Contemplate then Calculate

I’m taking a Structured Inquiry course this summer offered by New Visions for Public Schools. As part of the course, we are required to design a Contemplate then Calculate instructional routine and rehearse it with the class. David Wees and Kaitlin Ruggiero have both done some inspiring work with #CthenC routines and much of my motivation comes from them.

If you’re unaware of Contemplate then Calculate, the goal is to get students thinking about mathematical structure. The routine leverages student observations, honors their approaches, and highlights reflections on their own thinking. There is a public collection already created by David, Kaitlin, and the New Visions team.

Reflections:

• I initially planned for this to come at the end of an exponentials unit, but after some discussion, I think it would fit much better at the beginning of the unit – or at least before we discuss exponential equations. One thing I’m great at is having students get overly consumed with procedures, so my hope is to get them thinking instead about the structure of the equation and the relationships that exist within it before any mention of the term “exponential equation.” I want their mathematical insights to drive how we solve these equations…and the entire curriculum.
• To this end, I thought of four strategies that students may use, ranging from guess and check to equating exponents (excluding the one involving logarithms since the task would be presented before they appear). Interestingly, guess and check was not an approach that was adopted much during rehearsal (possibly because they were all math teachers).
• Since my phrasing of the number of solutions was ambiguous, Robin mentioned that students could use a graphical representation to address this. They could reason that only one solution exists because f(x) = 10^(x+5) -1 is one-to-one.
• I also considered how to represent the equation. Specifically, I thought about 100 = 10^(x-5),  10^(x-5) = 100, 100 – 10^(x-5) = 0, or using any other set of constants in place of 99 and 1. All of these alternatives have consequences for how students may approach the problem.
• During the rehearsal, I did a fairly poor job at annotating, which is a critical aspect of the activity because it models student thinking for the rest of the class. Although I anticipated all the strategies while planning, I rushed myself during the process and the result was unclear and unorganized.
• The reflection prompts should be tailored to the goals of the activity. The more specific they are, the more beneficial the reflection.
• I should omit the question (e.g. “find all values of x”) during the flash of the task. This move opens things up for more diverse observations and student thinking.
• An interesting extension was asking what if the 1 wasn’t there? How would this impact our strategy? This is a nice prelude to logarithms.
• Designing this activity made me think about mathematical structure like I never have before. Often times I take student thinking related to structure for granted. This activity helped me better value student understandings of mathematical structure – and how to leverage those understandings to enhance learning.
• One of the most beautiful aspects of this routine is that it goes beyond mere discovery learning. The goal isn’t for students to end up at the same strategy. The idea is to develop fluency and flexibility in their numeracy. It echoes the theme from my current book, Building Powerful Numeracy for Middle and High School Students by Pam Weber Harris.
• All of my planning materials can be found here, which include the anticipated strategies students will use, my planned annotations, the annotations I actually did during the rehearsal, and the slides.

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