Teacher Preparation in Statistics: Focusing on Variability through Attending to Precision

Abstract

The Common Core Standards for Mathematical Practice describe mathematical habits of mind foundational for mathematical thinking and understanding. The Statistical Education of Teachers (SET) report (Franklin et al., 2015), commissioned by the American Statistical Association (ASA), interprets the Mathematical Practices through a statistical lens. In this paper, we present tasks focused on variability that can be taught through the lens of Mathematical Practice 6: Attending to Precision (MP6). We show that teacher tasks that focus on variability and its quantification (by computing multiple different measures on different aspects of an entire data set and constructing arguments), prompts teachers to attend to statistical precision. Additionally, we make the connection that ideas related to variability are present in all components of the statistical investigative process: posing questions that anticipate variability, anticipating variability in data collection, analyzing the variability present, and interpreting the variability to make quantitative statements about error and tasks that aim towards understanding variability require the employment of Statistical MP6.

Key Words:

• Statistical Knowledge for Teaching
• Mathematical Practices
• understanding variability
• statistics education

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1. Introduction

Data are everywhere, and working with, understanding, and learning from data have become necessities in our daily lives. As such, it is imperative that students develop statistical literacy in school in order to make sense of the data-driven world around them (Bargagliotti et al., 2020; Levitt 2019; Keller et al. 2020). In 2010, the Common Core State Standards for Mathematics (CCSSM) added a heavy emphasis on statistics in the middle school years and continued that through high school (CCSSM 2010). Many states have since updated their standards from a decade ago and have included data science or data reasoning as a new content domain within their standards or included more data reasoning within their statistics standards (e.g., Oregon included an additional data reasoning strand throughout their K-12 mathematics standards (Oregon DoE 2021), Georgia included more data reasoning within their statistical reasoning strand (Georgia DoE 2021)). Overall, the study of data within K-12 has been expanding and teachers are increasingly being asked to work with data and teach about data in their own classrooms.

A central concept in working with data is understanding how to deal with variability in data. Cobb and Moore describe the discipline of statistics as a “methodological discipline that exists not for itself, but rather to offer other fields of study a coherent set of ideas and tools for dealing with data. The need for such a discipline arises from the omnipresence of variability” (Cobb and Moore 1997, p. 801). Therefore, a primary goal in data and statistical analyses is to recognize variability in data and work towards understanding and quantifying it. As noted in the Pre-K-12 Guidelines for Assessment and Instruction in Statistics Education II report (GAISE II), “statistical thinking, in large part, must deal with the omnipresence of variability in data (e.g., variability within a group, variability between groups, sample-to-sample variability in a statistic). Statistical problem solving and decision making depend on understanding, explaining, and quantifying variability in the data within the given context” (Bargagliotti et al., 2020, p. 7).

Although explaining variability lies at the heart of statistics and is a fundamentally important concept to grasp (Cobb 1992; Garfield and Ben-Zvi 2005; Utts 2003), students of all ages and teachers alike grapple with the topic (Aggarwal et al. 2021; Ben-Zvi 2004; Garfield and Ben-Zvi 2005; Konold and Pollatsek 2002; Peters 2011; Pfannkuch 2008; Roth and Temple 2010). Research indicates that students and teachers have difficulty identifying, dealing with, and accepting variability (delMas and Liu 2005; Kaplan et al. 2010; Konold et al. 1997; Vermette and Savard 2018). For example, Shaughnessy et al. (2004) surveyed 272 middle- and high-school students to investigate their conceptions of variability and found that only 8 out of the 272 students acknowledged variability when it was present. Working with teachers, Hammerman and Rubin (2004) noted that secondary teachers involved in professional development compared groups by discussing the variability in distributions using only segments and slices of the distributions and not the entire picture. Confrey and Makar (2002) found similar results with middle school teachers, who examined variability in distributions by focusing on single points instead of the distribution as a whole. Mooney et al. (2014) administered pre- and post-tests to pre-service teachers focused on variability in data displays, variability in sampling, and measures of variability. In interviewing a subset of 15 pre-service teachers that took the tests, they found that the teachers need more opportunities to collect and examine multiple samples to help them draw conclusions and patterns. While the pre-service teachers could discuss distributions with respect to its center, they need more practice recognizing the variability present in graphical displays of distributions.

Some evidence in the literature suggests possible ways to teach about variability in an effective manner (Chaphalkar and Wu 2020; Wells 2018; Wessels 2014). For example, Pfannkuch et al. (2014), worked with 21 students in their 11th year of school in New Zealand to support their development of their understanding of sampling variability. They developed specific instructional tasks that incorporated dynamic visualizations, mental process visualizations, and verbal descriptions. They found that engagement in these tasks led to emerging conceptualizations of sampling variability. While working with 78 teachers in Japan and 77 teachers in Thailand, Isoda, et al. (2018) presented nine tasks that covered various aspects of variability previously used in the literature. The comparative study found that there were cultural differences in the approach to variability of the teachers in Japan and those in Thailand. The developed tasks offered scaffolding which provided opportunities for teachers’ conceptual knowledge of variability to grow.

In this paper, we present tasks focused on variability that can be taught through the lens of Mathematical Practice 6: Attending to Precision (MP6), as stated in the American Statistical Association (ASA) commissioned Statistical Education of Teachers (SET) report (Franklin et al. 2015). We show that tasks focused on variability are tasks that require teachers to quantify variability in the data and explain their quantification. The tasks, developed by the NSF-funded Project-SET, a project aimed at developing Statistical Knowledge for Teaching (SKT), provide examples of tasks that teacher educators can use in teacher preparation to enhance teacher understanding of variability through teacher engagement with MP6. Note that author Bargagliotti used these tasks as activities to accompany a teacher learning progression on variability (Bargagliotti & Anderson, 2017). Such tasks are important for teacher educators as they provide a way to engage with the Mathematical Practices (MPs) while addressing important statistics content that is typically challenging for teachers and students alike.

2. Background

As statistics and data science are becoming more prominent in K-12 and standards related to data are mostly found as strands within mathematics standards, mathematics teachers are often charged with delivering such content to students. However, as noted by the Mathematical Education of Teachers II (MET II) report, there is a necessary and significant opportunity to enhance the education of mathematics teachers in the area of statistics by providing more comprehensive training (CBMS, 2012). In response to the MET II report, the ASA commissioned the SET report (Franklin et al., 2015) to unpack the statistics concepts that teachers should be exposed to. SET describes the teacher preparation recommendations of MET II by highlighting what statistics topics K-12 teachers need to know at each grade band. In addition to content recommendations, the SET report also interprets the MPs under a statistical lens. Differences in habits of mind needed in statistics with those needed in mathematics are highlighted in the SET report interpretation. In particular, as stated in the Guidelines for Assessment in Statistics Education PreK-12 (GAISE I) report, “a focus on variability in data [is what] sets apart statistics from mathematics” (Franklin et al. 2005, p. 6). This focus on variability in statistics requires heavy use of MP6. The description of MP6 provided in the CCSSM was adapted to statistics in the SET report in the following manner (condensed for brevity):

Statistically proficient students understand that precision in statistics is not just computational precision. In statistics, one must be precise about ambiguity and variability. Students understand that the statistical problem-solving process begins with the precise formulation of a statistical question that anticipates variability … Precision is also necessary in designing a data collection plan that acknowledges variability. … After the data have been collected, students are precise about choosing the appropriate analyses [of the data] … they are precise with their terminology and statistical language. … Students recognize the precision of this estimate depends partially upon the sample size––the larger the sample size, the smaller the margin of error. As students interpret statistical results, they connect the results back to the original statistical question and provide an answer that [allows for] the variability in the data … (Franklin et al 2015, p. 11-12).

To differentiate the MP6 given in the SET report from the MP6 given in the CCSSM, we will refer to the latter as “Mathematical MP6” and the former as “Statistical MP6.” The description of Statistical MP6 above discusses the statistical investigative process of formulating a question, collecting or considering data, analyzing data, and interpreting data; all while attending to variability. We see that ideas related to variability are present in all of these process components. One must pose a question that anticipates variability, the data one collects will have variability, the analyses must aim to understand the variability present, and the interpretation considers the variability to make quantitative statements about the error. Per this description, we see that understanding variability is intertwined with precision as described in Statistical MP6. Also noted in Statistical MP6 is the precision in calculating and analyzing margin of error. In this case, variability of an estimate can be quantified using a margin of error. While both Mathematical MP6 and Statistical MP6 require students to communicate precisely, problem solve, and be precise in computation and measurement, Statistical MP6 requires an additional component of being precise in addressing ambiguity and variability. We aim to show through comparison to standard statistical tasks that tasks that require precise measurement and statements quantifying variability offer viable and productive opportunities for teachers to conceptualize the notion of variability through Statistical MP6.

3. Teacher Understanding of Variability in Statistics

Teachers receive very little training in teaching statistics in their teacher preparation programs (Heaton and Mickelson, 2002). As a result, many teachers find statistics a difficult concept to teach and have negative attitudes towards statistics (Batanero and Lancaster 2011; Begg and Edwards 1999). Specifically with respect to variability, in addition to those cited in the introduction, several other studies have shown that teachers have trouble with the concept (Confrey and Makar 2002; Hammerman and Rubin 2004; Makar and Confrey 2004). For example, Leavy (2006) worked with pre-service teachers on comparing distributions and found that pre-service teachers found that drawing appropriate comparisons in light of the variability present in the data was a challenging task. Makar and Confrey (2004) gave teachers student performance data and asked them to compare performance between different types of students. Only a few teachers were able to make comparisons by discussing the variability in the distributions; most teachers focused on a single summary such as the mean or made very general statements about passing rates. Batanero et al. (2014) found similar results. When describing distributions, a sample of 208 pre-service teachers in Spain focused on individual observations and specific points rather than the distribution as a whole. This led teachers to make incorrect statements about sampling variability and incorrect predictions in random experiments. Peters (2009) worked with 16 teachers and found that only five were able to see a connection between study design and variability in data collected, understood that variability is something that should be explored in data, and when doing inference variability is something that can be modeled and expected.

4. Research Support for Pedagogical Aspects Related to Teaching Variability

Franklin (2013) proposes structuring professional development around the GAISE I framework to focus on the processes and practices involved in doing statistics. Several studies point to the importance of preparing teachers through the statistical investigative process of formulating a question, collecting or considering data, analyzing data, and interpreting results. Makar and Fielding-Wells (2011) show that teacher educators can encourage statistical understanding by implementing the statistical process into their teaching. Authors such as Cobb and Moore (1997) have repeatedly called for the teaching of statistics to be grounded in data and taught through the statistical process instead of with an emphasis on techniques and formulas. As such, several authors have presented tasks, lesson plans, projects, and classroom activities for all levels meant to enhance learners’ conception of variability. For example, Watson (2009) designed tasks that promoted students to consider variation. Diedorp et al. (2017) stimulated a group of 13 students, ages 17 to 18, to measure, describe, explain and investigate variability on a real-world task related to sports physiologist repeatedly measuring cardiac frequency measurements. Yet another task was proposed by Leher (2017) in which middle school students repeatedly measured the perimeter of a table in the classroom and then speculated about the measurement variability found. The Pre-K-12 GAISE II report also has a plethora of examples that engage learners in the statistical process to highlight variability in data (Bargagliotti et al., 2020).

Coupled with the statistical investigative process, a promising way to teach variability is through progressions from informal notions to more formal ideas of variability. Bargagliotti and Anderson (2017) used teacher learning progressions to guide teacher learning of sampling variability and found that the process of moving from informal to formal ideas through interactive activities that made heavy use of technology enabled teachers to connect ideas about variability in sampling distributions. Garfield, DelMas, and Chance (2007) found that college students have some intuitive ideas of variability, namely that variability is represented by overall spread and variability is represented by different values in data. They noted that activities highlighting these intuitions and asking students to draw comparisons in real world contexts could serve as a means for teachers to consider variability. Utilizing this prior research, professional development, curricula, activities and tasks can be developed to explicitly draw attention to variability.

5. Project-SET Professional Development

Project-SET was an NSF-funded project aimed at developing innovative curricular materials to enhance teachers’ SKT. Similarly to Mathematical Knowledge for Teaching (MKT, Ball et al. 2008), Groth (2013) articulated SKT. The SKT framework interweaves the MKT framework with the notion of Key Developmental Understandings (KDUs) introduced by Simon (2006) and Silverman and Thompson’s (2008) ideas regarding the development of pedagogical content knowledge. Groth mirrors the MKT framework by characterizing SKT into two overarching categories (subject matter knowledge and pedagogical content knowledge) and six subcategories of MKT: common content knowledge, specialized content knowledge, horizon content knowledge, knowledge of content and students, knowledge of content and teaching, and knowledge of content and curriculum (See Figure 1). A potentially promising way to improve teachers’ SKT is by having teachers engage in the statistical investigative process through rich tasks. For example, researchers have shown that teaching statistics through carefully designed sequential tasks that utilize technology promotes statistical understanding (Ben-Zvi and Amir 2005; Cobb et al. 2003; Pfannkuch 2006).

Based on this prior research, Project-SET developed materials focused on the teaching and learning of two fundamental topics emphasized in the CCSSM—sampling variability and regression––and developed a professional development (PD) around the materials. Central to the PD were a series of interactive tasks (see Bargagliotti and Anderson 2017 regarding results on the assessments). The goal of the tasks was to foster teachers’ SKT. A total of 13 tasks were developed—seven focused on sampling variability and six focused on regression. The tasks involved the use of dynamic statistical software and were designed to take approximately two hours each to complete. The PD was delivered in thirteen, 2.5-hour class meetings, where the teachers worked on tasks.

Three iterations of the PD were carried out—this paper focuses on the tasks that were refined from the first iteration and utilized in the second iteration of the PD, which was taught by Author 1. (See Bargagliotti and Anderson 2017 for results from the first iteration of the PD.) The second iteration of the PD was delivered as a graduate class to five participants: three full-time public school teachers one international student, and one private school teacher who did not adhere to the CCSSM. The private school teacher and one of the public school teachers had taught statistics before but had done so five years prior to taking the PD. The other two public school teachers had begun implementing the CCSSM within their classrooms and thus had taught some statistics units, however, they had never taught a full course in statistics. The international student had never taught any statistics. All of the teachers had over five years of experience in the classroom and the international student had two years of teaching experience in her home country. We use examples of these teachers’ work and these teachers’ discourse to illustrate how the tasks were successful in requiring teachers to focus on variability through their attention to Statistical MP6.

6. Which Tasks Emphasize Variability?

Our goal is to highlight features of the Project-SET tasks that elicited teachers’ knowledge of variability. We deemed tasks that elicit knowledge of variability as tasks that required teachers to quantify uncertainty and verify the quantification with explanation. Our main contribution in this paper is noticing that tasks that require teachers to explicitly consider variability, make sense of variability, and quantify variability also require teachers to use Statistical MP6. Teacher understanding of variability is thus intertwined with their ability to attend to precision as described in Statistical MP6, which is a pivotal distinction to the precision described in Mathematical MP6.

Next, we present three tasks, one beginning level (Example 1), one intermediate level (Example 2), and one advanced level (Example 3), to demonstrate the link between understanding variability and attending to Statistical MP6. For each example, we provide a standard task (Task A) that is typically used to teach a concept and a Project-SET task (Task B) to teach that same concept. As variability is a challenging concept to grasp, we argue that having the concept grounded in Statistical MP6 makes it more approachable for teachers.

Example 1: Descriptive Statistics. Introductory units in statistics typically cover descriptive statistics such as measures of center and spread. An initial Project-SET task focused on understanding appropriate descriptive statistics. Teachers were asked to consider data on commute times collected from a random sample of 98 commuters in Los Angeles and a random sample of 98 commuters in New Orleans. Consider two tasks in Table 1 that can be posed related to the data:

Two histograms in Figure 2 depict the distributions of commute times for the two samples. On the histograms, the green lines represent the means and the red lines represent the medians. Note that for the New Orleans data, the mean and the median are the same and thus the red line, the median, is pictured behind the mean, the green line.

Task 1A is computational and asks teachers to compute the mean, the median, and the standard deviation of each sample. This task does not require teachers to meaningfully consider the variability of commute times. Instead, this task, which is a standard task presented in K-12 curriculum and in teacher preparation, has a teacher compute the descriptive statistics of the commute times in each city. While this computation does require Mathematical MP6 (precise communication, accurate and careful calculations), it does not explicitly ask teachers to examine how the variability present in the distribution impacts the commute times. Having teachers explicitly and precisely communicate their understanding of the descriptive statistics, in particular, the variability that is present, necessitates the use of Statistical MP6. The computation will certainly require precision in computation and labeling of units, however, as noted in the SET report, Statistical MP6 “is not just computational precision.” Instead, Statistical MP6 requires one to “be precise about ambiguity and variability” (Franklin et al. 2015, p. 11).

On the other hand, a teacher responding to Task 1B, the Project-SET task, would need to understand that both of the distributions of commute times have variability present to answer the questions properly. Although the shapes of the distributions are different for the sample of 98 commute times sampled in Los Angeles and New Orleans, the typical commute time might be specified as approximately 55 minutes for both Los Angeles and New Orleans. This is because in the case of New Orleans, 55 is close to the mean and the median and in the case of Los Angeles, 55 is between the mean and the median. The Los Angeles distribution is skewed right and thus the mean will be larger than the median. In addition, in New Orleans, the values are approximately symmetric about the value 55. These considerations are due to the variability present in the data. The New Orleans commute appears to have less variability (standard deviation of 23) while in Los Angeles they have slightly more variability (standard deviation of 25) due mostly to the few people that are commuting approximately two hours. Understanding how the shape of the distribution and presence of variability impact the descriptive statistics are important when trying to decide what is “typical.” A teacher answering Task 1B must consider which measure of center best describes “typical” in the context of the variability present. Using the New Orleans data set, a teacher might respond that either the mean or the median would offer a good description of typical commute time due to the smaller amount of variability and the symmetry of the data about the mean commute time. However, for the Los Angeles commuters, the median offers a better representation of the typical value for a skewed distribution with larger variability.

The second part of Task 1B (describing what city you would rather be commuting in and why) also requires an understanding of the variability present in the data which aligns well with Statistical MP6. At first glance, one might state that they would rather commute in New Orleans since the Los Angeles mean commute time is higher than that for New Orleans. However, the difference in standard deviation between the two cities’ commute times requires teachers to think through what would be more desirable for commuting in terms of predictability of time spent on the road. While both Tasks 1A and 1B necessitate the computation of the standard deviation, only Task 1B requires the consideration of the variability in context of the problem and in turn, elicits Statistical MP6. In Task 1B, teachers are asked to reflect on the quantification of variability and communicate in a precise manner, thus employing Statistical MP6. In this problem, Statistical MP6 is used to precisely choose the analysis method to use and assess commute time, acknowledging the variability present.

This beginning level task for descriptive statistics illustrates how engagement with Statistical MP6 is necessary when teachers are asked to explicitly analyze and interpret variability. Both Tasks 1A and 1B require teachers to compute the mean, median, and the standard deviation, however, the additional open-ended questions included in Task 1B require teachers to specifically analyze and interpret variability with some level of precision, thus employing Statistical MP6. Beginning level tasks for descriptive statistics can be written in such a way that require explicit explanation of the variability present in data, by necessitating precise communication and description of the variability present, thus employing Statistical MP6. It is particularly important for teacher educators to phrase statistical tasks in such a way that highlights the entire distribution as this helps fosters a greater sense of understanding variability. For example, by asking teachers to construct an argument on which city they would rather commute in, teachers are asked to justify their thinking of how variability of the commute times affects their idea of what a typical commute time is.

Example 2: Curve Fitting. A main focus in middle and high school statistics as well as the Project-SET PD is regression. In the PD, teachers developed ideas related to general curve fitting and statistical modeling. An intermediate level task on curve fitting asked teachers to consider NFL quarterback salaries from 2009-2010. The data also included the quarterback’s pass completion rate, total number of passing touchdowns, and the average number of yards rushed per game¹. Consider two tasks in Table 2 that can be posed related to the data:

Task 2A requires teachers to make a scatterplot and then place a line of best fit onto the graphical display. This could be done informally using spaghetti (see Nagle et al. 2018), or technology may be used to help find the line of best fit according to the Ordinary Least Squares (OLS) method. Task 2B requires teachers to understand that exploring the relationship visually between two quantitative variables can be done using a scatterplot. The task lends itself to informally describing what type of relationship can be seen in the scatterplot. After informal exploration, a line of best fit might be estimated. For example, in the Project-SET PD one of the teachers captured the linear relationship between the pass completion rate and the quarterback salaries with a linear equation:$$\mathit{Estimated\; Quarterback\; Salary}=\mathrm{ }44375.9\mathrm{ }+\mathrm{ }244351.32\mathrm{ }\left(\mathit{Pass\; Completion\; Rate}\right)$$

While both tasks have teachers estimate the regression line, the answer to Task 2B is not complete until more is done. After estimating the regression line, teachers must measure the variability present on the scatterplot in order to assess the strength of the model by examining the residuals, computing and interpreting a correlation coefficient, and looking at the coefficient of determination. All of these additional measures have to be incorporated in order to provide a complete answer to Task 2B. For example, one teacher in the PD wrote:

There is a moderate (not totally linear) positive correlation between the salaries of the top paid quarterbacks and their percentage of complete passes. About 15.5% of the variability in the regression line can be explained by the relationship between salary and the percent completion.

While answering Task 2B necessitates several computations, the open-ended phrasing of the task requires a teacher to focus on and capture the variability present in the scatterplot by discussing the strength of the model, examining error patterns and justifying how the model explains variability. When teaching the PD, Author 1 continuously emphasized the modeling aspect of curve-fitting in statistics. This idea of regression as a model estimation resonated with the teachers. For example, one teacher in her response to Task 2B noted:

We modeled the data with scatterplots, then estimated and later calculated the line of best fit, which can be used to predict values. We interpreted the variables in our regression equation and discuss how they affect our data. We found that there are sampling distributions for the slope and y-intercept for the regression lines, so we can use inference to predict values.

This response demonstrates how teachers can engage in Statistical MP6 when given an open-ended task. This teacher acknowledged that there is a process that begins with a formulation of a question that then needs to be answered through the interpretation of variables. Because regression coefficient estimates have sampling distributions, the teacher quoted above noticed that there is variability in what the estimates could be depending on the sample. This idea again connects the phrasing of the task to Statistical MP6. By connecting to the sampling distribution, the teacher is attempting to be precise about sample-to-sample variability.

If we compare Task 2B to Task 2A, a similar task often given in a mathematics regression modeling unit, we can see that once the line of best fit is computed and a regression equation is found, it would then be accepted as the given model. Subsequently this model would be used to make predictions relating salary to the percentage of complete passes, without consideration of the variability of the data points. While prediction using the regression line is certainly a worthwhile task, discussion of the error between predicted values and actual values, namely the residuals, requires one to assess the strength and effectiveness of the model. The error inspection and what that means in the context of the problem requires teachers to think about the variability in the scatterplot. Small variability in the scatterplot will lead to small residuals and thus more precision in prediction.

As with Example 1, the differences between a typical task such as Task 2A and the Project-SET task (Task 2B), is that Task 2B requires the consideration of variability to fully answer the task. Note that while these tasks used an older data set, similar tasks can be developed around more contemporary and newer data sets that lend themselves to discussing regression and the line of best fit (e.g., as an example of such data sets, see https://skewthescript.org/ap-stats-curriculum/part-3).

Example 3: Construction of Confidence Intervals. Understanding variability is essential for understanding sampling distributions, an advanced topic that is informally introduced in middle school in the CCSSM and used in high school AP Statistics. An advanced level task that asks teachers to use the concept of sampling variability could focus on confidence intervals. Using data from a random sampling of 100 commuters in Los Angeles, consider the two tasks in Table 3 that can be posed related to the data:

A teacher answering Task 3A would precisely estimate the commute time by finding the sample mean commute time and adding and subtracting a margin of error in order to produce a confidence interval:$$(\underset{¯}{X}\mathrm{ }-\mathrm{ }2\mathrm{*}\mathit{Standard\; Error},\mathrm{ }\underset{¯}{\mathrm{ }X}\mathrm{ }+\mathrm{ }2\mathrm{*}\mathit{Standard\; Error})$$

where the standard error = $\sigma /\sqrt{n}$ is approximated by $s/\sqrt{n}$ .

While Task 3A elicits mathematical precision (Mathematical MP6), it falls short of the description of precision in statistics presented in the SET report. It does not require a teacher to recognize that in constructing the interval they are in fact accounting for variability in the data. Instead, Task 3B, the Project-SET task, brings attention to the meaning of and precision in creating the estimate, requiring teachers to provide an answer that allows for the variability in the data.

To answer Task 3B successfully, teachers need to examine the variability in the sampling distribution of the sample mean. For example, if the sample average commute times were all close to 30, then a teacher might estimate a plausible range for the population average commute time within a few minutes and be very confident that their range captured the actual population average commute time. If instead the sample average commute times were highly variable, then a teacher might provide a wider interval of plausible values for the population average commute time and be less optimistic about estimating the population average commute time in Los Angeles in a confident manner. The consideration of variability in this problem required teacher employment of Statistical MP6. In fact, teacher responses to the guiding questions reflected their engagement with Statistical MP6. An exemplary answer given by one of the teachers was:

There must be a way to measure how certain we are about our inferences—I think this might be called the standard error. If we have a representative sample, we can feel pretty certain about our inferences. If our sample is not representative of the population, then we can’t be very sure about the inferences we can make.

This teacher demonstrated precision as she alluded to the necessity of quantifying variability for making inferential statements. Interestingly, this teacher’s response directly aligns with Statistical MP6 as “one must be precise about ambiguity and variability” and “[teachers] are precise about choosing the appropriate analyses and representations that account for the variability in the data” (Franklin et al. 2015, p. 11-12). Another teacher’s written work in Figure 3 illustrated Statistical MP6 as she recognized that she can apply the Central Limit Theorem (CLT) and connect to the Empirical Rule to construct a confidence interval:

At the top of the teacher’s work, we can see the statement about 95% of the sample means being within 2 standard errors (SE) of the population mean. This is a reference to the Empirical Rule as well as a connection to the difficult concept of the CLT. This teacher created a range of plausible values by adding and subtracting 2*3.66 to the sample mean. The 3.66, although not explained in the teacher’s written work, represents the SE of the sampling distribution of the mean. Her estimate of the SE is an estimate of precision and variability. She then interprets the meaning of the interval, by stating “29.84-44.48 is a plausible range of values for the average commute time in LA. 95% percent of the sample means are within 2 SE of the population mean.” Another precise statement for the interval interpretation could also have been: “If we could sample repeatedly, then 95% of the time the intervals formed in this way would contain the population average commute time.”

Nonetheless, the interpretation the teacher provided illustrates her grappling with how to describe the variability present in the data in a precise, quantifiable manner through the computation and interpretation of the confidence interval. This teacher addressed the question about the average commute time in Los Angeles by quantifying variability, understanding the margin of error, and interpreting the precision of her estimate. This embodies the description of Statistical MP6.

In contrast to the typical computational Task 3A, the Project-SET task focused on conceptually understanding the mechanisms of creating a confidence interval and interpreting that interval. The scaffolding guided teachers to precisely understand the variability of the estimate that they were creating as an answer to the questions. Task 3B required teachers to consider, quantify, and be explicit about variability, which is exactly what teachers did in providing solutions to the task. This example thus illustrates how a task that requires teachers to consider variability of the data in context, is one that can elicit Statistical MP6.

7. Implications and Conclusions

Adhering to the call in the literature asking researchers to design courses, activities, and professional development that align with the statistical investigative process (de Ponte 2011), this paper provides insights into the types of tasks that can be used to foster mathematics teachers’ understanding of variability by attending to precision as described by Statistical MP6 in the SET report. Having teachers contend with variability in data through the statistical process is necessary to enhance teachers’ statistical knowledge for teaching (Cobb and Moore 1997). We note that ideas related to variability are present in all components of the statistical process: posing questions that anticipate variability, anticipating variability in data collection, analyzing the variability present, and interpreting the variability to make quantitative statements about error. Thus, we see understanding variability as being intertwined with attending to Statistical MP6.

In this paper, we highlight rich tasks that require precise consideration and quantification of variability, which lead to viable and productive opportunities for teachers to unpack this critical statistical topic. As statistics is the study of variability in data (Franklin et. al 2007), it is important to phrase statistical tasks in such a way that highlight variability. We found that tasks that require teachers to explicitly consider, make sense of, explain, and quantify variability are also those that elicit the use of Statistical MP6. Thus, teacher understanding of variability is intertwined with their ability to attend to precision in Statistical MP6. In the examples above, Tasks 1A, 2A, and 3A require teachers to compute or find specific values. On the other hand, Tasks 1B, 2B, and 3B ask teachers to consider the variability present by bringing in multiple aspects of the data to answer the task and provide an answer that allows for variability. Overall, our examples show that emphasis on measuring different aspects of the data and then putting a comprehensive analysis of the entire data set as a whole can engage teachers with understanding concepts of variability.

In addition to our noticing the role of variability in eliciting Statistical MP6, we view another contribution of this work to be the recognition that the MPs are represented differently in different content domains. As we consider teacher engagement with the MPs, it is important to understand how the MPs appear in tasks and problems in the different content areas. In our case, we found evidence that precision in statistics can be related to explicitly considering variability in data. We think it is important to start or continue conversations about the role of the MPs in the different content domains of mathematics as listed in current standards in order to expand teachers’ content and pedagogical knowledge. Specifically, we hope to encourage mathematics teacher educators to discuss and investigate best practices in incorporating the MPs into different content areas. This would potentially allow teachers to understand the subtle differences in the appearance of the MPs as related to the different content domains, which will better equip teachers to facilitate student learning across the different content domains. Ultimately, such discussions and connections could help refine theories, instruction, and curriculum development for mathematics teacher preparation.

Data Accessibility Statement

The authors confirm that the data supporting the findings of this study are available within the article [and/or] its supplementary materials.

Disclosure Statement

This work was supported by the National Science Foundation under NSF DRK-12 grant no. 1119016.

Conflict of Interest Statement

On behalf of all authors, the corresponding author states that there is no conflict of interest.

Table 1. Project-SET Task comparison for Example 1

1A. Standard Task What is the mean and the median commute time for the sample? What is the standard deviation of the commute times of the sample from each city?

1B. Project-SET Task What is the typical commute time in each city? Based only on these data, what city would you rather be commuting in and why? Support your answer by comparing the values of different measures of center and different measures of variability, and use multiple graphical displays to help discuss the shape of the distribution.

Table 2. Project-SET Task comparison for Example 2

2A. Standard Task Find the line of best fit using the scatterplot of the pass completion rate and the quarterback salaries.

2B. Project-SET Task What is the relationship between the pass completion rate and the quarterback salaries? To support your answer, 1. Describe the type of relationship (i.e., linear, exponential, etc.) and how strong that relationship is. 2. Use an equation to model the relationship observed and discuss the strength of the model by examining error patterns and looking at how much variability the model explains.

Table 3. Project-SET Task comparison for Example 3

3A. Standard Task Given a data set of 100 commute times in Los Angeles selected randomly from the population of commuters in Los Angeles, compute a 95% confidence interval for the mean commute time in Los Angeles.

3B. Project-SET Task What would a plausible range of values be for the average commute time in Los Angeles? How can you use the sampling distributions to help you determine which values might be plausible? Is it plausible for the average commute time to be 5 minutes? Is it plausible for the average commute time to be 30 minutes? Is it plausible for the average commute time to be 200 minutes? When answering these questions, think about the following: How can we estimate a population parameter and how can we decide if our estimate is plausible? How confident are you that your interval of plausible values captures the parameter?

Figure 1. Statistical Knowledge for Teaching (SKT) framework from Groth 2013 (p. 143).

Figure 2. Distributions of commute times in Los Angeles and New Orleans with means (green lines) and medians (red lines).

Figure 3. Teacher work on Task 3B illustrating the construction of a confidence interval

Notes

1 Note that while these tasks used an older data set, similar tasks can be developed around more contemporary and newer data sets that lend themselves to discussing regression and the line of best fit (e.g., as an example of such data sets, see https://skewthescript.org/ap-stats-curriculum/part-3