Facilitating Knowledge Development and Refinement in Elementary School Teachers

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

2. SEMT 2011

3. Teacher Education Context SEMT 2011 • 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

4. Teacher Education Program SEMT 2011 • 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

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

6. SEMT 2011

7. Goal SEMT 2011 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/

8. Mathematical Knowledge for Teaching SEMT 2011 • 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

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

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

11. A recent study of MKT SEMT 2011 • 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

12. SEMT 2011

13. SEMT 2011 • 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

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 SEMT 2011 Solve the task Hypothesize errors and causes of errors Propose instructional strategies to address the causes of the errors

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 SEMT 2011 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 student’s faulty procedural knowledge.

25. Findings, Cont’d. SEMT 2011 • 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.

26. Implication for teacher education SEMT 2011 • 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

27. Task Dialogues SEMT 2011 • 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?

28. Seeing the child through the mathematics SEMT 2011

29. Seeing the mathematics through the child SEMT 2011

30. Field Experience SEMT 2011 • 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 Hallman’s class: Do the task 3 weeks in a row with 3 different children to build MKT (study in progress)

31. Example Task SEMT 2011 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?

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

33. PST A dialogue excerpt SEMT 2011 Why don’t 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.

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

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

36. PST A dialogue excerpt SEMT 2011 T: Let’s 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?

37. PST B dialogue excerpt SEMT 2011 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 let’s 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 let’s 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?

38. Response 3 SEMT 2011 There will be 15 games: AB BC CD DE EF AC BD CE DF AD BE CF AE BF AF

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

40. PST B dialogue excerpt SEMT 2011 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?

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

42. Observations, cont’d. SEMT 2011 • 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

43. Conclusion SEMT 2011 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!)