Park City Mathematics Institute
Secondary School Teacher Program

Reflection on Practice Class: Day 2
Akihiko Takahashi

Aki started off our class by noting, "If you see a variety of methods in your table, naturally a discussion starts. One learns from others. This is what we really want your students to experience."

"Yesterday we talked about the problem and various solutions, using parallel strips. How are students actually discovering these solutions?"

"So far we have found a variety of relationships. Some are sophisticated; others aren't. Often, though, the lesson stops there -- after each of us found the solution. The most important part is explaining, comparing, and discussing. Which do we want to discuss at the beginning?" One of us responded with "consider the solutions which are contradictory." Aki: "What is a new idea? Which is the best to discuss? How do we lead students to a new idea?"

Our task today in our groups was "how to facilitate a discussion based on what we wrote down on our posters." "Where do we start?"

We discussed this in our groups for 20 - 30 minutes. After that, Aki asked for some of our ideas:

Responses included: "We wanted to start with the big ideas. We wanted this to be a 10th grade geometry class. We wanted to lead students to look at relationship between the size of angle and area of parallelogram. We wanted them to see a dependency relationship. Understanding this allows students to understand theorems that they may have seen. If we could do this well, trigonometry follows easily and naturally. Trig would be a natural extension of this." Aki: "What kind of previous knowledge would you expect students to have for this big idea?" Answer: "We wanted them to know the formula for area of a parallelogram, basic facts and properties of parallelograms."

Aki: "We need to specify what type of previous knowledge kids needed to have. What is our goal? What do we expect all the kids to understand at end of discussion?"

Responses: "Should we make the students justify their answer as we move on? Supposing they say that the area changes as the angle changes?"

"If our focus is to have student driven problems, I still feel that we aren't doing it correctly if we, as a teacher, decide which one to work on. How do we find the balance between allowing it to be student driven and still accomplishing our objective?"

"Deep planning is needed. If you plan you can anticipate what students will do when they see this statement."

"It seems to me that the ideas that are possible to get come from the things the students have come up with. You are really probing the capacity of students when you look at their responses." Aki: "You really need to know your students. What might be accessible? What might be a good idea? You need to know the original problem and how it might be appropriate to your students. What type of information do you want to include? What do you choose to exclude? You need to anticipate what type of discussion you want to have for your students. The teacher's goal is very different with this type of lesson."

At the end of class, Aki handed out lesson plans that he has used with different ages of students. He also showed us a Geometer's Sketchpad™ sketch which showed the change in area as the top of the parallelogram slid.

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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.