1. Reflecting on the Practice of Teaching
PCMI
Secondary School Teachers Program
July 2007

2. deLange, et al, 1993

3. The Teaching Principle
Effective teaching requires understanding what students know and need to learn and challenging and supporting them to learn it well. (NCTM, 2000)

4. As teacher learned to Choose tasks carefully
Listen to students(’) work
Students are often smarter than I am
Manage student responses carefully
Take risks
Let students do the work

5. Learning from Experience
Using new technology
Working with Japanese Colleagues
Developing Curriculum
Conducting Demonstration Lessons
Working with Preservice Students
Designing Professional Development

6. Learning from Experience
Working with Japanese Colleagues

7. Typical flow of a class United States Demonstrate a procedure
Assign similar problems to students as exercises
Homework assignment
Japan
Present a problem to the students without first demonstrating how to solve the problem
Individual or group problem solving
Compare and discuss multiple solution methods
Summary, exercises and homework assignment
Takahashi, 2005

8. The Lesson
Introduction: Hatsumon Thought provoking question Key question – shu hatsumon Individual or small group work Walking among the desks – kikan-shido Anticipated student solutions Student solutions - Noriage Massaging students’ ideas Summing up- Matome
In your opening, establish the relevancy of the topic to the audience. Give a brief preview of the presentation and establish value for the listeners. Take into account your audience’s interest and expertise in the topic when choosing your vocabulary, examples, and illustrations. Focus on the importance of the topic to your audience, and you will have more attentive listeners.
Bass et al, 2002

9. Which shape will hold the same amount of spaghetti and be the most economical?
Area of base
Surface area
Volume
Ratio of surface area to volume
Cylinder
Rectangular prism
Shape 3
MDoE, 2003

10. Learned the importance of
Being explicit about the math students are to learn
Anticipating student solutions
Lesson plan
Starting investigations with a “launch” that invites students into the math ……

11. Teaching means having “eyes” to see the mathematics
Can teacher identify the mathematical essential points of materials?
Does teacher deprive students’ of the opportunity to think mathematically?
Ikeda & Kuwahara, 2002

12. And “eyes” to see the students
Can teacher understand what students understand?
・Can students understand teacher’s asking questions?
・Does teacher ignore students’ ideas by his/her selfish reason?
・ Can teacher accept and evaluate students’ ideas appropriately?
・ Can students discuss cooperatively?
Ikeda & Kuwahara, 2002

13. Learning from Experience
Conducting Demonstration Lessons

14. Pencils cost 15 cents Erasers cost 25 Cents
How many pencils and erasers can you buy for $1.10?
For $1.50?
Kindt, et al, 1997

15. Number of pencils 15 40 25 3 2 1 1 2 3 4
Number of erasers

16. Number of pencils
155
115
140
165
100
125
150
175
200 60 85
110
135
160
185
210 45 70 95
120
145
170
195 30 55 80
105
130
180 15 40 65 90
190 25 50 75 3 2 1 1 2 3 4
Number of erasers

17. Number of pencils 15 40 25 3 2 1 1 2 3 4
Number of erasers

18. Number of pencils 3 2 1 1 2 3 4
Number of erasers

19. Preactivities leading to main goal
Scaffolding matters
Preactivities leading to main goal
1x25
2/25
3x25
4x25
5x25
1x15
2x15
4x15
5x15

20. What and how the work is recorded matters
2x x3
x2 x3
30 75
2x15 + 3x25

21. What and how the work is recorded matters
2x15 + 3x25 = = 105
3x15 + 2x25 = = 95
4x15 + 2x25 = = 110
x2 x3
30 75
Goal: Ax+By = C

22. Learned to deliberately think about:
How will students work?
What tools will be useful and how should they be made available?
How will the work be recorded?
How will they share their work?
How will I know what the students understand and do not understand?

23. Learning from Experience
Working with Preservice Students

24. Expect the Unexpected

25. Learned that Boards are disappearing
Modeling is not enough; need to be explicit
Preservice students are not really aware that others have different ways of thinking
Difficult to honor mistakes

26. Learning from Experience
Developing Curriculum

27. In the figure below, what fraction of the rectangle ABCD is shaded?
1/6
1/5
1/4
1/3
e) 1/2 C D
NCES, 1996

28. Dekker & Querele, 2002

29. Dekker & Querele, 2002

30. Comparing Quantities. Kindt et al, 2006

31. Learned to Pose tasks that go beyond routines
Ask what would happen if…? What should you do if you want…..
Frame a situation and let students comment
Collaborative work is better than individual - in doing math and in thinking about lessons

32. Teaching is a profession with a body of knowledge that can be learned and applied to improve the practice of enabling students to learn.

33. Research in mathematics education
Experimental-observation
Theories of learning - frameworks for thinking about teaching and learning
Quantitative Studies -experimental
-quasi-experimental
Qualitative Studies
--Case studies
--Ethnographic studies

34. Research findings Peer reviewed journals Synthesis of the literature
Nature of conclusions
-suggestions
-insights
-causal
Meta-analysis

35. Other sources of information
Visions - projections of what might/should be possible
Information from colleagues
Doctoral theses
Professional organizations
Lecture notes
Exhortions
Beliefs

36. Teaching involves Choosing and setting up tasks
Adaptation/modification
Implementation
Response to student questions
Discussion
Manage solution strategies
Probing for understanding
Evidence of learning

37. Our Work Formative Assessment Cognitive Demand/Scaffolding
Discussion/Questioning
Transfer/Learning for Understanding

38. Research Report Describe what the topic means and why it is important
Give three or four key findings and their relevance for teaching

39. Reflect on your own teaching
What are some questions you have?
What are one or two things about your teaching you would like to improve?
What would you like to learn about teaching?

40. Teaching is harder than it looks - making students come to life in the world of mathematics.
But we can learn not only from our own experience and that of our colleagues but from the research that helps explain and provides insights into teaching and learning math

41. References
Bass, H., Usiskin, Z, & Burrill, G. (Eds.) (2002). Classroom Practice as a Medium for Professional Development. Washington, DC: National Academy Press.
Dekker,T. & Querelle, N. (2002). Great assessment problems (and how to solve them). CATCH project
deLange, J., Romberg, T., Burrill, G., von Reeuwijk, M. (1993). Learning and testing mathematics in context: Data visualization. Los Angles CA:, Sunburst.
Ikeda, T. & Kuwahara, Y. (2003). Presentation at Park City Mathematics Institute International Panel.
Kindt, M., Abels, M., Meyer, M., Pligge, M. (1998). Comparing Quantities. From Mathematics in Context. Directed by Romberg, T. & deLange, J. Austin, TX: Holt, Rinehart, Winston
Michigan Department of Education. (2003). MMLA Lesson Study Project. Burrill, G., Ferry, D., & Verhey R. (Eds). Lansing, MI
National Council of Teachers of Mathematics. (2000). Principles and Standards for School Mathematics. Reston, VA: Author.

42. Takahashi, Akahito. (2005). Presentation at Annual Meeting of Association of Mathematics Teacher Educators.
Teachers for a New Era (2003). Michigan State University grant from Carnegie Foundation.
Third International Mathematics and Science Study (TIMSS). (1995). Released Item. National Center for Education Statistics. U.S. Department of Education. (1999).