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.

The Kind of Help My Son Wants

For the last few days my morning has begun at 6:00 on the dot. That’s when my son swings open the door to our bedroom and asks for help with a math question. Fielding questions from a small child who wakes you up and demands help an inch away from your face seems like a good way to sharpen your classroom skills. It’s like some bizarre emergency preparedness drill. This is a million dollar teacher-training idea, but for you I give it away for free.

On the other hand, all that time in the classroom is decent prep for dealing with some of these parental duties. And it gives those of us with young children an N = 1 perspective on a perennial teaching question: what are kids looking for when they ask for help? And I think the answer is that in that moment what they want most of all is to understand. They aren’t primarily interested in having their own original thinking validated. And the implication is that telling kids they’re wrong is mostly trouble when the kids aren’t able to quickly grasp what was incorrect.

This is a bit of a story, but stick with me.

When my son barged into our room this morning, he told us he had two math questions he needed help with, 16 x 9 and 16 x 12. I told him that this was great, but to be quiet so that he didn’t wake up the baby.

Once I picked up the baby (baby woke up) I told my son than I thought 16 x 10 would be a good place to start, since it was so close to 16 x 9. I asked, do you know 16 x 10? He told me it was 160.

Fantastic, now we’re cooking. This is a conversation I’ve had approximately ten billion times in my life as a 3rd and 4th Grade math teacher. I took the next step. Would 16 x 9 be bigger or smaller than that, I asked? He thought and said, smaller. Yes!

I went in for the kill. (Uhh so to speak.) 16 x 9 is smaller than 16 x 10. I asked, how much smaller? This is the hard part, the part that puts a lot of pressure on one’s conceptual understanding of multiplication. If 16 x 10 is ten groups of 16, then we can take away a 16 from 160. If 16 x 10 is sixteen groups of 10, then we have to think about what happens when all those 10s turn to 9s. Without practice thinking this way, kids tend to shrug and guess.

That’s what my son did. He said that we should take ten away from 160. He had zero confidence at all in this. He asked if he was right, and I said it wasn’t.

Now what? I knew that the thing to do would be to draw a picture. Unfortunately, I was horizontal and undressed. I started talking about groups of sixteens … I was shut down by my son who, after all, is six years old and who I recently overheard telling his kindergarten teacher that he is inspired by “vehicles.”

The boy started complaining that he already knew the answer anyway. Cool, I said. So you know the answer is 144. But then he got really annoyed at me, telling me I had spoiled the problem for him. And then I tried to walk this back. I realized that his own fragile competence was on the line. Also he gets loud when he’s worked up and the clock now read 6:01.

Anyway, when he came in he had also said he wanted help with 16 x 12, and I casually changed the subject to that problem. I dragged my sorry butt out of bed and to the table. I poured him a bowl of Cheerios and grabbed a piece of paper and a marker. Then I drew a rectangle in green. I told my son I was going to draw a picture of 16 x 12, but start with 16 x 10. Then I asked him which was bigger, 16 x 12 or 16 x 10, and then how much I would have to add to the green rectangle to make up the difference. This all went much better.

Why did our talk about 16 x 12 go better than 16 x 9? Clearly things started going south when I told him that subtracting 10 wasn’t right. But I also corrected him on things while we were talking about 16 x 12. He told me the wrong dimensions of that green rectangle at first, for instance.

There were two differences when we talked about 16 x 12. First, he was loading up on breakfast cereal and riding a blood sugar high. Second, he understood what I was talking about when I corrected him. That’s thanks to the image. A wonderful thing about people is we turn correction into self-correction. You can’t help but take some credit for recognizing your own mistake, even when someone else points it out.

But when I corrected my son the first time he had no idea why he was wrong, and because of that he freaked out a bit. It’s OK to tell my kid that he’s wrong, but if he doesn’t understand why it’s not going to go well. And it’s the same with students. Don’t focus on whether you’re honoring their original thinking or not — they wouldn’t ask for help if they didn’t want access to your expertise. Focus instead on whether you’re connecting what they don’t yet understand to what they already do.

Is this all there is to it? No. The same boy I’ve written about above just asked for help building something with his toy magnets. He was trying to build a porch, but it kept collapsing. I had an idea and showed it to him. I have to say, my solution was pretty clever. I’m great at toys. The boy took a look at it and decided it was “stupid” and “bad” and that I was “dumb.” Only some of which is true! (It was a great porch.)

It’s not all about understanding, but a lot of it is.

What if the world decided that every kid needed to learn how to solve crossword puzzles?

It was March 2020, we were stuck indoors, and I was looking for something to take my mind off of [aaaaaaaaaaaaaaaaah]. My wife is a big crossword fan, so I subscribed to the New York Times Crossword. And now? I’m hooked!

I’m not very good though. For the uninitiated, Monday is the easiest puzzle of the week. Each day is supposed to get a bit harder. Saturday is the hardest, arguably — Sunday is bigger but not trickier. I can get through Monday and Tuesday all on my own, but by Wednesday I almost always need help, either from my wife or “autocheck” (which tells you if you’ve made a mistake).

This has inevitably led me to this urgent question: What if instead of math and reading, every school in the country was tasked with teaching kids to solve crosswords?

In A Mathematician’s Lament Paul Lockhart has a similar thought experiment about music. “A musician wakes from a terrible nightmare. In his dream he finds himself in a society where music education has been made mandatory,” he writes. Lockhart is a present-tense dystopian and he teases out the likely result of mandatory music: drills, memorized rules, joyless schooling in “music.”

But that’s not what I’m interested in for this post. Sure, school would probably ruin crossword puzzles. But what if you really did want to help someone get better at solving them? What sorts of activities would you do?

Here is what I came up with.

Learning Activity #1: Learn the Most Useful Words

Schools currently teach vocabulary, and if we’re going to reorient the educational system around crossword puzzles we’re going to have to learn some new vocab. But which words?

Good news: not all words are equally likely to appear in a crossword. There are various online rankings of the words that most frequently appear in the NY Times puzzles, with “ERA,” “AREA,” “ERE,” “ONE” and “ELI” topping the list.

Those are fairly common words (sort of) but there are other ways of finding the most important words to teach our alt-world students. Noah Veltman has defined something called “crosswordiness,” which measures how much more frequently a word appears in crosswords than it does in books, in general. Topping Noah’s list are the words “ASEA,” “SMEE,” “URSA” and “SNEE.”

From Noah’s website

The best bang for our buck would be to design a curriculum that takes students through as many of these “crosswordy” words as possible. We would need to think carefully about how to group them in meaningful ways, but once the words and their meanings are introduced there are a lot of ways to practice.

The most crosswordy words must have dozens, scores — even hundreds of clues that hint at them. Students would need to practice these clues outside the context of the puzzles. Flashcards, quizzes, matching activities — whatever it takes to teach the crosswordiest words.

Learning Activity #2: Complete the Word

As a sort of experiment in self-improvement, Max Deutsch tried to spend a month getting much better at the NY Times crossword. One of the tools he created to do this (he’s a programmer and entrepreneur) was a “guess the missing letters” tool. (He calls it “a less exciting version of Hangman.”) Why is this useful? He writes:

For example, if I know a particular six-letter answer is I _ O N I _ , would I be able to guess, just based on word shape and my knowledge of the English language, that the last letter is most likely a C, giving me I _ O N I C, and then subsequently guess that the second letter is also a C, giving me ICONIC?

This seems useful. Let’s get every student in our alternate universe to do this too. We’ll also want to coach them in using this technique to push forward when they’re stuck on a puzzle. Which leads us too…

Learning Activity #3: Complete the Puzzle

Without a doubt, you have to try puzzles to get better at solving them. But if you always practice with blank puzzles, you’ll disproportionately practice the beginning of solving them. Why? No matter where you get stuck, you always will start the puzzle. If you are successful at the first stages of solving the puzzle then you’ll get practice on the middle and end of the game. But if you don’t get very far, you’ll still practice the beginnings — just never anything deeper. (Teachers: sound familiar?)

To get around this, teachers would assign students partially completed crosswords to solve. There is a limited amount of strategy that goes into solving a crossword. Starting with a puzzle that has already been partially filled in would give us a chance to talk about choices with kids.

We’d show students puzzles such as these and say things like: using the partially completed corner, complete 1 and 14 across, and then as many other clues as you can.

Of course, there are a lot of other things we would probably do. We’d ask kids to solve puzzles together on whiteboards, and we’d ask them to make their own puzzles at times. We’d have worksheets of puzzles and we’d give our honors classes 100 x 100 puzzles as extra credit. We’d field phone calls from parents worried that they tried to do a Tuesday — a Tuesday! — with their daughter and she wasn’t able to do any of it. There would be crossword tutors (“I have an Ivy League degree in Applied Puzzle Engineering”) and kids would write “LEE” on their finger nails before the big test.

Best of all, we’d get to read Op-Eds about whether kids really needed to learn crosswords these days: “Are Crosswords Outdated?” After all, in the 21st century most puzzles you need to solve don’t even have words at all…

In other words, this is an educational world that would likely be as dumb as our own. Which makes me wonder whether I really should try to get better at crosswords or not. Maybe part of why I enjoy them is because I don’t really care how good I am at them. So what if I suck? In life you need have things that you don’t have to be good at.

But, I have to tell you, I can’t stop looking at that list of “crosswordy” words.

Some books I loved in 2020

It feels like giving up to write a post like this with a week left to go in the calendar year. But what’s so bad about giving up? Anyway, I’m in the mood for comfort reading at the moment. That means I’m flipping between Kate Atkinson’s Life After Life, Rob Sheffield’s Dreaming the Beatles and Susanna Clarke’s The Ladies of Grace Adieu. No reason to put those on this list, though.

I’m typically a big subway reader which … well, you know. Even this fall, when I was back teaching in-person, I became an e-bike commuter unless the weather was truly awful. Now I’m on paternity leave for a few weeks, which demands its own sort of reading habits — morsels rather than feasts, is what works for me while caring for a baby.

OK, whatever, here are some books I loved reading in the order I read them.

The Power of the Dog, Don Winslow

Amazon.com: The Power of the Dog (9781400096930): Winslow, Don: Books

I think this review nails the situation with Winslow’s flawed but wonderful writing about America’s War on Drugs. “The book is trope-heavy, stereotype-heavy, occasionally (okay, often) one-dimensional,” which is true for all of Winslow’s writing. But why is it great? “Because it is a huge, meticulously researched book that comes at the end of a series 20 years in the making. Because it is a book that eschews flowery language for precision and quick action. Because the internal monologues of quiet men are often the ways we are given to understand their internal histories.”

As an aside, when my older son was born six years ago, I was reading The Jordan Rules by Sam Smith in the waiting room. This summer I was rereading Neil Gaiman’s Sandman when we had to rush into the hospital for my son’s birth. And three and a half years ago when my daughter was born I was reading Winslow’s The Cartel, a sequel to Power of the Dog, in the hospital.

Silas Marner, George Eliot

Silas Marner by George Eliot: 9780375757495 | PenguinRandomHouse.com: Books

I love moralizing fiction. What makes people bad? What makes them good? How do they get better (or worse)? I find it so easy to see myself as Silas Marner, emotionally wrapped up in myself and shut off from the concerns of others. In yeshiva we studied musar to reflect on our moral situation; Silas Marner is water drawn from the same well with a different vessel. Thanks to my wife for promising that I’d love this one.

Incidentally, this was the first book I was able to read to completion after lockdown back in March. Man, what a shitty year this has been in so many ways.

Ninth House, Leigh Bardugo

Ninth House (Alex Stern #1)

Do you hate Yale? I don’t, but maybe Leigh Bardugo does. Or maybe it’s just the moneyed elite that she hates, seeing the crimes she imagines them committing.

There are a lot of books that are “our world but magic” but this one is particularly audacious in the world it imagines. A friend who knows my enthusiasm for Stephen King lent this one to me. My advice? Make sure your friends know that you like Stephen King.

How To Tame a Fox (and Build a Dog), Lee Alan Dugatkin and Lyudmila Trut

How to Tame a Fox (and Build a Dog): Visionary Scientists and a Siberian  Tale of Jump-Started Evolution: Dugatkin, Lee Alan, Trut, Lyudmila:  9780226444185: Amazon.com: Books

I was so annoying while reading this book. I read long passages of it aloud to my wife. I couldn’t shut up about it to anyone I saw outside. “How are you guys holding up during these challenging times?” my friends would ask, and I’d reply with a long rant about all the physical traits that domesticated animals show.

OK, how do I get you to read this book? Put it like this: what does it mean to be domesticated? There is a real biological sense in which people are simply domesticated apes. But how does domestication happen? Who kicks off the domestication process?

This book tells the story of a group of Russian scientists who set out to study this question. Their experiment: to see if the famously undomesticated fox can be domesticated simply through selection for tameness towards humans. The experiment had to be carried out in secret thanks to the Soviet Union’s adoption of Lysenkoism, which held that natural selection to be a Western lie.

Biology is so cool. Science is wonderful. Keep it up, humans.

A Theory of Jerks and Other Philosophical Misadventures, Eric Schwitzgebel

A Theory of Jerks and Other Philosophical Misadventures by Eric  Schwitzgebel: 9780262539593 | PenguinRandomHouse.com: Books

Schwitzgebel has long been one of my favorite philosophy bloggers. His vision of philosophy is much broader than the as some grand debate between various -isms. He asks the best questions. Should we be surprised if ethicists are unethical? Is it OK to aim for moral mediocrity? What does it take for a machine to trigger our ethical impulses?

Each of the chapters is about blog-long, which feels like the perfect amount to bite off for each of these philosophical reflections. Just enough to read in one sitting and keep in the back of your mind for the rest of the day.

Let’s Talk About Love: A Journey to the End of Taste, Carl Wilson

The writing here is at times dense but it’s also lively and accessible. The question here is easy to state but hard to answer: if taste is subjective, what does it mean to have good taste?

More to the point, if Celine Dion sucks then why do so many people love her music?

I loved this book, but it’s not for everyone. You need to have a taste for theory and Wilson’s arguments are subtle and not straightforward. I think he comes out on the idea that artistic quality is always relative to a community. Within a community, there is something more or less objective that can be called “taste,” but that’s because the community has its own reference frame. Switch to another community and the rules of taste change. When a gap in taste between the critic and some other group shows up, that’s an opportunity to learn what makes that community tick such that they love the art that they love — really, a critic should do this.

This book is funny, big-hearted and big-brained. I should probably reread it in 2021.

A Series of Recently Published Famous Books that Everybody Likes that I Also Liked

Piranesi, Susanna Clarke

Station Eleven, Emily St. John Mandel

Circe, Madeline Miller

The Three-Body Problem, Cixin Liu

Exhalation, Ted Chiang

Station Eleven: Mandel, Emily St. John: 8601422213614: Amazon.com: Books
Madeline Miller - Circe
Amazon.com: The Three-Body Problem (9780765382030): Liu, Cixin, Liu, Ken:  Books
Exhalation: Stories: Chiang, Ted: 9781101947883: Amazon.com: Books

Some books that I didn’t love quite as much as the others, but still loved: The Great Secret, Jack Handey’s The Stench of Honolulu, George Bernard Shaw’s Man and Superman, Kitchen Confidential, Small Fry, Red White and Royal Blue, Very Short Intro to the American Revolution.

It’s impossible for me to review this objectively, but my friend Dr. Chavi Karkowsky wrote a truly remarkable book about her life as a doctor helping women with their high-risk pregnancies and deliveries.

High Risk, Chavi Eve Karkowsky MD

Even though it’s not really possible to be objective … the book is objectively great! And it’s not quite as terror-inducing as you might think. Chavi focuses on the systems — systems that save lives but also take on a life of their own and can feel at times oppressive for both doctors and patients both.

And here was my favorite more technical read of the year, one that I’m still in the middle of:

Ecology, Charles J. Krebs

Ecology: The Experimental Analysis of Distribution and Abundance | 6th  edition | Pearson

Like all textbooks this one costs 7 million dollars, but you can get the international edition used for far cheaper. (And if you are a determined googler … well, the internet contains all sorts of things, let’s leave it at that.)

This text has been a bit of a revelation to me. From start to finish, it’s mathematical. Are species more abundant when the species covers more territory? What happens when species compete with each other? How do populations grow?

The math is just accessible enough for me that I don’t need to break my head over it and can just enjoy the science. I’m looking forward to reading more of it and perhaps thinking about how I can bring some of it back for students.

Helping Students Make Good Mathematical Arguments

How do you teach someone how to make a good mathematical argument? Here’s my theory:

  1. Make sure they understand the stuff being argued about
  2. Show them what a good argument looks like
  3. Help them make good arguments on their own

I think I did a better job of this in this year’s geometry class that I have in the past, so I’ll share what I did.

First, I made sure that students understood what was being argued about. This is a plea for teachers not to teach mathematical concepts or skills through proof — a problem I see happening more at higher levels of math instruction. There’s simply too much that is new for students in a proof. It’s not the time to make sure students also understand every else.

In my class this year this happened both at the level of the course and that of the lesson. I started the unit with lots of time working with the sorts of diagrams that would feature later in proofs. But I also did this within a lesson. If I wanted to show students an example of a proof, I would make sure to launch the lesson with an activity that just involved analyzing that diagram.

Here is a diagram, I’d say. I just want you to tell me as many pairs congruent sides or angles in it as you can, and tell me how you know:

Then, after making a nice list of all the congruencies in the diagram, I’d preview the structure of the proof (thanks, Catrambone):

I would then model writing a proof along these lines using the congruencies the class found at the start of the lesson.

There’s a bit more that needs to happen at this stage to make sure students have a chance to really think about this. There are a few things that I’ve tried at this stage — asking students to turn and explain each step of the proof; removing a step from the proof and asking students to explain what it needs to be; asking everyone to take notes from memory with some or all of my proof erased. I’m still working on getting this stage of my little routine down.

Following this, I shared a new diagram and asked students to once again find congruencies. When I did this in class I used this new diagram and this is what students observed:

They came up with those questions as well. I asked the class to prove that the inner triangle was equilateral, using the structure that they had studied in that first proof.

So, the first stage is making sure students know what the proof is talking about. The above is how I’ve done a better job of that this year.

The last step is trickier than it may seem. Yes, it’s “obvious” that helping students make good argument is important for learning how to do exactly that. But the emphasis I’d put is on help and (especially) good. Because in my experience, there’s strong gravity that pulls students towards arguments that are intuitively good but not up to the rigorous standards of mathematical proof.

The issue is basically this: everyone knows a lot about shapes. In fact this is not an issue — this is fantastic. I love hearing people’s ideas about shapes. But if you don’t provide the right kind of help, you’ll confuse and anger your students. The reason is because your job, as a mathematics teacher, is to teach a very particular form of mathematical argument. And students will have many wonderful ideas about shapes all their own. In past years I fell into a tricky spot where my students were baffled why their own extremely reasonable arguments for why a pair of triangles must be the same size and shape aren’t acceptable as proofs.

In case you haven’t heard these things live and in person, the human eye is attracted to symmetry and change. If you ask a group of people why a pair of lengths are the same, they will be drawn to arguments that incorporate those sorts of features. “Those lengths have to be the same because the diagram is symmetric. If it weren’t symmetric then when you moved that angle over there, that other angle would have to change also. And then the whole diagram would be broken.”

What’s the issue here? Nothing, really. It’s beautiful. It’s just not rigorous enough to pass muster in a geometry class. And while some people might be tempted to push on students — how do you KNOW that it’s symmetrical? how do you KNOW that angle will change? — my experience is that this only angers teens, who grow disgruntled that their own reasoning has been invited to the party (finally!) and then rejected as unacceptable.

Far better, I now believe, to help students make GOOD arguments, where “GOOD” is defined as “whatever is acceptable from the point of view of the course.”

In practice, this means that I have started to include a lot of visual choices for students as they are asked to form justifications. Here is a snap shot from a problem set I assigned in class:

There are now choices. Your argument will look like one of these four choices — always. Learn to use them, and they’ll become part of your language of justification. (Thanks, self-explanation literature.)

In short, students’ choices are constrained. To put it one way, especially in the early stages of learning, students are allowed to choose a wrong argument but they are not allowed to make a bad argument. (Again by bad I don’t mean “bad,” just “not ideal within the context of this course.”)

I think I’ll be working on improving my teaching of proof until the day they drag me feet first out of a classroom, but these two ideas feel like progress to me.