Quadrilateral Hierarchies - Productive Struggle

An overarching goal for the semester is to help my pre-service teachers grow more comfortable with "productive struggle" and with persevering on challenging tasks. We worked at it for a long time earlier this semester on problems like the Chessboard Problem, the Cheesecake Task, and many others.

Last week, we moved into Geometry. After spending some time exploring the Van Hiele levels of geometric thought and the kinds of activities that help children progress to higher levels, it was time to put their own geometry knowledge to the test.

In this two-stage lesson, I first divided the class into five teams (of four) and assigned each team a shape class: rectangles, kites, rhombuses, parallelograms, trapezoids (inclusive definition). They were directed to produce a poster with a 2x2 grid with space for examples and properties of general and special members of their shape class.

Here's an example:

To support their work, I wrote the following list of sample properties to consider.

I checked in with the groups as they worked. We talked about things like why their "special example" (e.g. a sketch of a rhombus) was in fact an example of their assigned shape (a rhombus is a special parallelogram that happens to also have four congruent sides). So a parallelogram must have properties like "opposite sides congruent" and "opposite sides parallel", but a parallelogram might have the property "four sides congruent" (in which case, it is also a rhombus.)

When that phase was complete (after about 15-20 minutes), I used the jigsaw group work strategy and sent one representative from each original team to one of the four corners of the room, thereby forming four new groups. Each group was given the same charge: draw a quadrilateral hierarchy with the original five shape classes, together with "quadrilaterals" and "squares".

I reminded them they had already seen a hierarchy for {rectangles, squares, rhombuses, parallelograms} in one of their readings. A few students pulled that out and used it as a starting point.

With the pre-jigsaw work as a scaffold, each new group had an "expert" on each of the original five shape classes, and every person in each group had unique specialized knowledge the others did not have. It fostered a lot of communication and engagement, and made sure everyone's contributions were both valued and needed.

It was fun to watch them collaboratively think through the hierarchies. They compared notes to decide how the shape classes might fit together. They got stuck. They pushed through. Some made mistakes, while others pointed them out. There were great mathematical conversations, with students defending their thinking and explaining their reasoning to one another. They were creating new (to them) mathematical relationships by sharing and connecting their individual knowledge. And when they were done, they knew it. I didn't have to check their work -- they knew it was correct. 

The biggest challenge for most of the groups arose from the fact that the "quadrilateral family tree" is not based on a strict linear inheritance. You can't just go straight from quadrilateral to square. There are asymmetrical branches in this hierarchy, because quadrilaterals can be "special" in more than one way, and groups that tried to put their shapes in a strict linear relationship extending from "most general" to "most special" kept finding problems with their hierarchy--usually involving an issue with kites in some way--that could not be resolved by simply flipping two categories around. And they worked it out.

Here are their final results, which include refinements such as arrows to I ndicate the direction of increasing / decreasing "specialization"--the importance of which emerged out of our whole group debriefing conversation.

This first group came up with the idea of using a falling apple to connect the kite class to its quadrilateral roots without having to passthough the parallelogram trunk! Creative way to avoid having to redraw their "tree diagram!"

This second group was the first to indicate the direction of increasing generality. When I asked about the arrows, they interpreted them using the phrase "is a kind of." Because that was backwards from other groups, I invited them to label their arrows for clarity.

The third and fourth groups had trees that looked essentially like this one. One of these groups was the first to articulate the sense in which a square is the most "special". They said a shape is a special member of its shape-class if it has some properties that the more general shapes in that class do not have. In that sense, a square is a special kind of... everything!

Together, these preservice elementary teachers engaged in a productive mathematical discussion that, quite frankly, made me proud. We have a running joke about the #prouddad hashtag I had used on one of my Day 1 handouts after sharing a story about my son's clever mathematical reasoning. 

With that inside joke in mind, I posted the following to our class Facebook group the next day. 

#proudprof