Joan Garfield
Michelle Everson
University of Minnesota
Journal of Statistics Education Volume 17, Number 2 (2009), ww2.amstat.org/publications/jse/v17n2/garfield.html
Copyright © 2009 by Joan Garfield and Michelle Everson, all rights reserved. This text may be freely shared among individuals, but it may not be republished in any medium without express written consent from the authors and advance notification of the editor.
Key Words: Teacher development; Statistics education; GAISE.
This paper describes a unique graduatelevel course that prepares teachers of introductory statistics at the college and high school levels. The course was developed as part of a graduate degree program in statistics education. Although originally taught in a facetoface setting, the class has been converted to an online course to be accessible to more students. The course serves students who are pursuing graduate degrees in a variety of disciplines but who want to teach statistics as part of their careers. It also serves current teachers in high school who are teaching the Advanced Placement Statistics course as well as teachers at twoyear and fouryear colleges. The curriculum for the course is based on the theory that good teachers of statistics need to be developed, as opposed to being trained. Building on recent teacher preparation theory, we describe a course that models and builds specific knowledge about teaching and learning statistics. In addition, this course is organized around the six recommendations of the ASAendorsed Guidelines for Assessment and Instruction in Statistics Education (GAISE).
Preparing graduate students to teach, as teaching assistants or because some will follow academic careers, is a major concern for mathematics and statistics departments (Moore 2005). Moore writes, "Changing our understanding of what constitutes effective pedagogy, improving the use of technology, and placing an emphasis on working with data in elementary courses raise the standard that teaching assistants must meet. … Training programs should themselves be models of good pedagogy" (2005, p. 1). Moore also writes that competent teaching depends on the learned craft of teaching and he raises the issue of how to help graduate students learn the craft of teaching. A series of articles follow Moore's introduction that describe "the training programs" in place at several major statistics departments (e.g., Harkness and Rosenberg 2005; Froelich, Duckworth, and Stephenson 2005).
There are currently very few courses that are focused on the preparation of teachers of statistics, at any educational level (Garfield and BenZvi 2008). Most of these courses are offered in graduate programs in statistics, and focus on preparing graduate students in these departments to be good teaching assistants and future teachers (e.g., Gelman 2005). The focus of these courses is on teaching methods, which seem to primarily rely on giving good lectures. There are currently three books available that are focused on the teaching of introductory statistics at the college level (Gelman and Nolan 2002; Garfield 2005; Moore 2000) and sometimes, one of these books is recommended to graduate students as they prepare to become teachers of statistics. There are also many good articles both in print and online versions, but few graduate students are aware of these many resources to improve the teaching of statistics.
At our university, we have created a unique graduate course that has been offered yearly since 2002 called "Becoming a Teacher of Statistics." Although originally a facetoface course, it has recently been offered as an online course to make it accessible to a wider variety of teachers and graduate students. In our course, we strive to help preservice and inservice teachers become aware of the field of statistics education and its many resources, and to guide them to develop into knowledgeable, reflective, and effective teachers of this subject.
While most people agree that a good teacher needs to know his or her subject matter at a deep level, and needs to have good teaching skills, there is also another type of knowledge and related skills that have been defined: pedagogical content knowledge (PCK). This type of knowledge is more specific, as it involves knowledge about teaching of a particular content area. For statistics teachers, this is knowledge about ways to effectively teach important statistical ideas and skills, ways to help students use statistical software and technology tools, and ways to help prevent or overcome typical misunderstandings and misconnections about statistical concepts. Originally described by Shulman (1986), pedagogical content knowledge has been viewed as an important factor in good teaching and promoting student learning. A recent study by Hill, Ball and Schilling (2008) conceptualized and developed measures to assess mathematics teachers' PCK.
In their volume, Preparing Teachers for a Changing World, DarlingHammond and Bransford (2005) discuss the importance of preparing teachers to go beyond "covering the curriculum." Therefore, those who prepare teachers are encouraged to build on the growing knowledge base on learning and teaching to support teachers in meeting these demands. Their book provides a model of professional practice that includes as major components: Knowledge of subject matter and curriculum goals, knowledge of learners, and knowledge of teaching (the subject matter).
In thinking about preparing college level and high school level teachers of statistics, these components are also important to consider. While it is assumed that most graduate students preparing to teach introductory statistics have knowledge of the subject matter, they may be lacking the knowledge of curriculum, learners, and knowledge of teaching (both in general and specifically, of teaching statistics). Our course is designed to help prepare future teachers of statistics to become knowledgeable about each of these areas.
Any program to prepare teachers of statistics should include and be aligned with current guidelines for teaching statistics at the K12 and college level that have been approved by the American Statistical Association (see The Guidelines for Assessment and Instruction in Statistics Education – GAISE; amstat.org and Franklin and Garfield 2006). The guidelines pose challenges for novice or even experienced teachers and need to be carefully examined and used as a basis for instruction. That is why our course to prepare teachers of statistics is based on the GAISE guidelines.
These six recommendations are:
Our course was designed to both help students understand and align their teaching with the GAISE recommendations, and we therefore organize the course around these recommendations and at the same time address the three important types of knowledge: of curriculum, of learners, and of statistics.
Our broad goals for students who take the course are to:
A threecredit graduate level semesterlong course was designed that met once a weekfor three hoursto attract both fulltime graduate students as well as inservice teachers at the secondary and postsecondary levels. Each offering of the course has enrolled a variety of students representing different disciplinary backgrounds and teaching experience. Students have been required to read chapters in one or two textbooks (e.g., Moore 2000; Gelman and Nolan 2002) as well as weekly assigned readings.
Each class focuses on a different topic, and students are expected to prepare for class by having read the assigned readings. We discuss readings, watch demonstrations, and participate in various activities during class, many of which involve using technology. Students are expected to attend class and to actively participate in discussions and activities and make assigned presentations. Students are also encouraged to bring in materials, articles, and readings to share with the class.
Table 1 shows a list of some of the weekly topics, along with sample activities covered each week. Appendix A lists several assigned readings for students.
Table 1. Weekly Topics and Sample Activities
Topic 
Sample Activities 
Introductions and Overview 

Use real data 

Stress conceptual understanding rather than mere knowledge of procedures 

Emphasize statistical literacy and develop statistical thinking


Foster active learning in the classroom


Technology 

Assessment 

A Statistical Reasoning Learning Environment (SRLE) that builds on and integrates the GAISE guidelines. 

Innovative courses and curriculum development projects 

The Statistics Educational Community 

We now elaborate on how our course is built on the six GAISE recommendations.
This guideline challenges us to carefully examine intended outcomes of a first course in statistics. We do this in two ways. Students are assigned to read and reflect on writings by key statistics educators (e.g., Cobb and Moore 1997; Utts 2003; Wild and Pfannkuch 1999). We distinguish between definitions of statistical literacy, reasoning and thinking (see Garfield and BenZvi 2008) and examine ways to help students develop these outcomes. For example, we look at articles and graphs in the media and ways to have students discuss and critique them (statistical literacy) and ways to engage students in statistical thinking by considering the full process of statistical investigations and the use of student projects.
Data are at the heart of statistical work, and we try to make data the focus for statistical learning as well. In our classes, we examine many data sets as well as ideas for collecting data and consider how data may be used to motivate and engage students. Our teachers are challenged to explore and learn from data in ways we would like them to then have their own students explore data. We discuss the merits of data that can be gathered on a first day of class survey or body measurements (e.g., arm span, hand span, head circumference) that can easily be gathered in class using a measuring tape. We also discuss good types of data that can be gathered on the Internet (e.g., the accessible databases of the CensusAtSchool project at http://www.censusatschool.org/; and the datasets section of the Consortium for the Advancement of Undergraduate Statistics Education (CAUSE) at http://www.causeweb.org/). We even ask students to find and share datasets with their peers as part of a required class assignment.
The students read some articles from the research literature on some of the difficulties students have understanding important statistical ideas and reasoning about statistical concepts. They participate in activities and discussions where they present their ideas of the important learning goals and have to come to consensus in pairs and then groups of four students.
We focus on several key statistical ideas that we would like our students to understand at a deep conceptual level. Theses ideas have been studied in the research literature and our students investigate what has been learned about developing these ideas as we have them read a sample of the scholarly literature and experience activities that develop these concepts. Some of these big ideas include distribution, variability, model, sample, and inference.
An important part of our course is the use of and discussion about carefully designed activities that promote student learning through collaboration, interaction, discussion, data, and interesting problems. We discuss with our students the positive effects of active learning such as shortterm mastery, longterm retention, depth of understanding of course material, acquisition of critical thinking or creative problemposing and problemsolving skills, formation of positive attitudes toward the subject being taught, and level of confidence in knowledge or skills.
We draw our students' attention to two different models of class activities. The first engages students in making conjectures about a problem or a data set, as introduced in the preceding section on using real data. This method involves discussing conjectures, gathering or accessing the relevant data, using technology to test their conjectures, discussing the results, and then reflecting on their own actions and thoughts. Our students experience such activities and then discuss their implementation in class, in addition to benefits and possible obstacles to using such activities. An activity like this that we use involves setting up a blind taste test to determine if students can correctly identify Coke and Pepsi.
The second type of activity is based on cooperative learning, where two or more students are given questions to discuss or an open problem to solve as a group. For example, students could be given an activity involving a Web applet for bivariate data where they are asked to figure out a rule describing how individual points that seem to be outliers may affect the correlation and fitting of a regression line for set of bivariate data (e.g., the Least Squares Regression and Guess the Correlation applets in http://www.rossmanchance.com/applets/). They try different locations of a point and consider the resulting effect on the correlation coefficient and regression line.
We try to help our future teachers understand that when using cooperative learning activities, it is important that students work together as a group (and often in pairs using technology), not just compare their answers (Johnson, Johnson, and Smith 1998; Roseth, Garfield, and BenZvi 2008).
An important aspect of our course is exposing our students to innovative technology tools that can be used to explore and simulate data, test conjectures by analyzing data, and develop abstract concepts. Our students experience analyzing data sets using a variety of popular software tools (e.g. , Minitab, StatCrunch) which they then compare and discuss. They also examine TinkerPlots (Konold and Miller 2005; http://www.keypress.com/tinkerplots), a tool that allows students to build their own graphs and analyze data in ways that match their own intuitions. TinkerPlots is also introduced as a tool that can help them "see" the data hidden by graphs such as histograms and boxplots. They are also given experience using the Fathom software (Key Curriculum Press 2006; http://www.keypress.com/fathom), a flexible tool that allows them to easily explore data, as well as graphing calculators, simulation software, and Web applets.
Students in the course are provided with readings and resources on technology (e.g., Chance, BenZvi, Garfield, and Medina 2007) and are challenged to discuss the important ways technology may be incorporated into activities to enhance students' learning.
An important goal of the course to help our students become knowledgeable about alternative methods of assessment that provide formative information gathered during the process of learning (as opposed to summative information gathered primarily to provide a judgment of how well the student perfumed) should be to utilize that information to guide students' learning. They learn about student projects as a form of authentic assessment that allow students to pose or select a problem, gather or access appropriate data to answer the problem, analyze the data, and write up the results in a technical report and/or presentation. In many cases, projects allow students to collaborate with peers and professionals. Other forms of alternative assessment introduced and examined are assessments of students' statistical literacy (e.g., critique a graph in a newspaper), assessments of reasoning (e.g., write a meaningful short essay), or assessments that provide feedback to the instructor (e.g. , minute papers). We point our students to good assessment resources for creating quizzes and exams such as the ARTIST website (https://app.gen.umn.edu/artist), and consider ways to assess deeper levels of understanding, encouraging our students to move beyond items that focus on definitions, formulas, and computations.
We make the case that students will value what the teacher assesses. Therefore assessments need to be aligned with learning goals. We encourage our students to focus assessments on understanding key ideas and not just on skills, procedures, and computed answers. This should be done with formative assessments used during a course (e.g., quizzes, small projects, or observing and listening to students in class) as well as with summative evaluations (course grades). We attempt to model alternative assessment methods in the course by assessing our students in a variety of ways. For example, students are asked to work in a collaborative group to develop a class lesson plan, find and analyze a good data set, and present and explain a web resource to the class that they think would be a good resource to promote student learning.
Another important goal for the teachers in our courses is to develop an appreciation for the value of classroom statistical discourse. This is different from teachers asking questions and students responding. The kind of discourse we promote is dialogue where students learn to question each other, respond to each other's questions as well as defend their answers and databased arguments. The use of good activities and technology allows for a new form of classroom discourse. Cobb and McClain (2004) describe the characteristics of effective classroom discourse in which statistical arguments explain why the way in which the data have been organized gives rise to insights into the phenomenon under investigation; students engage in sustained exchanges that focus on significant statistical ideas. We try to model ways to create a classroom climate where our students feel safe expressing their views, even if they are tentative. We encourage them to express their conjectures, and ask other students to comment on these conjectures. Allowing questions that begin with "what do you think" or "what would happen if" can lead to good class discussions.
The final weeks of the course are devoted to student presentations of group and individual projects and assignments. The following section provides details on the required assignments and assessments.
Students are asked to complete one group project and two individual projects, all of which are presented in class. In addition, students write and submit four reflection papers throughout the semester. The following tables (2 and 3) provide details on the assignments and assessments. Points allocated to each assessment are listed below:
Total points for student assessments: 200
Table 2. Assignments for Presentation to the Class
Assignment/Assessment 
Description 
Presentation of a web resource 
Students find and share a favorite web resource that can be uses to support student learning in a unique way in a statistics course. 
Presentation of a data set 
Students find a multivariate data set that interests them and spend time exploring it and learning as much as they can about it. The data set is then shared with the class and discussion revolves around how the data set can be used in an introductory course. 
Group Lesson Plan Project 
Students work in small groups to develop a detailed lesson plan for one day in an introductory statistics course. 

Table 3: Focus of Assigned Reflection Papers
1. Experiences as a student of statistics and possibly as a teaching assistant or teacher of statistics 2. Description of one class observation of a first course of statistics in light of the GAISE recommendations 3. Summary and critique of three studentselected articles about teaching and/or learning statistics 4. Reflection on the "Becoming a Teacher of Statistics" course experience and the kind of statistics teacher they would like to become

After several years of teaching this course exclusively in a facetoface setting, we decided to adapt it to an online environment, in order to allow students outside of our local area to participate. The first online section of this course was offered during the spring of 2008, and a total of 15 students were enrolled in the online course. Although most of these students were graduate students from our university, three individuals from outside the campus (two community college instructors and one local high school teacher) were also enrolled in the course. A second offering of the online course enrolled 10 students, which is comparable to the number of students in the facetoface version of the course.
When the online course was designed, efforts were made to preserve the format and content of the facetoface version of the course. Our initial focus was on creating an online course where students would have many opportunities to collaborate and discuss course material, and to apply what they were learning about. Each week, students in the online course were asked to engage in smallgroup or wholeclass discussion by posting messages in designated discussion areas and responding to messages posted by their peers or by the instructor.
Discussions typically focused around the kinds of topics students would discuss in small or large groups in the classroom. Students posted reflections about the weekly readings, shared and critiqued ideas for classroom activities, discussed ways to implement the GAISE recommendations in the introductory statistics classroom (and talked about the challenges they might face in implementing these recommendations), and evaluated different types of assessment tools and statistical software packages. Other discussions were geared more toward producing products that could be shared with the whole class. For example, in one discussion assignment, students were asked to talk about their understanding of the terms statistical literacy, statistical reasoning, and statistical thinking, and to come up with a list of what they felt were the big ideas students should learn in the introductory statistics course. In a later discussion assignment, students were asked to revisit their list of big ideas and to create activities they might use in their own classrooms in order to help students better understand each of these big ideas.
The assignments students were asked to complete in the online course mirrored the types of assignments students completed in the classroombased course, but in some cases, the way in which the assignments were submitted or shared with classmates had to be altered. In the classroom, for instance, students typically give short oral presentations when they share web resources or data sets with the class. In the online course, special discussion rooms were set up for students to post and discuss the web resources and data sets they had found. Students were assigned to small groups to complete the group lesson plan project, and this too was shared by means of posting the project and accompanying materials in a special discussion area. When presenting their projects in the online course, students were asked to create Powerpoint presentations with detailed notes to share with their peers — much like the kinds of presentations they might give were they sharing their lessons at a conference.
In sum, the online version of "Becoming a Teacher of Statistics" retains most of the elements of the classroom version of the course but has a few differences. The main difference is that discussions among students and between students and the instructor are written, rather than oral. This both requires students to be more diligent about expressing themselves in a way that others will be able to read and understand, but at the same time allows students (and the instructor) to read comments and responses that would most likely not been "heard" in a face to face classroom. For example, if a teacher is talking with students who are working together on a group activity, she cannot hear what students in other groups are saying. Second, although almost all of the assignments from the classroom version of the course have been implemented in the online environment, some minor changes have been made to particular assignments. For example, some of the reflection papers have been turned into discussion assignments.
We use several different methods in both versions of the course to evaluate the impact of our course. For example, a midterm feedback form is used as well as an endofcourse evaluation to find out how students are perceiving the course and what they find to be the most valuable. Students in the course typically report on how amazed they are at the vast set of resources available for teachers of statistics and how appreciative they are of the course and what it offers them as future teachers. Most informative are the reflection papers the teachers write throughout the course and, particularly, their end of course teaching philosophy statement that usually documents their integrations of course learning goals. Here are a few comments from students' final reflection papers:
These reflection papers suggest that the students in the course take the teaching of statistics very seriously, have expanded their knowledge of statistics education, and have developed their awareness of good teaching and resources for teaching and learning statistics.
Despite the challenges in helping preservice and inservice teachers prepare to teach statistics, we have noticed many successes. Some of the teachers who have enrolled in the course have joined a local network of college statistics teachers who meet monthly to share teaching ideas and resources, and discuss the teaching of statistics. Other teachers have reported great success in introducing activities into their classes and encouraging their colleagues to also enroll in this class. Graduate students who have taken this course have won teaching awards and have taken good jobs in academic settings where teaching is valued. In 2006, a group of six students from this course attended the US Conference on Teaching Statistics (USCOTS)funded by an internal grantwhere they were excited and inspired to meet and learn from many people in the growing statistical education community. In summary, the course appears to be succeeding in developing knowledgeable, competent, and enthusiastic teachers of statistics.
An additional evaluation concern is whether the online course is as successful as the facetoface class, and if it is providing students with parallel experiences to those in the facetoface course. To answer this question, we have looked carefully at student performance on assignments and assessments, and we have examined anonymous midterm and endofterm feedback provided by the online students. Student engagement in discussion in the online course was high and students appeared to be taking the necessary time to reflect on assigned readings and think critically about their roles as teachers of statistics. Student performance on assignments and assessments in the online course was comparable to what we observe when teaching the course in a classroom setting, and feedback from students was equally positive. One student mentioned that not only did he/she learn more about teaching statistics, but he/she felt that the course served as a good model for teaching in the online environment. The one element of the online course that students felt could be improved related to the way in which discussion groups were configured. Four to six students were in a discussion group that worked together for the entire semester. Some students preferred to change groups during the semester. In the second offering of the online course, assigned discussion groups change at different points during the semester, and there are more opportunities for both smallgroup and wholegroup discussion.
This paper has described the use of the GAISE recommendations to design and teach courses to preservice and inservice teachers in two different settings: facetoface and online. Unlike some of the teaching assistant training courses or programs that focus primarily on administering large sections of introductory statistics, this course strives to prepare teachers to understand the challenges involved in helping students develop important learning outcomes, by developing both their pedagogical knowledge and their pedagogical content knowledge. We continue to evaluate and revise our classes as we teach them each year. We hope more faculty will explore ways to develop courses for precollege and postsecondary teachers and we encourage them to consider the different types of knowledge needed by competent teachers of statistics.
As other faculty begin to think about developing a similar course, we offer a few suggestions.
Moore (2005) offers four recommendations for preparing graduate students to teach statistics: Borrow from existing programs, model good instructional practices, provide students with written materials to study and carry away with them, and evaluate how well you are doing. We concur with these recommendations, and are happy to share our course materials with others who want to develop such a course. This includes lists of required and recommended readings, assignments, assessments, and evaluation methods. We encourage faculty who have the opportunity to work with graduate students interested in teaching statistics to consider developing and offering a course that goes beyond training to developing excellent teachers of statistics.
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Utts, J. (2003), "What Educated Citizens Should Know about Statistics and Probability," The American Statistician, 57, 74–79.
Wild, C.J., and Pfannkuch, M. (1999), "Statistical Thinking in Empirical Enquiry," International Statistical Review, 67, 223–265.
Joan Garfield
Quantitative Methods in Education
Department of Educational Psychology
University of Minnesota
250 Education Sciences Building
56 East River Road
Minneapolis, MN 55455
Phone: 6126250337
Fax: 6126248241
jbg@umn.edu
Michelle Everson
Quantitative Methods in Education
Department of Educational Psychology
University of Minnesota
250 Education Sciences Building
56 East River Road
Minneapolis, MN 55455
Phone: 6126240691
Fax: 6126248241
gaddy001@umn.edu
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