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.

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