** The reflection questions in italics below are welcomed to be used not only for yourself as a learner, but in facilitating the professional learning of others. **
When was the last time you were engaged in productive struggle while working on a mathematical task? (long pause…not a rhetorical question).
When was the last time you were engaged in un-productive struggle? (another long pause…really, think about it)
At NCSM 2016, Tim McCaffrey, Jill Bueckling and I decided to collaboratively work on deepening our high school content knowledge through Making Sense of Teaching Math for High School. Despite our geographic differences, we set a schedule for how much we would read and wanted to give the video-conference app Zoom (as recommended by Mike Flynn) a whirl on occasion. We all humbly approached with a growth mindset, knowing that we’ve all been taught in ways that did not encourage reasoning and problem-solving as high school students ourselves, and were working our own mathematical depth.
We are absolutely loving the book. If you’re an elementary or middle school teacher, and are interested in knowing where your content standards are headed, and what some of the big ideas to emphasize in your own classroom are, this is an excellent resource to try to piece some of that together. If you’re a high school teacher who understands the content, but could maybe use something to fill in the gaps as to how different concepts connect, this one is for you, too! If you’re somebody who would like to see high school students reasoning and problem-solving in live video…well, they’ve got that covered, too. Feel free to head on over to #MSMTHS and join the conversation, which includes the authors Juli Dixon, Ed Nolan, and Farshid Safi, as well as some promotional support by Solution Tree.
In doing some of the tasks presented in the book independently so far, I’ve engaged in some productive (and un-productive) struggle. For example, in the Geometry chapter I was working on the task below.
I noticed myself initially productively struggling. In other words, I was trying to make sense of the problem, testing out different ideas, drawing visual representations to match the context, conjecturing, looking for structure…sound like the Standards for Mathematical Practice?
Could it be that the threshold between productive and un-productive struggle lies in the Standards for Mathematical Practice? Is there a proportional relationship between the magnitude of engagement in the SMP’s and the magnitude of productive struggle?
However, my productive struggle became unproductive when I felt I had exhausted all of my options. As I tried to reach deep down into my own conceptual understandings, I had nothing else to pull from.
How often do your students get to this point?
How often do your students reach down into their own conceptual understandings, and feel they’ve exhausted all of their options?
What’s your role as the teacher in this situation?
How does your understanding of progressions of mathematical content come into play when students shift from productive struggle to un-productive struggle?
I found myself reflecting in that moment on what it’s like to be a student and not knowing where to go or what to do. Often, what students are told/asked in this moment include the following…
“What did you do first? What do you think you should do next?”
“Why don’t you try a different strategy?”
“Go back in and read the problem again.”
“Keep trying. You’ll get it.”
“Turn back to your notes and see what it says.”
None of these prompts would have helped me transition out of my unproductive struggle back into productive struggle. In this moment…
I wanted to hear somebody else’s thinking as they’re concurrently working on the task.
I wanted to see how others working on the task are representing the context.
I wanted to know that others, just like me, are struggling, too.
I wanted to hear what others are struggling with, what they’re thinking might work and why.
I wanted others’ thinking to be visible.
THESE ARE TEACHER ACTIONS THAT WOULD MOVE ME FROM UN-PRODUCTIVE TO PRODUCTIVE STRUGGLE.
I would’ve felt less capable and/or outright defeated if somebody who already deeply understood the content in the task was telling me where I’m going wrong, or what to do next. If that same person suggested trying another strategy, my response would have been simple. “I don’t have any other strategies to try.”
So what’s the point?
If I were a student sitting at my desk, trying to make sense, reason abstractly, look for and make use of structure and so on, and if I’m restricted from having social interaction where other students’ thinking (NOT the teachers’ thinking) is visible, my struggle remains un-productive. The mathematics lies in other students’ thinking, as we are all working on a common task…struggling together in many different ways. Our struggles become shared understandings, but only if they’re shared.
Tomorrow, I video-conference with 2 math friends on Zoom, where we will make each others’ thinking visible around a task that promotes reasoning and problem-solving by engaging in meaningful discourse, using and connecting each others’ representations and building conceptual understanding as a foundation for greater procedural fluency. We will grapple together, talking and trying to make sense of each others’ thinking through discourse and each others’ representations.
However, if I’m not allowed to talk to my peers, see and hear what they’re thinking, and/or connect their representations to my own, then math is something that does not make sense…and it’s just for other people. I’ll just sit, and wait for somebody to show me what to do, if I do anything at all.
Please substitute my own thoughts as I was struggling un-productively out of the text above and insert some of your students’ names.
Math is social. When we take away access for students to see each others’ thinking, we take away access to the mathematics. This access comes through opportunities for students to make their own and each others’ thinking visible in the form of meaningful discourse and making sense of other students’ representations of mathematical situations.