Additive, Transformative, and Subtractive Thinking

Yesterday I shared a problem that was presented at a conference I attended this summer. 

I teach AP Calculus BC, Discrete Math, and Geometry this year. For a variety of reasons, I presented this to Calculus and Geometry students. As expected, there were pretty different conversations. My Geometry class stared a bit and then someone asked, “What is the question?”. I replied, “That is the question.” My students, understandably, were pretty unimpressed by that response. I pushed a little and asked, “What is interesting to you? What do you want to know?” Some conversations ensued but I could tell that they were still pretty uncomfortable with the scenario. I then asked a student named Kaley to tell me how old she was. She’s 16, so I asked them to think about figure Kaley and then the ideas started flying. I told them that many of the questions they started asking/answering were the same ones that a room full of math teachers came up with as well. I think that the conversation ended up being positive and memorable. One student asked at the end of class, “What was that we were just doing? Would you call that problem solving?”

In Calculus the student were not at all bothered by the lack of question. They immediately dove in and started talking about building a formula to count the tiles or they talked about descriptions of the shape itself. I shared with them a quick story that I will share here. When we were presented this problem at the conference this summer – the summer meeting of the Pennsylvania Teachers of Mathematics (PCTM) – I finished the problem quickly and smugly leaned back in my chair. The graduate assistant who was working with the professor who ran the session looked over my shoulder and said, “Why don’t you try another way?” I had originally approached the problem in what I would call an additive way. I saw that there were four squares off to the right and one missing on the top left. I saw a rectangle in the core and described its shape in terms of figure number. I then added these pieces together. I shared this because almost every student I eavesdropped on was approaching the problem this way. I then posed the following question, after defining what I call an additive approach. I asked them to find a subtractive approach. In the summer session, after I was chided (correctly!) by the grad assistant, I saw the figure as a big square with negative space. Four or five teachers in the room presented their thinking about the problem and all but one took some additive approach. One teacher approached this as a transformative problem where she described moving pieces around. One of my students did this after my prompt! I had shared my square idea with my table mate and she asked me to present it to the group. It seemed odd to me that I, and all of the other teachers sharing ideas, took an approach of adding pieces together or moving them around. I was surprised that there was such consistency in our approaches. After I prompted my students with the question/challenge of describing a subtractive way of thinking one of my students, a boy named Patrick, proposed the square approach and talked about negative space in the diagram. Another student named Connor nodded appreciatively and said quietly, “I like that better.” As happy as I was with my top-flight Calculus students arriving at this conclusion, I was even more thrilled at the end of the day in Geometry when Kaley, for whom our mystery 16th figure was named, also came up with the square idea by saying (and I am paraphrasing here) that she wanted to think about what was not in the diagram as well as what was in the diagram.

Day two of the year and I am pretty thrilled with the level of conversations my students are willing to engage in .