Park City Mathematics Institute Secondary School Teacher Program Reflection on Practice Class: Day 14 Akihiko Takahashi

Aki started our class by teaching us how to make hats with PCMI envelopes. He emphasized that this was not an open ended problem.

Later, though, he showed us an origami based open-ended problem.

Fold a piece of origami paper along its diagonal. Fold that piece in half to make a right triangle. Cut the right angle off - by cutting across the midpoints of the two legs. What is the shape when the paper is unfolded?

Repeat this process - but fold one more time. Cut the right angle off in the same manner. What is the shape when the paper is unfolded?
Answer: a plus sign - or a cross.

Repeat this process - but fold one more time. Cut the right angle off in the same manner. What is the shape when the paper is unfolded?
Answer: a frame with four sub frames.

Repeat this process - but fold one more time. Cut the right angle off in the same manner. What is the shape when the paper is unfolded?
Answer: a lattice with four interior holes.

Each of the resulting figures have the same area - three fourths of the original square. This is true because you are always cutting off top triangle - which has area one fourth of original one.

Question: We've learned about open ended problem - and how to make them. How would you actually use open ended problems in your classes?

We discussed this question in our groups. Here are some of our answers:

• Look in lesson plans for places where we can insert open ended problems in place of boring ones.
• Look for what problems I can turn into open-ended ones.
• Take a "Problem of the Week" and turn into an open ended one.
• Use them as group worthy tasks to help students, especially freshmen, learn how to work in groups.
• Start our math department meetings by doing open-ended problems. This involves working with our colleagues.
• Use open-ended problems to facilitate talking about math.
• Use as a culminating event - or as an assessment event.

Aki: At any workshop new ways of thinking are introduced. The hard part is how to the new ideas. When you go back to your class, you are just one teacher. It is hard to implement new ideas and each of us is very busy. Everyday is so busy and it is hard to implement new ideas.
A solution: Lesson Study

Lesson study can help people learn how to implement new ideas. It was developed in Japan and has a 100 year history. One group used Lesson Study to help students become independent thinkers. That group took eight weeks to develop one lesson. This method enables one to look at students' learning to reflect back on your teaching. Lesson study requires changing culture. Once you start Lesson Study, teachers want to continue. It is very hard to use new ideas from workshops in our classroom. A collegial network will help teachers collaborate.

Aki: The real challenge for PCMI participants starts today.

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This material is based upon work supported by the National Science Foundation under DMS-0940733 and DMS-1441467. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.