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How My Mistake Turned into a Pivotal Shift in Teaching Math

One aspect of my job is engaging in mathematics with teachers. We do number talks, solve problems, utilize and analyze various strategies, among other things. There is a vulnerability in doing math together for all parties; teachers, students and even myself. It's always tugging at the back of my mind, 'What if I don't model this correctly?' You know what? It's happened recently, and I finished the day reflecting on how I had really messed this experience up for teachers. But, let's talk about why this might have been one of the better things that happened that day.

To set the context, I was spending the day with teachers uncovering big ideas around division and we were getting ready to move in a more abstract direction by utilizing the partial quotients strategy. I had written the following on the board.

I began the experience by asking the teachers how many groups of three we might be able to make. We had been connecting multiplication to division and I was hoping a participant might respond with something like, "well, we know we can at least take 10 groups of 3 because that makes 30." When they did, I recorded the multiplication equation off to the side while restating that we knew, which is that with 3, we can make 10 groups and have a total of 30 to which we can take out of the 97. I continued, "We can write our ten at the top to show the ten we took out and record our 67 below."

At this point, I am feeling flustered. It felt so much like moving to this abstract algorithm without really connecting prior experiences. We had been working with tiles and equal groups and grouping and sharing that this felt like we jumped to a whole new concept. As I imagined standing in front of students, I pictured their faces wondering why I was writing these number where I was writing them. At this point, I stopped what I was doing, in the middle of solving a problem! I sent teachers to break so I could regroup.

Earlier in the session, I had teachers read a letter from Marilyn Burns, one of the features of the Do The Math intervention program that has been a game changer in my teaching of mathematics. She reminds teachers that using context to introduce problems grounds the students in something familiar so as they work with new strategies, they will have more success.

When break was over, I tried again. This time, I asked teachers to imagine that in front of them were a pile of tricycle wheels, 97 of them in fact! (Insert math joke about 250 watermelons) I then asked; if we want to use the wheels to make tricycles, how many trikes could we make from the wheels available? Participants were asked to turn and talk to their neighbor about how many they think we could make. It was suggested that we could make at least 10 tricycles because we would use 30 wheels and we have enough from the 97.

As I recorded their responses, I did a think aloud to restate what they said. "So you're saying that you know we can make at least 10 tricycles because each trike takes 3 wheels, and 3 times 10 is 30. We would have used 30 of the wheels. I will record that equation off to the side to show how many wheels we have used so far, and I want to see how many wheels we have left. Let's subtract the number of wheels we have used from 97 so we can decide if we can make more tricycles." At this point, many teachers were commenting that we could, in fact, make 10 more trikes. One teacher raised her hand and added that she thought we could actually make at least 20 more trikes from the wheels we have left. I asked the room to think about that and decide if that was true and how they know. I then erased some of the words and continued recording their thinking.

As I recorded, I made sure to reiterate what each number stood for. "We can make 20 more tricycles because 20 trikes, times 3 wheels for each trike, uses 60 wheels. We have at least 60 wheels." We then decided as a group that we could make 2 more tricycles and there would be one extra wheel that we wouldn't use. Teachers then practiced this strategy with other numbers of wheels still making tricycles.

I breathed a sigh of relief after the experience pleased we had at least successfully practiced the partial quotients strategy. I knew that for some students, this would help them navigate the complex move from dividing concretely to eventually using a more standard algorithm and the teachers would feel empowered supporting them.

As I finally sat down after all of the participants had left for the day, I started reading through the exit tickets. I knew I had to face them even if it meant confronting a very uncomfortable experience from the day. The one on top read, "One experience that made an impact on me was when you took a break because you couldn't get your language right. It show the importance of language." After the shock wore off, I was grateful for the grace that the teachers had shown me from the experience. I thought about what the teacher had said about the language, and realized that in addition to that, using a context allowed me to provide accessible language. The context connected the mathematics to the written equation while making sense of what it means to divide. The context was key.

I recently watched an interview with *Cathy Fosnot who authored Contexts for Learning Mathematics and she stated that we provide contexts so students "don't get lost in an abstraction." She adds that it's important that teachers "choose the numbers carefully" so we can "see the strategies emerge." *Marilyn Burns states, in her letter above, that numbers have been "carefully and purposefully selected" not only for students to feel comfortable as they work towards more abstract concepts, but using the context of tricycles can help us get our language right as we connect concrete experiences to representational and abstract.

While the mistake was uncomfortable in the moment, the implications for teaching and learning mathematics will have a lasting impression. When I am teaching in classrooms, I remember that providing students with a familiar context will allow them to access the mathematics they are learning and transition from concrete and representational strategies to more abstract methods.

It's important to note; I've messed up many times, and I will definitely do it again!

Take care,


*Fosnot, Cathy. Interview. In Math, Context is Critical. Conducted by Heinemann. May 1, 2017

*Burns, Marilyn. Do The Math. Heinemann Publishing. Copyright 2008.

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