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Developing understanding of statistical variation: secondary statistics teachers’ perceptions and recollections of learning factors

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Abstract

This retrospective phenomenological study investigates activities and actions identified by secondary statistics teachers who exhibit robust understandings of variation as deepening their understandings of statistical variation. Using phenomenological methods and a frame of Mezirow’s transformation theory, analysis revealed learning factors that include their interests in statistics, motivation to encounter and resolve dilemmas, desires to have an overarching content framework, propensities for critical reflection, and actions on opportunities to engage in statistical learning activities and rationale discourse with more knowledgeable others. The extent to which these teachers embrace these opportunities distinguishes them from other teachers. Results from this study provide some basis for formulating hypotheses about secondary teachers’ statistics learning in general by contributing to understanding circumstances that may be conducive to developing deep understandings of statistical content. This study also advances the use of retrospective methods within a theoretical frame for adult learning to investigate teacher learning.

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Acknowledgments

This paper is based in part on the author’s doctoral dissertation, completed at The Pennsylvania State University under the direction of Rose Mary Zbiek, and supported in part by the National Science Foundation under Grant ESI-0426253 to The Pennsylvania State University and by a research initiation grant from The Pennsylvania State University College of Education Alumni Society. Any opinions, findings, and conclusions are those of the author and do not necessarily reflect the views of the National Science Foundation or the College of Education Alumni Society.

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Authors and Affiliations

  1. College of Education and Human Development, University of Louisville, 1905 S. First St., Louisville, KY, 40292, USA

    Susan A. Peters

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Correspondence to Susan A. Peters.

Appendices

Appendix 1: Event history calendar

How to enter an event

  • If your first experience is an undergraduate course, scroll down in the document until you see the year in which you were enrolled in the course. For example, if you were enrolled in an undergraduate statistics course for the fall semester of 1980, you would scroll down to 1980.

    • Type an “X” in the boxes that correspond with the seasons during which you were enrolled in the course. For example, if you were enrolled in an undergraduate statistics course for the fall semester of 1980, you would type an “X” in the third and fourth boxes for item 1 in the column for 1980.

    • You should also provide a brief description of the experience, such as “Intro Stats Course,” by typing the description in the second row of the block, as shown to the right.

    • If you feel the experience was pivotal to your development in learning statistics or as a statistician or as a statistics educator, mark the experience with a “*” in the description, as shown to the right.

The descriptions

  • After you record the timing and nature of your experience, you will need to record a short description of the experience. This description will be entered in a table different from the EHC but linked to the EHC.

  • If possible, you should include information about the people, places, and feelings associated with the experience.

  • Note those events that were particularly positive, such as events when you learned something new or realized something about your understanding of statistical concepts.

  • Also note those events that were particularly negative, such as events where you realized you did not have the knowledge of statistics to engage in productive dialogue about statistics.

Experiences to record in “other”

  • Statistics-related professional presentations, like NCTM sessions

  • Publications (statistics or statistics-related)

  • Statistics-related informal experiences

    • Conversations with colleagues

    • Reading statistics books

    • Reading professional journals

  • Pedagogical professional development that had an impact on statistics teaching

  • Any other experience that contributed to growth or learning in statistics or statistics teaching

Appendix 2: Critical incident description

Think about your experiences learning statistics, and to the extent possible, focus in particular on learning about variation. From your experiences, identify one particularly positive experience and one particularly negative experience related to your informal or formal study of statistics or variation. Please provide written responses to the information requested below. In general, your response to each experience should be approximately one single-spaced page long. You may be asked to expand upon your responses when we meet to discuss your experiences.

Describe one positive experience related to your informal or formal study of statistics or variation—an experience that you recall as being particularly good or that you feel resulted in significant learning on your part. Elaborate on this experience and the timing of the experience. To the extent possible, please address all of the questions listed below in your written response.

Describe one negative experience related to your informal or formal study of statistics or variation—an experience that you recall as being particularly bad or that you feel affected your perception of your understanding or knowledge of variation or statistics in a negative way. Elaborate on this experience and the timing of the experience. To the extent possible, please address all of the questions listed below in your written response.

List of questions for each incident

Details of the experience:

  1. 1.

    When, where, and for how long did the experience occur?*

  2. 2.

    What events or circumstances precipitated the experience or caused the experience to occur in the way in which it did?

  3. 3.

    What other people or circumstances played an influential role in the experience?

  4. 4.

    How did the experience end?

Reflections on the experience:

  1. 5.

    As you reflect on the experience, why do you think you viewed the events surrounding this experience positively or negatively?

  2. 6.

    What emotions did you recall feeling during the experience?*

  3. 7.

    In response to the experience, what actions did you take?

  4. 8.

    What do you think you learned from the experience?

Effects of the experience:

  1. 9.

    How has the experience affected your understanding of variation?

  2. 10.

    How has the experience affected your statistics teaching?

Beyond the experience:

  1. 11.

    If you could change past events surrounding the experience, what would you change and why?

  2. 12.

    If you were to encounter the experience under identical circumstances to those surrounding the original experience, what effect do you think the experience would have on you today?

* These questions overlap with some of the information requested in the event history calendar. If you already provided the information in your event history calendar, you don’t need to duplicate your responses to these questions.

Appendix 3: Abbreviated Context I interview schedule

Note: In general, whatever the participant responds, probe for details about the experience, particularly with respect to variation or experiences indicative of transformational learning related to statistics.

Before we talk about what you included in your calendar, I would like to know if you remembered any additional experiences you had with statistics that should be recorded on the calendar?

  • Could you tell me when the event occurred?

  • Over what time period did the event take place?

  • Please provide a brief description of the event, and the people, places, and feelings associated with the event.

  • Please describe any particularly salient characteristics of the experience.

General questions for each experience type listed in EHC

Think about the statistics courses you took (or other type of experience). Were there any in which you feel you learned a great deal about statistics or variation or in which you grew a great deal as a statistician or a statistics educator?

  • Which of the statistics courses you took was the course where you learned the most about statistics or variation or where you grew the most as a statistician or a statistics educator?

  • What did you learn in the course?

    In general, if the participant mentions variation, probe for details about what they learned about variation. If the person describes an event that may have triggered a disorienting dilemma, probe for details of the experience in light of potential indicators for transformation, including

    • a description of the disorienting dilemma,

    • self-examination,

    • critical assessment of assumptions,

    • recognition that others have had similar experiences,

    • exploring new roles through rational discourse with others,

    • planning a course of action,

    • constructing the knowledge and skills needed to enact the plan,

    • experimenting with new roles,

    • building a sense of competence, and

    • reintegrating into life based on new roles.

  • Three major areas of statistics are exploratory data analysis, study design, and inferential statistics. Which, if any, of these were part of this course?

    • What do you remember learning about the role of variation in [ exploratory data analysis, planning a study, inferential statistics ]?

    • How is what you learned in that course different from or similar to how you now see variation in [ exploratory data analysis, planning a study, inferential statistics ]?

  • As you reflect on your experiences in the course, what, if any, experiences seem particularly important to your development of knowledge about statistics and variation or important to your development as a statistician or as a statistics teacher?

    • Why do you think [ paraphrase experiences ] influenced your development of knowledge about statistics and variation or as a statistician or as a statistics teacher?

    • What development resulted from [ paraphrase experiences ]?

  • What do you remember feeling about [ paraphrase experiences ]?

  • What conversations did you have about [ paraphrase experience or characteristic ]?

    • In relation to the experience, what was the role of the person you had this conversation with?

  • What did you do in response to [ paraphrase experience or characteristic ]?

  • As a result of [ paraphrase experience or characteristic ], how, if at all, did you see a change in how you thought about variation?

  • In how you thought about statistics in general?

  • In how you view yourself as a statistician or statistics teacher?

How, if at all, does your knowledge of statistics now differ from your knowledge of statistics when you [ had this experience ]?

  • How does the way you now see variation differ from or agree with the way you saw variation when you [ had this experience ]?

Teaching question: How does the way you now teach variation differ from the way you taught variation when you first started teaching statistics?

  • How is your knowledge of pedagogy and use of pedagogical strategies affected by your knowledge of statistics?

Professional development question: How did you learn about this program, and what factors influenced your decision to attend this session [ program ]?

Questions related to critical incidents

If any of the requested questions were not addressed in the critical incident description, ask the participant to answer the unanswered questions from this list:

Details of the experience:

  • When, where, and for how long did the experience occur?*

  • What events or circumstances precipitated the experience or caused the experience to occur in the way in which it did?

  • What other people or circumstances played an influential role in the experience?

  • How did the experience end?

Reflections on the experience:

  • As you reflect on the experience, why do you think you viewed the events surrounding this experience positively or negatively?

  • What emotions did you recall feeling during the experience?*

  • In response to the experience, what actions did you take?

  • What do you think you learned from the experience?

Effects of the experience:

  • How has the experience affected your understanding of variation?

  • How has the experience affected your statistics teaching?

Beyond the experience:

  • If you could change past events surrounding the experience, what would you change and why?

  • If you were to encounter identical circumstances to those surrounding the experience, what affect do you think the experience would have on you today?

Additional Questions:After the incident occurred, what conversations did you have about the incident?

  • In relation to the experience, what was the role of the person you had this conversation with?

  • How, if at all, did those conversations impact your understanding of the incident?

  • Using the title of the person’s position or the person’s relationship to you, from whom, if anyone, did you seek input that you thought might differ from your perspective of the incident?

    • What perspective did they offer for the incident?

Describe how often you think about or thought about this incident.

  • About how often, and when did you reflect on the incident?

  • Describe the form of your reflection, e.g., thinking, writing, or talking.

    • How, if at all, did your reflection on the incident change your interpretation or understanding of the incident or strengthen your initial interpretation or understanding of the incident?

      • Why do you think your interpretation or understanding of the incident changed?

What, if any, relationship do you see between the critical incident you documented in your critical incident document and the other experiences you had surrounding this incident? [ Point to events on the EHC that surround the incident .]

Appendix 4: Abbreviated Context II interview schedule

Note: In general, whatever the participant responds, probe for details about the experience, particularly with respect to variation or experiences indicative of transformational learning.

For today’s session, I would like to explore some of the experiences we discussed last time. Since our last conversation and subsequent to your reflections on that conversation, what, if any, other thoughts, events, or experiences related to your learning of statistics do you feel should be added to your event history calendar?

  • Could you tell me when the [ event or experience ] occurred?

  • Over what time period did the [ event or experience ] take place?

  • Could you provide a brief description of the [ event or experience ] and the people, places, and feelings associated with the [event or experience ]?

  • Please describe any particularly salient characteristics of the experience.

We are now going to talk in-depth about some of the events you identified as pivotal on your calendar. Some of the questions that I ask may seem to be repetitive, but I want to be sure that we have touched upon important characteristics of some of your experiences.

Of all of your experiences with learning and teaching statistics, which two or three experience do you think precipitated the greatest change in your understanding of variation or resulted in the greatest change in your understanding of variation?

  • Did you view the [ name ] experience as pivotal at the time of the experience, and if so, how?

  • How do you currently view the experience?

  • What features of this experience do you think were essential for your learning?

    • Why do you think [ name feature ] was particularly effective for you?

    • Do you think [ name features ] might have a common effect among statistics teachers, and if so, why?

What features of this experience do you think were largely ineffective for your learning?

  • Why do you think [ name feature ] was particularly ineffective for you?

  • Do you think [ name features ] might have a common effect among statistics teachers, and if so, why?

In response to the experience, what actions did you take?

  • Why do you think you [describe the action ]?

  • How did [ describe the action ] affect your learning?

  • How did [ describe the action ] affect your teaching?

In our last session, you indicated that [ insert name of professional development/class/teaching experience ] was an experience that was particularly educative for you. In particular, you mentioned that you thought [ name experience or characteristic ] was instrumental in your learning about variation. Did you view the experience as pivotal at the time of the experience, and if so, how? Repeat the preceding series of questions for this experience.

Over the course of the past few months, you’ve spent a considerable amount of time reflecting on your teaching and learning in statistics. We’ve talked about each of these experiences individually, but as you’ve been reflecting on your experiences, did you feel there were any patterns in your experiences? If so, what are they?

Can you describe how you think this collective group of experiences might have contributed to your development of an understanding of variation?

  • Why do you think these collective experiences were particularly effective for you?

  • Which grouping(s) of these experiences do you think would have a similar learning effect on other statistics teachers, and why?

In your educational experiences, what hindrances to your development of an understanding of variation do you believe existed?

  • Why do you think these characteristics were a hindrance for you?

  • What personal characteristics do you think a person would need to have in order for these characteristics to not be a hindrance?

If you could change something in your learning experiences with statistics and with variation in particular, what would you change and why?

How, if at all, did your knowledge of variation change as a result of this collective group of experiences?

How, if at all, did you see a change in how you thought about variation?

  • Why do you think that is the case?

How, if at all, did you see a change in how you thought about statistics in general?

  • Why do you think that is the case?

How, if at all, did you see a change in how you view yourself as a statistician or statistics teacher?

  • Why do you think that is the case?

What else you would like to add about your experiences that would help me to understand your learning experiences related to variation?

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Peters, S.A. Developing understanding of statistical variation: secondary statistics teachers’ perceptions and recollections of learning factors. J Math Teacher Educ 17, 539–582 (2014). https://doi.org/10.1007/s10857-013-9242-7

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