Some people will have an allergic reaction to this post just from the title and the subheads. They’ll say, “well that’s just memorization, what I really care about is…” I can’t emphasize this enough: all learning happens in memory. If you’re a teacher who doesn’t emphasize memorization and only tries to support conceptual understanding or whatever else: those still happen in memory! Memory isn’t only for facts and figures. If you’ve ever begun class one day and discovered that your students don’t remember what you taught them the day before — and every teacher has — you can benefit from a better understanding of memory.

Every teacher has things they want students to remember. We can disagree on exactly what needs to be memorized and what doesn’t — that’s a really interesting topic I will return to in a future post — but there will always be stuff students will benefit from remembering. (I’m not going to engage with the “but you can Google anything…” or “but ChatGPT…” crowd, they’re wrong but that’s a topic for another post as well.) Here’s my goal: spending tons of time helping students remember stuff sucks. I want students to remember things, and I want that process to be as efficient as possible. I’m not arguing that all of math class should be memorization. Remembering things is just one important element of math class, and we should do it as efficiently as possible to save time for all the other stuff. We should also do it well, because when students see themselves learning and feeling successful in math class, it helps them show up motivated and ready to learn.

Here is a model of the mind during learning. This isn’t a perfect model. It leaves out or oversimplifies lots of things about the mind. But it’s helpful for me because it helps me understand memory without getting into myelination or the cerebellum or basal ganglia or whatever else.

Briefly: there’s the world and all of its stimuli coming into our brains through the blue arrow. There’s working memory, which is where thinking takes place but is limited in size. We can only think about a few things at once. When we think about stuff it creates a trace in long-term memory via the red arrow. Long-term memory is more or less unlimited in size. That stuff in long-term memory is organized and connected in lots of different ways. When we try to remember something we retrieve it from long-term memory with the purple arrow. Finally, once we have something in working memory we can answer questions and communicate with the world using the orange arrow.

This model of memory is useful because it helps me understand eight key facts about memory that shape my teaching.

Memory is the residue of thought (Daniel Willingham, Why Don’t Students Like School). Students learn what they think about. I can explain an idea all I want (the blue arrow), but it’s what’s happening in the student’s mind that causes learning (the red arrow). This leads to two corollaries. First, I need to constantly ask myself whether students are thinking. Thinking is hard, and it’s natural for students not to be thinking. If they aren’t thinking, they aren’t learning. Second, what are they thinking about? That fancy project where students make posters about integer operations might mean that students spend a lot of time thinking about coloring things and not a lot of time thinking about integers. The blue arrow is different from the red arrow — my goal is to get them thinking about things so those things move into long-term memory.

Remembering is the residue of remembering. If I want students to remember something they need practice remembering it. Once step one has happened — once I’ve gotten students thinking and they have an initial concept in their long-term memory — I need them to use that memory. If I want them to remember it, I need them to retrieve it, repeatedly and spaced over time, to strengthen the purple arrow. The red arrow isn’t enough — that gets the process started, but it’s practice retrieving with the purple arrow that creates a robust and durable memory. The more a student remembers something the more likely they’ll remember it in the future. The concise name for this process is retrieval practice. One key is that retrieval has to be successful to reinforce long-term memory. If a student tries to remember something, is unsuccessful, and looks it up instead, they’re not actually using the purple arrow. If they think about it they do use the red arrow which helps a bit, but isn’t a substitute for actual retrieval.

Repetition is not the same as remembering. I want to strengthen that purple arrow through retrieval practice. If I want students to remember how to find the circumference of a circle I might want to give them 40 circumference problems in a row. That way they have to use the purple arrow 40 times and it becomes strong, right? But if students are doing the exact same thing over and over again, they aren’t actually retrieving it from long-term memory 40 times. They’re retrieving it once at the beginning, and then holding that knowledge in working memory while they solve those 40 problems. They’re using the purple arrow once, and the orange arrow 39 times. This is why interleaved practice (mixing together different problem types), and spaced practice (practicing sessions spaced over days of weeks) are so effective. They require students to retrieve, and not only recycle what’s already in their working memory.

Understanding happens in memory. Some people will read my last few paragraphs and say, “well that’s all memorization. I don’t want students to memorize things, I teach for understanding.” Me too! But understanding needs to be remembered the same as anything else. Going back to the diagram, there are two types of things students need to remember. The first are facts, like multiplication facts or the formula for circumference or how to solve two-step equations. These are the dots in long-term memory. (These are important to remember, by the way, and I’ll come back to why later in this post.) The second type are the connections between facts. For instance, I want students to remember how to solve two-step equations. There are a few different types, and I could have them memorize each type separately. But a better approach is to use the common underlying principle of inverse operations to solve. That’s an idea that will serve students well over and over again down the road. That’s the connection between the dots. And students need practice remembering that understanding, the same way they need to practice facts. One way to do this is self-explanation — asking students to summarize in their own words an idea they have been learning. Self-explanation is like retrieval practice for understanding. Students have to retrieve the connections and context linking different things in long-term memory to build understanding. There are other ways to have students use retrieval practice for understanding, but self-explanation is a good model to start with.

Understanding helps knowledge stick. That network of connected ideas in long-term memory in the diagram? That’s what we aim for. The more connections there are, the more knowledge new stuff has to stick to. Rather than starting a new purple arrow from scratch, understanding lets us build off of all the other purple arrows we already have for related knowledge. When there’s already a lot of background knowledge on a topic in long-term memory, it acts as a primer, as if the arrow has a head start. This is the feeling when something “clicks” and makes sense in a way that didn’t moments before — your mind figured out where the connection is. If you’re familiar with the memory palace technique for remembering lists or similar everyday minutia, that memory strategy leans on connections with things that are already securely in long-term memory to remember new things. This is why teachers should begin with a review of relevant prior material to activate background knowledge, and emphasize the ways that new ideas connect to old ideas, or to contexts that students are familiar with.

Understanding acts as error correction. Knowledge fades over time. Inevitably, students will need to retrieve something and won’t be sure whether they’re remembering it correctly. We also want students to solve problems they haven’t seen before, where they are uncertain what knowledge to draw on. Understanding kicks in when students aren’t sure what to retrieve, or what that purple arrow should start from. Robust understanding can verify that what is being remembered makes sense and is consistent with other knowledge, or help decide between a few possibilities. This creates an opportunity for successful retrieval and the virtuous cycle of remembering begetting remembering. Without understanding, the risk is reinforcing incorrect knowledge or having to look something up and missing the opportunity for retrieval altogether.

Learning doesn’t work when working memory is overloaded. The last piece of the diagram is that little box in working memory. Working memory is severely limited. Humans can only hold a few items in working memory at once (the exact number is debated and seems to depend on context). The strength of that red arrow from working memory to long-term memory depends on how much of working memory is devoted to it. If I’m thinking about exactly one thing, I’m doing a better job of imprinting it into my memory than if I’m thinking about six different things. If I’m tangled up in problem solving, thinking about a ton of different things at once, any individual thing is unlikely to stick. Similarly, if I’m consumed by trying to make a joke to impress my friend, or wondering whether that other kid is going to tease me about my hair after class, that is taking away from learning.

Problem solving doesn’t work when working memory is overloaded. Imagine I have to solve a system of equations that includes the expression -5x + 8x. Ideally I can retrieve right away how to combine like terms, know that it’s equivalent to 3x, and move on to the next step. But if I have to go back to a number line strategy to find -5 + 8 or cast around for metaphors for positive and negative numbers, I’m bringing more stuff into working memory. Then, when I try and move on to actually solving the system of equations, I can’t hold everything in my mind, and I either make a mistake or feel overwhelmed and give up. This is why it’s important to commit foundational knowledge to memory. But there’s something tricky about this concept. When we have a lot of knowledge secure in long-term memory — and most teachers do — we don’t realize how critical it is for problem solving. We retrieve that knowledge, use it to move to the next step, and continue through the problem solving process without realizing how crucial that step was. It’s easy for experts to fool themselves into thinking foundational knowledge isn’t important, precisely because our foundational knowledge is so secure that we don’t realize it’s there.

In conclusion. This is a post about helping students remember things. There are two reasons I think all of this is important. The first is that memory has developed a negative connotation in education. My knowledge of memory has helped me see all the ways that deep understanding and problem solving are built off of memory. I don’t want to set up a false dichotomy between helping students remember things and helping students solve problems. Second, there’s plenty of mundane stuff that, while it’s not very exciting, is important for students to remember. If I use inefficient strategies to help them remember those things, like assigning 40 identical problems in a row, not using retrieval at all, or not building connected understanding, I’m wasting time that could be spent on more valuable classroom activities.

Here’s the diagram one more time. Again, this isn’t an exhaustive or perfect model of the mind. But it’s the understanding that links these ideas together, and helps me apply them in my classroom.

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