This post is by Dave Carter, Geometry teacher at Impact Academy of Arts and Technology, an Envision Education school located in Hayward, CA.
I teach geometry at Impact Academy, and have been in secondary math education for 12 years. In 2012, I shifted gears to become a math coach with the Oakland Unified School District (OUSD). I had the opportunity to develop my coaching skills with Elena Aguilar, author of The Art of Coaching and one of the strongest voices in instructional and leadership coaching today. After two years of working with Elena, this summer I had the opportunity to return to the classroom and resume my work teaching Geometry at Impact. I was thrilled to be back in the classroom and energized to apply my new learning to my teaching practice.
Recently, Gia Truong, Envision Education’s CEO and superintendent, asked me to what extent my work as an instructional coach has impacted my understanding of deeper learning and what that actually looks like in my classroom. It’s a great question. In the Envision model, deeper learning flourishes when students are active within projects that tie in numerous disciplines, that allow kids to consider a theme from varying angles, and that culminate in a public exhibition where kids defend their work--but these are not the only ways to generate a climate of deeper learning in the classroom. This also happens in smaller, more individual ways, and yes--even in the math classroom.
Deeper learning does not exist in a vacuum and cannot occur without developing deeper learners--learners who think deeply. The significant changes in my teaching practice since my return to the classroom are the result of honing in on the classroom activities that promote sustained student effort.
Much of the work that I did as an instructional coach revolved around mindfulness, and maintaining that mindfulness during coaching. I carry this mindfulness with me in my teaching practice. One simple way that this presents itself is in the form of wait time. Today, I am a more patient instructor, more willing to let unknowns in student discussions just sit. I consistently remind myself that for most of my students, many of whom come with significant math skill gaps, this content is new, perhaps foreign, and that kids literally need time to think about the content. Deeper learning happens in the gaps, in between the talk and during seemingly dead air. At times, when a student discussion comes to a pause, and kids are really grappling with an idea, I feel like I’m actually watching learning transpire.
In reflecting on my first couple of years of teaching, I can remember the anxiety I felt when I would hear or see an incorrect answer. More often than not, I would quickly ask a question--usually a leading question--that would produce the right answer. And then I would feel better. This is the opposite of a student-centered classroom: all of my actions were generated by my own internal anxiety. A student-centered classroom is one in which the student experience is at the center, and the teacher simply adjusts the climate so that kids are more frequently in contact with and generating the cool thoughts and ideas and mistakes and epiphanies that abound. It is important to acknowledge that the speed at which students problem-solve is most likely not the speed at which I problem-solve. It’s important for me to create open space in which students can think. And the most direct way to do this is simply to wait.
Recently my students were deriving the Pythagorean theorem using the area model below. Now perimeter, area, and volume are not taught until later in the course. So, my kids were manipulating this area model without any recent activation of prior knowledge. I asked one of the small groups, “Assuming all angles that appear to be right angles are, how wide is this figure?”
Student 1: “I don’t know. There are no numbers.”
Student 2: “It’s ab.”
Student 3: “It’s ab.”
[Pause - eight seconds of silent thinking, and then visible student discomfort with the silence.]
Student 4: “It’s a + b.”
Earlier in my teaching career, I would have interjected at this point, quickly targeting the students who misunderstood and offering them corrective instruction: “OK, if you are at one corner of the square and you walked across the top to the other corner, how far have you travelled?”
But here I didn’t. I waited, and waited, and waited longer. Initially, this feels weird for kids but the more that you do it, the easier it becomes for everyone. And it’s a powerful instructional stance. By protecting that thinking space, you are teaching. It’s important to notice whether kids are beginning to check out, whether the gears are still turning, or both.
Student 2: “It’s ab.”
Student 4: “No, it’s not. If a is 2 and b is 4, then you are saying multiply. The distance is not 8!”
Student 3: “Oh!!! It’s a + b”
Student 2: “That’s what I meant when I said ab; I was putting them together.”
Student 4: “Ok, but do you know what ab means?”
Student 2: “It means multiply ... oh, Ok. I get it.”
Student 4: “Can we all agree that it’s a + b.”
[Nodding heads]
Student 4: “OK, now how tall is it?
During this last interchange, I didn’t say anything, nor did I change my facial expression or body language. I actually remember dropping my shoulders and taking a long, deep breath. I was simply kneeling next to the desk, creating and protecting learning space, and remaining present for learning. The students got it right because they had the time and space to do so.
Another way my practice has changed since returning to the classroom is with respect to how my students engage with mathematics. Every math teacher has encountered a student who, when faced with a math problem, says, “I don’t get it. I tried, but I didn’t get it.” In my classroom, it is totally fine to not get it. But what’s not fine is to then fail to engage with the problem at all. So, I created an I don’t get it organizer. If students don’t get a problem, they must use this organizer, which asks them a series of questions about the problem. It requires them to break the problem into smaller chunks, consider the vocabulary in the problem, highlight the numerical values, and identify with precision the moment at which they began to not understand what was happening.
A couple of intended consequences emerged when I began to use this organizer in addition to some absolutely fantastic unintended ones. First, the organizer rewards perseverance. Instead of moving on to the next problem, the organizer rewards the act of wrestling with something difficult. Additionally, it exposes students to math vocabulary that they might otherwise have skipped over. Finally, it emphasizes the importance of locating the moment within a problem where students got lost. This is great stuff, but the unintended consequences are even more interesting. As I more frequently used this organizer, I noticed students opting back in to the original math problem because they saw it as a path of lesser resistance over filling out this page-long organizer. In a sense, the organizer is calling a student’s bluff - do you really not understand or are you just opting out? Secondarily, as students use this organizer, they start to not need it as much: they begin to internalize the questions asked of them and are better able to ask themselves in real time as they are solving a problem, without the aid of the organizer.
So how does deeper learning happen in my classroom? How can it happen in yours? I encourage you to wait longer amidst student discussions and to assess the specific behaviors you want to develop in your kids. You might be surprised by the outcomes. Think Einstein: “I never teach my pupils. I only attempt to provide the conditions in which they learn.”