Tag Archives: bongard problems

Before a Break: CCSS Math, Bogram Problems, and Peer Feedback

I spent the day in a room full of my colleagues as part of our school's official transition to using the Common Core standards for mathematics. While I've kept up to date with the development of CCSS and the roll-out from here in China, it was helpful to have some in-person explanation of the details from some experts who have been part of it in the US. Our guests were Dr. Patrick Callahan from the Illustrative Mathematics group and Jessica Balli, who is currently teaching and consulting in Northern California.

The presentation focused on three key areas. The first focused on modeling and Fermi problems. I've written previously about my experiences with the modeling cycle as part of the mathematical practice standards, so this element of the presentation was mainly review. Needless to say, however, SMP4 (Model with mathematics) is my favorite, so I love anything that generates conversation about it.

That said, one element of Jessica's modeling practice struck me by surprise, particularly given my enthusiasm for Dan Meyer's three-act framework. She writes about the details on her blog here, so go there for the long form. When she begins her school year with modeling activities, she leaves out Act 3.. Why?

Here's Jessica talking about the end of the modeling task:

Before excusing them for the day, I had a student raise their hand and ask, "So, what's the answer?" With all eyes on me, a quick shrug of my shoulders communicated to them that that was not my priority, and I was sticking to it (and, oh, by the way, I have no idea what time it will be fully charged). Some students left irritated, but overall, I think the students understood that this was not going to be a typical math class.
Mission accomplished.

Her whole goal is to break students of the 'answer-getting' mentality and focus on process. This is something we all try to do, but perhaps pay it more lip-service than we think by filling that need for Act 3. Something to consider for the future.

The other two elements, also mostly based in Jessica's teaching, went even further in developing other student skills.

I had never head of Bongard problems before Jessica introduced us to them. This involves looking at well defined sets of six examples and non-examples, and then writing a rule that describes each one.

Here's an example: Bongard Problem, #1:
p001

You can find the rest of Bongard's original problems here.

In Jessica's class, students share their written rules with classmates, get feedback, and then revise their rules based only on that feedback. Before today's session, if I were to do this, I would eventually get the class together and write an example rule with the whole class as an example. I'm probably doing my students the disservice by taking that short-cut, however, because Jessica doesn't do this. She relies on students to do the work of piecing together a solid rule that works in the end. She has a nicely scaffolded template to help students with this process, and spends a solid amount of time helping students understand what good feedback looks like. Though she helps them with vocabulary from time to time, she leaves it to the students to help each other.

Dr. Callahan also pointed out the importance of explicitly requiring students to write down their rules, not just talk about them. In his words, this forces students to focus on clarity to communicate that understanding.

You can check out Jessica's post about how she uses these problems here:
Building Definitions, Bongard Style

The final piece took the idea of peer feedback to the next level with another template for helping students workshop their explanations of process. This should not be a series of sentences about procedure, but instead mathematical reasoning. The full post deserves a read to find out the details, because it sounds engaging and effective:

"Where Do I Put P?" An Introduction to Peer Feedback

I want to focus on one highlight of the post that notes the student centered nature of this process:

I returned the papers to their original authors to read through the feedback and revise their arguments. Because I only had one paper per pair receive feedback, I had students work as pairs to brainstorm the best way to revise the original argument. Then, as individuals, students filled in the last part of the template on their own paper. Even if their argument did not receive any feedback, I thought that students had seen enough examples that would help them revise what they had originally written.

I've written about this fact before, but I have trouble staying out of student conversations. Making this written might be an effective way for me to provide verbal mathematical details (as Jessica said she needs to do periodically) but otherwise keep the focus on students going through the revision process themselves.

Overall, it was a great set of activities to get us thinking about SMP3 (Construct viable arguments and critique the reasoning of others) and attending to precision of ideas through use of mathematics. I'm glad to have a few days of rest ahead to let this all sink in before planning the last couple of months of the school year.