1. Randall E. Groth Salisbury University

3.  Generate: “To launch a research program, mathematics educators need to generate some ideas about the phenomena of interest so that they can begin to explore them. Those ideas might emerge from theoretical considerations, previous research, or observations of practice” (p. 5).  Frame: “A frame is seen as involving clarification of the goals of the research program and definition of the constructs it entails, formulation of tools and procedures for the measurement of those constructs, and consideration of the logistics needed to put the ideas into practice and study their feasibility” (p. 5).

4.  Examine: “to understand the phenomena better and to get indicators of what might work under which conditions” (p. 6).  Generalize: “generalization can address questions of scale (studying different populations or sites, using more comprehensive measures, examining different implementation conditions), or it can be used to refine the theory or reframe the entire research program” (p. 6).

5.  Extend: “A body of research that has yielded some generalizable outcomes can be extended in a variety of ways. Multiple studies can be synthesized; long-term effects can be examined; policies can be developed for effective implementation” (p. 7).  Cycling: Each component “has the possibility and potential to cycle back to any earlier component, and such cycling should be a conscious effort of the researchers. Progress in research is generally more of circular process than a linear one” (p. 7).

6. Generate Goals & Constructs Measurement Logistics & Feasibility Examine Generalize Extend

7.  Curriculum recommendations in school statistics (e.g., GAISE - Franklin et al., 2007; NCTM, 2000) and teachers’ need for content knowledge to implement them.  Conceptual and procedural knowledge (Hiebert & Lefevre, 1986) as an important distinction for teachers to understand.  Profound understanding of fundamental mathematics (Ma, 1999) marking out a specialized domain of teachers’ mathematical knowledge for teaching.

11.  SOLO Taxonomy (Biggs & Collis, 1982) – pre- service teachers’ conceptual knowledge of mean, median, & mode (Groth & Bergner, 2006). SOLO Taxonomy (Biggs & Collis, 1982) – pre- service teachers’ conceptual knowledge of mean, median, & mode (Groth & Bergner, 2006).  Spontaneous metaphors as an indicator of content knowledge and pedagogical content knowledge (Groth & Bergner, 2005). Spontaneous metaphors as an indicator of content knowledge and pedagogical content knowledge (Groth & Bergner, 2005).  Partially correct constructs (Ron, Dreyfus, & Hershkowitz, 2010) for SKT related to elementary categorical data analysis Partially correct constructs (Ron, Dreyfus, & Hershkowitz, 2010) for SKT related to elementary categorical data analysis  Participation in item-writing camps for LMT project – production of quantitative scales.

17.  Analysis and discussion of practitioner- oriented journal articles on NAEP results (Groth, 2009) ◦ Teachers’ personal frameworks, formed in practice, can differ markedly from those of researchers (e.g., debate on whether students should learn an algorithm for calculating the arithmetic mean without understanding its meaning) and be resistant to change.

18.  Analysis and discussion of written cases of teaching (Groth & Shihong, in press) ◦ Common knowledge of content need not be developed in isolation from other components of the knowledge base for teaching (e.g., prospective teachers developed knowledge of randomness, sampling, and independence while discussing students’ conceptions of the likelihood of making a long string of basketball shots (Merseth, 2000)). ◦ Conjecture for further research: It would be profitable to develop courses that do not compartmentalize the development of common knowledge of statistics from other components of SKT.

20.  On average, participants (n = 80) improved their scores on an LMT-designed CKT-M instrument on content knowledge for teaching statistics by 0.87 standard deviations.  The mean difference between pre- and post- test scores was statistically significant (M = 0.87, SD = 0.53), t (79) = 14.71, p <.0001, CI. 95 =.75,.99.

21.  Assess implementation of the SKT course at different sites (e.g., local community colleges).  Experimental design to compare performance of prospective teachers in SKT course and conventional introductory statistics.  Refinement of the LMT measure and development of equated forms.  Cycling back: Continued refinement of theory of SKT by gathering data on PreK-8 student learning outcomes of SKT course completers and other teachers.

22.  Retention studies – re-administration of SKT test and gathering of qualitative classroom observation data once completers of the SKT course begin teaching.  Observation of difficulties encountered as teachers in many different institutions begin to implement the SKT course, and development of interventions to address them.

23. Randall Groth Department of Education Specialties Salisbury University regroth@salisbury.edu

24.  Ball, D.L., Thames, M.H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59, 389-407.  Biggs, J. B., & Collis, K. F. (1982). Evaluating the quality of learning: The SOLO taxonomy. New York: Academic.  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. Alexandria, VA: American Statistical Association.  Groth, R.E. (2007). Toward a conceptualization of statistical knowledge for teaching. Journal for Research in Mathematics Education, 38, 427-437.  Groth, R.E. (2009). Characteristics of teachers' conversations about teaching mean, median, and mode. Teaching and Teacher Education, 25, 707-716.

25.  Groth, R.E., & Bergner, J.A. (2005). Preservice elementary school teachers' metaphors for the concept of statistical sample. Statistics Education Research Journal, 4 (2), 27- 42.  Groth, R.E. & Bergner, J.A. (2006). Preservice elementary teachers' conceptual and procedural knowledge of mean, median, and mode. Mathematical Thinking and Learning, 8, 37-63.  Groth, R.E., & Shihong, X. (in press). Preparing teachers through case analyses. In C. Batanero & G. Burrill (Eds.), Teaching statistics in school mathematics - Challenges for teaching and teacher education: A joint ICMI-IASE study. Dordrecht, The Netherlands: Springer.  Hiebert, J. & Lefevre, P. (1986). Conceptual and procedural knowledge in mathematics: An introductory analysis. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 1–28). Hillsdale, NJ: Lawrence Erlbaum Associates, Inc.

26.  Hill, H.C., Schilling, S.G., & Ball, D.L. (2004). Developing measures of teachers’ mathematical knowledge for teaching. Elementary School Journal, 105, 11-30.  Ma, L. (1999). Knowing and teaching elementary mathematics. Mahwah, NJ: Lawrence Erlbaum Associates, Inc.  Merseth, K.K. (2003). Windows on teaching math: Cases of middle and secondary classrooms. New York: Teachers College Press.  Ron, G., Dreyfus, T., & Hershkowitz, R. (2010). Partially correct constructs illuminate students’ inconsistent answers. Educational Studies in Mathematics, 75, 65-87.  Scheaffer, R., & Smith, W. B. (2007). Using statistics effectively in mathematics education research: A report from a series of workshops organized by the American Statistical Association with funding from the National Science Foundation. Alexandria, VA: American Statistical Association. [Online: http://www.amstat.org/research_grants/pdfs/SMERReport.pdf] http://www.amstat.org/research_grants/pdfs/SMERReport.pdf