Situation: we’re working on adding and subtracting integers in 7th grade. Students are having a hard time remembering how to add two negative numbers, like -4 + -7. What do I do? This might seem like a simple problem. Explain how to do it and have them practice it, right? But there are a ton of small decisions I need to make along the way. Answering this question led me to a more refined way of looking at human memory, a new technique to figure out what my students do and don’t know how to do, and a clearer idea of the role understanding and memory play in learning. Here we go:

**How to Remember Things**

I have not been able to remember the difference between affect and effect for my entire life. I always have to look it up. Recently I read this great post by Michael Pershan and I think I now understand why.

Here’s my summary of Michael’s post: The best way to remember something is to practice remembering is successfully. The heart of Michael’s post is asking students to practice multiplication facts with flashcards. When they don’t know a fact, they take that card and put it 2-3 cards back in their stack. That way, a few moments later, they have the chance to successfully remember it and start building that memory. The reason I could never remember affect/effect was that I never remembered it successfully. I would need to know it, I would look it up, but then by the time I needed to use it again I would have forgotten and need to look it up again. I was never remembering it successfully. So I set myself a few reminders over a few days to practice remembering the difference while I was working on this post. After a few days, I’ve got it! Fun!

**Mini Whiteboards**

Over the last month I have become a convert to using mini whiteboards as a teaching and formative assessment tool. I’d had them on my list of things to try for a while, and it was this video of Adam Boxer that pushed me to take the plunge. The core argument was that, when you ask a question and a student answers, you get one piece of information. When you ask a question and have every student answer on mini whiteboards, you get 25 pieces of information. Why not get 25? Additionally, mini whiteboards put you in a position to respond to that information right away.

So back to the situation from the beginning of this post. I have been using mini whiteboards in this unit to practice different types of integer addition and subtraction problems. I’m giving students problems to solve. Each student writes their answer on their mini whiteboard and, when I say go, they all hold them up for me to see. I can see every student’s answer, and I realize that much of the class is struggling with adding two negative numbers, like -4 + -7, often saying 3 or -3.

**Understanding**

Some people will say, “the real problem is understanding. They don’t understand the meaning of a negative number here. If they understand it, they won’t make that mistake.” Those people are wrong. Understanding plays a role, I will get to that later. But no matter how well you understand something, practice is necessary to make that understanding solid and durable.

Now my first response to the situation above is simple. Give a quick explanation about why adding two negatives works the way it does. (“Imagine I owe Jimmy $4 and Johnny $7…” or “Remember floats and anchors? If I have 4 anchors and then 7 more anchors…”) Then, give students a few quick chances to practice adding two negatives. Here almost all students will get it. The issue is, if I stop here and ask them again tomorrow, many of the same kids will make that mistake again. Why? (Hint: it’s not about understanding.)

**The Human Mind**

Here is a simplified model of the mind during practice like this.

There’s the world, which in this case is me asking questions and soliciting answers. There’s working memory, which is what students are thinking about at any given time. Then there’s long-term memory, where students are drawing knowledge from to bring into working memory. The blue arrow is me asking questions. The red arrow is learning — when I explain how to add two negative numbers, that arrow is the knowledge getting moved from working memory to long-term memory. The purple arrow is the student remembering how to do something — “ok, this is how I add two negative numbers.” The black arrow is the student writing that answer down and showing it to me.

Here is my key insight: telling a student how to add two negative numbers (the red arrow) is different from them remembering how to add two negative numbers (the purple arrow). Telling them and then having them do another problem right away is also different. In that situation the knowledge is already in working memory, so I’m only activating the black arrow and not the purple arrow. That would be like me looking up affect/effect and then quizzing myself five times right away. I’m not actually remembering it, it’s just floating around in my working memory.

**My Solution**

Back to the situation. Here’s the key modification. I give my explanation and we practice a few problems adding negative numbers. Then I give them a few different problems, ideally problems they know how to do (I don’t want to overload their working memory with too much challenge here). Maybe 5 + -3 and 8 + -2 and -1 + 2, nothing too crazy. Then I ask another question adding two negatives, like -3 + -5. Here I’m actually making students use the purple arrow — the knowledge left their working memory while they solved a few other problems, so they have to remember how to add two negatives again rather than repeat what they were doing moments before. This is the most important moment in the whole sequence. If lots of students get it wrong I can repeat the process again. If everyone gets it right I can focus on a new subskill, and throw in 2-3 more examples like this one as we move forward to activate that purple arrow again. If a small number get it wrong, I can check in with them later in class and try to figure out what the issue is.

Here’s the lesson: if I want students to remember something, they need practice remembering it. They might need another explanation, and it’s helpful to practice a few times right after the explanation to strengthen that red arrow. But it’s the purple arrow that matters most, and I want to design practice so I’m activating that purple arrow as often as possible.

**Understanding, Again**

I want to come back to the idea of understanding. I’m not against understanding. But I’ve emphasized understanding in my teaching forever, and students still have trouble remembering how to solve problems like this. Understanding is necessary, but not sufficient. Where does understanding fit into the diagram above? I have three ideas for how it supports learning.

First, understanding connects new learning to old learning. If I only say “when you add two negatives you do it like this” I’m missing an opportunity. Students understand lots of metaphors for negative numbers. There are some that they bring to school — owing money, temperature, maybe elevation. There are some that we talk about together — floats and anchors is my big one. Understanding is a chance to connect what we are learning to what students already know. That connection makes the learning more robust, and makes it more likely that the red arrow “sticks” because it has something to stick to.

Second, understanding builds the foundation for future learning. If students only remember that you add negative numbers this arbitrary way that Mr. Kane explained and drilled us on, they miss the opportunity to build more robust knowledge about negative numbers. Soon after this lesson students will grapple with subtracting a negative. The more metaphors they have for negative numbers, and the better they understand those metaphors, the better they will be able to assimilate a new and complex idea — that red arrow will have more to stick to.

Third, understanding acts as error correction. Later in the unit students will have to know how to add two negative numbers (-3 + -4), subtract a negative (5 – -6) and multiply two negatives (-6 * -8). It’s easy to mix those up. In that moment when they see one of those problems, when they think “how do I do this?” and reach for that purple arrow, understanding helps to ensure the arrow comes from the right place. “Ok, adding negatives is like adding debts, so it must be…”

**Finally**

Two final thoughts. First, this is the intellectual part of teaching I love the most. This is a a few decisions in a few minutes of class, but these decisions determine whether students can solve one of these problems, or if they struggle to remember and get frustrated over and over again. These micro-decisions are so important, but also under-discussed in teaching. Second, the model of the human mind I shared above is incomplete. Folks who know more than me could point to all sorts of places I’m oversimplifying or ignoring important parts of cognition. But teachers need simple models. Any model that captures the full complexity of human cognition is too complex to guide the moment-by-moment decisions teachers make every day.

revuluriAnother great post! Absolutely agree that there is a key step in making impact on student learning in going from robust, relevant research findings to tools and structures that allow implementation at scale, and simplicity is essential.

Michael’s previous post (http://notepad.michaelpershan.com/what-people-get-wrong-about-memorizing-math-facts) explicitly links to retrievalpractice.org, which is an AWESOME site getting into spaced practice and more.

And if you like what you find there, I’d strongly endorse both “Make It Stick” and “Powerful Teaching” — insightful, well-written, focused, and applicable.

dkane47Post authorI had never clicked through to retrievalpractice.org, thanks for sharing. A lot of good stuff to dig in to there!

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