Park City Mathematics Institute
Secondary School Teacher Program

Reflection on Practice Class: Day 11
Akihiko Takahashi

We reported on our discussions from Day 10.

Aki: Just asking a student to create their own problem isn't enough. It is impossible to use all of the student written problems. Problem: How to choose which problem to use with the whole class? We should compare and discuss how many varieties of problems can be done. One missing key point: often each student uses same method over and over. For example: guess and check. It is an important method, but each student needs to use a variety of methods. If we have a variety of solution methods - we have the flexibility. How to facilitate a discussion about different methods? The most important part: Comparing and discussing. How to integrate various methods? How can we use this process to enable benefits?

What might be the benefit of type 3 open ended problems for your students? Here are some of our responses:

  1. They begin to see connections between ideas - between algebra and geometry.
  2. Students become empowered when the problems become their own.
  3. It is much more likely that students will make problems which are appropriate for their classmates.
  4. Students will be able to generalize ideas - and add structure to the set of problems.
  5. This can create equity between student and teacher. They can have more control over their own learning.
  6. It may be that the teacher could become a learner - Maybe students will design a problem that the teacher can't do.
  7. Students can assess themselves via this process.

Our assignment for Thursday:
Iron chef week 3 - Create your own problem, working in groups of 2 or more.

Answer all four categories: Developing an open-ended problem for your students (end of Thursday)

  1. Determine if the problem is appropriate
    1. is the problem rich in mathematical content and valuable mathematically?
    2. Is the mathematical level of the problem appropriate for the students?
    3. Does the problem in clued some mathematical features that lead to further mathematical development?
  2. Anticipating students' responses to design a lesson
  3. Making the purpose of using the problem clear
  4. Make the problem as attractive as possible

We spent the rest of the hour working in groups on creating our problems.

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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.