Last week, I facilitated a session on responsive instruction in the math classroom. The structure of the session was to start with a Which One Doesn't Belong, because we always have to start with some math.
In these two responses, we made the connection between opening up and down and the coefficient of the x^2 term.
These two examples gave us an opportunity to talk about x-intercepts, as well as some representational limitations, with the quadrant picture highlighting that even though we can't see it in the picture, that the top right graph will eventually be in quadrant 2 as well as quadrant 1.
After we discussed our WODB answers, we got in groups to get to the heart of the session. Groups were asked to highlight areas that are consistently troublesome for students. Part of my responsive instruction was to anticipate participant responses, so I guessed that fractions and signed arithmetic would be two areas that were common problems. Sure enough, my participants agreed, adding also some trouble with creating and solving equations.
Our goal was to highlight trouble areas and come up with ways to help students before, during and after instruction. The desired outcome was to go back to GoFormative and draw a representation of how to teach the topics with responsive instruction in mind. To accomplish this, I provided participants with some resources to research their anticipated trouble spots. We used the Progressions Documents and Nix The Tricks.
The discussions that ensued with participants started to really blow me away. Participants went back and forth between the Progressions and Nix The Tricks. Questions surrounding signed arithmetic started popping up. Teachers found a lot of usefulness in representing the number line vertically, thinking it would be more intuitive for their students. Then came a struggle with how to use the number line to represent subtracting a negative. Here I pivoted the conversation to the order of operations and how students make sense of that. I referred participants to the PEMDAS section of Nix the Tricks. We talked about the advantages of switching to GEMA and getting students to understand the definition of subtraction as adding opposites. I was truly impressed by my participants openness and willingness to engage with these new thoughts on classic topics. As we debated the merits of the two approaches, we also incorporated the other aspects of GEMA: the lack of division, the idea that we don't need to go left to right (and by extension the commutative and associative properties), the broader aspect of Grouping symbols versus Parentheses, and the connection to inputs in the calculator. Here are some responses to this activity:
Working with my colleagues, seeing their excitement about some new approaches, has really invigorated me at the beginning of this summer, and I can't wait to check in with them during the school year to hear how these approaches are going!
