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Reflecting on the Practice of Teaching. PCMI Secondary School Teachers Program July 2007. deLange, et al, 1993. 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).

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reflecting on the practice of teaching

Reflecting on the Practice of Teaching

PCMI

Secondary School Teachers Program

July 2007

the teaching principle

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)

as teacher learned to

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

learning from experience
Learning from Experience

Using new technology

Working with Japanese Colleagues

Developing Curriculum

Conducting Demonstration Lessons

Working with Preservice Students

Designing Professional Development

learning from experience1
Learning from Experience

Working with Japanese Colleagues

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

Typical flow of a class

Takahashi, 2005

slide8

The Lesson

Introduction: HatsumonThought provoking questionKey question – shu hatsumonIndividual or small group work Walking among the desks – kikan-shido Anticipated student solutionsStudent solutions - NoriageMassaging students’ ideas Summing up- Matome

Bass et al, 2002

learned the importance of
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
  • ……
teaching means having eyes to see the mathematics
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

and eyes to see the students
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

learning from experience2
Learning from Experience

Conducting Demonstration Lessons

pencils cost 15 cents erasers cost 25 cents

Pencils cost 15 centsErasers cost 25 Cents

How many pencils and erasers can you buy for $1.10?

For $1.50?

Kindt, et al, 1997

slide15

Number of pencils

3

2

1

0

0

1

2

3

4

Number of erasers

slide16

Number of pencils

3

2

1

0

0

1

2

3

4

Number of erasers

slide17

Number of pencils

3

2

1

0

0

1

2

3

4

Number of erasers

slide18

Number of pencils

3

2

1

0

0

1

2

3

4

Number of erasers

scaffolding matters

Scaffolding matters

1x25

2/25

3x25

4x25

5x25

1x15

2x15

2x15

4x15

5x15

Preactivities leading to main goal

what and how the work is recorded matters

What and how the work is recorded matters

2x 15 + 25x3

15 25

x2x3

30 75

2x15 + 3x25

what and how the work is recorded matters1

What and how the work is recorded matters

2x15 + 3x25 = 30+ 75 = 105

3x15 + 2x25 = 45 + 50 = 95

4x15 + 2x25 = 60 + 50 = 110

15 25

x2x3

30 75

Goal: Ax+By = C

learned to deliberately think about
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?
learning from experience3
Learning from Experience

Working with Preservice Students

learned that
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
learning from experience4
Learning from Experience

Developing Curriculum

in the figure below what fraction of the rectangle abcd is shaded
In the figure below, what fraction of the rectangle ABCD is shaded?

A

B

1/6

1/5

1/4

1/3

e) 1/2

C

D

NCES, 1996

learned to
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
slide32
Teaching is a profession with a body of knowledge that can be learned and applied to improve the practice of enabling students to learn.
research in mathematics education
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

research findings
Research findings

Peer reviewed journals

Synthesis of the literature

Nature of conclusions

-suggestions

-insights

-causal

Meta-analysis

other sources of information
Other sources of information

Visions - projections of what might/should be possible

Information from colleagues

Doctoral theses

Professional organizations

Lecture notes

Exhortions

Beliefs

teaching involves
Teaching involves
  • Choosing and setting up tasks

Adaptation/modification

  • Implementation

Response to student questions

Discussion

Manage solution strategies

  • Probing for understanding

Evidence of learning

our work
Our Work
  • Formative Assessment
  • Cognitive Demand/Scaffolding
  • Discussion/Questioning
  • Transfer/Learning for Understanding
research report
Research Report
  • Describe what the topic means and why it is important
  • Give three or four key findings and their relevance for teaching
reflect on your own teaching
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?
slide40
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

references

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 www.fi.uu.nl/catch

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.

slide42

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).