Something different: for my post next week, I’m going to answer reader questions. Any questions you like, big or small, about this post, other posts, something else, whatever. Drop them in the comments or email me, and I’ll answer them in a mailbag post.
I’ve been thinking recently about what kind of teacher I am. I don’t like labels very much, and I don’t feel like I fit well into any box.
If I had to put myself in a box, the word I like is “empiricism.” I have tried just about everything under the sun in math education. I try to stick with what works.
In particular, I stick with what works for the broadest array of students. I’ve worked for three schools as a full-time teacher. If students are motivated, or the school has a strong culture, or other teachers do good work sending me students with skills and habits to be effective math students, I don’t have to get everything right. Without those affordances, I have to be on my game. There are lots of teaching strategies that work well in one school but fall apart in another. The best teaching strategies work no matter where you’re teaching.
Empiricism
The core of empiricism is checking for understanding.
In every class, at least once, I have students grab mini whiteboards. I ask them a question, have every student write their answer on their mini whiteboard, and hold them up on my signal so I can see every student’s answer. I do this to see if students understand the day’s lesson, remember what they learned yesterday, and more.
I’ve made a ton of changes to my teaching in the last few years, and this type of checking for understanding is the core of where those changes come from.
I realize it might sound trivial to say, “I stick with what works.” Doesn’t every teacher? Unfortunately, I don’t think so.
First, one of the unusual things about education as a field is that pretty much everyone experiences education as a student before becoming a teacher. Teachers often begin their career with very strong priors about what type of teacher they want to be. And the reality is, most teachers don’t do very much systematic, full-class checking for understanding. In that context, it’s easy to stick with your preconceptions, or focus on what seems to work for the most vocal students.
Second, I’m defining what works as what helps students learn. There are all sorts of incentives in school that push teachers to focus on all sorts of stuff besides learning. Classroom management. Keeping admin happy. Keeping parents happy. Getting students to like you. All of those priorities are competing for our attention at all times. It’s tough to filter out the noise and focus on learning.
I’m not trying to blame anyone here. I’m arguing that there’s this very strong status quo: some students learn, some students don’t. That seems like the best we can do, so we stick with our preconceptions or follow the path of least resistance. Checking for understanding is scary. It often reveals that many of our students haven’t learned what we thought they’ve learned. But checking for understanding is also the best way to figure out if changes to teaching are making a difference.
Cognitive Science
The second major pillar of my empiricism is cognitive science. Cognitive science provides the most parsimonious, practical explanations for why some teaching works and why other teaching doesn’t. This isn’t a full post on all of cognitive science, but here are the basic principles that I’ve found helpful:
Attention is a limited resource. If students aren’t paying attention to something, they won’t learn.
Working memory is where we think, but it’s easily overloaded. If I’m asking students to think about too much at once, they will become overwhelmed and learning will be compromised.
Long-term memory is the residue of thinking in working memory. The goal is to get students thinking, and what they think about is what they will learn.
Effortful processing leads to more durable learning. Not all thinking is equal. I want students to think hard, to think about the deep structure of what we’re learning, and to make connections between ideas. That type of thinking leads to the most useful learning.
Schemas help us to retain and apply learning. Long-term memory isn’t a pile of facts, it’s a network of connections. The goal of teaching is to strengthen those connections between ideas, so knowledge is more easier to apply in new situations.
Retrieval makes learning stick. The best way to strengthen a memory is to retrieve it — to reach into long-term memory and pull it out. That act of retrieval strengthens memory, and it’s critical to learning that lasts.
The Rest
Checking for understanding and cognitive science are great, but there’s more to teaching. Here are a few other practices I’ve found helpful.
Break learning down into small steps. Teaching works best when I teach one thing at a time. That means breaking learning down into small, manageable chunks. Teachers often try to teach multiple ideas at once without realizing it. I’m on a constant quest to break learning down into smaller steps and teach one step at a time.
Check for prerequisite knowledge. Learning is cumulative. But that doesn’t mean that students need to know every single thing they’ve learned to access the day’s lesson. Each new topic has specific prerequisites, and the best way to start a lesson is to check these prerequisites and do a quick reteach when necessary.
Connect to prior learning. The most powerful learning is a network of connected knowledge, linking what students already know what we want them to learn. I don’t want to leave these connections to chance.
Practice. Practice is important, but the structure of that practice matters. The best practice comes in multiple rounds of short chunks, with chances for feedback. Too much practice at once puts students on autopilot and stops thinking. Only practice something once and it’s not likely to stick. Effective practice finds a balance by spacing practice out to improve retention, but not practicing too much all at once for efficiency.
Obtain a high success rate. Keep practicing until students are successful. This seems obvious — if students can’t do the thing and you move on, it’s not very likely to stick. A high success rate also has an effect on motivation: students generally prefer doing things they feel good at, and continuing to practice until students are successful helps to support motivation.
Scaffold challenging tasks. There are two parts here. First, give students challenging tasks! It’s easy as a math teacher to focus on predictable, procedural skills. And sure, teach those procedural skills. But also help students to apply that knowledge in lots of different ways. The best way to scaffold these sorts of challenging tasks is to build gradually from what students know to what I want them to figure out. I wrote more about this in my Expansion post.
Retrieve. Practice is important, but it doesn’t stop once students have practiced a few times and achieved a high success rate. I keep asking students to retrieve what they’ve learned so it sticks in long-term memory. If students don’t remember something, I reteach and then start the cycle again. Retrieval was in the cognitive science category above and it connects to practice, but it’s so important it’s worth reiterating.
Hold students accountable. I try to communicate, in everything I do, that my classroom is a place where I expect students to participate and think hard about math. I ask students lots of questions, all the time. If a student isn’t following through, I want them to know that I care about their learning and I will hold them accountable for doing math every day. I wrote more about this in my High Accountability Teaching post.
This Is Boring
This all sounds incredibly boring. And hey, if you dig in the archives of my blog, you’ll find lots of clever fun stuff I do with students. That fun stuff is important. Students spend lots of time trapped in my classroom, it doesn’t need to be ruthlessly efficient every second. But what I’ve found is that it’s not the clever fun stuff that matters most to get students learning. It’s all the boring pieces. The connections between topics, the sequencing in the curriculum, the choices of problems in retrieval practice, the little teacher moves to send a message of accountability. All those little moves add up. It’s my job to help students learn math, and successful learning feels good. Lots of students hate math class not because it’s boring or irrelevant, but because trying to learn math makes them feel dumb. Successful teaching does the opposite, and while it’s not flashy it’s what a lot of students need.
These little moves sound simple, but they’re much harder than they sound in a blog post. With a full class in front of me, with the demands of the curriculum, with my limitations as one human, and new content to teach each day, every lesson is a little puzzle. That’s what I love about teaching, trying to figure out the best way to get lots of the regular, everyday elements of teaching to fit together in a way that helps every student learn.
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In his book Tools For Conviviality, technology philosopher and social critic Ivan Illich identifies these two critical moments, the optimistic arrival & the deadening industrialization, as watersheds of technological advent. Tools are first created to enhance our capacities to spend our energy more freely and in turn spend our days more freely, but as their industrialization increases, their manipulation & usurpation of society increases in tow
Illich poses convivial tools as directly opposed to this industrialized, radically-monopolized set of social systems. Similar to E.F. Schumacher’s concept of “intermediate technology” introduced in his 1973 book Small Is Beautiful: A Study of Economics As If People Mattered, convivial tools are sustainable, energy-efficient (though often labor intensive), local-first, and designed primarily to enhance the autonomy and creativity of their users.
