2. Does Theory Improve Practice? Can Participation in Research Based Workshops Improve Teachers’ Practice? Ronith Klein Kibbutzim College of Education, Tel Aviv This work is part of a doctorial dissertation which was carried out at Tel-Aviv University under the supervision of Prof. Dina Tirosh

3. A major goal of teacher education programs is to promote prospective and inservice teachers' knowledge of children's ways of thinking about the mathematic topics they are to teach. for example: The CGI program- Cognitively Guided Instruction:

4. Knowledge of students' ways of thinking can dramatically modify teachers' instructional methodologies The CGI program explored the impact of teachers ’ knowledge of students ’ thinking on actual teaching practices. Results show that:

5. Observed changes in teachers' practices are attributed to: (a) The participants learned the specific research-based model that formed the basis of the teacher development program (b) they received ongoing support while using this model in their classrooms

6. In our study we designed a workshop that was specifically designed for enhancing inservice teachers' knowledge of Students' ways of Thinking About Rational numbers. This workshop focused on students ’ misconceptions, possible incorrect responses and their sources and presented theories and research findings concerning students ’ ways of thinking.

7. Questions: 1)How does participation in a workshop that focuses on children's conceptions and misconceptions related to specific mathematics topics (with no support system in classrooms) influence inservice teachers’ knowledge of children's thinking about this topic? Our topic: multiplication and division word problems with rational numbers

8. 2) Do teachers take their knowledge of what children know and understand into account when designing and teaching specific topics? We examine the effects of such a teachers’ in-service program on lesson plans written by the teachers as an indication to possible changes in their teaching.

9. The Workshop (STAR) Students’ Ways of Thinking About Rational numbers 14 meetings of 4 hours each Participants: 14 experienced teachers with at least five years of teaching experience

10. All teachers, except one, taught mathematics in the fifth and sixth grades. Ten of the teachers graduated from teacher education colleges; four had university degrees.

11. Topics: Children's conceptions of rational numbers. Comparing decimals. Incorrect algorithms- calculating addition, subtraction, multiplication and division expressions.

12. Incorrect responses to multiplication and division word problems Theories related to students’ ways of thinking about rational numbers General teaching approaches for enhancing conceptual changes

13. A thorough discussion on lesson plans written by: 1) Teachers aware of students’ mistakes and took them into account. 2) Teachers not considering students’ ways of thinking in their planning. The participants examined the extent to which teachers' awareness of students' ways of thinking was reflected in the lesson plans.

14. The workshop was specifically designed for enhancing inservice teachers' pedagogic content knowledge- Knowledge of students' ways of thinking about rational numbers.

15. We describe the effect of the workshop on participants’ pedagogic content knowledge – Knowledge of students' ways of thinking about rational numbers and on the design of lesson plans on multiplication and division word problems involving rational numbers.

16. Data Collection and Procedure: A Diagnostic Questionnaire (DQ)- before and after the workshop Lesson plans before and after Interviews worksheets Workshops observations

17. The Diagnostic Questionnaire – Evaluated the changes in participants' Subject Matter Knowledge (SMK) of rational numbers and their knowledge of children's conceptions and misconceptions of rational numbers (PCK). Comparison of and operations with rational numbers Multiplication and division word problems involving rational numbers Beliefs about multiplication and division with rational numbers.

18. A typical item asked the teacher to: Solve the problem List common incorrect responses List possible sources to the incorrect responses. At the end of the course, participants responded to a modified version of the DQ, which was individually tailored for each teacher. Each teacher received the items he/she answered incorrectly on the questionnaire given at the first meeting.

19. Lesson plans- multiplication and division word problems- rational numbers: Main aims of the teaching unit Students' prior mathematical knowledge Teaching methods Manipulatives Lesson plan for one lesson How to teach students to solve a specific, given word problems.

20. The Teaching Unit PSYCHOLOGICAL ASPECTS OF MATHEMATICS TEACHING

21. Results Before Can compute- Can not give explanations Aware of typical mistakes- only on comparing fraction Knowledge of sources of students’ mistakes only on comparing fractions After Improvement in knowledge “why” Awareness of students’ typical mistakes Knowledge of sources of students’ mistakes

22. For example: Before the course, teachers could not explain why in division of fractions “ we invert and multiply ”. After STAR, SMK and PCK of all participants improved. The main improvement was observed on knowing “ why ” : Most teachers could explain the various steps of the standard algorithms.

23. Participants developed their lesson plans in three groups: one with four teachers and two of two teachers each. We shall describe, for each group, teachers ’ lesson plans before and after STAR, emphasizing observed changes in their consideration of students ’ ways of thinking.

24. Group one: from referring only to substitution mistakes to structured dealing with possible mistakes before they appear Before the course Using mapping tables to solve word problems ” S Students will only have problems placing numbers in the mapping tables "

25. Group one Behavioristic philosophy of learning Teaching an algorithm, “a magic rule”, will yield no students’ mistakes Good teaching- teaching the “right” procedures Lots of drill and practice Good teacher and good teaching  Students make no mistakes

26. “ No mistakes are expected when using mapping tables, the student will choose the right expression when solving the problem”. “ This procedure always works” it becomes technical and automatic, and there are no problems ”.

27. After the course: Group one- pair one Students’ intuitive beliefs were considered You have to tell students that “multiplication does not make bigger, division does not make smaller”.

28. After the course I am trying to understand why they answered as they did. Before, I never thought how integers affect fraction learning. I think I won’t emphasize that “multiplication makes bigger”, in lower grades Before, I never thought of students ’ mistakes and why they do them. Now it is interesting

29. After the course: Group one- pair two Students’ intuitive beliefs were considered Built structured activities in order to make students aware that “multiplication does not make bigger, division does not make smaller”.

30. The Excel graph for the following expressions: 1) 0.2x0.3=0.06 2) 3.2x0.3=0.96 The multiplication is The multiplication is smaller than each smaller than multiplier multiplier 1

32. Teachers are aware of and consider students’ possible errors Students make mistakes despite good teachers and good teaching Try to get to the source of mistakes Build the same structured activities for all students

33. “ We did not consider students’ mistakes ” Group two: from no reference to students’ mistakes to dealing with possible mistakes only if and when they appear Before the course Referred to different types of division word problems in their planning : Partitive division - Measurement division “Finding a part- when given the whole” “Finding the whole -when given a part”.

34. After the course Differentiated between high achieving, intermediate and low achieving students. Planned the lesson mostly for the intermediate group. High achievers learn without instruction, No instruction can help low achievers

35. Distinguished between easy and difficult word problems Considered possible student responses Were aware of students’ misconceptions Built the lessons on students’ answers Dealt with possible mistakes as they arise

36. Group three: from no reference to students’ mistakes to awareness of possible mistakes Before the course Sequenced activities from easy to difficult ones “Finding a part- when given the whole” “Finding the whole -when given a part”

37. After the course Were acquainted with the intuitive models of multiplication and division Believe in teaching for understanding Were aware of errors students are likely to make, but didn’t plan the lesson accordingly Were not sure how to deal with students’ possible mistakes

38. Before the course only a few written references to possible, common incorrect students’ responses were made in lesson plans. We hypothesized that after STAR teachers would adjust their lesson plans, taking account of common, systematic students’ conceptions and misconceptions when planning their instruction.

39. The study showed : Improvement in teachers’ mathematical and pedagocical content knowledge- Especially “knowledge why” All teachers were aware of students’ common errors but the ways and the extend to which they considered them varied.

40. Group one: Teachers believed in subject oriented instruction Teachers had university degrees in mathematics In favor of teaching algorithms Built structured activities, taking students’ ways of thinking into account Their lesson plans after STAR showed the biggest change.

41. Group two: Believed in students driven instruction Dealt with possible mistakes as they arose Group three: Believed in students driven instruction Aware of students’ mistakes No change in lesson plans

42. Participation in a course that focuses on children's ways of thinking emphasizes the importance of pedagogical content knowledge and its implementation in actual teaching, can improve in-service teachers’ mathematical content knowledge (“that” and “why”). It can also convince teachers of the importance of considering students’ ways of thinking in teaching and can effect the lesson plans written by the participating teachers.

43. Teachers beliefs about teaching are critical factors which determine their teaching (Lerman, 1999). Research shows that change in teachers’ knowledge without change in teachers’ beliefs is not significant (Fennema, Carpenter, Franke, Levi, Jacobs and Empson, 1996).

44. Our data show that teachers’ beliefs are an important factor that influence changes in teachers’ lesson plans.

45. The course had an impact mainly on teachers with substancial mathematical knowledge. Mathematical knowledge is very important for planning instruction.

46. Recommendation: Mathematics in elementary school should be taught by teachers specializing in mathematics teaching. Pre-service education programs for these teachers will emphasize mathematics and mathematics instruction for the purpose of enhancing the teachers’ mathematical knowledge.

47. In-service programs that focus on students' ways of thinking should be developed, implemented and evaluated.

48. We suggest that: More research is needed for examining the effect of mathematical knowledge on other aspects of teaching.

49. Our small sample included only three groups of teachers. We suggest that further studies with a larger numbers of participants be conducted to investigate the impact of inservice programs that focus on students’ ways of thinking on teachers’ lesson plans and on teachers’ actual teaching practices.

50. Author: Dr. Ronith Klein Director of Computer Applications in Education Kibbutzim College of Education, Tel Aviv E-mail: ronitk@macam.ac.il