I'm going to share how my two favorite geometry texts -- CME Geometry and Discovering Geometry -- use the handshake problem. The question is: what do their approaches to the problem say about their takes on geometry?
Discovering Geometry's Handshake Problem:
CME Geometry's Handshake Problem:
- Discovering Geometry breaks down the modeling process into eight steps, starting with completing a handshake table. CME doesn't offer any of those steps or supports.
- Discovering Geometry asks the student to model the handshake process with polygons and segments. CME presents the handshake problem and the diagonals problem separately, and then asks students to consider connections between the two questions.
- This isn't in the pictures, but the handshake problem is the first problem that appears in CME Geometry, while the handshake problem appears as subsection 2.4 in the Discovering text, titled "Mathematical Modeling."
I'm very curious to know what you folks out there think about these two presentations. My suspicion is that these two presentations speak to two different assumptions about how students learn to reason in mathematics. One assumption is that students need lots of informal experience that, with feedback and opportunity, will slowly shape their reasoning habits. The other assumption is that students need an explicit model of the reasoning process, either in addition to or at the beginning of their opportunities to reason on their own.
Your thoughts and criticisms are always appreciated, but I'll amplify the invitation for this post. I also know that authors of both of these texts hang out on the internet, and they could likely school us all a bit on their work.
Comments from the Bullpen:
Fawn and mrdardy want to see Discovering Geometry take away all that support for the kids, instead put it into a teacher's guide. They'd like to see CME get rid of the explicit connection between diagonals and handshakes, preferring texts that offer radically little upfront support for students.
l hodge finds CME's explicit call for a connection between diagonals and handshakes patronizing. Is there another way to push kids towards making that connection?
fivetwelvethirteen connects this with the CCSS standards of mathematical practice. How do you help kids get better at big skills like solving hard problems or mathematical modeling? My take: let's look to other fields, like literacy, because "problem solving" is as big a skill as "reading" or "writing."