Morning Math - Part II

I have already mentioned that morning math was one of the highlights of my PCMI experience. Over the course of the three weeks, I realized that this was because the purpose of the class was to give us opportunity to deepen and connect our mathematical knowledge.  This is in sharp contrast to the point being to finish problems. This made me wonder what were the conditions of this class that fostered this goal of us more deeply understanding math rather than completing problems? And of course, though I won’t address it in this post, how can I make this happen in my math classroom?

First, a couple of caveats given by Darryl on the last day:

- this way of structuring a class does not lend itself to getting everyone to the same place

- given the time and effort that goes into the creation of the problem sets, it is impractical  to write them every day for a school year

With that in mind, how did they foster this type of mathematical inquiry?

The Problem Sets

The problem sets each day were about four pages of math problems split into four different sections—the opener, the important stuff, the neat stuff, and the tough stuff. To directly quote Bowen and Darryl “Check out the Opener and the Important Stuff first. All the mathematics that is central to the course can be found and developed there. That’s why it’s Important Stuff. Everything else is just neat or tough.” It’s hard to describe the brilliance of these problem sets. Better to just check them out here.

Here are some of the attributes that I think made them so engaging to me:

- They let you think and struggle: There were many, many times that an important stuff problem made me think of a question.  Then, over the next couple of days, there were problems in each of the sections that were related to the question, but didn’t force me to answer the question by giving the exact scaffolding I needed. Eventually my question did show up in the important stuff at which point I was ready to answer it. The work I had done on the previous days had built my insights and understanding up to the point that I was ready to tackle that question myself. I had developed all the tools I needed—I didn’t need to be given anything.

- All of the important stuff questions were enterable with a basic high school understanding of math. In the first week, all the content was centered at Algebra I or earlier. Later on, they added conics and complex numbers. But there was no expectation that I had to remember formulas or laws or other easily memorized, easily forgotten math knowledge. Anything I needed I was able to construct or figure out myself as I was working.

- We got to make connections between seemingly disparate topics. Think about the question “Find the length, width, and height of a rectangular prism whose surface area is equal to its volume.” Now think about the question “Determine values of a, b, and c to make the following equation true: \(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{2}.” In many ways, those are the same question. That’s just really cool.

- There was something for everyone. I spent the majority of the time living in the first 3/4 of each day’s important stuff. I skimmed through the neat stuff and the tough stuff, but rarely had the chance to spend very much time on those problems. I felt totally fine with that because I was able to follow the progression and think about the big ideas that were being developed throughout the course. At many times the important stuff was simultaneously important, neat, and tough. But for people who moved more quickly through the important stuff there was always something else to think and be excited about.

 - Finally, I can’t talk about the problem sets without at least mentioning the jokes. The problem sets didn’t take themselves too seriously, so neither did we. On the papers where I was doing the math, I put in exclamation marks when I found something particularly cool or satisfying. I’ve never done that before. I think part of it is because there was personality and voice in the problem sets. Therefore, I felt like there was a place for my personality and voice in my own math work.

Norms

I learned the most math and enjoyed my work time the most when I was with a table that really embodied the norms. Here are the norms that Darryl and Bowen set:

- don’t worry about answering all the questions

- don’t worry about getting to a certain problem number

- stop and smell the roses

- be excellent to each other

- teach only if you have to

- each day has its stuff

On top of these norms, Darryl and Bowen reinforced a couple of things about morning math at the beginning of the second week. One of them was that we should not be working on these problem sets outside of class. This message was really for the people who were feeling bad that they hadn’t finished whatever they felt like they should have finished in the previous class. They mentioned this in order to try to rid people of guilt about “not having gotten far enough” and it made me feel so much better about how I was choosing to spend time in class. I think they would have been fine with people continuing to do work on the problem sets outside of class—but only if it was because people were so excited and interested in continuing their investigations. Not because of guilt or living up to someone else’s expectations.

These norms meant that I was really able to explore my preferences for how to work during morning math. I certainly fell into the group of participants who liked to explore “rabbit holes.” Often when an idea was introduced for the first time, it was through a problem that was pretty straightforward to answer, but the answer illuminated something that was interesting or surprising. For example, executing a transformation that ends up creating a non-similar figure, but the area of the transformed figure is always 10 times the area of the original figure (see Day 6, problem 5). The problem set moved on at this point, but I was left with the question why does this happen? What about this transformation algebraically and/or geometrically scales the area by a scale factor of 10? How can I prove that this always happens? (I am still left with all of these questions, by the way). At this point, instead of continuing on with a new problem, I really enjoyed trying to answer my why questions. Sometimes I was able to answer them that day and sometimes I wasn’t. They often came up in the neat/tough stuff or in the important stuff in the next couple days. But even when they didn’t, I still felt like it was a good use of my time.

And that gets to ultimately what I felt was so valuable about this experience for me. The problem sets, the norms, the whole environment was designed so that the focus was on deepening our mathematical understanding, not finishing problems. And while I have experienced this before, it has been in the context of deepening my mathematical understanding in order to make instructional decisions. I have been cut off in my thinking about a specific math problem in order to think about how what I am doing relates to my students. And that is certainly worthwhile. But this allowed me to deepen my mathematical understanding for myself, in the directions that I was interested in and excited about. While I certainly don’t think this will make me a worse teacher in any way, it was an extremely freeing experience to just be able to focus on myself and the joys of thinking mathematically.