A few years ago, upon Jill Gough’s recommendation, I read How the Brain Learns Mathematics by David Sousa. In 2012, we got to hear Sousa present the opening session at the T^3 International Conference in Chicago. Jill writes briefly about the session here.
I still make sure teachers with whom I work know about the Primacy-Recency Effect, which shows that in a learning episode, we learn best what we learn first. We learn second best what we learn last, and we learn least what is in the middle. Sousa’s research suggests that 20-minute learning episodes offer the most “prime time” learning, with the least amount of down time for students. So even in our 90 minute block periods, we try to focus on 20-minute learning episodes.
Jill writes about a 20-minute experiment in which she participated at The Westminster Schools a few years ago.
I try to provide students some cognitive process every 20 minutes of class so that they can think about what they are learning and what their questions are. I don’t formally ask that question every class period, but as we are coming to the end of our unit on Rigid Motions, I asked students to reflect on what they have learned and what their questions are. The actual prompt was:
Name one thing you have learned in this unit on transformations that you did not know prior to this year. List a question that you still have about transformations that you need to get answered before your summative assessment.
I exported the student responses so that they were easy to organize. I’ve included a few below.
I have learned how to generalize transformations.
My question is is there any easier way to do a rotation. 1
I have learned how to rotate a point either clockwise or counterclockwise.
My question is how do you remember which direction y=x and x=y go? 1
I have learned why shapes are congruent, not because they look the same but because they can be mapped using rigid motion.
My question is nothing. 1
I have learned that a shape does not have only one way of deciding if it is congruent.
My question is…What is the easiest way to map a rotation without memorizing the (x,y) things we learned 1
I have learned to rotate a point.
My question is what is orientation. 1
I have learned how to rotate the point around the orgin.
My question is 1
I have learned . that more things can be congruent than just size and shape
My question is . how do you find rules for rigid motions 1
I have learned what vectors are and more information about transformations.
My question is are there different types of vectors. 1
I have learned what was orientation.
My question is can a vector be used for anything else? 1
I have learned how to use vectors correctly.
My question is without a grid, how would you accurately use the vector? Would you just move it to a relative location? 1
I have learned how to reflect over lines y=a, x=b, x=0, and y=0.
My question is-Why do we need a question? 1
The responses that I got help me know where to focus on our last class day before the students’ summative assessment.
But I have to admit I am most excited about the first response I saw, highlighted below. This response gives me confidence that our focus on how we are solving problems – our focus on the Standards of Mathematical Practice – is getting us somewhere.
Make sense of problems and persevere in solving them at its best.
And so the journey continues ….