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 Using new technology Working with Japanese Colleagues Developing Curriculum Conducting Demonstration Lessons Working with Preservice Students Designing Professional Development

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

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

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

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

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

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

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

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

What and how the work is recorded matters 2x 15 + 25x3 15 25 x2 x3 30 75 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 x2 x3 30 75 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

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? 1/6 1/5 1/4 1/3 e) 1/2 C D NCES, 1996

Dekker & Querele, 2002

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 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 Peer reviewed journals Synthesis of the literature Nature of conclusions -suggestions -insights -causal Meta-analysis

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 Implementation Response to student questions Discussion Manage solution strategies Probing for understanding Evidence 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).