Learning Distributions Redux

I still think sometimes about the ideas behind this old post of mine. I was wondering which distribution I should strive for in my teaching:

I’m still not sure I have a clear answer. But another layer I’m thinking about is how a linear measure of “student learning” oversimplifies how I want to support students. Content knowledge is one thing, but students who have struggled in math class in the past often have negative beliefs about themselves and about their learning. Those beliefs do a lot to undermine learning in the long term. If students show up to class believing that they’re bad at math and math class is pointless they’re likely to continue having a hard time.

One distribution to think about is which students learn the most — is it the students who have done well in math classes in the past, or students who have had a hard time? A second distribution is how I influence student beliefs and dispositions about their own learning. This is an area where I want to prioritize students who have struggled in the past.

I can try to facilitate those beliefs a few different ways. One is having a constant barometer of how successful students feel. If students feel like they are constantly getting things wrong they are likely to internalize that as a part of their identity in math class. If I can start class with tasks that are accessible to every student and scaffold success in challenging work I can help more students to develop positive beliefs. I want to have a barometer on success for all students but pay particular attention to students whose beliefs I want to influence the most.

A second strategy is assigning competence. When I ask a student to share a great idea or a new connection I am publicly signaling that student’s competence. It’s often only a few high-achieving students in a class who regularly have their ideas highlighted. I can keep a special eye out for high-quality work from students who often struggle. It’s important not to tokenize or compromise; students can see through me when I’m celebrating someone without good reason. But when students do great work I want to celebrate it, and I want to prioritize celebrating students who I know often struggle.

I’d love to think more about other dimensions of what I hope students take away from my class. It’s easy to generalize about a group of students — “they learned this” or “they didn’t learn that” — but there’s a wide range elided by those statements. I want to get better at thinking about and acting on that complexity in ways that support all students.

Never Do Anything a Kid Can Do

We must not fool ourselves, as for years I fooled myself, into thinking that guiding children to answers by carefully chosen leading questions is in any important respect different from just telling them the answers in the first place.

-John Holt, How Children Fail

I think “never say anything a kid can say” is bad advice. I realize that it’s a common refrain in some education circles and there was that popular NCTM article. But some of my worst teaching has come from a desire to get kids to say things so I don’t have to. It’s not a terrible idea in the abstract, but it often leads to games of “guess what’s in my head” that don’t lead anywhere. Having students talk in class is great. Having students talk so that they can say the magic words that I decide are important is a waste of time.

Here’s a try at better advice: never do any math a kid can do. Math class should be about students doing math. If they spend all their time watching me do math they’re not learning very much. I think that first advice, never say anything a kid can say, leads to students playing fill in the blanks bit by bit, which is a hollow shell of doing math. Instead I should ask myself, “what is the least I can do to set students up for success doing this math?” When I’m doing this problem, could they be doing it instead? If I try to make students do everything they won’t have the tools to access it they won’t do anything at all. But I want to be deliberate in putting all of the work on students that they can manage. Then I can have them talk with each other about the math they did. They’ll say lots of great stuff, they’ll do math, and we won’t spend any time playing “guess what’s in my head.”

Maybe for some people, “never say anything a kid can say” means “say as little as you can to set students up to do some math then have students talk about that math, wash, rinse, repeat.” That’s not what the linked article above says, and I don’t think that’s the most common interpretation of the phrase. But that’s the advice I wish I’d gotten years ago, and it’s advice that sets students up to learn math and not just to learn whatever I have in my head for them to guess.


I enjoy learning about the human mind and human memory. It’s a huge rabbit hole. A few times I’ve started and stopped writing a longer piece summarizing what I understand about memory. But each time I end up making it really complicated. Memory is complicated, but the lessons I take from it as a teacher are pretty simple. This is my attempt to distill those ideas into a few principles I use in the classroom.

  • Working memory is where thinking happens
  • Thinking about things puts them into long-term memory
  • We can only think about a few things at a time
  • Overloading working memory with too many things at once makes it less likely anything will end up in long-term memory
  • We make mistakes when working memory is overloaded, and also when we mistake two similar things
  • Long-term memory is what we think with
  • Long-term memory doesn’t have a limit
  • Having more stuff in long-term memory makes thinking easier
  • Long-term memory isn’t just facts and figures — it’s also skills, strategies, and feelings and literally everything else we think about and do
  • The more often we use something the more easily we can retrieve it from long-term memory, and this works best when we space out those retrievals
  • Stuff tends to go out the way it comes in: we remember things best in the context that we learned it
  • It’s hard to transfer knowledge from one context to another, but this becomes more likely when we have experience thinking about that knowledge in different ways and different contexts

12 bullets. Not simple, but not the most complex thing in the world either. I’m fascinated by memory and I’m sure I’ll keep trying to learn more about it. There’s lots more to memory than these bullets, but the more I read the more I come back to the same few ideas that feel relevant to teaching and learning.

Taking Hands

I have largely stopped asking students question and calling on a raised hand. And for the most part when I do it it’s because I didn’t plan that part of class very well. I’d be happy if I never took hands for questions I ask again.

One caveat — I’m not saying I don’t let students raise a hand to ask a question. I’m talking about students raising hands to answer questions I have.

My basic issue is that calling on raised hands warps everyone’s perception of learning. Let’s say a student gives the “right” answer. I have no idea what that means. I’m calling on someone who has volunteered their answer, so it’s a biased sample of the class and I have no idea what other students know or don’t know. Let’s say a student gives the “wrong” answer. I could interpret that as a sign that more students are confused but I don’t really know. What’s worse, giving a wrong answer in math class is pretty fraught socially. Now I need to take some time to help that student understand where the confusion came from. Maybe that’s helpful for other students but maybe not, I don’t know.

Here are a bunch of things I do instead of taking hands:

  • Give students a problem or several problems to solve, on their own, with a partner, or with a group. Circulate and pick out one or several students to share with the class.
  • Same situation, but pick out a common mistake to address. I can choose to share it anonymously or have a student share depending on where we are with classroom culture and how socially safe that feels.
  • Same situation, but just offer some summarizing thoughts myself and give students a chance to check their answers.
  • Have students study some worked examples with the full solution written out. Discuss with a partner what they notice, take questions, then have them try one on their own.
  • Spend less time modeling and doing “guided practice” up front. Instead, break instruction up into small chunks where I give explicit instruction and student try problems on their own, and debrief those problems using one of the strategies above.
  • Recognize when an activity isn’t doing well — usually because I underestimated the knowledge students need to be successful — and stop the activity entirely to change course rather than trying to rescue it with lots of questioning.

I could go on. But moral of the story is I find that classic of the math classroom, working through a problem at the board while asking students to raise hands to help me out along the way, a relic of my past teaching and something to avoid whenever I can.

What Gets Measured Gets Valued

One thing I’ve learned about teaching during the pandemic will stay with me: what gets measured gets valued.

Where I live, and in much of the rest of the country, the official criteria for exposure to Covid-19 are simple: if you were within 6 feet of someone for more than 15 minutes in the two days before they test positive for Covid-19, you are considered a close contact and have to go into quarantine.

But the coronavirus doesn’t care exactly how many feet apart you are or exactly how long two people are close together. Distance and time are two important variables in transmission of respiratory viruses, but they’re just two of many. It also matters whether people are wearing masks, the quality of those masks, how many people are present, how confined the space is, whether they are indoors or outdoors, the quality or lack of ventilation, whether the infected individual is coughing or sneezing, exactly how close people are and how long they are together beyond a binary 6 feet/15 minutes, and more.

But once we decide to measure something, that something becomes the focus above whatever we don’t measure. A video came out over the summer of a school board member in Georgia suggesting that their schools can be in compliance with that rule by having students change seats every 14 minutes.

That’s the example that got public attention, but those same conversations have been happening behind closed doors all year. The 6 feet/15 minutes rule has narrowed attention on those two variables, and has distracted from other factors worth thinking about.

The close contact rule takes a complex, messy, uncertain situation and tries to make it objective. And I understand why the rule is what it is! I understand that the rule needs to be simple to enforce. I understand that masks aren’t a factor so that indoor dining to continue with distancing to help keep restaurants afloat. But regardless of anyone’s good intentions, the rule causes people to believe that you can only get Covid-19 if you are within six feet of someone else.

That same thing happens in education. Here’s an example. Teachers often grade for completeness and take points off for lateness. We do so with the best intentions — we don’t have time to look through every answer of every assignment, and late work is a pain to manage. But when we focus on measuring completeness and late work, we send a message that completeness and late work are what we value.

This phenomenon is everywhere. It would be easy to tee off here on standardized tests. Right now the AP Calculus test is on my mind. I need to make sure students can answer those contrived Mean Value Theorem problems and have a ton of practice interpreting a function written as an integral of a piecewise function. But standardized testing is too easy of a target. What we measure becomes what we value in ways large and small, every time we try to take some complex situation and distill it into a simple metric. Learning is always complex and schools are full of incentives; every measurement distorts those incentives and creates new ways to be successful that don’t actually involve learning.

There’s no easy solution, but it’s important to be honest about the ways that measurement distorts values. When we pretend that measurements don’t influence what we intend to measure we amplify the distortions. Measurements are often mathematical, and math is often held up as a higher truth. Those measurements can be taken at face value when they shouldn’t. It’s important to strip away that veneer of objectivity and be honest about what measurements are actually measuring.

For All/There Exists

Two different statements come up often in mathematical proofs: “for all” and “there exists.” “For all” makes a statement that is true in every case. For instance, for all numbers divisible by both 2 and 3, they are also divisible by 6. “There exists” makes a statement that is true in at least one case, but need not be true in every case. For instance, there exist whole numbers not divisible by any whole number except for 1 and itself.

I think a similar distinction is useful when talking about teaching. There are some things in the “for all” category — things that should always be true, that I should strive to do every class, every day. Academic safety falls into that category. Every student should feel like they can take risks, share ideas, and be wrong, all with unconditional support. I should strive never to compromise on academic safety. I fall short on this all the time, but I need to set “for all” as my goal and work to help every student feel safe every day.

There are other things that fall in the “there exists” category — things that students should experience but don’t need to happen every day. Learning through discovery falls here for me. I think every student should experience mathematical discovery. And it’s hard to get discovery right, so this can’t be a one-off every few months. At the same time, I don’t believe students need to discover something themselves to understand it, or need to experience mathematical discovery every class.

There are other examples. Practice is a “for all” — I value practice, and I want students to practice every mathematical concept they encounter. Representation is “there exists” — I can’t show every student a mathematician who looks like them every day, but I can strive to share mathematicians who represent each of my students several times over the course of our time together.

Discourse around teaching can get lost when we confuse “for all” with “there exists.” I need to hold myself to a high standard around academic safety, every day and for every student. But it would be easy to get defensive and say, “but this other student feels safe, so x student should feel safe too!” Teachers face a constant onslaught of decisions and information; I have to avoid cherry-picking examples to fit my narrative. And it’s easy to make the opposite mistake, to take something that should exist somewhere and assume it has to exist everywhere. The value I place on discovery doesn’t mean that every lesson has to be a discovery lesson, and doing so risks losing sight of my true goals. One of the biggest challenges of teaching is how many decisions I have to make a day, and how quickly I have to make them. I hope distinction can help me to better live out my values and avoid lazy shortcuts.


Here’s a tension I think a lot about:

As a teacher I only have so much time. Time per class, classes per semester. I can’t easily fabricate more. I have to make a lot of decisions about time: what can I get done next class? Is there room for that in this unit? Do I go deeper with fewer ideas or try to touch on every little thing?

But constantly worrying about time can make teaching feel industrial. I can justify a lot of shitty teaching by saying “we just didn’t have time” or “its time to move on.”

I don’t have a great answer. It would be nice to ignore the limits on my time entirely, but that would come back to bite me worse later. I think my biggest takeaway is around how students feel in class. I don’t want students to feel rushed, or to feel like they need to get through something to get to the next thing, or that they are being shortchanged because of the limitations of the school schedule. If my students don’t feel like they’re cogs in a machine sprinting from one topic to the next, I’ll take it as a small victory.

Familiar Territory

After I graduated college I road-tripped across the country, stopping in different national parks. After my first year of teaching I backpacked the famous John Muir Trail in California. I had a long list of spectacular and well-known destinations I wanted to see, in the US and beyond. A year later I moved to Leadville, Colorado, excited to climb tall mountains and tick destinations off of my list.

Almost six years later I still love spending time outside, but I have different priorities. I don’t often visit nearby national parks — too crowded and too much time driving. I almost never climb the tallest peaks or ski the big resorts. I’d rather know my home better and better than do lightning tours of the “biggest and best” of the west. Leadville is a bit out of the way but still has great sights to see, and I enjoy exploring new corners of the abandoned mining district east of town, or getting to know a quiet new trail nearby, or finding a new perspective to appreciate what I see every day.

I try to see math the same way. There are spectacular puzzles and fields of math that I find exhilarating. There’s a time and place to share math’s great theorems. But there’s also magic in the hidden corners of everyday school curriculum, digging deeper in familiar territory. In the last weeks I’ve enjoyed finding patterns in equivalent forms of sine functions, exploring all the ways an arithmetic series can sum to 60 or 105, and finding clever new rational functions to graph. These aren’t the great theorems of mathematics and aren’t very interesting to professional mathematicians. But they’re fun for me. I enjoy seeing familiar terrain from a new perspective. And I think finding new puzzles in the everyday content I teach students is a great way to spark student curiosity. Math’s “greatest hits” are beautiful to me but not to everyone else, and especially not to students who have a hard time with math class.

Four Types of Problems

I’ve got an idea that I’m pretty sure is wrong, but is also maybe useful.

I love problems, and I love trying to better understand what types of problems help students learn math. Here’s a rough way to categorize problems:

Two axes. First, does the problem require a moment of insight, or is it something that one can solve by grinding, sticking with the problem, and trying different things? Second, does the problem have one strategy or several possible strategies?

Problems in the top left I would describe as puzzles. Catriona Agg’s geometry problems typically fall in this category. I learn a ton from seeing different ways to solve them, but if I don’t have that moment of insight I won’t figure it out. These problems can have value in the classroom, but only if students have support. If I leave someone without help as they struggle to find the right insight I’m setting them up for frustration.

Problems in the bottom left I would describe as riddles. I often think of the wolf, goat, and cabbage problem as the prototypical riddle, but plenty of everyday math problems turn into riddles when students don’t have the right tools. I’ve developed a pretty strong distaste for riddles. They tend to make people feel stupid, and they don’t lead to much interesting discussion because you either end up in the right place or flail, and there’s not much in between.

Problems in the bottom right I would describe as exercises. Exercises get a bad reputation, but I give my students exercises every day. They’re an important part of a balanced mathematical diet. They’re best in regular but relatively small doses.

Problems in the top right I would describe as discussion problems. These are everywhere, from a derivative problem where students can choose to use the product rule or simplify first, to everyday quadratic graphing problems that students can solve by factoring, making a table, or completing the square. When different students approach the same problem in different ways, and most of the class can work through the problem successfully, that problem lays the groundwork for some great learning. I had a misconception in the past that a problem needs to be rich and unique to be useful for discussion. But there are lots of problems I consider exercises until a student finds a new method, presenting an opportunity for a fun discussion.

One important caveat is that this distinction depends on students’ prior knowledge. A problem that requires insight for one student is accessible for another with different knowledge. An important but often neglected role of the teacher is to give students the knowledge they need to access problems, rather than hoping the problems will do the teaching on their own.

I want to say again that I’m pretty sure this is wrong. The world of problems is too complex to simplify into a neat little two-dimensional diagram. But I do think this can be a useful representation of the problems I give students. I think math class should involve all of the categories except for riddles. Giving students problems is always a delicate balance. I want problems to be accessible so students can feel successful and recognize their mathematical talents. But I also want to give problems that provoke curiosity and give students a glimpse of the beauty of mathematics. I think this way of conceptualizing problems can help me to manage that balance while giving students a rich experience in math class.

The Crest of the Peacock

I recently read The Crest of the Peacock: Non-European Roots of Mathematics. It’s a great read! I’d definitely recommend it. I’ve been trying to blog about it for a few weeks, but I’ve had trouble summarizing what I’ve learned into some neat and tidy lessons. Maybe that’s the point. The history of math is messy.

Two themes of the book are interesting to me. First, the history of math isn’t linear. It’s full of false starts and dead ends. One example is calculus; Chinese and Indian mathematicians anticipated calculus concepts centuries before Newton and Leibniz. That two European mathematicians get credit for inventing calculus is mostly an accident of history. There are tons of other examples of non-European cultures discovering ideas that Europeans later get credit for. We ignore them because they don’t fit into a simple trajectory.

Second, different cultures — including Europe — often tiptoed around ideas we now think of as obvious, seemingly almost there but not quite reaching a modern concept or notation. Place value, limits, functions, algebra, and more were all hinted at for centuries or millenia, across different cultures and contexts. Looking back now it sometimes feels cringeworthy. But everything is obvious in hindsight, and mathematicians were doing just fine with the concepts they had.

I see these themes as useful metaphors for how students learn math. Student learning is often less linear than teachers like to pretend. It happens in fits and starts. While we emphasize the ways mathematical ideas build on each other, that sequence is often arbitrary. Some students grasp “advanced” topics before “simpler” ones. Part of teaching is engaging with that complexity. Students also often seem like they’re close to understanding an idea, but take longer to get there than might be convenient. And that’s normal. Everything is obvious to a teacher, but some of the conceptual leaps that we take for granted are much larger than we realize.

The history of math is fascinating. It’s easy to tell a story about the Greeks, the Middle Ages, and modern European mathematics. It’s neat and tidy and confirms our preconceptions. But that history silences incredible mathematical learning in cultures around the world, learning that doesn’t fit into a little box. And that history also paints a portrait of math that sands away the rough edges and oversimplifies complexity. I think engaging with that complexity is a great way to see math with new eyes and new perspective.