Curriculum recommendations for statistics and probability in grades K-8 are prevalent (e.g., Franklin et al., 2007; NCTM, 2000; National Governor’s Association for Best Practices and Council of Chief State School Officers, 2010). These recommendations hold the potential to improve statistical literacy in society, but they also pose a dilemma: teachers often lack the knowledge to carry them out (Stohl, 2005). Researchers have described problems with teachers’ knowledge in areas such as measures of center (Groth & Bergner, 2006), informal inference (Leavy, 2010), and data displays (Jacobbe & Horton, 2010). Given the present state of affairs, there is a pressing need to design and test new approaches to developing statistical knowledge for teaching (SKT).

            Knowledge of teachers’ cognition provides a foundation for improving teacher education efforts (Oliveira & Hannula, 2008). The purpose of the present study was to identify critical cognitive stages in SKT development. The guiding research questions were: What cognitive differences exist between a pre-service teacher who exhibited great improvement in SKT during a one-semester course and one who did not? What cognitive similarities exist between them? The answers to the research questions were expected to highlight important considerations in designing interventions to improve pre-service teachers’ SKT. 

            Theories about SKT are in beginning stages of development (Groth, 2007). The present study used the work of Hill, Ball, and Schilling (2008) as an organizing frame. Hill, Ball, and Schilling characterized subject matter knowledge and pedagogical content knowledge as two primary components of content knowledge for teaching. Subject matter knowledge can be discussed in terms of three sub-components: common knowledge, specialized knowledge, and horizon knowledge. Pedagogical content knowledge can also be discussed in terms of three sub-components: knowledge of content and students, knowledge of content and teaching, and curriculum knowledge. These components and their sub-components serve to distinguish the content knowledge needed by teachers from that needed by other individuals.

            The two participants studied as contrasting cases were enrolled in a semester-long SKT course taught by the researcher. This positioning of the researcher provided a first-person perspective (Ball, 2000) on participants’ learning, allowing research strategies and interpretations of data to be grounded in prolonged interaction with the participants. The two participants, Annie and Shelly (pseudonyms), each attended class regularly, submitted assignments, took tests, and participated in class as asked. At the beginning and end of the semester, each took a test of SKT designed by the Learning Mathematics for Teaching (LMT) project (Hill, Ball, & Schilling, 2008). They had similar scores on the pre-test. On the post-test, Annie had a slight decrease in score, and Shelly’s score increased by 1.58 standard deviation units (the average standard deviation unit increase for the class was 0.64). Shelly’s written work and class participation during the semester was also superior to Annie’s, providing further confirmation of their differences in SKT development.

            Data on each participant’s SKT were collected at several points during the semester. Class examinations included probability and statistics items from past administrations of the National Assessment of Educational Progress (NAEP). The NAEP items were relevant to the grade levels the participants would be certified to teach, and consisted of free response and multiple choice items. Practitioner-oriented articles from NCTM journals were assigned during the semester. Participants completed five reading comprehension prompts related to the components of SKT (Hill, Ball, & Schilling, 2008) for each of the 12 assigned articles. Participants also completed selected multiple choice items from the Assessment Resource Tools for Improving Statistical Thinking (ARTIST) database (delMas, Garfield, Ooms, & Chance, 2007) pertaining to common knowledge. Additionally, the researcher designed examination items to probe specific difficulties in SKT development observed during class. Responses to all of the above items were assembled into a case study database for analysis.

            The theoretical notion of cognitive obstacle (Herscovics, 1989) was used to analyze the data and characterize similarities and differences between the two participants. Cognitive obstacles occur when previously held knowledge conflicts with new information. During the first stage of data analysis, an initial list of difficulties participants encountered during the course was generated. However, not all difficulties were considered cognitive obstacles. For instance, responses that were incomplete or lacking clarity of expression were flagged as difficulties during the initial reading of the data but ultimately not considered to indicate cognitive obstacles. Only evidence of existing ideas that blocked the development of one or more SKT components was considered to suggest a cognitive obstacle. Applying this criterion helped narrow the broad set of difficulties down to a subset of cognitive obstacles for each participant.

            Four types of cognitive obstacles emerged during data analysis. The obstacles were associated with: everyday language, everyday experiences, pre-college mathematics, and pre-college statistics. The cognitive obstacles that occurred pertained to both subject matter knowledge and pedagogical content knowledge. Annie and Shelly differed in terms of some of the obstacles they encountered. For instance, Annie adhered to everyday language meanings of statistical terms rather than preferred technical meanings to a greater extent than Shelly. Annie also showed more evidence of resilient cognitive obstacles related to her pre-college statistics instruction. The two participants did experience some common obstacles. In regard to pedagogical content knowledge development, for example, both had instances of adhering to transmission-oriented pedagogy, typical of many pre-college mathematics classes in the U.S. (Jacobs et al., 2006), rather than the inquiry-oriented instruction promoted during the course.   

            To catalyze conversations during the discussion portion of the session, the following questions will be posed:

• What are the important components of SKT? How do they compare to the model of SKT that guided the study?
• Which comparisons between Annie and Shelly are most likely to be helpful in the design of teacher education curricula? Least helpful? Why? Are there additional types of comparisons that should be made between the two participants?
• Aside from the notion of cognitive obstacle, what other theoretical tools should be employed to characterize important transition points in the development of SKT?
• What additional data collection techniques and instruments could be used to investigate the development of SKT?

References

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delMas, R., Garfield, J., Ooms, A., & Chance, B. (2007). Assessing students’ conceptual understanding after a first course in statistics. Statistics Education Research Journal, 6(2), 28-53. Retrieved from http://www.stat.auckland.ac.nz/~iase/serj/SERJ6%282%29_delMas.pdf

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Conceptualizing and measuring teachers’ topic-specific knowledge of students. Journal for Research in Mathematics Education, 39, 372-400.

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National Governor’s Association for Best Practices and Council of Chief State School Officers. (2010). Common core state standards for mathematics. Retrieved from http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf

Oliveira, H., & Hannula, M.S. (2008). Individual prospective mathematics teachers. In K. Krainer & T. Wood (Eds.), The international handbook of mathematics teacher education (vol. 3) (pp. 13-34). Rotterdam, The Netherlands: Sense Publishers.

Stohl, H. (2005). Probability in teacher education and development. In G.A. Jones (Ed.), Exploring probability in school: Challenges for teaching and learning (pp. 345-366). New York: Springer.