Discrete Math update - engaging students in proof writing

The inaugural full year discrete math class is rapidly closing in on the half-way mark. The content is working out that for the first semester we'll get through the counting and graph theory units. The counting unit test became our mid-term and the graph theory test will be our final exam.

I have a colleague who is teaching discrete math for the first time this year. It is also her first time teaching an inquiry-based course. She has been a tremendous help in making suggestions and helping to clarify meaning of teaching suggestions for different lessons.

It has also been helpful having three sections of students taking the class. I have been able to clean up typos, re-word confusing question prompts, and make corrections in printed solutions. In total, there are now over 250 pages of textbook, teacher notes, and solutions.

As for the classes, we are now looking at conjectures and proofs related to Euler and Hamilton paths and circuits. As in past years, students struggle with communicating precisely what they see that leads them to believe something is either true or false.

I had attended an AP Computer Science Principles training this summer that focused on cooperative learning strategies. Seeing some students trying to work through their explanations while others sat watching, I decided to mix things up a bit.

Students had been asked to prove or provide counterexamples for four different statements. I told students that every group was going to put their proof for the first claim on the board. I divided my whiteboards into sections and assigned each group to a different section. After students wrote their proofs on the board the fun began.

First, I told the class that when I read a proof, not only theirs but any proof, I look for clear supporting evidence of any claims made. Basically, I ask myself if this is a proof or if the statements are basically saying, "Trust me, it works." With that preface, I asked students to consider the first proof. I asked the class is this a proof or is the group saying, "Trust me." The class concurred it was a trust me statement.

I pointed out that what was on the board was a good start. However, a proof cannot stop there. There needs to be statements of why the statements are being made. A proof needs to go into the "because" realm to explain the features that support the claim.

One proof actually went beyond a simple "trust me" it works statement. Students recognized that the proof was more than simply a trust me statement but that it still wasn't totally convincing.

I asked students to consider what they just did and to attempt a proof of the second claim. I prepared the board for a second round of proof writing while the students worked in their groups.

The second time through there was a marked improvement in what they wrote for proofs.We went through these proofs and students could see the improvement. While there was only one trust me statement, students also recognized that none of the proofs were totally convincing. We discussed how to take the proofs further.

Besides helping students to better differentiate trust me statements versus a proof, I had more engagement from all students in the proof writing process. Different students were required to go up to the board and write down their thinking. I had other students check what was being written, watching to make sure the proof wasn't simply a trust me statement. This checking led students into discussion about what should be said and how to state it clearly.

I actually had students who normally don't participate be the ones from their groups to go up and start writing their proof on the board. During the second go around, I hadn't asked students to start writing their proofs on the boards. Groups just got up and started writing when they were done at their tables. Other groups would see the activity at the board and start working to try and finish quicker.

I felt like this was a big step in the proof writing journey that we are taking during the school year. I recognize that this won't necessarily make it easier for students to get their thoughts down on paper, but I think they will recognize when they are tending toward a "trust me" proof. My hope is this recognition will spur them to push further in getting additional support and trying to be more specific in their explanations.