1. 1 Contrasting Cases in the Development of Statistical Knowledge for Teaching 2012 NCTM Research Presession Randall E. Groth regroth@salisbury.edu 2012 NCTM Research Presession Randall E. Groth regroth@salisbury.edu

2. 2 Background  Recommendations for statistics and probability in the Pre-K-8 curriculum point to the need to foster statistical knowledge for teaching (SKT).  There is a pressing need to design and test new approaches to developing SKT (more than just statistical knowledge).  This research is set in the context of a one- semester undergraduate statistics course I taught designed to build the SKT of prospective Pre-K-8 teachers.  Recommendations for statistics and probability in the Pre-K-8 curriculum point to the need to foster statistical knowledge for teaching (SKT).  There is a pressing need to design and test new approaches to developing SKT (more than just statistical knowledge).  This research is set in the context of a one- semester undergraduate statistics course I taught designed to build the SKT of prospective Pre-K-8 teachers.

3. SKT Conceptualization  Subject matter knowledge: common knowledge, specialized knowledge, and horizon knowledge.  Pedagogical content knowledge: knowledge of content and students, knowledge of content and teaching, and curriculum knowledge (Hill, Ball, & Schilling, 2008).  SKT is distinct from MKT because of the roles of variation and non-mathematical elements in statistics (Groth, 2007).  Subject matter knowledge: common knowledge, specialized knowledge, and horizon knowledge.  Pedagogical content knowledge: knowledge of content and students, knowledge of content and teaching, and curriculum knowledge (Hill, Ball, & Schilling, 2008).  SKT is distinct from MKT because of the roles of variation and non-mathematical elements in statistics (Groth, 2007). 3

4. Research Questions  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.  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. 4

5. Participants  Annie and Shelly each attended class regularly, submitted assignments, took tests, and participated in class as asked.  They had similar scores on a pre-test designed by the Learning Mathematics for Teaching Project. 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).  Annie and Shelly each attended class regularly, submitted assignments, took tests, and participated in class as asked.  They had similar scores on a pre-test designed by the Learning Mathematics for Teaching Project. 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). 5

6. Data Sources  Class examinations included probability and statistics items from past administrations of the National Assessment of Educational Progress (NAEP).  Participants completed 5 writing prompts for each of 12 assigned journal articles.  I designed additional examination items to probe specific difficulties in SKT development observed during class.  Class examinations included probability and statistics items from past administrations of the National Assessment of Educational Progress (NAEP).  Participants completed 5 writing prompts for each of 12 assigned journal articles.  I designed additional examination items to probe specific difficulties in SKT development observed during class. 6

7. Content Focus  Participants’ SKT was compared in regard to content that has been identified as problematic for teachers in past research:  Mean and median (Groth & Bergner, 2006)  Sampling (Groth & Bergner, 2005)  Categorical and quantitative data (Leavy, 2010)  Participants’ SKT was compared in regard to content that has been identified as problematic for teachers in past research:  Mean and median (Groth & Bergner, 2006)  Sampling (Groth & Bergner, 2005)  Categorical and quantitative data (Leavy, 2010) 7

8. Data Analysis  Data generated by the participants related to the identified content were used to produce Partially Correct Construct (PaCC) maps (Ron, Dreyfus, & Hershkowitz, 2010).  PaCC types:  missing element (in terms of SKT categories)  incompatible element  disconnected element  Participants’ PaCC maps were then compared and contrasted with one another.  Data generated by the participants related to the identified content were used to produce Partially Correct Construct (PaCC) maps (Ron, Dreyfus, & Hershkowitz, 2010).  PaCC types:  missing element (in terms of SKT categories)  incompatible element  disconnected element  Participants’ PaCC maps were then compared and contrasted with one another. 8

9. NAEP item on mean and median 9

10. Writing prompt item on PCK for choosing between mean and median  Explain why some students believe the mean is always a better indicator of typical value than the median. How might you convince these students that the median is more appropriate in some cases? 10

11. Mean and Median: Annie’s Map 11 Median is suitable for describing typical values in some data sets Mean is suitable for describing typical values in some data sets Outliers influence the value of the mean but not median Children sometimes favor the mean when it is not appropriate Would direct children to use median every time outliers exist Shelly Blue rectangles: Subject Matter Knowledge Green rectangles: Pedagogical Content Knowledge

12. NAEP item response - Annie  I would choose the median to describe the typical daily attendance for the 5 days at Theater A because there was a very low attendance on day 4 which brought the mean way down, but median is resistant to outliers.  I would choose the mean to describe the typical daily attendance for the 5 days at Theater B because there are no major outliers in the data and the mean takes into consideration all the days.  I would choose the median to describe the typical daily attendance for the 5 days at Theater A because there was a very low attendance on day 4 which brought the mean way down, but median is resistant to outliers.  I would choose the mean to describe the typical daily attendance for the 5 days at Theater B because there are no major outliers in the data and the mean takes into consideration all the days. 12

13. Writing prompt response - Annie  Some students may believe the mean is always a better indicator of typical value than the median because the mean does take into consideration ALL the data, so the students may think that it would be more accurate than just the middle number (the median). I would tell the students that although the mean is sometimes more accurate than the median, the mean will be skewed if the data contains an outlier, a number far away from the rest, and the median will not be affected by outliers. I would advise the students to use the median whenever there is an outlier in the data. 13

14. Mean and Median: Shelly’s Map 14 Median is suitable for describing typical values in some data sets Outliers influence the mean but not the median Mean is suitable for describing typical values in some data sets Children sometimes favor the mean when it is not appropriate Would direct children to use the median when there is a “long list of numbers” Blue rectangles: Subject Matter Knowledge Green rectangles: Pedagogical Content Knowledge Sampling

15. NAEP item response - Shelly  You would use the mean because you are trying to find the average number of people who attended for 5 days at Theater A and Theater B. You wouldn’t use the median because you aren’t finding the number of people in the middle of those 5 days. You would use the mean to describe the typical daily attendance for the 5 days at both Theatre A and Theater B. 15

16. Writing prompt response - Shelly  Some students believe that the mean is always a better indicator of typical value than the median because it has to do with finding the average. I could convince students that in some cases when you have a long list of numbers then that is when you find the median. 16

17. Sampling Test Item  A group of elementary school children wanted to determine the percentage of students in school who would like to have pizza in the cafeteria on Fridays. Students suggested several different sampling strategies. Critique each strategy described below:  Johnny wants to set up a booth outside of the cafeteria. In the booth, he will have a ballot on which students can vote for whether or not they want cafeteria pizza on Fridays. Students who wish to vote can come up to the booth and cast a ballot.  Patricia wants to walk through the hallways and select people to ask whether or not they want cafeteria pizza on Fridays. As she selects people, she will try to keep a balance between the number of boys and girls who are asked.  Suzie would like to distribute ballots to all of the members of the cooking club at the school. On the ballots, they will indicate whether or not they want to have cafeteria pizza on Fridays.  A group of elementary school children wanted to determine the percentage of students in school who would like to have pizza in the cafeteria on Fridays. Students suggested several different sampling strategies. Critique each strategy described below:  Johnny wants to set up a booth outside of the cafeteria. In the booth, he will have a ballot on which students can vote for whether or not they want cafeteria pizza on Fridays. Students who wish to vote can come up to the booth and cast a ballot.  Patricia wants to walk through the hallways and select people to ask whether or not they want cafeteria pizza on Fridays. As she selects people, she will try to keep a balance between the number of boys and girls who are asked.  Suzie would like to distribute ballots to all of the members of the cooking club at the school. On the ballots, they will indicate whether or not they want to have cafeteria pizza on Fridays. 17

18. Sampling Writing Prompts  Write three of your own original scenarios about sampling. The first should involve random sampling, the second should involve restricted sampling, and the third should involve self- selected sampling. Explain why each scenario fits each category.  In 200-250 words, describe a general strategy you would use for teaching young students about survey sampling and how you would assess their understanding. Then provide a rationale for your general strategy.  Write three of your own original scenarios about sampling. The first should involve random sampling, the second should involve restricted sampling, and the third should involve self- selected sampling. Explain why each scenario fits each category.  In 200-250 words, describe a general strategy you would use for teaching young students about survey sampling and how you would assess their understanding. Then provide a rationale for your general strategy. 18

19. Sampling – Annie’s Map 19 Distinguishes random sampling from subjective sampling Sees restricted sampling as inclusive of only a specific segment of the population Sees self-selected sampling as limited because only those interested in the issue participate Identifies limitations of a child’s decision to use subjective sampling Identifies limitations of a child’s decision to use restricted sampling Identifies limitations of a child’s decision to use self-selected sampling Proposes lecture and discussion as a means of teaching sampling Blue rectangles: Subject Matter Knowledge Green rectangles: Pedagogical Content Knowledge Blue-green rectangles: Possible blending of specialized and KCS Shelly

20. Test item response part A - Annie  Johnny’s strategy [self-selected sampling] is not the best because he is only asking people who want to be involved in the survey so he might only select people from one particular group. For example, students who really want pizza may be more likely to take the survey than people who don’t want pizza. Also, maybe not even every student goes to lunch in the cafeteria. 20

21. Test item response part B - Annie  Patricia’s strategy [subjective sampling] is not the best either because it is not random sampling. She will not get the random group she wants because she is selecting the students which makes the selection bias even if she does not mean for it to be. Also, not every student is going to be walking in the hallway so those students have no way of being selected. 21

22. Test item response part C - Annie  Suzie’s strategy [restricted sampling] is not good because the point of the survey is not to determine the percent of students in the cooking club who want pizza on Friday. It is the percent of the entire school. Suzie’s strategy is selective and bias when a good strategy is random and unbias. 22

23. Teaching sampling - Annie  First, before ever exampling sampling I would ask the students to raise their hand if they think they know what a sample is. Then I would pick on a few or all the students who raised their hand (depending on how many there were). From that I would get a feel of what some of the students already know or think they know about sampling. After that, I would give them a definition of sampling (one that they would understand). Then, I would probably talk about survey sampling specifically and ask if anyone has ever taken or conducted a survey and what they thought about it (if they liked it or not). By asking the students questions I will be keeping them involved, helping them realized that they already know some things about this topic. (response continues)… 23

24. Sampling – Shelly’s Map 24 Distinguishes random sampling from subjective sampling Sees restricted sampling as inclusive of only a specific segment of the population Identifies limitations of a child’s decision to use restricted sampling Proposes lecture as a means of teaching sampling Subjective sampling believed to be limited only because it does not include entire population Believes self-selected sampling to be when a survey-taker decides when to stop accepting responses Finds no weaknesses in a child’s choice to use self-selected sampling Blue rectangles: Subject Matter Knowledge Green rectangles: Pedagogical Content Knowledge Blue-green rectangles: Possible blending of specialized and KCS Categorical data

25. Test item response part A - Shelly  I think this is a good strategy [self-selected sampling] because he will catch everyone who eats in the cafeteria. If they want to vote, they can, but if they don’t then they have that choice. 25

26. Test item response part B - Shelly  I don’t think this strategy [subjective sampling] is the best because she may not be able to get everyone’s votes. Also, not everyone may like or eat pizza. 26

27. Test item response part C - Shelly  This isn’t a good strategy [restricted sampling] because Suzie is just getting the opinions from one particular group of people. What about all of the other students who aren’t in the cooking club. 27

28. Teaching sampling - Shelly  I would teach the young students about the random, restricted, and self-selected sampling by going over the vocabulary and relating examples to real life events. I believe that if you relate things to everyday life the students can understand it better then if you were to make up an event. I would explain that in random sampling methods, everyone has an equal chance of being selected. I would then tell them that if they were to work on a project with a group of people. If they chose names out of a hat to find out their partners, that would be random sampling because everyone has the same chance of being selected. I would then explain to them that restricted sampling methods is when a specific population is asked and that this method could make the results bias…(continues) 28

29. Categorical data item 29

30. Categorical data item 30

31. Annie’s Categorical Data PaCC 31 Category frequencies are not quantitative data Nominal categorical data cannot be ordered Mode is suitable for nominal categorical data Mean is not suitable for nominal categorical data Median is not suitable for nominal categorical data A common student error is to compute mean of frequencies A common student error is to attempt to order categories in nominal data to determine a median A common student error is to compute median of frequencies Employs teaching strategy of attempting to directly transmit knowledge to students in class Blue rectangles: Subject Matter Knowledge Green rectangles: Pedagogical Content Knowledge Blue-green rectangles: Possible blending of specialized and KCS

32. Shelly’s categorical data PaCC 32 Mode is suitable for nominal categorical data Mean is not suitable for nominal categorical data Median is not suitable for nominal categorical data Category frequencies are not quantitative data Nominal categorical data cannot be ordered Students are correct to compute the mean of frequencies for nominal categorical data Students are correct to put frequencies in order to determine the median for nominal categorical data Transmission- oriented teaching strategies that are likely to mislead students about content Blue rectangles: Subject Matter Knowledge Green rectangles: Pedagogical Content Knowledge Blue-green rectangles: Possible blending of specialized and KCS

33. Differences between the cases: PaCC types  Missing, conflicting, and unintended knowledge elements were more prevalent in Shelly’s thinking than in Annie’s  Conjecture to be tested: Participants’ metacognition may be sparked if asked to analyze PaCC maps that represent their thinking alongside artifacts they produce during a course.  Missing, conflicting, and unintended knowledge elements were more prevalent in Shelly’s thinking than in Annie’s  Conjecture to be tested: Participants’ metacognition may be sparked if asked to analyze PaCC maps that represent their thinking alongside artifacts they produce during a course. 33

34. Similarities between the cases: Disconnected element PaCCs  Both participants had some disconnected element PaCCs – where relevant subject matter knowledge and PCK were not used to build viable teaching strategies.  Conjecture to be tested: These types of disconnected element PaCCs are most resistant to change because they are rooted in the transmission-oriented teaching culture in the U.S (Jacobs et al., 2006).  Both participants had some disconnected element PaCCs – where relevant subject matter knowledge and PCK were not used to build viable teaching strategies.  Conjecture to be tested: These types of disconnected element PaCCs are most resistant to change because they are rooted in the transmission-oriented teaching culture in the U.S (Jacobs et al., 2006). 34

35. Next steps for Annie and Shelly  Annie’s needed next steps as a learner can be described in terms of “decentering” (Silverman & Thompson, 2008) – a need to translate personal content knowledge to “pedagogically powerful ideas.”  Shelly’s needed next steps as a learner necessitate developing deeper subject matter knowledge – as this knowledge aspect caused her to devise strategies likely to mislead students.  Annie’s needed next steps as a learner can be described in terms of “decentering” (Silverman & Thompson, 2008) – a need to translate personal content knowledge to “pedagogically powerful ideas.”  Shelly’s needed next steps as a learner necessitate developing deeper subject matter knowledge – as this knowledge aspect caused her to devise strategies likely to mislead students. 35

36. Potential discussion questions  Aside from PaCCs, what other theoretical tools should be employed to characterize the structure and development of SKT?  What additional data collection techniques and instruments could be used to investigate SKT?  Aside from PaCCs, what other theoretical tools should be employed to characterize the structure and development of SKT?  What additional data collection techniques and instruments could be used to investigate SKT? 36

37. Potential discussion questions  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?  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? 37

38. References  Groth, R.E. (2007). Toward a conceptualization of statistical knowledge for teaching. Journal for Research in Mathematics Education, 38, 427-437.  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., & 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. (2007). Toward a conceptualization of statistical knowledge for teaching. Journal for Research in Mathematics Education, 38, 427-437.  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., & Bergner, J.A. (2005). Preservice elementary school teachers' metaphors for the concept of statistical sample. Statistics Education Research Journal, 4 (2), 27- 42. 38

39. References  Hill, H.C., Ball, D.L., & Schilling, S.G. (2008). Unpacking pedagogical content knowledge: Conceptualizing and measuring teachers’ topic-specific knowledge of students. Journal for Research in Mathematics Education, 39, 372- 400.  Jacobs, J.K., Hiebert, J., Givvin, K., Hollingsworth, H., Garnier, H., & Wearne, D. (2006). Does eighth-grade teaching in the United States align with the NCTM Standards? Results from the TIMSS 1995 and 1999 video studies. Journal for Research in Mathematics Education, 37, 5-32.  Hill, H.C., Ball, D.L., & Schilling, S.G. (2008). Unpacking pedagogical content knowledge: Conceptualizing and measuring teachers’ topic-specific knowledge of students. Journal for Research in Mathematics Education, 39, 372- 400.  Jacobs, J.K., Hiebert, J., Givvin, K., Hollingsworth, H., Garnier, H., & Wearne, D. (2006). Does eighth-grade teaching in the United States align with the NCTM Standards? Results from the TIMSS 1995 and 1999 video studies. Journal for Research in Mathematics Education, 37, 5-32. 39

40. References  Leavy, A.M. (2010). The challenge of preparing preservice teachers to teach informal inferential reasoning. Statistics Education Research Journal, 9(1), 46-67.  Ron, G., Dreyfus, T., & Hershkowitz, R. (2010). Partially correct constructs illuminate students’ inconsistent answers. Educational Studies in Mathematics, 75, 65-87.  Silverman, J., & Thompson, P.W. (2008). Toward a framework for the development of mathematical knowledge for teaching. Journal of Mathematics Teacher Education, 11, 499-511.  Leavy, A.M. (2010). The challenge of preparing preservice teachers to teach informal inferential reasoning. Statistics Education Research Journal, 9(1), 46-67.  Ron, G., Dreyfus, T., & Hershkowitz, R. (2010). Partially correct constructs illuminate students’ inconsistent answers. Educational Studies in Mathematics, 75, 65-87.  Silverman, J., & Thompson, P.W. (2008). Toward a framework for the development of mathematical knowledge for teaching. Journal of Mathematics Teacher Education, 11, 499-511. 40