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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). - PowerPoint PPT Presentation
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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)
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
Working with Japanese Colleagues
Working with Preservice Students
Conducting Demonstration Lessons
Developing Curriculum
Designing Professional Development
Using new technology
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
Takahashi, 2005
Introduction: HatsumonThought provoking question
Key question – shu hatsumonIndividual or small group work
Walking among the desks – kikan-shidoAnticipated student solutions
Student solutions - NoriageMassaging students’ ideas
Summing up- Matome Bass et al, 2002
The Lesson
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
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
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
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 Experience
Conducting Demonstration Lessons
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
15 40
0 25
Number of erasers
0 1 2 3 4
0
1
2
3
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 80 105 130 155 180
15 40 65 90 115 140 165 190
0 25 50 75 100 125 150 175
Number of erasers
0 1 2 3 4
0
1
2
3
Number of pencils
15 40
0 25
Number of erasers
0 1 2 3 4
0
1
2
3
Number of pencils
Number of erasers
0 1 2 3 4
0
1
2
3
Number of pencils
Scaffolding matters
Preactivities leading to main goal
1x25
2/25
3x25
4x25
5x25
1x15
2x15
2x15
4x15
5x15
What and how the work is recorded matters
2x 15 + 25x3 15
25
x2x3
3075
2x15 + 3x25
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
3075
Goal: Ax+By = C
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 Experience
Working with Preservice Students
Expect the Unexpected
Change
0
50
100
150
200
250
300
350
1 3 5 7 9 11 13 15
Change
Person
Linear (Change)
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 Experience
Developing Curriculum
In the figure below, what fraction of the rectangle ABCD is shaded?
A B
D C
a) 1/6
b) 1/5
c) 1/4
d) 1/3
e) 1/2
NCES, 1996
Dekker & Querele, 2002
Comparing Quantities. Kindt et al, 2006
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
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
Quantitative Studies -experimental -quasi-experimental
Experimental-observation
Theories of learning - frameworks for thinking about teaching and learning
Qualitative Studies--Case studies--Ethnographic studies
Research findings
Synthesis of the literature
Nature of conclusions-suggestions-insights-causal
Meta-analysis
Peer reviewed journals
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
• Choosing and setting up tasks Adaptation/modification
• ImplementationResponse to student questionsDiscussionManage solution strategies
• Probing for understandingEvidence of learning
Our Work
• Formative Assessment
• Cognitive Demand/Scaffolding
• Discussion/Questioning
• Transfer/Learning for Understanding
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
• 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?
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 •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.
•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).