This week I registered for my last two courses in my Masters of Arts in Teaching Mathematics, from Mount Holyoke College. Realizing that this program was almost over brought up a lot of emotions. Pride and satisfaction for sure but also disappointment – this amazing journey was coming to an end. I miss Cohort 7 already! A collection of educators from all over, varying in age, assignments, experiences and teaching environments, all working together to improve our understanding of mathematics and our ability to create the conditions that help children succeed in this amazing subject. I will never meet most of these educators in person – but I feel that the experiences we have shared will bind us together as a lasting learning community. I started looking back over some of the reflections I wrote after different classes and courses. Here is one reflection in a group I call, Play, Persist, Prove.
Today was another awesome experience in my mathematics journey. Hours later I am still buzzing about how transformative these experiences have been. So transformative in fact, that I feel compelled to write about it.
Being a math teacher implies a certain level of success and know-how from my early days as a student. Why else would I or could I become a math teacher? Being of a certain age also implies the type of math education I received: memorize your facts, watch what I do, now you do the same. What most people know very little about is that math is taught very differently now. Or they may know math is taught differently but may not know much, if anything, about the how or the why. While some people (and even some math teachers) may think, “I learned it this way – and I got it! Why can’t everyone still learn it this way?” Those truly interested in effective math education know that, like everything else, when practice follows research, everyone wins!
Today, like most days with Cohort 7, we dug deep into a math investigation. The context wasn’t beyond my grasp – we were discussing the sum of the interior angles in polygons. But, like so many other math topics, I had a long before memorized solution, formula, or fact. Today, the formula was just out of reach. I hadn’t used it in a while. It wasn’t something I ever thought about. When I learned about this concept way back in junior high, the teacher just told me the formula. I believed them. I applied it. We moved on. What was it again? I knew it was something to do with 180°. Maybe the side measure of the polygon was in there? Minus 2? Guess. Check. Confirm. Got it! 180(n-2). But I wasn’t going to get away with reciting a formula today! And I didn’t want to.
Out came the pattern blocks, the power polygons, and the virtual manipulatives. The directions: figure out the interior angles of these polygons, be convincing, don’t use a protractor. I was excited. Time to play. Time to persist. Then time to prove. There was no rush. Work on your own, discuss with your group as you go or when you are ready. I created a couple of models to assist in my investigation. Can you see what I was trying to do here? How could these models help or do they? What do you think my explanation was that accompanied these models? What would you create?
The investigation alone was powerful enough. But the seasoned facilitators, Jan Szymaszek and Zak Champagne, did not stop there. After comparing notes and models in small groups, each group created a few slides to share with everyone. Together as a class we scrolled through the gallery of representations, asked questions of the creators, noticed what was common, and combined any new information with our own. It was awesome. I saw another group’s model and relief and understanding washed over me. I was finally able to connect the formula I have learned and relearned so many times to a picture for figuring out interior angles. Because that formula corresponds to a picture – that connection is solid. I will never forget it again. It is so powerful to be in that student role and make a discovery that is deeply meaningful to me. I have that wonderful satisfaction and contentment that things do, in fact, make sense. That everything is connected. I don’t need to memorize it – I KNOW it. What was interesting too is that my colleagues had similar experiences but not about the same models. Seeing and hearing about lots of ways was neccessary for the group to be satisfied.
How can I make sure my students have these experiences too? Have the feeling that they know something – not that they were told it or heard it; the knowledge and connection has to be theirs. That is what I am left to ponder. While I don’t have that all figured out just yet…I did take notes on the “teacher moves” of our facilitators. What did they do that made this lesson such a success? Here’s what I wrote as tips to myself about being a better teacher: Listen and listen hard. Stop talking so much and really listen. Don’t be so quick to jump to conclusions about what kids do and do not know. Ask questions. Put the life preserver away. They’ve got this. Give students a safe space to play, lots of tools and encouragement so they can persist when something doesn’t work right away and the opportunity to share, discuss and defend their proof. Your job is to find cool problems to investigate with lots of hands-on irregularly wonderful mathy things – then be present at the moments of insight and an enthusiastic witness to their awesomeness.
Hmmmm. Let my students Play, Persist, Prove. Yup…I can do that.