Ayesha Delpish, Ph.D.

Elon University

Department of Mathematics

adelpish@elon.edu

Introduction

Statistics as a discipline demands to be taught in an applied context. One way to accomplish this is through the use of inquiry-based techniques, such as case studies, that help students more critically consider the relationship between theory and application. The need for new approaches to teaching mathematics has become more evident as both the student and the Statistics landscape has changed. In recent years, new movements in Statistics education have been motivated by the American Statistical Association (ASA).

The ASA endorsed report entitled The Guidelines for Assessment and Instruction in Statistics Education (GAISE)[1] gives six recommendations for teachers of college-level statistics:

1.  Emphasize statistical literacy and develop statistical thinking.

2.  Use real data.

3.  Stress conceptual understanding rather than mere knowledge of procedures.

4.  Foster active learning in the classroom.

5.  Use technology for developing concepts and analyzing data.

6.  Use assessments to improve and evaluate student learning.

But literacy, thinking and learning are all truly complex ideas, so how can I “foster” them with my students? Is “understanding” mastery? How do students come to know the material? Am I facilitating learning in my classroom? How do students learn? These questions, and many others, led me to this project.

Statement of Problem

My overall goal for the project was to arrive at a useful understanding of the developmental stages involved in solving authentic problems involving statistics. The purpose of the research was thus to investigate critical thinking/problem solving development skills among statistics students with secondary focus on the effectiveness of the case-method approach as a teaching and learning tool.

At the beginning of the ETLP project, my primary research questions were:

1. Does teaching using the case method give evidence of student learning?
2. Does this method lead to more authentic problem solving?
3. What skills do students use when faced with a problem?

With the knowledge gained through the program, my research questions evolved to include the following:

1. How do students in statistics solve problems?
    -  How do they sort information?
    -  How do they start the problem?

2. How do students approach critical thinking situations within an inquiry-based context?
    -  What does effective problem-solving look like in the case method?
    -  How do students develop “statistical thinking”?

These questions are of interest not only to the statistics education community, but to all communities of learners who may consider approaches such as the case study method as means of strengthening student content knowledge and their own understanding of learning.

Previous Resesarch

While research into problem-solving techniques has been conducted for many years, the statistics community has only recently begun exploring the idea of “statistical thinking” (not to be confused with the often researched “mathematical thinking”). The most relevant work to this project is that of Wild & Pfannkuch [3] who define statistical thinking as “the incarnation of ‘common sense’…we know it when we see it…its absence is often glaringly obvious” (p. 223). In investigating the thought processes involved in solving real world statistics problems, the authors propose a framework for statistical thinking that involves four-dimensions:
a.    The investigative cycle—problem specification, planning, collection, analysis, conclusions
b.    The types of thinking—strategic planning, seeking explanations, modeling, applying techniques, need for data, variation, integration of statistics and context
c.    The interrogative cycle—generating, seeking, interpreting, criticizing, judging
d.    Dispositions—skepticism, imagination, curiosity, a propensity to seek deeper meaning

References

[1] >> Guidelines for Assessment and Instruction in Statistics Education (GAISE) College Report.

 
[2] Naumes, W., & Nuames, M. J. (2006).  The Art and Craft of Case Writing.  Thousand Oaks: Sage Publications.

 
[3] Wild, C. J., & Pfannkuch, M. (1999). Statitsical Thinking in Empirical Enquiry. International Statistical Review, 67(3), pp. 223-248.

Implementation

In the fall semester of the project year, I did not teach the introductory level course so I could not directly collect data on student problem solving techniques in introductory statistics. However, I used that semester instead to pilot my methods in an advanced statistics course.  During the semester, I investigated using video and audio tapes of the students as the main data collection technique and found that audio tape, while easier to transcribe, did not provide as much information as the videos which often highlighted affective student responses as they worked through the problems. I also found that it was not useful to video- or audio-tape the students at each class period especially since the problems often ran from one course period to another.

In the spring of 2009, the project was implemented into my introductory level statistics course. The main data collection method involved videotaping students as they worked to solve cases in class. Thus there was no interruption of routine. Written consent as required by the Institutional Review Board was obtained from each of the 31 students enrolled in the course, thereby making the videotaping process easier. Based on the observations from the pilot class, students were videotaped only on the days on which a new case or set of cases were started. At the beginning of this project, I had also planned to collect individual ‘think-alouds’ from individual students throughout the semester. However, as my research questions evolved, I opted not to explore this avenue just yet and focus on the information provided by the larger groups.

Students were also required to keep individual reflective journals detailing their individual approach and reaction to the cases, group dynamics and other reflections on their own learning. In groups, they were also required to write reports at the end of each case. The report-writing aspect has implications in fields other than Statistics since statistical tools are used in many disciplines—students from almost every discipline have a need to collect and/or analyze data collected from their research or class projects.

Preliminary Evidences of Learning

It wass anticipated that students will have a better understanding of how statistical analyses can be applied to real data which in turn will help students put their statistical knowledge to use in future classes.

There were thus many indications or evidences of student learning including the following:
1.    Student writings on processes involved
2.    Student ability to use tools in different settings
3.    Instructor observations during classes
4.    Videotaped recordings as detailed above

For this project, focus lay on the analysis of the vide0taped evidences. Videotape recordings were transcribed and responses were thematically coded to identify trends. As the investigator watched the play-back video, attention was also paid to observing how “successful” groups began their case versus those who were not as “successful.” This allowed the researcher to begin to identify possible barriers or helpful problem solving strategies employed. The preliminary results of this analysis are presented in the following section.

Results

For this project, 2 groups of students were video-taped 4 times during the semester at the very beginning of a case or set of cases. Groups contained 3 to 4 students, with self-selected membership.  The choice of the two groups occurred during the second week of school, after I had observed the dynamics of all the formed groups.

The two groups were selected because they were made up of two clearly personality types:
•    Group 1 consisted of a 4 students who were all strong personality types and were high achievers. Three of the 4 students were honors students. In general when presented with a case, these students went right to work on getting it done (the “doers”).
•    Group 2 consisted of 3 students who individually were less comfortable overall and from the start of the class, always looked to the instructor for explicit directions. When given a new case, these students without a fail asked for clarification on exactly what they were supposed to do (the “askers), versus at least reading through the case once and trying to think through it.

At the beginning of the work period, each group was required to select a “writer” who was responsible for recording the group’s official solutions. Preliminary analysis revealed a number of apparent differences in the operation of these groups, especially as it relates to the group’s success at problem-solving.

Question Asking & Group Organization

There was a marked difference in the type and number of questions that the doers asked compared to the askers. Particularly in the earlier tapings, the askers tended to almost begin the case with a sense of panic—“What are we supposed to be doing?”—while the doers approached the case more directly—Okay, what information do we have, and what are we trying to do?” From the first, the doers group started a case by having each member read the case, and then discussing what each person understood before brainstorming their options for addressing the problem.

This was very different from the askers group, where in the first case the students all simply sat around saying “do you know what to do? What chapter is this on?” It was also not clear from the tape whether each member had in fact read the entire case nor understood. In one doers tape, it appeared that two of the three members started looking through their notes before even reading through the case or coming to some understanding of what was expected of them.Once the doers had agreed on a plan, they then spoke to each other as they worked through analyses options. As the semester progressed, the askers group also became a little more organized but never quite achieved the fluidity of the doers group. I believe that this overall lack of organization and consensus often led to the askers group being less successful not only in terms of not completing cases, but also in terms of thinking through the statistical concepts.

There was also a difference noted in the behavior of the writers for the two groups, and in the behavior of the groups themselves. In the doers group, each member had their individual papers on which they worked, while in the askers group, only the writer ever wrote on a paper separate from the assignment sheet.

The writer for the doers group worked through the problem with everyone, but only wrote on the group paper when the group was satisfied with the direction or conclusion. After writing each section, the writer would read the recorded response out-loud and the group would either agree with the statements as written, or offer changes for clarification. They would not proceed to a new question or portion of the case until they, as a group, were satisfied with the write-up for the previous section. The writer for the askers group however, tended to write as the group thought through the problems and there was also some arbitrary division of labor where one student would flip through their notes constantly while the other appeared to simply be trying things out on his calculator. The asker writer during this time simply sat and waited. As a result, the group write-ups for the asker groups were disorganized and often incomplete. 

In one instance that was not video-taped, each of the groups came to my office and sat with me as I read and graded their case write-up. At numerous instances during the read-through, I would question the askers group about what was written and the “leader” would often be able to correctly explain their technique, but it was often not consistent with what was written on the paper. This would then lead to the writer apologizing because he/she didn’t understand what they were supposed to write. Even after addressing the need for the group to work as a unit, the group never quite corrected itself.

A lot of the confusion that appears on the tape for the askers group was noted by the instructor during the classes. As I walked around to the groups asking leading questions or for clarifications, the askers would tend to simply say “I don’t know” to any question that I ask, or “We’re confused—we don’t remember doing this before”. As the semester progressed, one member of the three-person group did evolve into a “leader” role where he would often figure things out by himself and then explain it to the others. The other two members genuinely didn’t appear capable of independent thought and were quickly overwhelmed as the case materials became more extensive. Any questioned member of the doers group could often explain the logic behind their case and analysis strategy.

Reflections

In the doers group, it was very interesting to watch the non-honors student. I was concerned that her quiet nature would be over-run by the group, but as the semester progressed you could visibly see her becoming more confident in her ideas and with her own voice. This was not observed in the asker group where the quietest personality (who also happened to often be the writer), remained detached in spite of my attempts to engage him. This led me to believe that the group dynamic plays a bigger role than I had previously thought in group problem solving success.

It is truly too early in the investigation to make definitive statements on the nature of student learning and problem solving.  Instead of the “does the case method work” question that I started out with, as the semester progressed it became apparent that the case method of teaching can be used to simply provide context for further research into how students begin to solve problems in statistics.

This project will be continued for at least two more years which will allow me to collect more data on which to base my findings. In particular, I’m interested in looking more closely at how different groups start to solve problems across time. Having only formally taped 2 of my 10 groups, I know that there are many different variations in group dynamics that may affect the problem solving techniques employed.

The findings of this research will be of interest not only to the statistics education community, but to all communities of learners who may consider methods such as the case method of teaching as a means of strengthening student content knowledge and their understanding of learning. The ETLP program provided me with the opportunity to share/obtain perspectives on the advantages and issues with using case studies and an inquiry-based learning approach to teaching. The ETLP program has helped me to refine my research questions and finally get to a point where I can identify the type of evidences that I need to collect. The introductory training given on coding and analyzing video-taped data has been useful, though I am nowhere close to addressing all of the interesting things observed on the student tapes.

Overall, I am happy with using the case study approach in this course. Not only are the students analyzing real data, but the activity-based nature of the course helps ensure that they are engaged in the learning process as they consider collection and production issues. In this way conceptual understanding may be stressed rather than mere knowledge of procedures and/or rote calculations—all recommendations of the GAISE. Students are thus afforded the opportunity to discover, construct and analyze statistical ideas.

Future Work

Questions of interest for future work involves looking at whether students get better over time at starting problems, negotiating within the groups, and the overall development of learning strategies. In terms of data collection, it would also be useful to this extended project to collect data of experts solving problems to compare with the novice approaches. In the end, I’m interested in developing a problem-solving rubric that can be used to understand how students learn statistical thinking.