1. Facilitating Knowledge Development and Refinement in Elementary School Teachers Denise A. Spangler University of Georgia USA SEMT 2011

3. Teacher Education Context Elementary in the US means different things in different places. K-5 (ages 5-11) K-6 (ages 5-12) K-8 (ages 5-14) Rarely K-9 (kindergarten through first year of high school) Increasingly includes prekindergarten (PreK = age 4) University of Georgia: PreK-5 SEMT 2011

4. Teacher Education Program All courses are 3 semester hours or 45 contact hours 2 years liberal arts curriculum–60 semester hours 2 mathematics classes (modeling, precalculus, statistics, calculus) 1 mathematics course for elementary teachers (number and operations) Special education, foundations of education, educational psychology, diversity/equity SEMT 2011

5. TE Program, continued 2 years of the teacher education program 2 mathematics courses for elementary teachers Geometry, measurement Algebra, statistics 2 mathematics methods courses Childrens mathematical thinking with respect to numbers and operations (whole & rational) Curriculum, assessment, teaching of other content areas (geometry, measurement, algebra…) SEMT 2011

6. First semester ECE Community experience Math content (geometry/ measurement) Math methods Second semester ECE ECE field experience Math methods Reading methods Language arts methods Social studies methods Third semester ECE ECE field experiences Math content (algebra) Reading methods Language arts methods Science methods SEMT 2011

7. Goal Examine where the mathematical knowledge needed for teaching in elementary schools comes into play. Look at an effort to assess it in preservice upper elementary teachers. Look at an effort to develop it in preservice elementary teachers/ SEMT 2011

8. Mathematical Knowledge for Teaching Knowing mathematics to pass a test knowing mathematics in the ways needed to teach it. Teaching mathematics involves knowing Representations Analogies Illustrations Examples Explanations Demonstrations Shulman, 1986 SEMT 2011

9. And also Knowing What makes a topic easy or hard Students preconceptions and misconceptions Strategies to address misconceptions Shulman, 1986 SEMT 2011

10. MKT as defined by Ball, Thames & Phelps 2008 SEMT 2011

11. A recent study of MKT Conducted by Jisun Kim, University of Georgia, 2011 Preservice teachers of mathematics, grades 4-8 Calculus I Numbers & Operations Content course and methods course Geometry & Measurement Content course and methods course During study Goal: investigate multiple aspects of MKT SEMT 2011

13. 5 geometry/measurement tasks, given one at a time over a semester Preservice teachers had to Solve the task Examine 4-5 student solutions (from research projects) to determine if they were correct Identify causes of errors Propose instructional strategies to address the causes of the errors SEMT 2011

14. Area of triangles SEMT 2011

15. Types of Triangles SEMT 2011

16. Similar Figures SEMT 2011

17. Volume SEMT 2011

18. Area/Perimeter SEMT 2011

19. Your turn Solve the task Hypothesize errors and causes of errors Propose instructional strategies to address the causes of the errors SEMT 2011

20. Student response 1 SEMT 2011

21. Student response 2 SEMT 2011

22. Student response 3 SEMT 2011

23. Student response 4 SEMT 2011

24. Findings of the study Preservice teachers generally got the correct answer themselves, but in some cases they exhibited the same misconceptions as the students even though the topic had been addressed in a course. focused on the answer, not the solution path. (Students 3 & 4) attributed errors mostly to students faulty procedural knowledge. SEMT 2011

25. Findings, Contd. Diagnoses did not match prescriptions. Example Diagnosis: only thinking about squares Prescription: review area formula Small, weak repertoire of instructional strategies; often wanted to tell students the correct answer/formula. Used examples from class for instructional strategies. SEMT 2011

26. Implication for teacher education Focus on planning but shift Away from lesson planning Toward task planning Task dialogues (Crespo, Oslund, & Parks, 2011) Create plausible teacher/student conversation Equal sign a balance point vs. do something 5 + 3 = _____ + 7 Common student answer is 8 SEMT 2011

27. Task Dialogues Task–we solve it in class and discuss multiple solution strategies I give them possible student solutions 1 correct 2-3 incorrect or incomplete What mathematical thinking could be behind that response? What question could I ask next to test whether or not that is what the child was thinking? How would the child respond if it was or was not what she was thinking? What is my next move? SEMT 2011

28. Seeing the child through the mathematics SEMT 2011

29. Seeing the mathematics through the child SEMT 2011

30. Field Experience They do the task dialogue task with children They also do other tasks that day, and in their plans THEY have to posit student responses Two formats My class: Do the task one week with one child Allyson Hallmans class: Do the task 3 weeks in a row with 3 different children to build MKT (study in progress) SEMT 2011

31. Example Task In a soccer championship there are 6 teams. If all teams are going to play each other, how many games will there be in the championship? SEMT 2011

32. Response 1 There will be 3 games because Team A will play Team B Team C will play Team D Team E will play Team F. SEMT 2011

33. PST A dialogue excerpt Why dont you set up your diagram like this: A B C D E F And now make sure team A plays all the teams. Team B, team C, and team D also play all the teams. SEMT 2011

34. PST B dialogue excerpt You have the right idea by going in order, but maybe you should try looking at just one team at a time so that its easier to see the number of games they play. SEMT 2011

35. Response 2 There will be 30 games because each team plays 5 other teams. There are 6 teams so 5 + 5 + 5 + 5 + 5 + 5 = 30. SEMT 2011

36. PST A dialogue excerpt T: Lets draw out your diagram. S: Okay. T: Now, if each team can only play each team one time does your diagram still work? S: Yes. T: Do you agree that Team 1 vs. Team 2 is the same as Team 2 vs. Team 1? SEMT 2011

37. PST B dialogue excerpt T: Do you think you could draw a picture to show that? S: Yeah, I could. It would look like this [draws picture]. T: You did a great job on your math and adding correctly. But lets look back at what a tournament means. S: It means every team plays all the other teams. T: Good, but there was one other part of it, too. Do you remember? S: Oh, yeah. They only play each other once. T: So lets take a look at your picture. Did we stick to the rules of that kind of tournament? S: (after looking at the picture) No. There are repeats. T: So, you definitely have the right idea with the way you solved the problem. What do you think we could do to solve the issue with the repeats? SEMT 2011

38. Response 3 There will be 15 games: ABBCCDDEEF ACBDCEDF ADBECF AEBF AF SEMT 2011

39. PST A dialogue excerpt Can you show me another way of getting there? SEMT 2011

40. PST B dialogue excerpt What if there were 10 teams? What about 20 teams? Do you see a pattern in the solutions for 6 teams and 10 teams that would help you solve 20 teams without writing them all out? What if I told you there were 45 games. How many teams would that be? SEMT 2011

41. Observations PST with lower content knowledge tend to Have difficulty seeing childrens mathematical thinking, especially when its different from their own Assume they know what children are thinking and do not ask Push children to do it their way (the PSTs way) Ask bite-sized questions, leading/directive questions Start over rather than building from existing ideas Dont push on correct answers Dont make an effort to connect solution strategies SEMT 2011

42. Observations, contd. PST with higher content knowledge tend to Ask more open questions Try to get students to figure things out for themselves Push students to analyze their solutions and go on from there rather than starting over Pay attention to process as much as final answer Link solution strategies Extend correct solutions to push for generalizations SEMT 2011

43. Conclusion Focusing on preservice or inservice teachers content knowledge is necessary but not sufficient. Need to develop OUR repertoire of tasks/activities to tap into the application of that content knowledge (and study them!) SEMT 2011