error for an estimated population mean. A response that typified this theme was,

I feel least confident here [margin of error] because I have only touched upon this in one of my early classes post-secondary, and it was not in the context of teaching it, but learning it. I could do research and learn it again so I could teach it, but right now I would not feel confident in teaching this.

Like many PSMTs, she identified she had learned this material in her previous college course, and although she did not feel like she remembered the content, she could review and teach it to herself and then feel confident in teaching it.

When discussing items PSMTs were most confident to teach in the open-ended responses, 89% of the responses also referred to their statistical knowledge. Some PSMTs continued to make short statements and did not provide much information about previous experiences. However, in more than half of the most confident responses that referred to statistical knowledge, PSMTs discussed their previous experiences to support their confidence. For example, “I feel very confident with histograms because that is mostly what my statistics class covered.”

As the interviews were focused on PSMTs’ experiences with statistics, and all interviewees were asked to discuss their college statistics courses, the interviewees all discussed that their statistical knowledge impacted the confidence to teach statistics. Approximately one third of the 25 PSMTs interviewed discussed that their university statistics courses were procedurally focused and they stated they did not develop a deep understanding of statistics because of the nature of the course. For example, “Even the statistics content I do feel comfortable with was more of a memorize and regurgitate the process and repeat it kind of approach instead of understanding it.”

All 25 interviewees were also asked about their experiences using simulations to develop a sampling distribution and test for statistical significance (a common item on SETS indicated as low confidence), yet only one reported experience learning this material they will be expected to teach (CCSSM standards HSS.IC.B.4, HSS.IC.B.5). An interviewee felt that the reason she and other PSMTs did not have experience with simulations was because “A lot of teachers we had are, I don’t want to say old fashion, but old fashion in a way that they don’t use a lot of technology and they don’t get the creativeness in using simulations.”

Almost all interviewees (n = 21) felt most confident to teach topics with which they have had extended experiences. For instance, in discussing confidence to teach creating histograms, one said, “Histograms have been incorporated into the regular math classes. Pretty much any math class I have ever been in since middle school has included a stat unit and histograms have been a part of it.” Thus, it is not surprising that PSMTs felt confident in areas such as creating graphical representations and computing measures of center because of extended experiences with them in middle and high school.

Role of pedagogical knowledge

Approximately 19% of PSMTs’ justifications of items they were least confident to teach were categorized as indicating a concern about their pedagogical knowledge. The majority of time, these responses still included PSMTs reflecting on their statistical knowledge, as well as their pedagogical statistical knowledge. From these responses, two themes emerged: knowledge of appropriate teaching strategies and knowledge of students’ approaches and difficulties in statistics.

PSMTs frequently discussed that they did not know how to teach or explain a topic when justifying their lack of confidence. As previously mentioned, often this stemmed from their content knowledge, but it was expressed in terms of pedagogical statistical knowledge, often because PSMTs lacked confidence to teach topics when they were lacking teaching strategies and resources. For example, “I would be least confident in this because I wouldn’t know how to go about showing this to students explicitly without just saying it.” Similarly, many PSMTs discussed they would need to learn more about the best approach to teaching a topic beforehand.

PSMTs also often expressed lack of confidence because they did not know different ways students could approach a topic. One PSMT discussed her lack of confidence in teaching variability because she felt “it’s difficult to prepare for a solution path that could occur if you are unable to think in the same manner as that particular student.” Another PSMT discussed what he thinks he knows about students’ difficulties with association that are impacting his confidence to teach it, “I think students would have a difficult time understanding how association and cause and effect are not exactly the same thing.” Both of these types of justifications pointed to PSMTs recognizing that pedagogical knowledge for teaching statistics specifically was important, but that they were not well-equipped with knowledge of students’ understandings and misconceptions regarding statistical topics.

When discussing items PSMTs were most confident to teach, 35% of the instances were categorized into pedagogical statistical knowledge. In just under a third of these responses, PSMTs discussed their teaching experience where they had an opportunity to teach others a statistical topic. This idea came up as well during interviews when PSMTs were asked what other knowledge or experiences they need to feel prepared to teach statistics. Several PSMTs discussed needing more experiences “actually discussing more of how to do it [teach statistics] with students.” PSMTs recognized they need to know appropriate teaching strategies and resources to feel more prepared to teach statistics.

To investigate PSMTs’ experiences in developing pedagogical statistical knowledge, all interviewees were asked about their field experiences and tutoring, mathematics education methods courses, and opportunities to learn about students’ understandings and misconceptions. Regarding field experiences and tutoring experiences, five of the 25 interviewees said that they had tutoring or teaching experience with statistical topics. These PSMTs felt more confident in statistical topics they had experience teaching and described the value of such experience. Perhaps one of the most intriguing comments was as follows:

I wish I would have [had tutoring or teaching experience in statistics] now because I am going to teach statistics in a month and I have no experience what-so-ever. It could have been a valuable learning opportunity for me to get comfortable with it in a one-on-one setting before I have 30 teenagers staring at me.

In general, PSMTs seem acutely aware that practical teaching experiences would be immensely helpful in building their confidence to teach statistics.

When asked about their mathematics methods course, PSMTs that were interviewed felt like their mathematics methods courses prepared them well to teach secondary mathematics. When probed to discuss how the mathematics methods course prepared them to teach statistics, over half of the PSMTs interviewed explicitly stated that their mathematics methods course(s) did not prepare them to teach statistics. The other PSMTs felt that their mathematics methods courses were a “nice start” or “it was just like scratching the surface.” PSMTs reflected that one reason they feel their mathematics education course(s) did not prepare them to teach statistics was that these courses were largely focused on preparing them to teach algebra. For example, “When we are secondary teachers and statistics is going to be such a big focus now. I feel like more time should definitely be spent on it. It shouldn’t be something you stick in for one week”.

Another interviewee felt like she could “tell you more about how abstract algebra relates to teaching algebra than I can about some basic statistics topics.” Not only across different universities did PSMTs express an emphasis on algebra, but they also expressed that the statistics unit (often 1 week) in their mathematics methods courses mostly occurred near the end of the semester and “was thrown in at the end and cut off early.”

Interviewees were also asked if students’ understandings and misconceptions were discussed in their mathematics methods courses. All interviewees discussed that students’ understandings and misconceptions were a part of the mathematics methods course, but only half recalled understandings and misconceptions specific to teaching statistics being discussed. One interviewee stated, “we [PSMTs at her university] studied so much about how students think about geometry . . . but statistics, I don’t know how they think about it. I think that would be really helpful.” Another interviewee expressed similar feelings that “it would be helpful to know more about how kids think about statistics. Because I don’t know anything about the misconceptions students would have.” Methods courses for secondary mathematics are perceived by PSMTs as preparing them holistically for teaching mathematics and giving them strategies and understanding of students’ conceptions in topics like algebra and geometry, but lacking in statistics.

Impact of using technology

The use of graphing calculators and dynamic statistical software such as TinkerPlots and Fathom emerged in PSMTs’ least and most confident responses. In only 4.0% of instances did PSMTs justify their lack of confidence because of their previous experiences with technology. These responses focused on not having the knowledge or experience using technology, even the graphing calculator, to fit linear models or calculating the correlation coefficient. One example is from a PSMT who expressed lack of confidence to teach fitting linear models as, “I have little knowledge on this content material or understanding for how to apply it to various types of technology; even the graphing calculator.” Thus, a few PSMTs seem to recognize they should have more agile abilities and knowledge of how to use technology for these concepts.

In approximately 9% of responses regarding topics they were most confident to teach, PSMT also reflected on their experiences with technology. In these instances, two themes emerged: PSMTs viewed using technology as a computational tool that they were confident in using, such as computational statistical software. One PSMT noted, “It’s [calculating correlation coefficient] easy. The software should just pop this value out,” and another stated, “I know how to click the right button on Fathom or TinkerPlots that will give me the fit for a model. I am confident that I can show students these same buttons.” Thus, some PSMTs referenced work with technology to support their confidence because they were comfortable quickly generating computations, but made no reference of using dynamic statistical tools for addressing conceptual issues.

Some PSMTs, however, did discuss experiences they had with technology that helped them develop a deeper understanding of statistical topics. This emerged only in responses when PSMTs discussed their confidence to teaching graphical representations, typified by a PSMT’s confidence in teaching comparing distributions, “I have had great training in TinkerPlots and Fathom, this is a great way to teach students about the spread of data.” Some PSMTs went more in depth in describing the knowledge they gained through working with technology:

We not only learned multiple technologies that can assist us in our lessons, but we had to work through the material ourselves. I feel that this has really helped us to develop a better understanding of the material and which aspects may be difficult for our students. Thus, it helps us see how to form a lesson that can help our students understand the material better as well.

To further understand the impact of dynamic technology on PSMTs’ confidence, interviewees were asked to reflect on their experiences with technologies, what topics they examined using those technologies, and to discuss how useful they view those technologies for teaching.

Seven of the 25 PSMTs elaborated on their lack of experiences using technology and acknowledged they needed to know more about technology for learning statistics and how to use in their lessons. These seven PSMTs had no previous experiences with dynamic statistical software; they had only used a graphing calculator or statistical computing software (e.g., SAS). One interviewee commented that she needed a program “like GeoGebra for statistics” to be able to teach it because “technology is most important in statistics because you can’t do it without any technology.” These seven PSMTs recognized the power of technology in teaching, especially statistics, and were aware that they were lacking experiences with technology to prepare to teach statistics.

Sixteen of the 25 PSMTs interviewed reported previous experiences using dynamic statistical software in a mathematics methods course. These interviewees were asked about topics they had experience with, and if and how they would incorporate it into their future classrooms. They mentioned experiences creating graphical representations, examining measures of center, and fitting linear models, and several described how experiences using TinkerPlots and Fathom provided experiences to explore data, therefore increasing their confidence. All 16 PSMTs with experiences using dynamic statistical software expressed that their experiences using those technologies deepened their understanding and they would use those technologies to provide their future students with similar opportunities. For example, one interviewee discussed that TinkerPlots is “a good visual for them [students]” and would engage her future students in “exploratory lessons like we did” in her methods courses.

View of statistics

The PSMTs’ view of statistics emerged as another category that influenced why they were least or most confident to teach a statistical topic. In PSMTs’ responses regarding topics they were least confident to teach, 4.0% of instances referred to some topics in statistics as being a “gray area,” being “uncertain” in the correct answer, or no “algorithm” to do it. This occurred most frequently in PSMTs’ Level A responses regarding variability, sampling, and informal inference. The following response shows how a PSMT acknowledged her view of statistics, “Most of the other tasks were more objective—making plots, comparing plots. They were more technical in nature. Recognizing limitations, is more obscure, . . . it is an art.” Here, she attributes her lack of confidence to teach recognizing the limitations in making inference as an area of uncertainty, instead of a procedure like creating a graphical representation.

In PSMTs’ responses regarding topics they were most confident to teach, two themes emerged: viewing statistical topics as procedures and viewing statistics as algebra. Viewing statistical topics as procedures emerged in 15.1% responses regarding PSMTs’ confidence in teaching graphical representations and computing measures. An example response that typifies confidence in teaching graphical representations is, “it [creating histograms] is a simple idea and is somewhat procedural. It does not require a lot of cognitive demand to create a histogram of data that you’re given.” Many PSMTs discussed their confidence in teaching how to compute interquartile range because “this is more procedural and easy computations, which is why I feel the most confident to teach this.” When PSMTs responded in this way, they were reflecting on topics being solely procedural and did not consider other nonprocedural aspects such as determining bin width for intervals of a histogram based on problem context.

In responses where PSMTs were viewing statistics as algebra, PSMTs were comparing statistical topics with algebraic ones or referencing they had learned the topic in high school algebra courses. This most frequently occurred when PSMTs justified their confidence to teach identifying the slope and y-intercept in a linear model and interpreting them in the context of data, because “I feel most confident about teaching this topic since it directly relates to algebra which I am very familiar with.” Often PSMTs focused on identifying slope and y-intercept, and did not comment on interpreting these in the context of data when justifying their confidence.

PSMTs’ view of statistics also emerged during interviews. Eight of the 25 interviewees discussed that the statistical topic they felt least confident to teach was different from mathematics because it did not have a procedure or a right answer. An interviewee discussed in her previous experiences with statistics she was “always looking for a right answer” and lacked confidence in herself because she “could never come up with the right answer.” Another PSMT stated, “I would assume so many people are uncomfortable with statistics because so much of it is centered around thinking on your own rather than following a set procedure.” Thus, PSMTs seem very uncomfortable in areas of statistics that require statistical thinking that are inherently built on uncertainty.

When interviewees discussed topics they were most confident to teach, 20 of 25 PSMT interviewees described the topics as “simple,” “procedural,” or “computational.” A typical sentiment was because “either you know interquartile range or you don’t. There is not that much understanding to computing it. I feel like I can tell someone how to do a procedure and they can do it.” This PSMT focused not just on the computational nature, but that teaching a computation was perceived as easy to do. The theme of viewing statistics as algebra emerged as well when eight of 25 interviewees mentioned that this particular item “sounds like algebra, it is easy to do. It is just math not statistics.” Thus, overall PSMTs are confident when they view an aspect of statistics as procedural or algebra-like, but less confident when it involves uncertainty.

Discussion and Recommendations

With increased emphasis on statistics in high school mathematics, our cross-institutional study investigated a cohort of PSMTs’ common statistical knowledge and statistics teaching efficacy as they completed their teacher education program. While these data give broad overviews, the intent of our third research question was to unpack the experiences and factors that contribute to PSMTs’ preparedness to teach statistics. Together, the results from the quantitative and qualitative analysis provide evidence to support recommendations for how teacher education programs should consider improvements to their programs given the increased emphasis on statistics in the Common Core standards.

PSMTs’ Statistical Knowledge

By using an instrument designed to assess high school students’ understanding of statistics commonly included in CCSSM-informed curricula, our results illuminate the current landscape of PSMTs’ knowledge of the statistical topics they will soon need to teach. As we have shown, PSMTs’ statistical understandings diminish as the sophistication of topics increase from GAISE Level B to C. This indicates that PSMTs are more knowledgeable about content taught in middle school, but much less knowledgeable of topics typically taught in high school. Previous research has shown a similar trend with inservice teachers demonstrating stronger understandings of GAISE Level B statistical knowledge measured by LOCUS (Jacobbe, 2015). Quite simply, it appears that PSMTs may not be getting enough opportunities to learn the statistics content they are going to teach in high school.

PSMTs’ Statistics Teaching Efficacy

As we have shown, PSMTs’ confidence diminishes as the difficulty of topics increases from GAISE Level A, to B, to C, and thus, they are least confident in teaching typical high school content in the CCSSM. PSMTs felt much more confident to teach algebra and geometry than statistics and, accordingly, felt more confident to teach aspects of statistics they perceived as procedural (e.g., creating graphical representations, computing measures, describing normal distributions, identifying slope and y-intercept). Areas that PSMTs felt least confident to teach include selecting appropriate graphical representations, measures of association, two-way tables, and ideas of formal inference using simulation. PSMTs’ confidence to teach creating graphical representations and computing measures, and lack of confidence to teach interpreting measures of association, is consistent with previous research (Harrell-Williams et al., 2015). These findings provide a clear focus on areas that should be emphasized in courses that could strengthen PSMTs’ preparedness to teach statistics.

Recommendations for Enhancing PSMTs’ Preparedness

Our results show that PSMTs’ confidence or lack of confidence to teach statistics stems largely from their experiences and knowledge gained, or not gained, in their university statistics courses and mathematics education courses. Considering the specific areas of PSMTs’ strength and weaknesses, and the associated factors and experiences the PSMTs identified, we have developed several recommendations that can assist secondary mathematics teacher education programs.

University statistics courses

Despite over a decade of reform efforts placing a greater importance on statistics education and efforts to change statistics teaching, PSMTs report that their college courses were typically traditional lecture that did not incorporate active learning, were procedurally focused, and did not use simulation approaches to inference. Taught in this way, these courses are not deepening PSMTs’ statistical knowledge or engaging them in statistical reasoning, and do not seem to be implementing the GAISE college-level standards (Garfield et al., 2007). Thus, if PSMTs continue to experience statistics as procedural and “algebra-like,” it will be difficult to have more wide-scale change on how teachers perceive and understand statistical content they are expected to teach.

Widespread efforts like the Consortium for the Advancement of Undergraduate Statistics Education (www.causeweb.org) have been advocating for changes to statistics courses for over a decade, but our results indicate that they may not be implemented on a large scale. Mathematics teacher education and statistics faculty should work together to reexamine the statistics courses PSMTs are required to take to align with updated guidelines for statistics courses from the GAISE college report and SET recommendations (Franklin et al., 2015), that include an emphasis on active learning and use of technology tools for learning and doing statistics. PSMTs should have opportunities in their university statistics courses to engage with dynamic statistical software to build conceptual understandings of topics that are now included in secondary mathematics curricula such as two-way tables, correlation, variability, sampling distribution, margin of error, and simulation-based approaches to inference.

Mathematics methods courses

In mathematics methods courses, there should be a greater emphasis on statistics and statistics pedagogy for PSMTs to feel prepared to teach it. The PSMTs themselves identified this as missing and yearned for more opportunities to learn how to teach statistics. These courses should include topics that PSMTs feel comfortable with, such as creating graphical representations, and build from this knowledge to develop a deeper understanding through engaging in the statistical investigative cycle. Also, mathematics methods courses should provide PSMTs with experiences in learning and teaching other areas of statistics, such as selecting appropriate graphs, measures of association, two-way tables, and simulation-based inference as PSMTs do not feel confident to teach these areas.

PSMTs in this study understand the importance of learning about teaching strategies and student misconceptions to effectively teach mathematics. Therefore, PSMTs feel less prepared to teach statistics than algebra and geometry because they have not had opportunities to learn about pedagogical issues in statistics like they have had for algebra and geometry. This result points to the need for secondary mathematics teacher programs to incorporate substantial units into methods courses that focus on developing pedagogical knowledge for topics difficult to understand and teach such as variability, sampling distributions, and simulation-based inference, all of which benefit from the use of dynamic statistical software. This, of course, points to the need for mathematics teacher education faculty to be well versed in these strategies and technology tools, and to have access to appropriate materials for use in their methods courses. We also wonder about the preparedness of mathematics teacher education faculty to teach others to teach statistics, as much of their own education may not have focused on teaching and learning statistics.

As seen in our results, PSMTs are not being provided opportunities to experience teaching statistics in their fieldwork or through writing lesson plans. Mathematics teacher education programs should build opportunities for PSMTs to experience teaching a statistical topic. Suggestions include requiring PSMTs to develop a lesson plan on a statistical topic, and arranging for PSMTs to serve as tutors or teaching assistants for introductory statistics classes, providing PSMTs opportunities to explain statistics to others. Finally, programs could ensure that PSMTs are observing a variety of topics, including statistics, during their field experiences.

Limitations

There were a number of limitations, which should be taken into consideration when interpreting the findings. First, this study was a purposeful sample of PSMTs from U.S. institutions to serve as a critical case and was not a random sample. While most studies in teacher education have been conducted at one institution or a small number of institutions with small sample sizes, our study includes 18 institutions varying in program size and location, which may expand the potential usefulness of results. Another limitation is the SETS instrument focuses largely on procedural aspects of statistics; therefore, many of PSMTs’ responses reflected on their confidence to teach procedures as well. To capture PSMTs’ true statistics teaching efficacy for how statistics should be taught, changes should be made to the instrument to better align teaching conceptual understanding of statistics. Teachers need to engage students in formulating questions, collecting data, analyzing data, and interpreting results, but the instrument focuses heavily on confidence to teach students to analyze data.

Conclusion

Teacher preparation in advanced disciplinary content for secondary teachers (math, science, English, social studies) have always struggled to balance the need to advance teachers’ understanding of content in the domain, as well as increasing pedagogical knowledge. Our results suggest that while PSMTs recognize the importance of, and seem to yearn for more, understandings of pedagogical strategies, they are also very concerned with their conceptual understanding of statistics. The PSMTs in this study generally do not exhibit strong common content knowledge or confidence to teach many aspects of statistics needed for teaching high school students. Studies examining mathematics teacher educators’ preparedness and current practices in methods courses would aid in fully understanding this important issue. Our results, however, suggest a critical need for mathematics teacher education programs to consider strategic ways they can increase PSMTs’ preparedness to teach statistics, while also maintaining their strong attention to other content areas taught in secondary mathematics. Our results provide fruitful directions for programs to consider for how they can increase the emphasis on statistics and statistics teaching, ideally in collaboration with faculty who teach statistics content courses for teachers. If secondary mathematics teacher education programs continue to not emphasize the importance of statistics in high school curricula and lack opportunities that provide PSMTs the content and pedagogical approaches they need, we fear that new teachers entering the field will continue to be underprepared and perhaps devalue statistics in their curriculum and instruction. It is time to rise to meet this challenge.

Footnotes

Authors’ Note

The opinions, findings, and conclusions or recommendations are those of the authors, and do not necessarily reflect the views of the Foundation.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This article is based on the doctoral dissertation of the first author, under the direction of the second author, and was partially supported by the Preparing to Teach Mathematics With Technology project, funded by the National Science Foundation with grants to North Carolina State University (DUE 0442319, DUE 0817253, and DUE 1123001).

Notes

Author Biographies

Jennifer Lovett is an assistant professor of mathematics education in the department of Mathematical Sciences at Middle Tennessee State University. Her research focused on the teaching and learning of mathematics and statistics using technology tools.

Hollylynne Lee is a professor of mathematics and statistics education in the department of STEM Education at NC State. Her research focuses on the teaching and learning of statistics and probability, particularly with technology tools. Many projects include the development of educational resources for statistics teacher education and K-12 classroom materials.

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