Memory

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.

Two Things

I love to read research about education. I try to use research to inform my teaching. But the reality is that research is full of correlations, contradictions, and vague advice that doesn’t feel applicable to classrooms.

If I were to summarize the biggest theme in education research relevant to math teachers it would be simple: spaced practice is the most important ingredient in learning. The first thing teachers should prioritize is practice, and whenever possible to space that practice out over time.

If I were to summarize the biggest thing I have learned about teaching that, while there is not much research behind it, is critical for learning it would also be pretty simple: have students practice in as many different forms as possible. Have them notice, wonder, calculate, predict, estimate, guess, check, reflect, describe, draw, move, argue, read, interpret, analyze, and more.

There’s more to teaching than practice. But these days when I think about planning I ask myself, “how do I want students to practice this math?” Then, I ask, “what is the best and fastest way to set them up for success with that practice?”

Homework Is Hard

My daughter gets a list of words to practice for HW. I quiz her on them for five minutes. She goes to class and feels successful because she knows the words. The class can move forward. Everyone is happy. Why do people want to eliminate homework again?

— Zach Groshell (he/him) (@MrZachG) January 7, 2023

I found this tweet really thought-provoking. Homework is a hard problem to figure out! There are no easy solutions. But I did have a few specific reactions to this:

First, if I have an idea for something that takes 5 minutes and guarantees that students will be successful the next day, I will do it in class. I don’t ever want to give homework that is critical to students’ success in my class. I think that’s a good mantra for teaching in general: if it’s important, do it in class.

Second, something I learned from Michael Pershan’s review of the research on homework is that many families want homework similar to what Zach describes in his tweet. Something that is short, consistent, and helps them understand what’s happening in school. It helps families feel like they understand their kids’ school experience without causing undue stress.

Third, the crux of the challenge for me is time. In math, there does not exist an assignment that takes 5 minutes for everyone unless I simply say “spend 5 minutes on X,” which has its own challenges. There are some assignments that will take one student 5 minutes and another student 45. I teach 7th grade, and I am very confident that giving 45 minutes of homework for a single subject at that age is unproductive. Most families don’t want tons and tons of homework. Michael’s research review validates this: there is not a strong association between the amount of homework given and the learning benefits. Giving homework consistently, however, is associated with learning benefits.

So there’s the pickle. I want to do important things in class, so homework can’t be critical to student success. Families want homework to understand what’s happening in school. But I need to keep the time constrained so it doesn’t take forever for some students. I don’t know the best answer.

Memory, Remembering, and a Few Minutes of Teaching

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.

The Benefits of Traditional Assessment

This post is a bit tongue in cheek. I’m going to argue that there are some advantages of traditional, tests and quizzes, points-based assessment. My goal isn’t to convince people to use traditional assessment over something else. I’m using a pretty traditional system at the moment and I wrote about my philosophy here, but I’ve also used versions of standards-based grading in the past and felt fine about it. In short, I don’t find the results of fancy new assessment systems worth the effort. (I’m also constrained by a policy at my school that pushes me toward something more traditional.)

My goal is to push back on what I see as lazy thinking. The thinking is: X assessment system isn’t traditional, so X must be good. Traditional assessment systems are an easy punching bag, but there are a few things they do well. I worry that too many teachers are building a system to be “not traditional” rather than “to build the best assessment system possible.” They end up successful at being “not traditional” but failing at a bunch of other important things.

I realize that “traditional assessment” is a bit vague. I’m speaking generally about points-based, quizzes and tests, this category is this percent of your grade types of things. It might be a bit vague, but I’ve seen plenty of teachers reject anything that seems even a little similar to those things because it’s “traditional,” and that’s what I want to argue against.

Here are some benefits of traditional assessment:

Spacing. Spaced practice is an important ingredient in long-term retention. With spacing, students don’t only understand proportions today, they understand proportions in a year or five years. In traditional assessment there might be a quiz at the end of the week, a test at the end of the month, and a final at the end of the semester, each with a bit of review beforehand. This is an effective schedule of spacing.

Avoiding binaries of learned/haven’t learned. Traditional assessment doesn’t try to stamp a particular skill or topic as “learned/didn’t learn” but instead gives a number that is more or less a percentage of accuracy, averaged in some way between quizzes, tests, a final, and maybe some other stuff. Trying to say someone has “mastered” a topic (a common theme in nontraditional grading) misunderstands the role of forgetting, false positives, and accuracy. Traditional grading provides more information about the student’s learning and doesn’t mislead students into stamping something “mastered” when they actually haven’t, or when they’re likely to forget it the next day.

Students understand traditional grading. Imagine you walk into a random classroom and ask a random student to explain how the teacher grades them in that class. First, I believe many students — many more than most teachers would think — would be at least a little confused about how they’re assessed. We’re all worse at explaining this stuff than we think. But students generally understand traditional assessment for the simple reason that it’s the default across most of their school experience. I’ve seen too many teachers come up with some fancy assessment system, but it fails because students just don’t understand it very well and don’t take advantage of it. (I take back this argument for any department or school with a consistent and clearly communicated policy for nontraditional assessment across multiple classes.)

Points communicate value. Students spend lots of hours in school, take a bunch of classes at once, complete lots of assignments every week, and constantly have to make decisions about what to prioritize. Many teachers argue that points are meaningless and artificial. They are definitely artificial, but one important function they serve is to communicate what we value. We can’t value everything equally; points communicate which tasks and assignments students should prioritize for the good of their learning.

Traditional grading can better measure effort. Lots of alternative grading systems try to zero in on how much students have learned. They say, “points are silly because they don’t actually mean anything, what even is a point?” And that’s true: points are arbitrary and don’t have a clear meaning. But that flexibility can be a benefit when teachers create ways to get points based on effort. Now these are never perfect, and often lead to “studenting” behaviors that are not linked to learning. But if an assessment system only captures how much students have learned, it can be discouraging for students who are behind or have a hard time in math. Building in some ways to do well based on effort can be an important motivator and signpost for students to see their progress, even if their progress falls short of the course goals or grade-level standards.

My argument is not that all teachers should use a traditional assessment system. Other systems can be great. It’s perfectly possible to address all of these concerns in an alternative system Lots of traditional assessment can also be bad. My argument is that, whatever assessment system you use, don’t only do it because it’s not “traditional.” There are also plenty of bad reasons to use traditional assessment. Some teachers use it because they don’t know of anything different. Others see points and quizzes and tests as tools to coerce or punish students. Others say, “well you’ll have to take a final in college, so you should get practice taking one now.” Whatever your system is, have a good reason for using it, and do your best to use it well.

Make Ten

This tweet caused a bit of a stir among math teachers:

My sons math homework. I simply can’t roll my eyes enough. Why are we adding needless work instead of teaching simple math. pic.twitter.com/Jaraw9w0hd

— Seth Lawrence (@lawrence_farms) December 18, 2022

The replies are full of teachers defending this assignment and explaining why the “make ten” strategy is an important one that most people use without realizing it.

I agree that this is a good assignment, and is a good way to help students practice mental arithmetic and number bonds. But in my view it’s inappropriate as homework. I think too many educators were too quick to defend this. Just because it’s a good assignment doesn’t mean it’s a good assignment to send students home with. My stance is that homework should never require students to use a specific method or strategy. Homework, if it’s used at all in lower grades, should have as few restrictions as possible. Teachers need to remember that homework can be a fraught time for families. We should keep homework assignments as simple as possible to make homework a painless habit rather than a stressful minefield to navigate. I would absolutely give this assignment to students in lower grades, but in class where students have support to understand what it’s asking and connect this strategy to other strategies they are using.

Teachers should also recognize that, while this assignment is one of the good ones, there is plenty of garbage math that has gone home in the Common Core era in the name of “new math” or whatever. Parents have good reason to respond with skepticism. Math has a public relations problem right now. Sending home assignments that require a specific strategy which parents might not be familiar with exacerbates those problems. If you can’t do that, just don’t give homework at all.

Again, I support the assignment. But it should happen in class, and I don’t support giving this assignment or assignments like it as homework.

Artificial Intelligence and Teaching

“ChatGPT will change everything.”

(In case you missed it, OpenAI released an artificially intelligent chatbot called ChatGPT. It’s pretty easy to set up an account and try it out here. It can do some crazy stuff, from writing essays and recommendation letters to giving ideas for recipes with ingredients I have lying around to answering complex questions with some level of detail and nuance. It’s not perfect and makes plenty of mistakes, but it’s also way beyond anything else widely accessible before.)

I’m skeptical that this will “change everything.” SparkNotes started in 1999. Photomath was released in 2014. Have these tools changed teaching? To some degree, yes. But have they changed everything? I don’t think so.

I’m skeptical that we’ll look back in a few years and say “ChatGPT was the moment that education changed forever.” But I do have one wish. We live in a world where computers can do a large majority of the tasks we give students. Solving a math problem, writing a paragraph or essay on many topics, explaining a concept, making an inference. ChatGPT can probably do it.

For a long time the default paradigm in teaching has been that during class the teacher delivers information, and after class the students practice or produce something. In math class the teacher explains a concept, after class students do problems 1-31 odd for homework. In English class the teacher leads a discussion of the chapter, after class students write about what they’re reading. In science class the teacher explains the anatomy of amphibians, after class students practice labeling diagrams. Sure there are exceptions, but they’re exceptions because it’s so common to deliver information in class and have students produce for homework.

My wish is that more of that independent work happens in class with support from the teacher. There can still be homework, but not “ok class is over, remember to write those three paragraphs for class tomorrow” or “ok class is over, complete the DeltaMath homework before class tomorrow.” Instead I’d love to see, “ok now that we’ve spent 20 minutes outlining and drafting make sure those three paragraphs are complete for class tomorrow” or “ok it looks like everyone’s gotten a good start with their DeltaMath the last 10 minutes, finish this for class tomorrow and bring any questions you have.” Cheating will always exist. But for some students, cheating is a natural response to feeling confused and dumb while working on those assignments outside of class. They aren’t sure how to solve the problems for math class, or they don’t know where to start for their English essay. If ChatGPT pushes more teachers to bring that work into their classroom, with systems for support and accountability, it will be a good thing.

One response to ChatGPT is more surveillance, trying to catch students cheating. My preferred response: if it’s important for their learning, have students do (most of) it in class. It’s not surveillance, it’s support.

Cutting Some Standards

I’d like to reduce the number of standards math teachers are required to teach. The less we have to teach, the better we can teach what’s left. Technology continues to advance. It doesn’t make math irrelevant because of computers or AI or whatever. But the same way we left slide rules behind decades ago, it’s time to let a few more things go so we can do what’s left better.

If someone gave me a magic wand to change K-12 math standards I would have two goals: reduce the time students spend practicing complicated calculations, and make math less sequential.

Here are three changes I would advocate for in the elementary/middle grades that are relevant to my current position:

Some people argue that students shouldn’t have to learn things that calculators can do for them. I disagree. Understanding arithmetic and knowing how to use different algorithms create fluency with foundational skills and an understanding of place value. But that doesn’t mean students should spend their time perfecting more and more complicated algorithms in third, fourth, fifth, and sixth grades. I think the whole-number algorithms should be part of the curriculum, but addition and subtraction fluency should end with numbers up to 100. Students should see how to extend the algorithm for larger numbers but shouldn’t spend time practicing it. Multiplication by hand should end at one digit by three digits and two digits by two digits. Those algorithms help students see place value at work, and nothing bigger is necessary. Finally, long division should end with one-digit divisors and three-digit dividends. The goal with each of these algorithms should be to help students understand how to calculate, to reinforce their understanding of place value, and to give them intuition so they recognize when an answer does or doesn’t make sense.

Next, fractions. Fractions are important. They are the foundation for proportions, slope, and more. But calculating with fractions often devolves into minutiae that real humans never use. The first thing we need are small, cheap, four-function calculators that can 1) perform operations with fractions, and 2) do so in an intuitive and easy way. Imagine a four-function calculator with an extra row of keys for entering fractions, and a display that formats those fractions accurately. Once we have those calculators, I think we should keep many fraction operations but reduce the complexity students need to do without a calculator. Adding and subtraction fractions should emphasize common denominators, denominators where one is a multiple of the others, and denominators of 2, 3, and 4. No more convoluted problems adding fractions with denominators of 6 and 8 or 7 and 10. Similarly, mixed numbers should play a much smaller part in the curriculum, with no mixed number operations with unlike denominators and no mixed number multiplication or division. Finally, division of fractions should be limited to divisors that are unit fractions. All of those other operations are fair game when calculators are allowed — where it just assesses whether a student knows how to use a tool, not whether they remember an algorithm most adults will never use. The goal is for students to understand how fractions operations work without getting lost in unnecessary calculations. Too many students never understand what fraction division actually means, or are too busy converting mixed numbers to improper fractions to think about what they mean. These narrower goals would help build number sense for fractions while giving students the tools to move forward even if they struggle with a few of the pieces.

Finally, where fractions are used. Fractions come up in the real world. I see fractions all the time in recipes. But in too many cases our current regime of tests and accountability throws fractions in everywhere. In proportions problems, as coefficients in complicated equations that require combining like terms and distributing, in systems of equations, and more. I think fractions should be radically reduced on contexts that are not intentionally assessing fractions. Some fractions are still appropriate, but they should primarily be unit fractions and fractions with denominators of 2, 3, and 4 to both match what is most likely to come up in the real world, and avoid assessing fractions skills when we really want to assess other concepts. Finally, calculators capable of operating on fractions should be allowed, always. The goal is to assess the skill in question, not a bunch of unrelated fraction and arithmetic skills. Fractions should come up — in particular, they are critical to understanding slope. Let’s not get rid of fractions everywhere. But we can be judicious in where fractions are appropriate, and give students tools so that fractions aren’t an unstoppable roadblock.

Math is sequential. That’s inevitable. But I worry that we overemphasize how sequential math is. Sure, plenty of things are hard to learn if you lack the right foundation. But it’s not true that, in order to learn 8th grade math, a student has to have mastered every skill in 7th grade math. Ubiquitous technology means we can outsource more (although not all) of those foundational skills to machines that can calculate for us. School should reflect this reality. Too many students fall behind in math and feel like it is impossible to catch up. These students constantly encounter barriers communicating that math isn’t for them. The less math class relies on previous learning, the more students we can engage.

Asking ChatGPT About 7th Grade Math

I thought it would be fun to ask ChatGPT, a conversational AI that everyone is talking about this week, some questions about 7th grade math. I picked a few ideas that are often tricky for 7th graders to understand and that I often have trouble explaining clearly. I found ChatGPT’s answers really interesting. Some are spot on, and some are completely nonsensical.

First, percents:

Really clear. Probably better than I could do for this one.

Next, constant of proportionality:

That’s a pretty good explanation! A little algebra-heavy for my taste but no issues that I see.

Next, negatives:

Now I recognize that my question isn’t totally clear — multiplying two negatives makes a positive, but adding two negatives does not. Still, I’d expect the AI to explain that distinction rather than spewing some nonsense about how algebra and arithmetic are different and then incorrectly adding -5 and -3.

Next, equations:

It starts with a valid explanation: the difference has to do with the order of operations. But the computer made a few mistakes while solving both equations and doesn’t seem to understand that the goal is to find a value of x that makes the equation true.

Next, the triangle inequality theorem:

Lol. It doesn’t even know how to apply the Pythagorean Theorem, which is irrelevant anyway! Disappointing.

Next, probability:

This is perfect. For some reason this AI crushes percents, proportions and probability, but struggles with most of the rest of 7th grade math.

Some Thing I Don’t Do

I’m probably a typical math teacher in lots of ways. But there are some things I don’t do that I think distinguish me from many other teachers. I’m not trying to be counterculture, and I’m not even convinced this is the right way to teach or anything like that. But this is how I teach and the way I make math class work for me.

I also don’t blame any teacher for teaching the way they’ve seen teaching done before. These are all things I used to do but don’t anymore. I did them because that’s how I was taught or told to teach. There are probably tons of other things I do not. because they’re a great idea, but only because I’ve never thought about doing them differently.

I don’t break problems into little pieces and question students through them bit by bit. Imagine some multi-step percent problem, like how much does the sweater cost after a 10% discount with 7% sales tax. I could be like, ok class, what is 10% as a decimal? Jimmy? Ok great, then how do I find 10% of the cost of the sweater? Jenny? Ok great, now does a discount mean I add or subtract… and so on. I don’t do that, or anything like it. First, I find it stressful, for me and students. It puts students on the spot, it creates lots of opportunities for confusing tangents, and it’s easy to convince myself that a student understands when they don’t. If I want to explain a concept I try to make my explanation as clear and concise as possible. If I want to see students try a problem I have them all try it themselves and circulate around the room. If I want to explore student mistakes I select and sequence that mistake ahead of time, not by taking a hand and engaging in a back-and-forth in front of the class.

I don’t call on lots of raised hands. It’s a biased sample. Just because a student who raised their hand knows something doesn’t mean everyone else knows it. I could cold call — and I do cold call under some very specific circumstances — but in general I don’t cold call and I don’t find taking hands a very effective strategy. Some students will never raise their hand. Rather than spending my time coercing them or begging them or bribing them to raise their hand, I focus on having students solve problems, looking at their thinking, and sharing that thinking with the class more purposefully than taking hands.

I don’t badger kids about showing their work. Don’t get me wrong, showing your work can be important, especially for certain types of problems. But I find that math teacher mantras about showing your work focus more on control than learning. It doesn’t mean “show your work,” it means “show the work I want you to show.” I do often ask students to explain their thinking and practice showing their thinking clearly. But I avoid doing it as a “do this every single time” type of thing.

I don’t badger kids about asking questions when they’re stuck. I often see teachers put all of the responsibility on students to ask a question if they are confused. I do want my students to advocate for themselves, no question. But teachers often overestimate students’ ability to do so. Often students don’t know what questions to ask. And some students feel confused almost every day in math. Should they raise their hand every time? Should they say, “I have no idea what’s going on, can you explain that again?” There’s a social cost to being that student always asking for help. While I want students to advocate for themselves, I also want to hold myself accountable for checking in with students, starting conversations about how I can best support them, and helping them in ways that don’t require outing oneself as confused in front of the class.

I don’t give much feedback on assessments. The majority of students won’t read it. Some students will actually do more mathematical thinking by knowing they got an answer wrong and trying to figure out what they screwed up. I am available if students want to discuss their mistakes and I create space for students to ask those questions. To be clear: I’m not saying no students read comments teachers write. I know some do. I’m saying that very few students learn from comments teachers write.

I don’t make kids work in pencil. I don’t understand where this comes from. Mistakes are worth learning from and not erasing. Even if you don’t like that, you can always scribble something out.

I don’t give lots of notes. I find it a waste of time. Students aren’t thinking when they’re copying off the board. It’s sometimes useful to have something to reference, but I find those types of notes — at least for middle school math — are useful way less often than math teachers would like to think. I’d rather prioritize giving pre-made notes, sharing worked examples, and posting anchor charts. Then, all that time students would spend copying things down they spend solving problems.