Content is not important to this post, but it sets the context. We looked at derivatives of exponential functions on Friday, starting with the derivative of y = ex. We do this by looking at values of f(x) and f'(x) and then calculating the ratio of f'(x) to f(x). Today, we explored finding derivatives of exponential functions from first principles. I therefore didn't think there would be any issues when I asked them all to complete this:
And I might not have known that this was neither clear nor obvious had it not been for Popsicle sticks.
This is my tin of Popsicle sticks for my morning class. I have one for each of my afternoon classes as well. I generally go with the "no hands up except to ask a question" rule which means that the one or two extroverts in the class who have all the answers are not the only voices heard. I choose a name randomly to answer a question and then quite often choose another name to add thoughts to the first response. A response of "I don't know" is okay and is often followed by several other "I don't know"s which tells me that we all need to take a step back.
This morning's Popsicle sticks told me a lot. I went through a lot of sticks - there was a huge pile outside the tin - which means that there were many answers (some right, some not) to my questions and much "What do you think?" from me as I chose another name following each answer. When this happen I have them talk in their groups to see if they can make connections together before trying again.
The Popsicle sticks help make my classroom a learning space where everyone has a voice and every voice is important. I do my utmost to make it a safe place where making mistakes is not only okay, but important. In addition to that, Popsicle sticks help me be a better teacher. They help me gauge the understanding in the room (I do thumbs up-sideways-down a lot, also) and help adjust the pace and choose what we need to practice in the moment. This is still a work in progress for me, but one that I think is important to help me grow as a teacher and to ensure that my students truly understand what we are doing, not merely mimic completed examples.
I first read about using Popsicle sticks in Dylan Wiliam's Embedded Formative Assessment. Here is a video about this strategy and here is his website.