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.