What it’s like to not know something

I’ve been reading the book “Metazoa” by Peter Goddfrey-Smith, and I was struck by his description of the “molecular storm,” the setting for cellular life:

“Inside a cell, events occur on the nanoscale, the scale at which objects are measured in millionths of a millimeter, and the medium in which things happen is one of water. Matter in this environment behaves differently from anything in our midsize, dry-land world. At this scale, activity arises spontaneously, without having to be made to happen. In a phrase due to the biophysicist Peter Hoffmann, within any cell is a “molecular storm,” a ceaseless turmoil of collisions, attractions, and repulsions.”

Peter Hoffmann wrote a book that looks just as fascinating, titled “Life’s Ratchet: How Molecular Machines Extract Order from Chaos.” More on the molecular storm, from the description of his book:

“The secret of life, he says, is not some “vital force,” but the unique operations of the second law of thermodynamics at the nanoscale, where molecular machines from kinesins to DNA synthase, fueled by ATP, can harness the energy of the “molecular storm”—the random bombardment of water molecules at jet-plane speeds—to move and work. Hoffman convincingly demonstrates how such “motors” could have evolved from simpler self-assembling structures, but admits that how all these cellular components came to regulate one another so precisely is still a mystery.”

Later, Goddfrey-Smith points to the storm as a crucial difference between the functioning of a biological system like a cell and a computer:

“In the days of ubiquitous computers and AI, it is natural, almost inevitable, to ask about the relationships between living systems and these artifacts. Do organisms and computers do essentially the same thing with different materials? Similarities between the two do arise, often unexpectedly, but it’s also important to recognize dissimilarities. One difference is that much of what a cell does, its main business, is something a computer never has to do. A great deal of the activity in a cell is concerned with maintaining itself, keeping energy coming in, keeping a pattern of activity going despite decay and turnover in materials. Within living systems, the activities that look like the things computers also do — electrical switching and “information processing” — are always embedded within a sea, a mini-ecology, of other chemical processes. In cells, everything that happens takes place in a liquid medium, subject to the vicissitudes of the molecular storm and all the chemical digressions that living systems engage in. When we build a computer, we build something whose operation is more regular and uniform; we build something that will be distracted as little as possible by the undirected ruminations of its chemistry.”

All of this has me thinking about knowledge and its absence. Just as cells are constantly bombarded with molecules, we exist in a storm of images, sounds, smells, and thoughts. Of course we don’t experience it this way — thank you brains! — and instead experience it in a reasonably orderly manner. This order is thanks to two things: the biological structure of our minds, and the structure provided by what we learn.

But the chaos is still lurking beneath the surface. I think you can see this when people make mistakes, especially when they are experiencing cognitive overload. In those moments the structure of knowledge seems to bend under the stress of the challenge, and we get to see some of the random dynamic churning of association.

You can see this at times when students are pushed to apply numerical skills to a new context. You could maybe, maybe, find a way to explain how a student multiplied 0.8 by 1.6 and got 8.0. But to me, that misses the point. The point is that there is no explanation. It’s just the intellectual storm of experience and association bubbling across the mind.

CR parallelogram

Mistakes in math are usually like this: the result of having insufficient structure, not of having the wrong structure. Whatever structures are present are not strong enough to withstand the storm of experience.

This is why I bristle when people talk about student misconceptions. Not because I’m sheepish about judging student ideas as right or wrong. (I’m not into “alternate conceptions.”) And I’m not totally skeptical that misconceptions matter for learning, particularly in science or history. But with math, I think the situation is different. There aren’t a lot of ways to have conceptions about some of these abstract topics. There aren’t a lot of “folk theories” in math, I don’t think.

And when I personally don’t understand something in mathematics, it usually feels like the storm to me. Undifferentiated thoughts, slipping and sliding between different intellectual currents. I’m searching for something to hold on to! And I think our students often feel similarly.

A lot of math teachers feel that every mistake needs to be addressed and explained. If every mistake were a misconception — a faulty or limited mental structure — then I think this would make sense. But mistakes aren’t usually like this. They are chaotic, a snapshot of the storm. And because of this, they can only be addressed is by learning, which is to say building something new.

Start class with the good stuff

Not much is going very well in this most cursed school year, but I like the way I rewrote this lesson today.

Here is the original activity, from the Illustrative Mathematics Algebra 1 course:

I like the idea, but I saw room for improvement. Here were my issues:

  • The problems were somewhat repetitive.
  • Students could answer the questions without using the function expressions.
  • This might make it so they’d miss the main ideas until the last two questions.
  • I like creating situations where students have a principle that they try to apply to a lot of different situations. The way this is designed, they wouldn’t be applying any sort of general mathematical principle until the very last question.

For my redesign, I wanted to get a generalization out in front. I started by showing a parabola and asking students to identify the x-intercepts and the y-intercepts.

Then, I displayed a worked example showing how someone could use the standard form formula to find the y-intercept. I asked students to read all the text that was in green and to put their thumbs up when they were done reading. Then, knowing I had some deeper questions coming up, I explained each line of the example. “Because the y-intercept is always on the y-axis, we know that x will equal … say it aloud,” I gestured for everyone to chime in. “That’s right, x will equal 0. If you input that into the function, your output will be the y-coordinate of the y-intercept.”

After briefly explaining, I revealed those deeper questions:

First, a “what if” question that swapped the numbers from the original function around. Instead of “-5x + 4” I asked about “+4x – 5.” What would the y-intercept be if the function looked like that? This question is asking students to take a step towards a generalization, and that second question goes all the way towards generalizing.

The third question, “What about the factored form formula?”, is a bit of an extension question. It was a more challenging question than I could have asked at this stage if I hadn’t lead off with the example. I didn’t expect every student would necessarily arrive at a correct answer to this question, but every student could benefit from trying to apply what they’d learned to factored form.

As it happened, there was a nice little debate about whether the y-intercept could be recovered from factored form. One student said, only if you turn factored form back into standard form by multiplying. Another chimed in to say that they noticed that -1 x -4 = 4, which connects to the y-intercept, but they weren’t sure if it were a coincidence. Then a third kid pointed out that -1 x -4 would always be part of the binomial expansion, so it wasn’t a coincidence. And then I pointed out that the same logic as before — input x = 0, your output is the rest of the y-intercept — applies to factored form, and (0 – 1)(0 – 4) = 4. Good talk.

Then, it was time to apply this knowledge. I started with a question that was probably a bit too easy, but I wanted a chance to connect factored form to the graph one more time. I asked students to put a thumb up when they’d found the x-intercepts of this function:

Good, good, I asked them to shout out the x-intercepts and I pointed out that they could have gotten this just from the equations. Ditto for the y-intercepts.

Anyway, on to the good stuff. I assigned partners and asked students to talk together about what the x- and y-intercepts were of this function:

Surprisingly challenging for the kids, even given all that we’d talked about! Most got the x-intercepts, the y-intercept required a bit more discussion. All were interested in the reveal, where I removed the grey box.

Moving along, I asked students to talk to their partners about the x- and y-intercepts of this graph:

And then this one:

This one threw kids for a loop! There was much discussion among pairs as to what the “other” x-intercept was going to be. And then there audible “ooohs” from a few kids when I revealed the graph, and there was only one x-intercept.

I feel as if it’s important to mention at this point that this was a class of pretty strong students. Their reaction to this graph says as much about them as it does about me. I want to be real, I don’t often elicit “ooohs” from my teaching. I don’t even know if I want to be a teacher that gets a lot of “ooohs.” But I want to tell you the truth, and the truth is that there were “ooohs.” I could be making that up, but I’m not.

Finally came the “Are you ready for more?” problem from Illustrative Math:

The lesson as written by Illustrative Math was pretty good, but it definitely fell into a paradigm that we might call “Notice, Notice, Generalize, Apply.” To give students enough mathematical data to make the generalization the lesson asks kids to do something a bit repetitive — to find a lot of x-intercepts and y-intercepts. Because students don’t yet know how to use formulas to do this, the graphs need to be provided. Because the graphs are provided, kids will probably just use them. Only at the end, when they’re prompted to generalize, will kids go back and try to connect their answers with the formulas. Kids then are asked to apply that generalization to a single new case where a graph wasn’t provided.

My rewrite follows a different paradigm, something I’ve sometimes called “Analyze, Explain, Apply,” but might also be called “Generalize, Apply, Apply, Apply.” The generalization is front-loaded in the lesson, which means that I was able to turn some of those “notice” problems into chances to apply the big idea about using the formulas to find the intercepts. I hope it’s clear that this hardly eliminated the challenge for students. If anything, it deepened our mathematical discourse.

The generalization is the thing we want to help students think about. Sometimes it can be nice to give students a chance to form that generalization on their own. But the tradeoff is that they then have less time to try out that new mathematical idea with new problems. If you put that big idea towards the top of the lesson, they get many more chances to apply the idea to problems. Of course, that means that you have to provide that new idea to students, either in the form of an example or an explanation. That’s a trade that I’m increasingly willing to make.

Some basic questions about college admissions

  1. Who should be allowed to go to college?
  2. Should everyone?
  3. Who benefits from attending college?
  4. Suppose there were a test that could predict perfectly whether you would successfully graduate from a particular college or not. Would that be the fairest way to determine whether you should be allowed to attend a particular school?
  5. Suppose you had a number that could perfectly summarize a student’s prior achievement. Colleges could admit students above a certain number X of prior achievement. Would that be fair?
  6. If every student who graduated high school could choose to enroll in free public college, what percentage of students would? How many students now seek to attend college but aren’t able to?
  7. What percentage of students would like to attend a more selective college but are not able to, either because of money or rejected applications?
  8. Why does it benefit students to attend more selective colleges?
  9. Should there be elite private colleges?
  10. Who should be allowed to elite private colleges?
  11. Who benefits from attending elite private colleges? And how?
  12. Are there benefits to attending elite public colleges?
  13. Should there be elite public colleges?
  14. Who should decide who goes to elite private colleges?
  15. Do the “official admissions criteria” impact who private colleges admit? Or does it just force them to use different criteria to create demographically identical classes?

I find all of this confusing. I found it more confusing after reading this and this, both of which I recommend reading. Feel free to answer these questions. Are these even the right questions? Feel free to ask better questions.

A final question: “Just like you can’t actually diversify the British royal family, you can’t make colleges tools of equality; creating hierarchy is their very function.” Is this correct?

Good explanations connect particulars to principles

What should count as an explanation in math? Even though most people in education think that explanations are important, they often struggle to clearly define what an explanation is. This has led to a situation where it’s relatively easy to make fun of mathematical explanations: You want kids to explain why 2 + 3 is 5? Solve the problem, what other explanation do you need!

In support of the pro-explanation camp, there is a large body of evidence that finds that it helps people when you prompt them to explain things to themselves during their learning. And this indeed does seem to settle the case, until we ask the exact same question as before: what does it mean to explain something to yourself?

Making it even more confusing? The use of “self-explanation” by researchers is a moving target. Sometimes it is supposed to be self-talk, a coherent explanation to yourself; other times it’s just any old inference you’ve made, even if it doesn’t add up to an “explanation.” Sometimes it counts as self-explanation if you say it aloud. Other times self-explanation is supposed to be an exclusively internal phenomenon.

I indeed found this extremely confusing, but felt much clearer after I read a fantastic piece by Alexander Renkl and Alexander Eitel that lays the whole situation out in a coherent way. First, they note that the situation is indeed very confusing and the meaning of “self-explanation” has changed over time:

As use of this construct has become so widespread, its meaning has changed. For example, Chi (2000) now sees (at least some) self-explanations as the process by which learners revise and improve their yet imperfect mental models. Presumably due to the widely varying use of this construct, its recent characterizations are rather general such as “inferences by the learner that go beyond the given information” (Rittle-Johnson et al., 2017) or explaining “the content of a lesson to themselves by elaborating upon the instructional material presented” (Fiorella & Mayer, 2015, p. 125). If one relies on such definitions, it is hard to draw the boundaries between other established constructs in research on learning such as “elaborative inferences” (in text learning; Singer, 1994) or “elaboration strategies” (Weinstein & Mayer, 1986). It is also questionable whether all the phenomena labeled self-explanations in previous research can be justifiably called explanation (for discussions about the concept of explanation, see, e.g., Keil, 2006; Kiel, 1999; Lombrozo, 2012).

Then they say something very sensible, which is that self-explanation is probably not one but rather many phenomena, related but distinct:

Given that even the initial characterization of self-explanation already included relatively heterogeneous phenomena (see the four findings reported by Van Lehn et al., 1992) and given that the subsequent extensions led to very general characterizations, there would appear to be little justification for discussing the learning effects of self-explanations and their use in (classroom) instruction on a general level. It is highly probable that different types of self-explanations have different functions, lead to better learning via different mechanisms, and should not be regarded as a unitary construct when providing practice recommendations.

They then announce that they are going to focus in on just one particular form of self-explanation. They call it “principle-based self-explanations” but that isn’t terribly evocative. What they’re describing is connecting the particular details of whatever it is we’re looking at to generally applicable patterns or rules. This form of self-explanation is a form of explanation; it’s explanation by way of connecting these specifics to that general rule:

We focus on principle-based self-explanations (Renkl, 1997): Learners “self-explain” a step in a problem solution (e.g., physics problem) or a feature of an object (e.g., the appearance of an animal) in reference to an underlying principle (e.g., one of Newton’s laws or mimicry). Such self-explanations were part of the initial self-explanation concept (e.g., VanLehn et al., 1992). They can also reasonably be called explanations. Although the concept of explanation can have different meanings (e.g., Keil, 2006; Lombrozo, 2012), explaining a case in terms of underlying principles is a quite prototypical case of explanation. This type of explanation fits the subsumption and unification accounts of explanation (e.g., Lombrozo, 2012). These accounts regard a case as explained when it is subsumed under a more general pattern (e.g., Newton’s laws or the general strategy of mimicry).

Let’s come back to 2 + 3 = 5. How could you connect this to some general principles? It wouldn’t just be taking two things and three more things and counting them. That’s of course valuable, but to get the benefits of what Renkl and Eitel are talking about it would have to be something more abstract and general. It probably shouldn’t involve the words “2” or “3,” or at least that’s not the important part. Maybe you’d say, “Whenever you are adding you can count on from one number by the other number.” Or maybe “You can always put both numbers on your fingers and count them together.” The point is the always. You are seeking to connect the particulars of this to a general pattern that repeats a lot.

The benefits of this kind of self-explanation, Renkl and Eitel write, are to be better prepared to apply that generalization to future cases. This makes sense since you are articulating a generalization, in the process abstracting some example or solution and turning it into a generally useful principle, procedure, strategy, structure.

They even have a nifty diagram:

When you see a new problem it may remind you of a similar problem you’ve seen and then you can access the general principle — that’s how noobs would do it. Once you’re better at something, you’ll tend to go straight to the principle to solve a new problem.

I think this is the simplest way to think about what a mathematical explanation is, or what it should be aiming for. It’s simply asking students to make a generalization. Articulating a generalization, if one is able to, is pretty clearly going to help students handle future cases that fall under that general pattern.

Just as equally, if students aren’t yet able to articulate a generalization then they probably won’t get much out of being asked to explain. A lot of the cases that are easiest to make fun of are situations where students just don’t yet know how to articulate those generalizations. If students don’t know how to explain 2 + 3 = 5, they can be taught what those general principles are explicitly and then asked to apply that to other problems such as 4 + 5 = 9 or whatever.

One of my favorite techniques I’ve been playing with this year is giving students two generally applicable principles and asking which better explains a given solution. This removes the burden of articulation from the kids, and instead asks them to focus on deciding which abstraction best matches the particulars. And if it’s sometimes ambiguous (as arguably the choices below are) that can be OK too, since it’s all still about connecting particulars to general principles.


Practically, this means that teachers can be clearer to students about what we’re looking for. We can ask students to tell us something that is always true, or that will always work. We can teach that great explanations aren’t just about this problem, but that they show how a particular instance is part of a larger pattern. In this way we can teach students how to give good explanations, at first to others. But my guess is with time this habit is internalized, and you start giving better explanations to yourself.

What is “fading out a worked example” and why would it work and also when?

The basic theory is that people often learn things via a two-step process: they watch what someone else does and try to do it on their own. This isn’t the only way that people learn, but in its broad strokes covers a surprisingly large number of cases, at many different scales of learning. Parent shows baby how to play with ball, baby tries to bounce ball. New kid sees cool kids wear trendy shoes, learns to imitate their interests and affectations. Elizabethan poet-playwright writes awesome murder play, contemporary screenwriter writes mediocre murder movie.

Sure, but what about when this is very hard? If it was easy to imitate excellence, everyone would do it. It helps if you can impose some sort of structure on the effort. Sometimes it’s useful to focus on imitating just one step at a time. Other times you need to better understand what that other person did. Hopefully this is starting to sound familiar, it is typically called “teaching.”

Specifically in the context of math or science and learning procedures, there is a method for structuring this kind of learning called “fading a worked example.” I don’t remember where I first heard of it, but it’s strongly associated with Alexander Renkl’s research. Here’s how it works:

  • Show someone a task and perform all of it for them.
  • Then, present a new task, and perform almost all of it for them, everything but the last step. They do they last step on their own.
  • Present another new task, and let them handle a bit more of it on their own. Ask them to do the last two steps.
  • Etc., until the person is performing the entire task on their own.

Last week I did this sort of activity with my calculus class. It started with a fully worked example, that I prepared before class and asked students to analyze:


Then, I started knocking out steps from the procedure, starting at the end. This lets kids practice just that step, but in a meaningful context. (That is, the context of the integration problem.)


And then I knock out some more steps.


And by the end, they should be able to try the problem all on their own.


Here is another fading activity I ran in class last week, this time with my 8th Graders. I was teaching writing expressions for linear situations. At first I gave them the expression and asked them to think about why the expression fits the scenario:

But soon after I was leaving out the expression and asking them to provide it:

As I describe in my book Teaching Math With Examples, these sorts of tasks do much more for students than I first expected them to:

“As my class got to work, I was surprised by how challenging the assignment was. Students were asking me good questions about mechanical issues I hadn’t even considered to be problematic for them. It turns out that lurking beneath their overall struggles had been a variety of smaller difficulties. The faded-out exercise gave them a chance to systematically work on all of them.”

I find myself using these faded worked examples the day after a rough class. One where every student seemed to do great right up to the moment when I asked them to try the problem on their own. It’s for those days when the class seemed nervous about even starting, not because the problem was opaque but because the solution seemed like just SO MUCH.

It’s worth pointing out that this is not rocket science. Which makes sense, because teaching is not rocket science. It’s not science? It might be fair to say that teaching is maybe cognitive science plus other educational research plus a lot of random practical knowledge multiplied by a ton of emotional intelligence. The point is that you don’t need a researcher to invent something like fading a worked example.

One the other hand, I learned about this from reading research. And then, after I read about it in research, I sat on that knowledge for a few years. I wasn’t sure if it would fit in with the rest of my random practical knowledge about teaching. I also was worried that it would be dull and repetitive to my students, throwing off the mood in the room. (I shouldn’t have worried, it’s great.) Now that I’ve tried it a lot in my teaching, I’m happy to share the idea with enthusiasm.

Research is great, but that will only ever be a part of what it takes for a good idea to make its way into practice. The people who use the idea need to chew on it for a while and make sure it fits in with everything else. It would be good if researchers understood this, because it implies different sort of studies than what is frequently done. (“We Did This One-On-One With a Bunch of Undergrads Which Reveals Universal Teaching Techniques and Now Go Ahead and Use It.”)

Anyway, if you’ve got a long-ish procedure and it’s a topic that is easy to create many similar-ish examples for, you might consider writing a worked example and fading out the steps, starting at the finish.

My Billion Dollar Idea

I recently had a billion dollar idea. It’s for an education business, but there’s no way I could pull it off. I figured I’d write it up here, and hopefully someone else can create it, because it really should exist.

OK, so here’s the problem: there are young children who know a lot more math than their peers. The smaller the school, the bigger a problem this can be. Get five kids who know a lot of math together, you’ve got yourself a small math group. But if you just have one kid in your 1st Grade class who can confidently and fluently do 5th Grade whole number arithmetic? You’ve got yourself a problem.

Long-time readers know that we’ve discussed this before: What should a school do with an advanced 1st Grader?

But that was before the pandemic. And now I have a much better understanding of the pros and cons of online teaching. And I think that an online math class for young students — one that was integrated with their school day but didn’t focus on just advancing them further down the path of arithmetic — could really work.

The big downside of online teaching is that a demotivated student can slip through the cracks with the greatest ease. Kids who are excited by the material and eager to engage online, in my experience, have been able to handle the digital setting quite well. Even kindergarteners. (I’ve watched my son get quite good at using his favorite apps and websites.)

Going forward the most promising use of online teaching in K-12 is for increasing access to electives and advanced courses for motivated students. China has experimented, seemingly successfully, with beaming math courses with knowledgeable teachers to students in underserved rural areas:

New technology is helping close the gap. Since 2016, 248 under-served high schools in the poor areas have tuned into “live streaming classrooms” hosted by Chengdu No.7 High School, one of the top high schools in China. Over the livestream, the program puts together two groups of students that could not be more different — upper-middle class students and their peers from the country’s most educationally under-served families.

Similar experiments are happening in the United States:

That’s why there’s considerable excitement about the free program bringing AP physics to Mississippi this school year, courtesy of the Global Teaching Project, a Washington D.C.-based education company that is part of a nonprofit consortium in the state. A few years ago, the Holmes County school district offered a few college-level AP courses at only one of its three high schools. After the three schools consolidated during the 2014-15 school year, the newly formed Holmes County Central High School was able to offer five classes, including AP calculus, English language and English literature.

And last summer, I personally was involved with BEAM’s attempt to take their math enrichment camp for underserved students fully online. Of course, this move was a necessity of the pandemic. It was not easy. Read the director of BEAM, Dan Zaharopol, describe how difficult it was to provide reliable internet access and proper devices to every student admitted into the program. (They delivered and troubleshooted a lot of hotspots.)

On the teaching end, it wasn’t always easy either. Some kids didn’t really end up getting emotionally connected to the teaching. They sat through classes with their cameras off and didn’t truly engage. But by the end of camp it was clear that our teaching made a very real difference to a great number of students — students who would not have had this opportunity otherwise.

So here’s the money-making idea: a country-wide math class for elementary school students who know a lot of math. It would be online and (here is the crucial part) it would have nothing to do with the math they are learning in school.

Graph theory. Probability. Games. Puzzles. Logic. Incompleteness. Fractals. Anything! There are a million facts of mathematics and what we cover in school is just one, narrow slice of the mathematical world. If a student is ready for more, mathematically speaking, the solution is more mathematics. Yes, but what mathematics?

You can accelerate a student through many years of elementary math fairly quickly. Facility with arithmetic compounds; I’ve seen 3rd and 4th Graders that can do truly impressive things with numbers. But at a certain point the race forward itself becomes a problem. You can put a young student in a class full of teenagers, but do you really want to? The emotional needs and social environment is quite different. Anyway, kids like being with their friends. And all sorts of messy inequities emerge when some kids race forward and other kids are left to handle the grade level material.

But! What if you just teach kids who know a lot of math different math? In time, the gap between the knows and the know-nots will narrow, it will be less absurd for advanced students to share a 5th Grade math class than in 1st Grade.

And, with motivated students, you could totally do this online.

People would pay for it. Districts would pay for it, but individual parents might pay more. This is a way to keep their kids interested in math but without creating a schooling problem. It would happen during the day, a few times a week, with a device and headphones while the rest of the class is building up their basic number skills. The three ingredients are: young children; math that isn’t arithmetic; online learning. We can do this.

If you end up taking this idea and building your own business, it will take time. You’ll have to slowly develop a curriculum that is both accessible and advanced. You will need to learn how to help very young students do math over a platform like Zoom. You’ll need to gain a reputation and win the hearts of parents and the faith of a school district. This will be a steady, modest way of building a business, one that will enable you to slowly build up a wealth of capital.

At that point, you sell the business. You take the money, you put half in Bitcoin and the other half in Dogecoin. It’s too late to get in on the ground floor, but if you act quickly there is still a fortune to be made. Then you sit back and just wait.

That’s my billion dollar idea.

Teachers are afraid of hybrid learning


I’ve been puzzled by reports that some teachers and their unions, even after being fully vaccinated, are reluctant to return to in-person teaching. (For example, in Portland.)

My wife and I are both teachers. We’ve both been teaching in-person all year, and recently got our second shots. Over dinner a few nights ago we were hashing this out. If you’re vaccinated, why wouldn’t you want to come back? If we take these teachers/unions at their word, it’s all about lingering safety concerns. Or is it just trying to hold on to the “perk” of working from home for a bit longer?

I bumped into an elementary teacher friend yesterday who I admire a great deal. She has been fully vaccinated and I know she cares a lot for her students. She understands that vaccines are effective. She works hard for kids. Still, she’s praying that they don’t return in-person. And she even said she has colleagues who are afraid of getting their shots, for fear that they will have to come back to schools. This is seemingly crazy — sure, ventilation is awful in a lot of places, but are they less safe than not being vaccinated?

But after talking to my friend, I think I understand the situation much better: teachers are scared of hybrid teaching. There are safety concerns, but they aren’t the primary source of anxiety. After talking with teachers on twitter about this last night, I think I got some confirmation of this line of thought. And while this theory has the downside of not taking union rhetoric at face value, it does have the benefit of being an entirely reasonable concern about working conditions. It is not a crazy thing for teachers or their unions to worry about.


For people who either aren’t teaching or aren’t deeply thinking about this, they might be baffled as to why there’s hybrid teaching at all. Just let kids back — all of them! And teachers will teach their kids, and not have to worry about Zoom or Google Classroom or anything.

There are two big things that make it difficult, if not technically impossible, to bring every kid back in person:

  1. Parental choice
  2. CDC requirements

The first challenge is parental choice. At this point, anyone paying attention knows that a large percentage of parents don’t want to send their kids back to physical school buildings, and that disproportionately it is white parents who are eager to get their kids in-person. The exact number of parents will vary from place to place. The point is that it doesn’t take that many parents to present school systems with a problem: what do you do with kids who aren’t coming in to school?

Schools have a few different ways to handle this situation. The idea of matching fully-online students and fully-online teachers seems reasonable, and some places have managed to pull that off it seems. Most places haven’t done this though. Why not? I think because it necessarily means that kids are not in contact with their friends from school — they are instead placed in new classrooms with teachers who may not even be from the school. (This only works if you have a large pool of fully online teachers, so it has to happen at the district level.)

The other thing is that this is expensive, especially since every in-person teacher needs to have the capacity to teach online simultaneously anyway. Everyone who teaches in-person has had students drifting in and out of the computer this year as they are quarantined, sick, tired, or because their parents are taking them to visit family or whatever. If you’re building that capacity in classrooms anyway, it’s hard to make the case that you should also spend money to create a dedicated online teacher corp. (Brookline, MA schools had a Remote Teaching Academy this year, but are phasing it out next year because of cost.)

As long as a large percentage of parents are choosing not to send their children in-person, teachers will then be forced to teach students simultaneously online and in-person. Which (as we’ll get to soon) hardcore sucks.

The other factor is CDC requirements. Now it’s important to note that states don’t have to follow the CDC. Florida is not. If you ignore what the CDC says, then what the CDC says is not a problem. Due to a variety of American pathologies, this is more likely to happen in Republican-governed states than in Democrat ones.

But if you do follow CDC requirements, then schools have a problem. Which is that you can’t fit all the kids in school, and it’s not even close. Again, there are various more or less expensive and reasonable ways of handling this. But the point is that if all kids were in school, then all kids couldn’t sit six feet apart, and the CDC says that they have to do this. This necessarily involves complicated scheduling that dictates which kids can be in school on a given day.

How do you handle the teaching side of this? Some schools give students dedicated online and in-person teachers for a given day. But it seems to me that most places are not doing this[citation needed]. Instead most places are asking all teachers — even and especially those who come in in-person — to do either of these two things:

  • Create a day full of activities for students who are in-person as well as a slate of online activities that students can do at home on that day.
  • Asking teachers to simultaneously teach students in-person and online at the same time.

Both of these are a lot of work, and simultaneously teaching both on a computer and in a classroom is especially a lot more work. My main point above is that if schools ask teachers to come back, then they will in almost all cases be asked to do some version of hybrid teaching — either managing multiple cohorts on a given day or teaching simultaneously online and in-person.


These forms of “hybrid” or “simultaneous” or “concurrent” teaching all suck, and they are all more or less frustrating, and they are certainly asking teachers to do a great deal more work. And as such, they are completely valid concerns for teachers to raise about returning to school, whether or not they have been vaccinated.

Speaking personally on this is tricky for me. I currently teach at a relatively wealthy private school where class sizes are in the teens, not the twenties or (god help me) thirties. I have been going in in-person since the start of the year. (Biking from Washington Heights to Brooklyn earlier, now that I’m vaccinated I’ve been back on the subway.) The school schedule at my place is fairly complicated, but because of parents keeping their kids home I’ve been teaching simultaneously online and in-person.

And it’s very hard but here’s what I’ll say — I only feel dumb about it when very few kids are at school. When most kids are in the classroom and a few are online, well, it’s not more effective for the kids who are online. But at least I feel like my presence at school is worthwhile. I can more easily help kids with things, I can keep an eye on everyone, and more important it feels like there’s a real social environment. Yeah, it’s really hard for the kids who are online and it’s even harder to pay attention to them.

I think this distinction is maybe missed on people? Hybrid with a few kids in-person is really, really hard. Hybrid with a few kids online is manageable.

I know of lots of teachers who are facing situations where three or four kids are coming into their physical classroom while they have many more students Zooming in online. That’s a recipe for frustration. That feels like you might as well have everyone stay at home, since you’re basically teaching online anyway — but you also have to watch these kids. (Especially elementary school teachers, since you’re really supposed to be in charge of these kids but you can’t! It can feel like impossible babysitting while also doing your job.)

Teachers are being asked to do this now. And so it really would be reasonable for teachers to worry about this, at least until the CDC relaxes restrictions (perhaps because cases have fallen), until more parents are comfortable sending their kids into school, until weather is warmer and outdoor spaces can be used, or until a more robust promise can be secured that bad teaching that is double the work won’t be required.

In many places, this won’t happen until vaccines are widely available not just to teachers but to all adults. That means that a lot of places should realistically be aiming to open in the fall, not this spring.


I’m not trying to be totally dismissive to the safety concerns — like I said, I was nervous about the subway earlier this year, and actually my three-year old caught asymptomatic Covid from her preschool teacher (thankfully vaccines work!) — but I think once teachers have been vaccinated the situation really should be quite different.

Vaccines are never 100% effective, but science is starting to come in, and it looks like (as expected, honestly) the vaccines take a HUGE bite out of transmission. You are well protected not just from severe Covid but from any Covid symptoms. And if you do get infected, there is a huge reduction in how likely you are to develop symptoms. And then there is a huge reduction in your ability to transmit the disease to others, even if you are infected.

I don’t think these are additive … this means that ~90% of transmission is reduced once you have been fully vaccinated with one of these mRNA vaccines. In other words, when you are fully vaccinated you are about 10 times safer to those around you than you were before.

There are risks that you could give Covid to somebody else even when you’re vaccinated — a child, a loved one, or an elderly parent. But once you’re vaccinated it gets much safer. Here’s a reasonable way to think about this I think: once cases are down low enough that you aren’t 10 times more likely to get Covid at school than you are just going about your daily business, then you should be OK with the risks you pose to others.

I have also heard teachers mention concerns about kids and Covid, since kids aren’t getting vaccinated now. It’s true, but there’s a good reason that vaccines weren’t tested on kids yet: kids don’t really need a Covid vaccine. The risks to kids are very small compared to the risks of things that kids get every winter.

Normal, seasonal flu is a good comparison. “It’s just the flu” was a dismissive thing that people said last March about Covid, meaning i.e. Covid is not as bad as people are saying. But when it comes to kids, “it’s just the flu” is wrong in the opposite direction — Covid is not anywhere as dangerous for kids as the flu.

According to the CDC for kids 5-17 there are about 38 hospitalizations per 100000 kids who get the flu. But for Covid that number is more like 8 per 100000.


It is reasonable to be concerned that schools will contribute to the overall spread in a community. This is the issue that has been discussed endlessly throughout the pandemic and I find myself with absolutely no energy for it. (I think Matt Barnum has done a great job sifting through it all while keeping everyone honest.)

My point is this: if you’re concerned about safety, I think the only thing you should be worried about is whether keeping schools open will contribute to overall spread of the disease to adults. And if this is your main concern, you should no longer be very worried about this once either cases are very low or once all adults have access to a vaccine that will protect them in the case of infection. That will probably happen by the summer, so I think that all teachers should be supportive of a return to schools this fall.


To sum up:

  • I think the most likely explanation for teachers/unions not wanting to come to schools even after vaccination is for fear of hybrid learning.
  • Hybrid learning is in fact a natural consequence of having in-person learning, at least this spring, because of parents choosing to keep kids home and CDC requirements.
  • Hybrid learning in fact does suck for all involved, and it typically creates far more work for teachers — making this a legitimate concern for unions.
  • Hybrid learning particularly sucks in the situation where a few kids are in physical school but most are online. That’s a recipe for intense teacher frustration.
  • While in general I think post-vaccines the safety concerns aren’t a big deal, they really shouldn’t be a concern at all once all adults have access to the vaccine.

I wanted to write all this up for a few reasons. The first is to lay out why I think teachers have some legitimate grievances about the return to in-person learning, even post-vaccination.

A second reason is to make clear that parental choices and CDC requirements are a major obstacle to having every kid back in school buildings. And in fact in places where parents are enthusiastic about getting their kids in buildings and states are enthusiastic about making fun of the CDC requirements that is happening — see Florida and Texas and others.

Another is to explain why I’m still extremely optimistic that school in the fall will be pretty much in-person. The safety concerns will almost certainly be reduced as cases plummet and every adult has access to effective vaccines. Science about reductions in transmission will get firmed up and popularized. And even if a certain amount of hybrid is necessary, I think it can be the less annoying kind — the kind where a few kids whose parents aren’t ready to send them back Zoom in to class, with most students present in person.

I’m also more optimistic about the possibility of fully online academies working in the fall. At that point, parents who aren’t sending their kids might be just those in extenuating circumstances and would be willing to “switch schools” to a fully online experience. And the costs could be more manageable with more students in-person. And I think as cases drop the CDC regulations will be more clearly in favor of school even if kids can’t be six feet apart. (Someone told me online that they might already be OK with that? I haven’t read the document carefully.)

But finally this is a note to teachers and unions: just explain the situation with hybrid. Parents will get it. Parents appreciate what we do for kids, and they know we work hard. If the concerns are about hybrid learning, there is a way to communicate that it’s twice the work for half the benefit, as a teacher put it to me. The safety concerns come off as vague and unconvincing. (“It just feels kind of icky,” a teacher says in this article.)

Whereas the troubles with hybrid learning are quite clearer and easy to point to. If the real concerns are with hybrid let’s talk about that, because the concerns about increased workload with reduced benefits have the benefit of being absolutely true.

P.S. I think this piece is broadly connected to this post of mine from a few months ago reporting on some teacher polling.

Was it a mistake to call KIPP the top recommended charity in the United States?

Who should you donate your money to? Not all charities are equally good at helping people. If your choice is between “The Pershan Family Vacation Fund” and “Give Food to Starving Children” you should definitely give money to hungry children. Then again, how do you know if the kids really are starving? And maybe the charity is corrupt? Say what you will about that Pershan Fund, the money really does go where they say it will.

These examples are silly, but reality isn’t that much different and it gets complicated very quickly. It’s good to give to charity, but where should you give? This is why Givewell exists — “We search for the charities that save or improve lives the most per dollar” is their motto, which is reassuring in its frank dorkiness. They are optimizing a rate, which means they are institutionally committed to solving a calculus problem.

For years, I’ve used Givewell to guide my giving. I’ve followed their advice in donating to various anti-malaria groups. Their ethical advice in has seemed to me entirely sound, and I’ve been grateful for their guidance.

But over the past few months, something has been bothering me, which is that in 2011 their top recommended charity for the United States was … KIPP, the charter school network. In particular, KIPP Houston, which they thought was especially in need of additional funding. (Givewell no longer lists top US charities in the same way, so the recommendation isn’t current.) The more I think about it, the weaker this recommendation seems. But I trust and admire their work in other areas so much, I don’t know what to make of the KIPP recommendation. And I still don’t.

One of the many admirable things about Givewell is that they make their reasoning public, so the case for donating to KIPP is crystal clear. The case begins by citing a trio of studies that find KIPP schools effective at improving test scores. They note concerns about students leaving and retention, but the studies seem to cover these issues. Basically, there is evidence that KIPP increases test scores and that is the basis for their recommendation.

If I’m reading correctly, the Givewell evaluators seemed to have two major concerns with KIPP: teacher burnout as a proxy for KIPP’s sustainability and capacity for growth, and whether KIPP really could do more with more funding.

It seems to me that the really obvious concern is not raised in their evaluation, which is “how sure are you that it’s very important to raise children’s test scores?” I find it genuinely confusing that this question wasn’t raised in the report? They write, “Note that we have focused our analysis on KIPP’s impact on middle-school test scores and not longer-term impacts such as high school and college graduation or adult earnings because we have not seen this data.” That’s the whole point, though!

What’s especially surprising is that even KIPP doesn’t treat test scores as their measurable endpoint. Their self-professed endpoint was reducing poverty by helping more low-income students get through college and towards higher-paying jobs. While Givewell labeled KIPP the top US charity, KIPP went through an effort to increase their college graduation rate, a crucial step in the “do well on dumb tests, go to college, and reduce poverty” plan. How is that going?:

The college graduation rate for KIPP alumni is about 35 percent, above the national average for low-income students but not nearly as high as its founders had envisioned. After years of attempts to help KIPP alumni graduate, the network is proposing new solutions, which it hopes other schools will emulate.

I raise this not to criticize KIPP or the way that it’s handled this. I wish more schools thought hard about how their students do in college. But if you’re an organization that recommends charities and you’re evaluating a anti-poverty project by how well it increases test scores, something is off. It’s like if instead of measuring the impact of the Anti-Malaria Foundation by how many lives it saved you measured it by how many mosquito nets they purchased — precisely the sort of mistake that Givewell is typically so good at avoiding.

There are other concerns that you could have raised in 2011 about KIPP. Research over the past decade has made it clearer that charters might really impose costs on surrounding public schools. Matt Barnum writes:

Charter schools really do divert money from school districts. Those districts can make up for that by cutting costs over time. But the process of doing so is often fraught, especially because the most straightforward way to reduce costs is to close schools.

The systemic issues are important, but they don’t seem as basic as the most basic point: did Givewell evaluate the value of increased test scores for poor children? No I don’t think they did? In which case what is the basis of their recommendation of KIPP as a top charity for the US, when I could give money instead to a local food shelter?

I’m trying to think this through in a non-hysterical way for two reasons. First, because I’m a teacher with teacher readers and it’s very easy for us to play Whack-a-Charter. I don’t want to do that. Second, because I genuinely admire the people at Givewell and I am inclined to respect their recommendations in general. And yet I can’t help but think that some sort of large mistake was made with KIPP.

If there was a mistake, what was its nature? Here are some options, and I don’t know which is true:

  • They didn’t put much effort into evaluating US charities because people in the US are so much wealthier and better off than those in areas their top charities target. In other words, the KIPP recommendation was lazy.
  • Charters were quite popular in a bipartisan way in the late 2000s and there was a general consensus that increasing test scores would decrease poverty. The mistake of Givewell was not critically examining this consensus.
  • Givewell’s recommendations are in general not as reliable as they seem. I noticed this one because I work in education, but people in international development would also easily recognize mistakes in their giving recommendations overseas. This would be a case for expertise in a field being important in addition to a general commitment to an open, transparent, evidence-based process.
  • I am wrong, and they did seriously consider the tradeoffs with KIPP and I’m just missing something.

What makes most sense to me is that it’s some sort of combination of the above. Maybe I’m being a bit unfair using my 2021 brain to evaluate a 2011 claim about charters. That movie “Waiting for ‘Superman'” came out in 2010 and it was a pretty blatant piece of pro-charter propaganda that generated a lot of discussion — such were the times. And maybe Givewell didn’t give the evaluation the full Givewell analytic effort. I see that a volunteer graduate student in plant biology evaluated the KIPP recommendation favorably. Would someone working in education have been able to raise concerns? Probably. And it’s to their credit that eventually Givewell probably realized that recommending US charities that weren’t really their recommendation (since what they actually recommend is giving internationally) wasn’t their core competency, and they stopped doing this.

Still, this gnaws at me. Is there a lesson here? And is the lesson, be very careful when thinking outside your expertise?

Teaching Yourself and Others

In his wonderful new book “Proof and the Art of Mathematics”, Joel David Hamkins asks a question about irrational numbers that stopped me in my tracks:

Prove that the irrational real numbers are exactly those real numbers that are a different distance from every rational number.

I reacted to this problem in three stages:

  1. OK, sure.
  2. Wait what?
  3. Oh cool! Why?

It took me a moment to understand what he was asking, then a moment longer to think about how to approach a solution. In the end, what helped me make sense of it was a problem that I’ve used to teach fractions to my 3rd and 4th Graders. These sort of connections between the math of young students and of sophisticated adults is mathematically exciting. It gives me the same sort of “oh man that’s beautiful” buzz that mathematicians sometimes use to describe their realizations.

The question comes from Marilyn Burns’ fraction books: Put two fractions on the number line, maybe 1/4 and 1/2. What number is exactly halfway between them?

Lots of students would say “1/3.” This isn’t how fractions work, though. (The jumps from 1/4 to 1/3 to 1/2 aren’t constant, an early example of non-linear growth.) This problem often catches my students in stages just as the irrational number question caught me: Oh, it’s obvious! Wait what now?

The question mark is half of the distance from 1/4 to 1/2. That makes it 1/8 away from 1/4, and 1/8 + 1/4 = 3/8.

It’s possible to generalize this result. I have good memories of pre-teen students at math camp, huddled around a chalkboard and trying to express this result algebraically.

The actual solution to this isn’t particularly important at the moment. The point is that you can always find the distance between two fractions, and that distance can always be expressed as a fraction. And half of that distance? Again, a fraction. Pick the lower place on the number line, jump ahead by that half-distance and what do you get? Fraction, fraction, fraction. It’s fractions all down the line. As long as you start with two fractions, you get one in the middle.

Fractions are also known as “rational numbers.”[1] So let’s take another step closer to the original problem, which was about the distance between an irrational number and any rational numbers.

Start with a number on the number line. If it’s rational, then you can write it as a fraction. Then reach out in one direction to another fractional/rational number. What’s waiting for you in the other direction?

Again, you can compute the answer using the distance between 2/5 and 1/2, What’s significant is that this other number will also be … wait for it … a fraction. No way out, they’re everywhere. You’ll be able to express the distance between 2/5 and 1/2 as a fraction, you subtract that distance from 2/5 and … you get the picture. Fractions, across the sky. But what does it mean?

This time, start with an irrational number – something like the square root of 2 – and stick it in the middle slot. Then, stretch out in one direction to reach a rational number. What sort of number is waiting for us an equal distance in the opposite direction?

Suppose for a moment that, as in the 4th Grade question, the other number turns out to be a rational number. Then at the left and the right are rationals. Remember the pre-teen version of the question: if the left and right numbers are expressable as fractions, there is a formula for finding a rational number exactly halfway between them. And that would mean the square root of 2 is rational. Which, no. So! That other number is irrational.

Another, more precise way of saying the above: Let m be an irrational number and suppose it is the same distance to two rational numbers. Then m is exactly halfway between two rational numbers, and based on Marilyn Burns that too is rational…hey no way, man, it’s irrational! So irrational numbers are a different distance away from every rational number, which means if a rational number is on an irrational’s right side, there isn’t a rational number an equal distance to the left. And vice versa.

(Strictly speaking this is only half of what Hamkins asked us to prove. The other half is showing that every number that is a different distance to every rational number is irrational.)

A lot of mathematics seems obvious in retrospect, and this was one of those times for me. That’s also the case for the connections to what I teach my elementary and middle school students. Duh, fractions are fractions and this was a problem about fractions. (Fractions, EVERYWHERE. Let that haunt you.)

What I find wonderful is that when you’re teaching you never know what seeds you’re planting. That’s supposed to be a truism about teaching kids, but it also seems true when applied to yourself. Every new idea you share with someone is an idea that might be useful to you, the teacher, in some new and unexpected context. In a very real sense, teaching is also sometimes teaching yourself.

[1] There is probably a nit to pick about rational numbers being more precise a term than fractions in this context, but I’m going to juuuuuust slip right away from that conversation if you don’t mind.

People Actually Really Like Mathematics

Sure, people tell you that they don’t like math. They’ll say they’re bad at it, that they hate it. That they’ll do anything to avoid it, that the very thought of it gives them the heebie-jeebies.

Don’t believe them for a second, though, because actually? Everybody likes math. The proof is that even people who profess to hate mathematics do a lot of it by choice, for fun. The issue is what people see as “doing mathematics,” and how disconnected what they enjoy is from what happens at school.

There may be other examples of popular mathematics, but mostly I’m thinking of games and puzzles.

Mathematics is the art of necessity, but not everything is necessarily true. Some things just happen to be true. My name is “Michael” though my parents could have named me “Marvin” or “Melrose.” Other things have to be true; they’re forced to be the way they are. These are the inescapable facts of existence, the ones we can’t wriggle out of. So it’s not some sort of coincidence that ever since I’ve had children, I’ve been a parent. And you may not be shocked to learn that ever since I’ve been a resident of New York City, I’ve lived here. Much as I wish at times, there’s no escaping your home as long as you live there.

One of my favorite games to play with students is Mastermind. It’s a game of code-breaking and logic. It’s a game that puts you in direct contact with necessity and possibility, and I use it in class to help students grasp the difference between those two kinds of reasoning.

It’s not a “math” game. It’s a board game, the kind of thing my friends’ parents would have in their basements when I was a kid. But it’s all about logical necessity. That’s not science. It’s not a sport. If anything deserves to be called math, these sorts of games do. What else would they be?

And once we’re on the lookout for logical necessity, it’s all around us. It’s printed in the newspaper (KenKen, Sodoku), sold at the toy store (Guess Who?), and built into our computers (Minesweeper). The reason it’s everywhere? People love this stuff. They love how it makes them feel and think. It’s mathematics; they like it.

The point of course is that very few people recognize this in their school mathematics, which is dominated by a different experience: that of carefully following steps.

Now is that so bad? Absolutely not, people also love carefully following steps. They love assembling LEGO cars and (even if they won’t admit it) IKEA furniture. There’s joy in successfully executing a tricky procedure…

…but it’s not quite the same as thinking logically, is it? And why can’t people experience both in school?

I’m getting a bit too close to preaching to the choir for my own personal comfort. What I’m trying to emphasize is that there’s no reason to define “mathematics” as an activity that is identical with the math that people study in school. That’s entirely artificial. True, all definitions here are pretty artificial. Still what’s distinctive about mathematics to me is its obsession with what is logically guaranteed to be true. And that’s an obsession shared by billions of game players and puzzle solvers around the world.

Mathematicians and educators can sometimes be heard asking, how do we get more people interested in mathematics? And I think the answer is, mostly they already are. The question is whether we and our institutions are interested in their mathematics. And mostly, we aren’t.