Developing Integrated Reasoning about Statistical Variation

Our data-filled, technological world has created a time when statistical reasoning is important for responsible citizenship (Franklin, Kader, Mewborn, Moreno, Peck, Perry, & Scheaffer, 2007). Statistical reasoning includes understanding and explaining statistical processes and interpreting statistical results (Ben-Zvi & Garfield, 2004). To facilitate students' development of abilities to reason statistically and advance educational goals of developing responsible citizens, statistics content features prominently in school curriculum recommendations (e.g., CCSSI, 2010).

Visions of graduating students who reason statistically can only be realized if students are taught by competent reasoners, as implied by research that connects teacher knowledge with student achievement (e.g., Hill, Rowan, & Ball, 2005). Researchers opine, however, that many teachers do not have sufficient statistical knowledge or experiences to facilitate students' development of sophisticated statistical reasoning (e.g., Shaughnessy, 2007). Although researchers are making progress in uncovering characteristics of experiences that result in teachers' statistical learning (e.g., Madden, 2011), the work is only beginning to reveal characteristics that lead to sophisticated reasoning about and with statistical concepts.

This retrospective study with teachers who exhibit sophisticated reasoning offers a viable, complementary, and timely means to uncover characteristics of experiences associated with learning. Given the centrality of variation to statistics (Wild & Pfannkuch, 1999), this study focuses on characteristics of experiences that contributed to teachers' development of integrated reasoning about variation, which arguably provides insights into statistical learning in general. This study answers the question: For secondary statistics teachers who exhibit integrated reasoning about variation, what are the activities and actions that contributed to their current understandings of variation as reflected in their perceptions and recollections?

This study uses phenomenological methods (Moustakas, 1994) to explore the phenomenon of secondary teachers' development of integrated reasoning about statistical variation with a purposeful sample of 16 secondary statistics teachers from 14 states across the continental United States. To determine the sophistication of teachers' reasoning, the researcher conducted a 90- to 120-minute task-based interview with each teacher. Indicators of integrated reasoning emerged from teachers' responses to variation-related tasks, research about students' reasoning related to variation (e.g., delMas & Liu, 2005), and expositions on sophisticated reasoning about statistical variation (e.g., Garfield & Ben-Zvi, 2005). In combination with the Structure of the Observed Learning Outcomes Model (Biggs & Collis, 1982), the indicators formed the basis for a perspectives framework of reasoning about variation. Sophisticated reasoning is indicated from integrated reasoning about variation within design, data-centric or modeling perspectives or across design, data-centric, and modeling perspectives. Data sources to investigate common characteristics of activities and actions that contributed to teachers' development of integrated reasoning about variation include questionnaires, two learning-experience interviews, event history calendars, and critical incident descriptions.

Teachers' learning experiences are viewed through the lens of transformative learning theory (Mezirow, 2000)—a theory of adult learning that provides explanatory power for teacher learning. The main phases of transformative learning are critical reflection, rational discourse, and action. Transformative learning typically begins with one or more events that trigger dilemmas or examination of assumptions. An individual may become aware of previously implicit assumptions and then assess those assumptions through critical reflection. Upon critical reflection, the individual may explore options for new roles and actions by engaging in rational discourse with others in preparation for developing and acting on a plan to resolve the dilemma.

Data analysis follows the detailed, systemic procedures recommended for phenomenological studies (Moustakas, 1994) and focuses on teachers' learning experiences related to reasoning from perspectives for which integrated reasoning was evident. First, the researcher sought evidence of elements related to transformative learning, including evidence of experiences that triggered dilemmas, critical reflection, rational discourse with self or others, planning to learn statistics content, experimenting with new roles, or changing beliefs and assumptions related to the teaching and learning of statistics. Interview passages for each teacher were then grouped by elements of perspective transformation. During the next phase, the researcher read and reread teachers' statements, organized statements into recurring themes, and grouped them to produce a factual description of each teacher's phenomenon using elements of transformative learning. For each element, the researcher extracted the essence of the experience using divergent perspectives, including another mathematics educator's input, and continually reread the descriptions to gain further insight into teachers' experiences. Descriptions from this phase of analysis focused on how teachers experienced the phenomenon. The final stage of analysis involves integrating factual descriptions of teachers' experiences into a composite description that combines evidence related to each element of transformational learning for the larger group of teachers to form a synthesized description that characterizes the overall essence of teachers' experiences of developing integrated reasoning about statistical variation. This stage of analysis is ongoing.

This study is part of a larger study that identified teachers who exhibited robust understandings of variation and investigated their perceptions of characteristics of the activities and actions that contributed to their development of robustness. Analysis revealed factors including interest in the field of statistics, desire for an overarching content framework for learning and teaching statistics, sufficient foundational knowledge upon which to deepen understandings, and opportunities to engage in rational discourse and potential learning experiences. This study builds on the previous study by examining characteristics of activities and actions related to reasoning within and across perspectives and includes data from teachers who exhibited integrated reasoning without necessarily exhibiting reasoning characteristic of overall robust understanding. Preliminary analyses suggest subtleties in characteristics of learning experiences between perspectives. For example, the focus of rational discourse in which teachers engaged differed when reasoning about variation from the design and modeling perspectives. When reasoning from the design perspective, the benefits of rational discourse came about from hearing alternative perspectives when considering factors in designing experiments or considering the design of enacted experiments. When reasoning from the modeling perspective, discourse focused more on discerning the underlying statistical principles illustrated during physical and computer simulations.

This poster contributes to understanding circumstances conducive to deepening understandings of statistical content and advances the use of retrospective methods within a theoretical frame for adult learning to investigate teacher learning.

References

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Common Core State Standards Initiative. (2010). Common Core State Standards for Mathematics. Common Core State Standards (College- and Career-Readiness Standards and K–12 Standards in English Language Arts and Math). Washington, D.C.: National Governors Association Center for Best Practices and the Council of Chief State School Officers. Downloaded from http://www.corestandards.org

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Franklin, C., Kader, G., Mewborn, D., Moreno, J., Peck, R., Perry, M., & Scheaffer, R. (2007). Guidelines for assessment and instruction in statistics education (GAISE) report: A pre-K-12 curriculum framework. Alexandria, VA: American Statistical Association.

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Madden, S. R. (2011). Statistically, technologically, and contextually provocative tasks: Supporting teachers' informal inferential reasoning. Mathematical Thinking and Learning, 13(1), 109-131.

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Shaughnessy, J. M. (2007). Research on statistics learning and reasoning. In F. K. Lester, Jr. (Ed.), Handbook of research on mathematics teaching and learning (2nd ed., pp. 957-1009). Greenwich, CT: Information Age.

Wild, C. J., & Pfannkuch, M. (1999). Statistical thinking in empirical enquiry. International Statistical Review, 67(3), 223-265.