The last unit for my senior class (Advanced Math Decision Making) is Networks and Graph Theory. Win for me (and hopefully the students)! My student teachers and I are taking the opportunity to teach a little more like I want to next year (my student teachers are amazing). Here’s how we’ve done, so far.

**Day 1: **We started by challenging the students to find a Eulerian path and circuit through the basement (we didn’t use that language). And then took them to the basement.

They explored in groups of ~3, at first with only the criteria for a path/circuit and then with a map. Once students had found the path and explained to a teacher why a circuit doesn’t exist, we sent them upstairs to work on representing the basement diagrammatically and then “fixing” the basement so that it did have a circuit. All the groups finished their diagrams. A few created a circuit in the basement rather trivially by eliminating the middle hallway.

**Day 2:** We started this day by naming our circuits and paths. Students came up with “S/E” for circuits (because you Start and End in the same place) and “GPS” for paths (because a GPS takes you from one place to another). Students then articulated in writing why no circuit exists in the basement, and gave each other feedback on their reasoning. Finally students were tasked with creating a non-trivial circuits. The methods for doing so were pretty awesome — some groups figured out that you could create a circuit by drawing anything without lifting your pencil. One student started drawing the iPhone (?) dots and making shapes. Once groups created an S/E, they traded with another group and proved that the other group’s graph was an S/E. By the end of the day, we had a collection of S/Es and GPSs, all student generated.

**Day 3:** We sorted 12 graphs into S/E, GPS, and neither. Yay sorts!

**What’s working**

– Students are engaged. One student who has been rather checked out for a few weeks was one of the most determined doing the sort today.

– Students feel ownership. One of the reasons I had students name the circuits/paths was to model what happens when people do mathematics – they need to name things, and they can choose whatever names make sense! Additionally, the students have created as many graphs as teachers. All the graphs on the wall, except one, were created by students.

– Students are making their own sense. They physically experienced the basement. Both the strategies for making circuits mentioned above came from students. Today, a student went to the board to show how she would go around an S/E from the sort and another student interrupted her to tell her to start somewhere else. The original student stuck to her guns and then the other student showed his S/E route, leading to the question of whether you could start anywhere when walking an S/E. (We haven’t finished answering that.)

**What I still need to work on**

Ohhhh whole-class discussions. How hard you are for me to lead! How to balance argument/excited participation with listening? How to balance desire to go to lunch with listening? Discussions, I aspire to be best buddies with you one day in the near future. Please be open to our future, awesome relationship. Love, Sarah.

Next week … onto the Bridges of Konigsberg and plowing city streets!

*SL*