Modeling teachers reflecting on their experience with the slinky lab
More views of their whiteboards
Obviously, I'm not saying that we should be throwing away the Three Act framework. Clearly, tasks with a fun and engaging hook that sets up a conflict or ones where a student might engage quickly with a guess are popular with students and provide excellent "needs" that prompt the development and application of mathematical tools. But, I do think that there is a place for tasks or activities in which the goal is just to tinker and think about how something works, where there is no resolution or ultimate reveal. I also make the bold claim that thinking of modeling only or primarily within the Three Act structure does a disservice to students in its focus on getting the right answer, especially if the purpose is to more closely mimic how actual mathematics is done.
I was recently reading a biography of Terry Tao, a leading mathematician and Fields medalist, and was struck by his description of doing mathematics:
From "The Singular Mind of Terry Tao" by Gareth Cook for the New York Times
His analogy to doing mathematics as being similar to being a jazz musician really struck me, as well as the notion that mathematicians are not handed problems to solve. I don't disregard the ability of Three Act tasks to hook and engage students, but I do hope that there is balance with the types of activities and tasks we ask students to do, and that there is also inclusion of more open investigations, an opening of our students' minds to curiosity and wonder about how things work and how we might describe and understand them better using mathematics, not just because it might give us correct answers, but because it's interesting and engaging to try to know and understand something beautiful.