# I kind of like these congruent triangle problems I made

For each of these the prompt is, “find as many missing things as you can.” And what I like about them is that it gave my students to use congruent triangles alongside other sorts of deductive moves. We have ways of finding missing angles in a diagram, we have other ways of finding missing lengths. A congruent triangle argument is just like that — it’s another way of using what you know to know a little bit more.

The other nice thing is that in each of these diagrams, there is information that we can’t know. I think it was good for my students to experience that too.

For each of these my routine was to show the image, state the prompt, ask for a raised thumb when you figured something out, assign partners, task them with finding more things out, quickly listing some of the lower-hanging fruit on the board, then discussing whatever I found most interesting in each diagram.

I thought it might be nice to make a set of practice problems to follow-up on these, but I haven’t gotten around to it yet.

Like practically every teacher I’ve ever met, I am deeply skeptical of replacing classroom learning with personalized learning software. But since the Chan Zuckerberg Initiative is in the middle of a long, sloppy kiss with personalization advocates, we’re apparently going to have to put up with it for a while.

In particular, Summit Learning is essentially a CZI initiative, and — surprise! — it’s having precisely the same issues that everyone who has ever attempted to do this has had, i.e. open rebellion:

Brooklyn teens are protesting their high school’s adoption of an online program spawned by Facebook, saying it forces them to stare at computers for hours and “teach ourselves.”

“It’s annoying to just sit there staring at one screen for so long,” said freshman Mitchel Storman, 14, who spends close to five hours a day on Summit classes in algebra, biology, English, world history, and physics. “You have to teach yourself.”

Look, I haven’t done a ton of research about Summit Learning or the particulars of what’s happening at this school. It’s ridiculously easy to hate on personalization and that’s not what I came here to do.

No, I came here to talk about DeltaMath.

DeltaMath is dead simple. It’s a robot that puts different numbers into math problems and tells kids if they solved them correctly. There are examples to study. Nothing fancy here, though there are many different types of problems the teacher can assign. The algebra sequence is especially well-covered. I choose what kids do for homework. I see their responses. It’s free for now, and God help us let it be free forever.

I’d known about DeltaMath for years and dismissed it. After all, it’s just a robot that puts different numbers into math problems and tells kids if they’ve done them correctly or not. There are all sorts of fancier robots out there. And there are well-documented problems with simple robots. Why get excited about this one?

Not to beat up on DeltaMath, but it is literally the least inspiring idea in education. It brings simple, repetitive practice with right/wrong feedback to homework. Wheeeeeeee.

And yet: today was parent teacher conferences, and parent after parent thanked me for using DeltaMath as homework. Thank you, they said, over and over again.

And my kids, the algebra students? They love it too. One kid: “DeltaMath has changed my life.”

I mean this is ridiculous, right? But I swear, it’s true.

And it’s not because it’s perfect. No no no, not at all. The point is that it’s marginally better than conventional homework for every party. I know, not exactly the sort of thing that will get you billionaire money, but here are its advantages over conventional homework:

• For the kids: They get simple information about whether their answer was accurate. Sometimes kids get frustrated when the computer doesn’t get their input and marks them wrong…but they all seem to recognize that the alternative is no feedback while working on their homework. This is better.
• For the parents: A lot of what we do in class is difficult to communicate, but this is simple. Parents appreciate the simple clarity of understanding just a bit of what their kids are working on. It’s an improvement over being totally confused by what their kids are working on.
• For me: Homework is a relatively low-yield instructional activity, as far as I’m concerned. I’m not there to help or observe so it’s hard to trust what kids bring in. It can be worthwhile for kids, but it’s definitely worth less of my work hours than what I put together for the classroom. And that’s the thing: a homework worksheet takes too long to make for its contribution to learning. The software both improves the practice a conscientious kid can get from doing homework while drastically cutting the time it takes to create a homework assignment.

Insert quote from literally anything Larry Cuban has written about educational technology here. Maybe a line from Tinkering Toward Utopia, maybe. My copy is in the other room, so I’ll just make up a Cuban-ish quote:

“Despite the sky-high promises of would-be reformers, schooling has a strong conservative tendency. This is not to suggest, however, that teachers have not embraced technology. They have — though often not in the ways reformers intended.”

Just to be clear, that quote is entirely made up. Do not cite that.

That’s the thing with personalization software, though. In a few years when this all plays out and Chan Zuckerberg compliment themselves on having taken a big swing and on not being afraid of failure, classroom learning will be more or less intact. But I have no doubt there’s going to be a role for cheap software that improves learning at the margins. And now that I’ve seen how it’s playing out in my algebra classes, I’m much more willing to support software that replaces paper homework.

Seriously: the robots are ready for homework.

# Why it’s so hard for online math communities to do stuff in 2018

I.

Have you heard of Nix the Tricks? It’s a great example of the way you used to be able to get things done on the internet.

In 2013 Tina Cardone was part of a Twitter conversation about bad mathematical shortcuts. (I personally dislike the “two negatives make a positive” shortcut because kids use it for adding/subtracting and you can just say “multiplying/dividing by a negative changes the sign.” I digress.)

Tina had the insight to take that twitter conversation and turn it into a collaborative google document. Tons of teachers on twitter contributed, and soon they had what was (after Tina organized it) a book full of tricks, examples of when those tricks go badly, and suggestions for replacements. (They weren’t considered tricks at that point; a good trick is not a trick.)

If you were trying to explain to someone what “crowdsourcing” meant, you couldn’t point to a clearer example.

Should we consider it a coincidence that a famous TED talk that popularized the concept of crowdsourcing was filmed in March 2012?

The key to understanding online math edu communities (I’m thinking of one in particular, MTBoS) is that they are totally subject to every trend that the rest of the internet is subject to. Crowdsourcing was big in 2012-2013, partly because internet culture was totally ready for that.

Online math teacher communities are part of internet culture.

So if you want to understand how to change the way that community operates, you have to understand why the old methods for getting stuff done online no longer really work.

II.

Here is a history of the internet over the past 20 years based only on my recollections. I’m basically not looking anything up here — just going based on memory.

• 2001 – 2005: Forums are a big thing. Blogs are increasingly a thing, but they’re mostly something you do with a fake name. I had a high school friend who was in love with another friend and blogged about it. Online was a place to anonymously post your secrets. Blogs are a joke.
• 2005 – 2010: Blogs are totally ascendant. Blogs break significant news. Blogs become a way to start a famous career. It’s totally respectable (but odd) to blog, and increasing numbers of people do it. I used to slavishly read the bloggers on The Atlantic back when I was in college. Nate Silver, Ezra Klein, Matt Yglesias, Andrew Sullivan, this is that time.
• 2008- 2012: Video: YouTube, Vlogging, these all become popular and merge into the mainstream — more precisely, become a path towards internet stardom. John and Hank Green crossed my experience in this era. This is the era of the ascendance of TED talks. This is also the time when crowdsourcing became a big, hot thing. I’m worried about getting the timing wrong about this one, but I remember reading this New Yorker article about TED talks and it was published around 2012 so let’s go with that.
• 2012 – 2015: The rise of social media, the decline of blogs, the death of Google Reader. I remember the death of Google Reader was around this time, because it was life-sustaining for me when I got right out of college. All of the sudden, though, it was gone, and everyone was on Twitter. Everyone.
• 2015ish: Everyone talks about the death of blogs.
• 2016ish: Everyone talks about how blogs are making a comeback.
• 2017-Now: Blogs are definitely mostly dead in a certain sense, but like every other internet trend of the past 18 years it is thriving if you look in the right places: mainstream content producers. Blogs are alive in the sense that it’s where established people share shorter pieces that didn’t make it into a longer piece, or it’s a place for product announcements. Likewise nearly every ever other web trend is also alive in the sense that it has merged with the mainstream. Twitter is no longer a Wild West where anyone can rise to prominence — it’s a place where prominent people create content, and the vast majority of people’s activity is in response to that content. You follow a journalist or a celebrity, and then your fundamental activity is resharing and liking.

And here is a completely parallel history of the MTBoS, a particular online math edu community:

• 2006-2010: The rise of bloggers like Dan Meyer, Kate Nowak, Sam Shah show that you can do this thing with your name and people will love it. I started teaching in 2010 and these blogs were my main outside influence. These are bloggers that gained a kind of mainstream prominence in the math edu world via their blogging.
• 2010-2012: Video! Dan’s popularity really kicks into gear when he gives a fabulously popular TED talk and he shared high-resolution media resources (WCYDWT?) For a few years the MTBoS reflects this trend — Andrew Stadel, Timon, others, oh god I’m forgetting everybody.
• 2012 – 2015: The decline of blogs, rise of Twitter. People start posting less frequently, starting posts with phrases like “Is this thing still on?”
• 2015 – Now: All of the previous internet activities are alive, but in the same limited sense that the rest of the internet displays. There are a lot of people who create content but they are mostly people in the mainstream. Blogging is something that people mostly do if (a) have leftover stuff from their larger projects (b) have announcements or updates (c) feel nostalgic or (d) are dorks who love writing.  And twitter activity is mostly reacting and responding to more prominent people. (Though, like the rest of twitter, sometimes individual tweets go viral. In general this doesn’t lead to new people becoming prominent any longer.)
• 2016 – 2018: People start talking about all the problems with MTBoS Twitter.

So the thing is that MTBoS or any other community is not a thing apart from the rest of the culture. If you want to understand the changes that MTBoS has made over the years, the clearest information comes from the rest of the internet’s evolution.

III.

There clearly are problems facing MTBoS. Two recent ones that came across my radar:

• Kent Haines wishing that more practical, nuts-and-bolts teaching advice got shared and discussed.
• Tina Cardone, Marian Dingle and Anne Schwartz correctly pointing out that the MTBoS in all its manifestations is an overwhelmingly white online space. Black people especially feel this.

My contribution to this discussion is to say: don’t think that the old way of doing things will be able to change the culture.

Don’t rely on crowdsourcing.

Don’t even rely on attempting to change the culture through exhorting individuals to change what they do. I’m not saying this is bad, I’m saying I don’t think this works in 2018. The internet is too big, and the MTBoS is also too big. I’m not saying this isn’t valuable, I’m saying I don’t think ultimately this will make a community either less-overwhelmingly white or more politically engaged.

(By the way, I think those are two separate goals — inclusion and political activity — that often get conflated in this discussion. You can have a white space that is engaged in anti-racist, progressive work. You can have a racially inclusive space that only talks about math. I’m not convinced that if you get one, you get the other.)

But I’m not pessimistic. Here’s what I wrote over at Sam Shah’s most recent “State of the MTBoS” post:

I think it’s generally true that a lot of conversations are happening on Slacks or within teams instead of in public right now, and that this is because some of the most interesting online presences from the first/second generation of MTBoS-ers are working for Desmos, IM, writing for Stenhouse, etc.

I am as guilty as anyone for making a big stink about this, but I think what I’m realizing now is that this just is. People change, careers change.

I think we’re past the the time when we could hope that the conversations that we need to have just emerge from the froth and slosh of online activity. This is what happened in 2008 – 2013 or so, but then a lot of things happened: blogs changed, RSS got abandoned, Twitter got huge, teachers moved on, etc.

The first stage of MTBoS was about the excitement of this new thing we all had. Then came a kind of order emerging from the chaos. Very recently, it feels like the benefits of that order are being consolidated and harvested for mainstream consumption.

I think we’re maybe entering a new era of MTBoS and online activity now, and it’s a time of ACTIVE ORGANIZATION of the new. The stage that we’re beginning to see is a time when spaces can’t just be taken for granted — we need to cultivate the sorts of spaces we want to have. This is a change, but it’s a necessary one if we’re going to keep moving.

So if we’re feeling that there’s not enough energy around curricular discussions, Kevin, I think we need to bust out our rolodexes and start organizing. Who else do we know who’s interested in Algebra 1? What sort of a project might they be interested in? Can we put together an online gathering? An in-person gathering? Who can we connect with, and how?

I think we’re going to be asking ourselves those questions more and more in the years to come.

This is a lot to ask from your average community member, and that’s not what I mean to suggest. But I think we need to start thinking about online communities in the internet that we have, not the internet a community was born in.

IV.

So there are some tough, hard problems to work on. In 2018, Twitter clearly has problems. It generates a mono-culture that doesn’t satisfy everybody. It’s not a hidden corner of the internet, and everything you post is visible to your employers. It’s not a forum where everyone can safely talk about important but sensitive topics.

This might require some people to do the hard, interpersonal work of building a discussion forum outside of Twitter that can focus on sensitive but important topics. Topics of discussion like classroom management, coaching, racist colleagues, etc.

The internet has left behind the resource-sharing days of the past. But there are still people who want to write curriculum with challenging extensions, practice ideas. There is surely a way to help people who are interested in this find each other. Someone will need to do that.

And if MTBoS is an overwhelmingly white space, that inertia will probably need to have a solution outside of Twitter, where things have already ossified into a real hierarchy — just as non-teacher Twitter has. Any real solution will involve hard work outside of Twitter — finding people who don’t fit the white (progressive, coastal) norm and partnering with them in significant ways. You can’t get there, I don’t think, by making it easier to enter the online space. You need to find non-Twitter ways of connecting with people, and then inviting them in.

And you’ll probably need to make it safe for people to enter this overwhelming white space by helping them meaningful connect and form community first, as I wrote about in this post.

***

Every community makes the mistake of thinking that their history isn’t subject to larger forces. (The Talmud says about Jews, “there is no constellation for the Jewish people.”) But you can’t crowdsource the changes that online communities need in 2018 — you can’t just share an idea, and hope that people jump on it, and one thing leads to another and suddenly BAM that thing exists. The internet hasn’t worked like that for a while.

It is hard work, and it’s people work, but change can happen in 2018. If you think something ought to be different online and you think you have energy for it, here are the steps I think we should all take:

1. Email or call friends, until you have 3-5 people signed up for the project.
2. Start the project, and once you’ve figured things out, try to invite more people to get involved.
3. Figure out a way to share it with the rest of the online community.

This is not to discredit anything else that anyone else is doing, but it’s this sort of change that I think the times call for.

# The absolute best way to practice flash cards with elementary students

I will show you my favorite way to ask kids to practice with flash cards.

But first, some logistical tips that I’ve picked up.

Tip #1: Everybody gets a hard plastic case.

Tip #2: Everyone gets one and only one color of flashcards. I repeat, DO NOT LET YOUR KIDS MIX COLORS OF FLASHCARDS. CHAOS WILL ENSUE AND THEY WILL NEVER BE ABLE TO FIND THEIR CARDS IF THEY GET MIXED UP WITH SOMEONE ELSE’S DECK.

I KNOW WHAT YOU’RE THINKING AND, YES, SOMETIMES KIDS WITH SAME-COLORED CARDS END UP PRACTICING NEAR EACH OTHER AND, YES, THEIR DECKS GET MIXED UP. IT’S HONESTLY NOT THAT BIG A DEAL, BUT I’LL DO WHAT I CAN TO PREVENT CONFUSION RIGHT UP TO THE POINT WHERE I’M TELLING NINE-YEAR OLDS THEY CAN’T PRACTICE WITH A FRIEND BECAUSE THEY HAVE SAME-COLORED FLASH CARDS.

Tip #3: If kids have a hard time with a card, let them write little helper problems at the bottom of a card. Put them in pencil so you can erase them when you don’t need them any longer.

Tip #4: Have kids make their own cards, but don’t have make solving all the problems a pre-condition for making all the cards. In other words just give them the answers with the problem. The whole point of this is to practice with feedback, so don’t be shy about giving out answers at first.

To illustrate, I gave out this sheet to my third graders:

OK so that’s the logistical tips. Now, on to the best practice set-up with flash cards.

The thing to remember is that there are all sorts of problems involved with asking two kids to practice together, even though it’s very fun to practice with a friend.

Meaning suppose that you’re doing what I call forward practice, i.e. you’re looking at the problem.

How do you make sure both kids get something out of that exercise? Inevitably one kid goes and yells out the answer in excitement before the other one has a chance to finish thinking. This I think can make kids feel less than their partners, mathematically, and also can make the activity a waste of their time.

So you tell everyone to make sure they each have a chance to raise a thumb or some sort of other check-in to make sure both partners are ready to check the other side, but these are third-graders we’re talking about and this isn’t something particularly easy for them to remember. So you remind them, again and again, and the practice starts to feel exhausting and not as much of a fun game any longer.

That’s an advantage to what I call backwards practice, which is looking at the ‘answer’ and then trying to think about what problem is on the other side. So here’s 48, what’s the problem? Jeopardy style.

And this is better, because there are multiple answers and both partners can contribute…but honestly it’s got the same problems as forward practice when it’s done with partners.

Which brings me to the best way to practice with cards but also one FINAL logistical tip.

OK here it goes: one partner does forward practice, the other does backward practice with the same card, held between them.

Here is a picture illustrating the basic dynamic:

And you probably don’t realize the best part about this, which is that IF I SHOUT OUT MY ANSWER IT DOES NOT TAKE AWAY MY PARTNER’S CHANCE TO THINK.

Here is that best part of this best way to practice, illustrated via dialogue.

***

Card: Kid A’s side says 6 x 8, Kid B’s side says 48.

Kid B: You are excellent. Also yes that is in fact what my side of the card says. Good work. Now, does your side say 12 x 4?

Kid A: No it does not.

Kid B: Hmm. Can I have a hint?

Kid A: You’re right that there’s multiplication. Also it has an 8.

Kid B: Oh, OK. How about 6 x 8?

Kid A: Now you are correct.

***

Tip #5: Have the kids write with pencil on their cards so that the numbers don’t bleed through to the other side of the notecard, thereby ruining the best way of practicing with flash cards.

(Though a few inventive children have found that if you insert a card into the translucent decks it obscures the backside while leaving your side visible. Children are genius.)

***

There are two other interesting ways to practice that I’ve come up with. I won’t spend much time explaining them because they don’t deserve it; they aren’t the best.

• Place a bunch of cards out in front of view, and try to pick a few that get you as close to some target number (e.g. 100) as possible.
• Pick the same three cards from two different decks (two different colors, please!). In one set place the problems facing up, with the other place the answers facing up. Try to match them.

But both of these require more set up or more clean up or more time than the very best way, which I’ve already detailed extensively above. That is all.

# Practicing equation solving at the level of the move

My 8th Grade students need a lot of practice with some of the basic moves of algebra. But I’m not finding practice resources that organize things along the lines that I feel they need.

A lot of the existing practice resources categorize equations in big amorphous clumps like “one-step” or “two-step” equations — but there is a whole world of variety between different types of “one-step” equations!

I think this is a confusing way of asking kids to practice equations. It groups equations based on how the equations look. It would be better to group practice along the lines of how you can think about them.

What I really want — what I think my students need — is practice that focuses on moves, and that includes chances to use those moves on situations that look very different from each other.

Here’s something I whipped up for class today. I haven’t tried it yet, but this is the sort of thing I’m thinking of:

Questions:

• Do you think this is a useful way to organize practice (example/targeted practice/mixed)?
• Do you agree that this (adding to both sides) is a useful chunk of a skill to teach? (I know that, for my students, it is, because I’d found that kids very easily see opportunities to remove something from both sides but not to add.)
• What other moves in algebra do you think would be useful to practice in this way? (Next up for my students, I think, would be super-focusing on equations/inequalities that come down to Ax = B.)

# Writing good questions for fast finishers

Here’s the original activity, from Illustrative Math’s 8th Grade curriculum:

I love the main task. I don’t really know what to do with that “Are You Ready For More?” extension, though.

It’s definitely challenging and interesting, but it’s not really connected deeply with the math that comes before it. And really solving that HANGER + HANGER + HANGER = ALGEBRA problem likely would require a bunch of time — where is that time coming from? Usually I don’t need something that’ll take kids ~15 minutes to solve, I need something interested a fast-finisher can think about for ~3 minutes.

Put it like this: I’m not sure it would be worthwhile (or even possible) to talk about that HANGER/ALGEBRA puzzle with the whole class, if only a couple students even got to it. But if I’m not going to be willing to honor those sorts of problems with airtime, why would a kid ever dig into it?

I don’t mean to give the Illustrative Math curriculum a hard time, as I’m a huge fan. I’m really thinking more about how to improve the problems I’ve prepared for students who finish quickly. How can I make it clear that they really are part of our class?

I’ve really been playing with these ideas this year. Here was my replacement for the HANGER/ALGEBRA puzzle:

The math is close enough to the task itself that I really felt like I could talk about these problems with everyone — even briefly — and it was valuable. Kids who try it will get some airtime. I’m trying to bring the extensions closer to the main task.

I’ve also been trying this with geometry, where I built an activity on top of some Don Steward practice problems:

Understandably, Illustrative Math doesn’t have extensions like these ready for every problem. But something like this is an easy way to make an activity more useful for a broader range of students (which also gives kids who need it more time).

But Illustrative Math is a free and openly licensed curriculum. They’re actively seeking user contributions to their curriculum — all this seems like a nice match for the sorts of revisions I’m talking about here.

How long would it take to write these things? How many could 20 teachers write in an afternoon? One for each 8th Grade task?

A teacher can dream!

# Is there a good way that schools could better serve their most advanced students?

This essay is co-authored with TracingWoodgrains, and is the result of our adversarial collaboration project.

TW and I have deep disagreements about education. I’m a pessimist about personalization software and our ability to gamify learning; TW is an optimist. TW thinks there very well might be a deep reserve of untapped potential at the upper ends of student achievement; I don’t really think this is true. I don’t think a radical restructuring of schools would change much unless you could simultaneously change the thoughts and desires of parents and students; TW thinks we should seriously consider rebuilding schooling on the grounds of mastery learning without age-grading.

With all this disagreement, we set out to see if we could write something coherent about our areas of agreement. Not easy.

We also set out to win a cool \$1K from the blog SlateStarCodex, who hosted a contest for the best adversarial collaboration and has published our essay but has not yet told us if we won.

tl;dr THIS THING IS LONG. Here is the section breakdown:

Section One: Gifted underachievement is real, but there are no easy answers.

Section Two: Expanding or deepening tracking is unlikely to help even top students without deeper curricular modification.

Section Three: Acceleration could, though.

Section Four: The thing is that schools are not in the business of maximizing learning at all costs.

Section Five: Personalization software, as it exists today, is almost universally bad. The potential for software to help learning comes when it pays attention to the social, motivational aspects of learning.

Section Six: Our recommendations for parents, educators, kids.

Please skip around. Skip to Section 5 if you care about edtech and personalization software. If you only read one section, I’d personally prefer if you read Section 4, but hey, that’s just me. TW would (I’m guessing) want you to read Section 1.

Also this was written for a non-teacher audience, i.e. not my blog. So apologize if the tone is off. Also half of this is TW’s voice, so it probably reads different than what you’re used to hearing from me. Anyway, enjoy, and please comment with thoughts, disagreements, agreements, etc.

***

“What do America’s brightest students hear? Every year, across the nation, students who should be moved ahead at their natural pace of learning are told to stay put. Thousands of students are told to lower their expectations, and put their dreams on hold. Whatever they want to do, their teachers say, it can wait.”  – A Nation Deceived, p.3

“There is an apparent preference among donors for studying the needs and supporting the welfare of the weak, the vicious, and the incompetent, and a negative disregard of the highly intelligent, leaving them to “shift for themselves.” Hollingworth, 1926

1. Eager to Learn and Underachieving

Pretend you’re a teacher. With 25 students, who gets your attention during class?

There’s the kid who ask for it, whose hand is constantly up. There’s also the quiet kid in the corner who never says a word, but has been lost in math since October, who will fail if you don’t do something. There’s the student in the middle of the pack, flowing along. Finally, there’s the kid who finishes everything quickly. She’s looking around and wondering, what am I supposed to do now?

In a survey of teachers from 2008, just 23% reported that advanced students were a top priority for them, while 63% reported giving struggling students in their classes the most attention. A 2005 study found the same trend in middle schools, where struggling students receive the bulk of instructional modification and special arrangements. This was true even while 73% agreed that advanced students were too often bored and under-challenged in school. While teachers, it seems, are sympathetic to the smart bored kid, that’s just not a priority for them.

This isn’t to blame teachers who are under all sorts of pressure to carry low-performing students over the threshold and who, in any event, are only trying to do what’s best for their kids. Which is the most urgent concern? If you don’t equip a kid with the skills they need, next year’s class might be a disaster for them. Or maybe they’ll fail out of school. And behavior problems? Often those begin with academic struggles. Gifted children, on the other hand — they’re on the way to becoming gifted adults. They can take care of themselves, for a minute, the logic goes. More often than not, the teacher will encourage the early finisher to go read a book, or start homework, or do anything at all while the teacher works to help the quiet, lost kid in the corner.

If the kids are just a little bored, that’s nothing strange. It’s hard to find someone who wasn’t bored in school sometimes. For many top students, already poised for achievement, this turns out just fine. And yet, there are persistent stories of how the lack of challenge can turn into something more serious.

One version of the story goes like this: from a young age, a student finds the work in school easy. It doesn’t take long for them to expect school to be easy for them — it becomes a point of pride. Over years of floating through school, an identity takes hold. Then, one day, maybe after years of schooling, something finally becomes challenging for the student… but there’s nothing nice about this challenge. The challenge is now a threat. The student begins to find school challenging, and their world falls apart. They feel isolated and misunderstood at school. They lash out. They hate it, and they can’t wait to get out.

When we asked Reddit users and blog readers to describe their experience of school, we heard versions of this story:

• Miserable waste of time, was almost never offered opportunities to learn. Largely ignored teachers and read books during class. I felt like it was a profound injustice that I was punished for doing so. I now have kids of my own and will be home-schooling them.
• I was bored. The pace was too slow and work was not interesting. Being forced by law to get up early and go somewhere to learn things I already know means permanent and firm dislike.
• I went to local public schools for kindergarten through high school, and the experience wasn’t good. Academically, the classes were slow and poorly taught. Even the AP classes were taught at the speed of the slowest student, which made the experience excruciating. The honors and regular classes were even worse: I was consistently one or more grades ahead of the rest of the class in every non-AP class except honors math. I learned not to bother studying or doing homework even in the AP classes which probably wasn’t great for my work ethic.

The stories of student pain and underachievement in school get more intense as we consider cases of extremely precocious children. The pressures on the student increase, and without help a student often experiences isolation from their peers and a whole other host of difficult feelings. Miraca Gross studied students like these in Australia and found that precocious students were often suffering in silence. Speaking particularly about precocious students who underachieve, she writes:

The majority of the extremely gifted young people in my study state frankly that for substantial periods in their school careers they have deliberately concealed their abilities or significantly moderate their scholastic achievement in an attempt to reduce their classmates’ and teachers’ resentment of them. In almost every case, the parents of children retained in the regular classroom with age peers report that the drive to achieve, the delight in intellectual exploration, and the joyful seeking after new knowledge, which characterized their children in the early years, has seriously diminished or disappeared completely. These children display disturbingly low levels of motivation and social self-esteem. They are also more likely to report social rejection by their classmates and state that they frequently underachieve in attempts to gain acceptance by age peers and teachers. Unfortunately, rather than investigating the cause of this, the schools attended by these children have tended to view their decreased motivation, with the attendant drop in academic attainment, as indicators that the child has “leveled out” and is no longer gifted.

What do we make of these stories? How common are such experiences?

From the literature on “gifted underachievement” we get partial confirmation — underachievement is a real phenomenon, supported by numerous case studies. According to a survey of various school practitioners, underachievement is the top concern when it comes to gifted students. By definition, advanced students are only a small percent of each student body, so few are affected in any given place, but on a national scale it becomes a more serious problem.

This is not just a problem for the affluent. It has persistent impacts on Black students, poor students, and students who are learning English, who are less often recommended for gifted programs or special accommodations. Here’s one way this manifests itself: in one study, 44% of poor students identified as gifted in reading in 1st Grade were no longer academically exceptional by 5th Grade. For higher-income families, only 31% of 1st Graders experience this slide.

The lack of attention to this group extends to the research. It’s difficult to pin down the number of students impacted. While underachievement is a real phenomenon, current research doesn’t tell us very much about the factors contributing to gifted underachievement. What studies have been done tend to focus almost entirely on things like whether students with ADHD or unsupportive families underachieve, rather than looking at controllable factors like the sort of teaching students experience in school.

Schools are the institutions in charge of educating kids. Those who rush into school, eager to learn, should not walk out feeling rebuffed and ignored. This is doubly true for talented kids from at-risk populations, who may not have the support structure outside of school to ensure their success if school has no time for them. It’s clear, though, that we cannot degrade the experience of other students to help those who already have an academic leg up. Is there a feasible approach to address this problem without making things worse?

We have good reason to think that personalized attention makes a huge difference to a student’s learning. Research suggests that tutoring that supplements a student’s coursework is a very effective educational intervention. Benjamin Bloom caught people’s attention with the idea of a 2 standard deviation effect in the 1980s. More recent research has lowered that sky-high estimate to more realistic numbers, and a meta-analysis found an effect size of 0.36, still a powerful impact, enough to take a student from the 50th percentile of achievement to the 64th.

If supplemental tutoring works, the dream goes, what if we replaced classroom work entirely with tutoring? Can’t we just do that for gifted underachievers and precocious students? We have tantalizing success stories of this kind in the education for precocious children. In a famous case, John Stuart Mill’s father decided that the philosophy of utilitarianism needed an advocate, and planned a demanding course for him. Mill didn’t underachieve: he learned Greek at age 3, Latin at age 8, and flourished as a philosopher. László Polgár declared he had discovered the secret of raising “geniuses” and went about showing it by tutoring his daughters in chess from the age of 3. It’s hard to argue with his results: two grandmasters and an international master, one of whom became the 8th ranked chess player in the world and the only woman ever to take a game off the reigning world champion.

Though this sort of tutoring seems like a dream come true for underachieving gifted students, in practice it’s a non-starter in schools. (It lives on in homeschooling, to an extent). In a world where schools are struggling to help every kid learn to read, the ethics of only assigning tutors to gifted students is dubious and almost certainly a political impossibility. The cost of assigning a tutor to every child, meanwhile, would do something special to property taxes. This simple answer, then, can lead to a clearer understanding of the complexity of educational questions: It’s possible to focus on simple practices that work while disregarding nonacademic concerns and political feasibility.

To be useful, educational ideas should be effective, politically feasible, and economical. If tutoring for gifted underachievers isn’t workable, might there be some other way to approximate the benefits of personal, human attention? Here are three of the most common tools that advocates for gifted education propose:

What follows is an evaluation of how promising each of these tools is, both in theory and in practice.

Our favorite one-stop reading on tutoring: this.

1. Ability Grouping (a.k.a Tracking)

The case for placing students of similar abilities together in a classroom seems like it ought to be as simple as the case for tutoring. Teachers will be more effective if their students have similar pacing needs. So, group kids who need more time in one class and those who need less time in another. It’s not tutoring, but it should be the next best thing.

Things in education research are rarely that simple, though.

Bob Slavin, a psychologist who studies education, is one of the most-cited education researchers around. He seems like a compulsively busy fellow. He writes, he runs research centers, he designs programs for schools. (He blogs.) A journalist from The Guardian once asked Slavin for his likes and dislikes, and in case you were wondering he likes work and dislikes complacency.

In the late ‘80s and early ‘90s, Slavin performed a series of meta-analyses of the existing literature on tracking and between-class ability grouping. Overall, he found no significant benefits from ability grouping, even for “top track” students across elementary, middle, and high schools.

But the other surprising finding of Slavin’s was that nobody was academically hurt by ability grouping — not even the lowest track students. Slavin argued that when you consider all the non-academic concerns, the scales weigh in favor of detracking, i.e. avoiding ability grouping.

What are those non-academic concerns? In the conclusion of his review of the evidence from elementary schools, he writes:

“Ability grouping plans in all forms are repugnant to many educators, who feel uncomfortable making decisions about elementary-aged students that could have long-term effects on their self-esteem and life chances. In desegregated schools, the possibility that ability grouping may create racially identifiable groups or classes is of great concern.” (p.327)

That’s Slavin’s view. So, where is the debate?

One thing that is decidedly not up for debate in the literature is that Slavin’s non-academic concerns are real. Opponents and defenders of tracking alike agree that low-track classes are often chaotic, poorly taught environments where bad behavior is endemic, and that this is a major problem. Tom Loveless is a contemporary defender of tracking, and writes that “even under the best of conditions, low tracks are difficult classrooms. The low tracks that focus on academics often try to remediate through dull, repetitious seatwork.” Jeannie Oakes made a name for herself by carefully documenting the lousiness of a lot of low track classes.

Some tracked schools seem to have done better with their low tracks. Gamoran, an opponent of tracking, speaks highly of how some Catholic schools handle lower tracks. Gutierrez identifies several tracked schools with strong commitments to helping students across the school advance in mathematics, and concludes that “tracking is not the pivotal policy on which student advancement in mathematics depends.” Making these experiences better is an important goal. These difficult dynamics are a genuine and widespread issue, though, and educators are rightly concerned about them.

Slavin’s concerns about exacerbating racism in schools are relatively uncontroversial as well. It’s not so much that race is a factor in track placement. Using a large nationally representative sample and controlling for prior achievement, Lucas and Gamoran found that race wasn’t a factor in track placement. (Though Dauber et al, found that race was a factor in track placement in Baltimore schools, so maybe sometimes racism is a factor in placement.)

But because of existing achievement gaps between e.g. Black and white students, there’s the potential in a racially mixed school that ability groups will effectively sort Black students into the lowest track and expose them to a lot of dynamics that are difficult to quantitatively measure but frequently discussed in education. A school where being Black is associated with poor performance and misbehavior will, according to many educators and researchers, lead to lower expectations and academic self-esteem for all Black students.

(Good news for people who like bad news: school segregation is getting worse, so the interaction between tracking and race is getting better.)

The main controversy surrounds Slavin’s claims about the academic impact of ability grouping. His meta-analyses were part of an extended back-and-forth with Chen-Lin & James Kulik, who wrote several competing analyses on the ability grouping literature. Slavin and the Kuliks each criticized the other’s methodology, but the core point the Kuliks made was that ability grouping did have positive effects on gifted students as long as curriculum was enhanced or accelerated to match, and that this typically did happen in dedicated gifted and talented programs. The Kuliks pointed out that both they and Slavin largely agreed on the data both analyzed, but that Slavin excluded studies of gifted programs from his research while the Kuliks made those studies a focus.

Tom Loveless, senior fellow at the Brookings Institution, summarized one important aspect of their dispute, which is that their debate centers more on values than their read of the extant evidence:

Slavin and Kulik are more sharply opposed on the tracking issue than their other points of agreement would imply. Slavin states that he is philosophically opposed to tracking, regarding it as inegalitarian and anti-democratic. Unless schools can demonstrate that tracking helps someone, Slavin reasons, they should quit using it. Kulik’s position is that since tracking benefits high achieving students and harms no one, its abolition would be a mistake (p.17)

Betts notes the studies the Kuliks and Slavin reviewed in their meta-analyses had some flaws, with relatively small N and non–nationally representative data. Using more nationally representative samples, a number of researchers (Hoffer, Gamoran and Mare, Argys, Rees and Brewer) came to the conclusion that conventional tracking benefits students in the high tracks and hurts those in the low tracks. But it’s really hard to control for the right factors in these definitely non-experimental studies, and Betts and Shkolnik raise questions about the results of these papers. And there was also a recent big meta-meta-analysis that found no benefits for between-class grouping, echoing Slavin, but that did find benefits for special grouping for gifted students, echoing the Kuliks.

Just to mess with everybody, Figlio and Page argue that by attracting stronger students to the school (because parents seek tracking) students in low-tracks benefit, secondarily.

So, in summary, what should we make of all this? Betts, an economist, says in a review of the literature that when it comes to the average impact of tracking or the distribution of achievement “this literature does not provide compelling evidence.” Loveless doesn’t disagree, but notes that for high achievers, the situation is clearer:

“The evidence does not support the charge that tracking is inherently harmful, and there is no clear evidence that abandoning tracking for heterogeneously grouped classes would provide a better education for any student. This being said, tracking’s ardent defenders cannot call on a wealth of research to support their position either. The evidence does not support the claim that tracking benefits most students or that heterogeneous grouping depresses achievement. High achieving students are the exception. For them, tracked classes with an accelerated or enriched curriculum are superior to heterogeneously grouped classes.” (p.22)

At the end of the day, all academic impacts of tracking are mediated by teaching and the curriculum. If a teacher doesn’t change what they teach or how they teach it, no grouping decision will help or hurt a student academically in a significant way. Tracking only could benefit gifted students if it came with some sort of curricular modification.

This is a conclusion with wide-reaching support. Even Slavin, who so staunchly opposed conventional ability grouping, was extremely impressed by something called the Joplin Plan, which involves three core features:

• Regular testing and regrouping of students on the basis of the tests
• A different curriculum for each group of students

Slavin, the Kuliks, and everyone else seemed to agree that students in the plan — at all ability levels — tended to get 2-3 months ahead of students in typical programs over a year of instruction. The Joplin plan involves ability grouping — the good kind of ability grouping.

So in 1986, when the Baltimore School Superintendent turned to Bob Slavin to design a program that would improve the city’s most dysfunctional schools, guess how Slavin grouped students?

Slavin worked with research scientist Nancy Madden (they’re married) to design Success for All for Baltimore, and it’s a prominent program in the school improvement world, implemented in thousands of schools and spreading. Those three features of the Joplin plan — assessment, regrouping along the lines of ability and targeted teaching — are core features of their program.

Success for All isn’t the only example of a successful curriculum implementing these ideas. Direct Instruction was created by Siegfried Engelmann and Wesley Becker in the 1960s, and it also groups students according to their current levels in reading and math while frequently reassessing and regrouping. DI has a strong body of research supporting its efficacy (for one, it was the winner of the famous-in-education Follow Through experiment), but fell largely out of favor outside of remedial classrooms. In early 2018, a new meta-analysis spanning 50 years of research reinvigorated conversation around Direct Instruction. It found an average effect size of 0.51 to 0.66 in English and math over 328 studies (p<0.001), — strong evidence that the program works.

While its effect on student performance is rarely disputed, the program remains controversial. Historian of education Jack Schneider writes: “Direct Instruction works, and I’d never send my kids to a school that uses it. The program narrows the aims of education and leaves little room for creativity, spontaneity and play in the classroom. Although test scores may go up, the improvement is not without a cost.” Ed Realist worries that its pedagogy is unsavory, has not been shown to work for older students, that wealthier parents are voting with their feet against the curriculum, and that DI could exacerbate gaps between students. Supporters, by contrast, paint the picture of a robust, effective system that has been ignored and disregarded.

Success for All and Direct Instruction are not simple programs for schools to adopt. Implementing them amounts to a major organizational change, and pushes at the extremely resilient notion that children in school should be grouped by their ages. Comprehensive ability grouping programs such as these seem to work, but in practice they are rarely used.

Our favorite one-stop source for reading on ability grouping: here, or maybe here to get a broader picture of the controversy.

1. Acceleration

Forget the comprehensive approach, then. Does it work to simply move an individual student (e.g. an underchallenged and frustrated student) through the curriculum at whatever pace seems to make sense?

There are a few different ways schools can help some students access the curriculum more quickly. A kid can skip a full grade, or several grades in extreme cases. They can stay in their grade for some classes, but join higher grade levels for some parts of the day. They might be assigned to two classes in one year (e.g. Algebra 1 and Geometry). Or, in some cases, a young student might start school at an even younger age than is typical.

If a child is ready for a higher level within a subject and studies it instead of the lower level, it’s almost a given that they’ll learn more. The real research questions are (a) from an academic standpoint whether accelerated children do tend to be ready, or if they do poorly in classes post-acceleration) and (b) whether acceleration exposes students to non-academic harm (e.g. stress, demotivation, loss of love for subject, poor self-esteem).

The Study of Mathematically Precocious Youth (SMPY) is an ongoing longitudinal study examining thousands of mathematically gifted students. In one SMPY study, researchers compared the professional STEM accomplishments of mathematically gifted students who skipped a grade to those who remained at grade level. They found that, controlling for a student’s academic profile in a pretty sophisticated way, students who skipped a grade tended to be ahead of the non-skippers in terms of degrees earned, publications, citations accrued, and patents received. From this work it seems skipping a grade in the SMPY cohort did nothing to hurt a kid’s learning or enthusiasm for their passions.

Acceleration has been one of the focuses of SMPY studies. A 1993 piece about SMPY findings reported “there is no evidence that acceleration harms willing students either academically or psychosocially.” This is supported by various meta-analyses, going back to the 1984 Kulik & Kulik paper and confirmed by more recent work such as a 2011 analysis of existing studies. Beyond the “does no harm” findings, these meta-analyses also report academic benefits to students.

It can be confusing, when reading these studies, to keep track of just how gifted the students happen to be. For example, SMPY has studied five cohorts so far, ranging from students who assessed in the top 3% to those who assessed in the top 0.01%. As we consider students farther away from the mean of achievement, the need for acceleration becomes more acute.

Lots of teachers encounter “1 in 100” students every year, but the education of “off the charts” students is necessarily more a matter of feel than policy. Still, there are success stories to learn from, and they show a remarkable sensitivity to both the academic and social well-being of the student.

Terence Tao is a famous success story of this kind. He surprised his parents by discovering how to read before turning two, and as a child he started climbing through math at a blistering rate. He was identified as profoundly gifted from a young age, and his education was carefully tracked by Miraca Gross as part of her longitudinal study of profoundly gifted children:

His parents investigated a number of local schools, seeking one with a principal who would have the necessary flexibility and open-mindedness to accept Terry within the program structure they had in mind.

This set the pattern for the ‘integrated,’ multi-grade acceleration program which his parents had envisaged and which was adopted, after much thought and discussion, by the school. By early 1982, when Terry was 6 years 6 months old, he was attending grades 3, 4, 6 and 7 for different subjects. On his way through school, he was able to work and socialize with children at each grade level and, because he was progressing at his own pace in each subject, without formal “grade-skipping,” gaps in his subject knowledge were avoided.

His education continued in much the same fashion, culminating in a Ph.D. by the age of 21 and a remarkable and balanced life since. He has since given his own advice on gifted education.

Given the success of acceleration, are we accelerating enough? On the one hand, it appears that acceleration is a widely used tool for giving gifted students what they need. When looking at the top 1 in 10000 students in terms of mathematical ability as identified by the SMPY, nearly half of the group skipped grades, and almost all of them had some form of acceleration, whether that meant advanced classes, early college placement, or other tools. About two-thirds reported being satisfied with their acceleration, rating it favorably across many categories:

[Source: SMPY]

The dissatisfied third of those 1 in 10000 students, for the most part, reported wishing they had been offered more acceleration. And advocates for gifted education strongly endorse the notion that acceleration is under-used. A Nation Deceived is premised on this idea — though besides for “more” the report doesn’t get specific concerning how many students ought to be accelerated, and the report mostly makes a cultural argument in favor of acceleration, citing stories like Martin Luther King Jr. graduating high school at 15.

We wanted to know more about how educators think about acceleration, so we surveyed (via twitter) twenty-one teachers, academic coaches, tutors and administrators. The survey prompted educators to respond to the following scenario:

In your school there is currently a 1st Grader who does math above grade level, e.g. he performs long division in his head. His parents initiated contact with the teacher after hearing their child complain that math at school was boring. They’re concerned that he isn’t being challenged. The classroom teacher knows that he is above grade-level in math, and is trying to meet his needs in class. The parents, however, do not think the current situation is working. The teacher reports that the student is difficult to engage during math class, and that sometimes he misbehaves during math.

From their responses, it certainly seems that acceleration was on the table, but almost always the last option after a number of in-class or non-classroom options (e.g. after school clubs) were explored. That acceleration in math should be a “break in case of emergency” response is also the line offered by the National Council of Teachers in Math: tracking is morally indefensible, acceleration should be viewed with suspicion but can sometimes be appropriate.

In many ways, mainstream education is living in Bob Slavin’s world. He was a leading opponent of tracking, but was impressed by certain forms of ability grouping. He took the research on ability grouping that actually works (through assessment, frequent regrouping, and curricular modification) and used it to create a program for failing schools. He expresses suspicion about acceleration of gifted students in general, but agrees that at times it is a useful and necessary tool. If you broach the conversation about acceleration with your child’s teachers, you might hear some version of Bob Slavin’s take.

There is more to say about where this skepticism comes from. But it’s important to note that just because a student could be accelerated doesn’t always mean that they should. While some gifted students fit the profile we sketched above — frustrated with school, bored and underchallenged, and finding it hard to connect to peers — many equally capable students are happy in their school lives. (We heard some, but not many, happy stories from online commenters.) If a child is happy and successful without acceleration, they are likely to remain happy and successful regardless of whether they are accelerated, and if they don’t want to accelerate, it should not be forced on them. At least some of the suspicion towards acceleration comes from parents who inappropriately push schools to accelerate their happy, satisfied children.

Acceleration is also not the only option. There is much more to learn than is taught in regular courses. Even in a normal class, a well-designed curriculum or an experienced teacher can create “extensions” to the main activity, so that students who are ready for more have something valuable to engage with. Enhancement or exposure to new, similar topics can serve students as well. A student who has jumped ahead in arithmetic may be entranced by a glance at Pascal’s triangle and number theory. One who is fascinated by English might find similar joy in learning Spanish or Chinese. Both of these, alongside acceleration, follow a simple principle: if a child wants to learn more and is able to do so, let them learn more. Overall, the balance of evidence suggests that acceleration is a practical and resource-effective way to help gifted, underchallenged students flourish in schools.

Our favorite one-stop source for reading on acceleration: here.

1. Educational Goals in Conflict

Through acceleration, tutoring, or ability grouping, some kids could learn more. Why aren’t schools aggressively pursuing that? Shouldn’t they be working to teach kids as much as possible? Isn’t that what a school supposed to do? That educators are skeptical of ability grouping or acceleration can be maddening from the perspective of learning maximization: Why are schools leaving learning on the table?

Here’s something we don’t talk about nearly enough: schools are simply not in the learning-maximization business. It turns out that parents, taxpayers and politicians call on schools to perform many jobs. At times, there are trade-offs between the educational goals schools are asked to pursue, and educators are forced to make tough choices.

Historian David Labaree has one way of thinking about these conflicting educational goals, which he expands on at length in Someone Has to Fail. For Labaree, there are three competing educational goals that are responsible for creating system-wide tensions:

• democratic equality (“education as a mechanism for producing capable citizens”)
• social efficiency (“education as a mechanism for developing productive workers”)
• social mobility (“education as a way for individuals to reinforce or improve their social position”)

As Labaree tells it, these goals end up in tension all the time. A lot of things that seem like gross ineptitude or organizational dysfunction are really the result of the mutual exclusivity of these goals:

These educational goals represent the contradictions embedded in any liberal democracy, contradictions that cannot be resolved without removing either the society’s liberalism or its democracy … We ask it to promote social equality, but we want it to do so in a way that doesn’t threaten individual liberty or private interests. We ask it to promote individual opportunity, but we want it to do so in a way that doesn’t threaten the integrity of the nation or the inefficiency of the economy. As a result, the educational system is an abject failure in achieving any one of its primary social goals … The apparent dysfunctional outcomes of the school system, therefore, are not necessarily the result of bad planning, bad administration, or bad teaching; they are an expression of the contradictions in the liberal democratic mind.

Ability grouping and acceleration fit nicely within the tensions Labaree exposes. These learning-maximizing approaches could find support from those who see education as a national investment in our defense or economy. Of course, the strongest demand for acceleration in schools can come from parents, who want schools to give their children every possible opportunity to be upwardly mobile. (“We want to make sure they can go to a good college.”)

Those act as forces in favor of ability grouping and acceleration. But schools also know that they are held responsible for producing equitable outcomes for a citizenry that sees each other as equals. A program that raises achievement for top students without harming others has an appeal an economist could love, but within schools this can count as a problem.

The way this plays out in practice is that many schools are inundated with requests to accelerate a kid. Parents — especially financially well-off, well-connected parents — can typically find ways to apply pressure to schools in hopes of helping their children reach some level of distinction. They’ll sometimes do this even when it wouldn’t benefit a child’s education (it would be educationally inefficient), or when it would exacerbate inequality (by e.g. letting anyone with a rich, pushy parent take Algebra 1 early).

In short, from a school’s standpoint those are two problems with acceleration. First, parents will push for it even when it’s not academically or socially appropriate. Second, it can exacerbate inequalities. That could explain where the culture of skepticism within education comes from.

This is meant entirely in terms of explaining the dynamic. The way this plays out can be incredibly painful. Systems designed to moderate parental demand can keep a kid in a depressing and frustrating situation:

My older son wanted to move up to a more advanced math course for next year. He took two final exams for next year’s course in February and answered all but 1/2 of one question on each. So roughly 90% on both and his request to skip the course was denied. (source)

Districts sometimes have extensive policies that can be incredibly painful to navigate when trying to get a student who truly needs acceleration out of a bad classroom situation. We heard from one educator who had a very young student expressing suicidal ideations. It was all getting exacerbated by the classroom situation — the kid said he felt his teachers and peers hated him because he loved math. The parents and the educator tried to find a better classroom for the child, and were met with all the Labaree-ian layers of resistance. Off the record, the educator advised the parents to get out of dodge and into a local private school that would be more responsive to his needs.

A happy ending: the 4th Grader moved to a private school where he was placed in an 8th Grade Honors class. He likes math class now. He seems happier, he’s growing interested in street art and social justice work.

But without a doubt, there are some unhappy endings out there.

1. Personalization Software

[source: Larry Cuban]

“Ours is an age of science fiction,” Bryan Caplan writes in The Case Against Education.  “Almost everyone in rich countries — and about half of the earth’s population — can access machines that answer virtually any question and teach virtually any subject … The Internet provides not just stream-of-consciousness enlightenment, but outstanding formal coursework.”

The dream of using the Internet to replace brick-and-mortar classrooms is a dream that is entirely in sync with the times. This is reflected in the enormous enthusiasm directed towards online learning and personalization software. Bill Gates, Elon Musk, and Mark Zuckerberg have all invested heavily in personalization and teaching software. And the industry as a whole is flush with funding, raising some 8 billion dollars of venture capital in 2017, while reaching 17.7 billion in revenue.

Finally — a way out of the school system and its knot of compromises! If schools are institutions whose goals are in tension with learning-maximization… then let’s stay away from schools and their tensions and give the children the unfettered learning they want. Let’s create the ideal tutor as a piece of software.

This dream isn’t just in sync with our times — it has a long history. This history is particularly well-documented by historian Larry Cuban (author of Teachers and Machines and Tinkering Toward Utopia) and by Audrey Watters (she’s writing a book about it). Watters’ talk “The History of The Future of Education” is as good a representative as any of the major thesis: that the dream is larger than any particular piece of technology. Motion pictures, radio, television, each of these was at times promoted as an educational innovation, able one day to free students from lockstep movement through school and into a personalized education. From Thomas Edison to B.F. Skinner, tech advocates have long envisioned the future that (at least according to Caplan) we’re living in now.

Then again, tech advocates in the past also thought they were living in the age of personalized learning. In 1965, a classroom that used a program called Individually Prescribed Instruction was described this way:

Each pupil sets his own pace. He is listening to records and completing workbooks. When he has completed a unit of work, he is tested, the test is corrected immediately, and if he gets a grade of 85% or better he moves on. If not, the teacher offers a series of alternative activities to correct the weakness, including individual tutoring.

For comparison, here is the NYTimes in 2017, and the headline is A New Kind of Classroom:

Students work at their own pace through worksheets, online lessons and in small group discussions with teachers. They get frequent updates on skills they have learned and those they need to acquire.

The similarity between modern day and historical personalization rhetoric doesn’t settle the matter — in a lot of ways, clearly the Internet is different — but personalization software seems to have arrived at a lot of familiar, very human frustrations.

Anyone who has gone online to learn has, at some point, come face to face with this dilemma: On the internet, you can study almost all human knowledge, but usually you don’t. In a world with virtually every MIT course fully online for free, a world with Khan Academy and Coursera and countless other tools to aid learning, why has the heralded learning revolution not yet arrived?

In a way, the revolution has arrived — it just hasn’t improved things much. Rocketship Schools, a California charter using online learning for about half of its instruction, has had solid results. Lately, though, they’ve moved away from some of their bigger bets on personalization and rediscovered teachers, saying “We’ve seen success with models that get online learning into classrooms where the best teachers are.” School of One was a widely hyped high school model in NYC that was preparing to scale up its offerings… until a fuller picture of the results came in and it was pilloried. Online charter schools, meanwhile, seem to actively depress learning.

Part of the problem is that it’s hard to get solid research on the efficacy of various ed tech products. Many tools, particularly those sold directly to schools or used by online charters, are proprietary and stuck behind paywalls, selectively presenting their best data and limited demos. The ed tech sector in general seems to deliver mixed results to students.

Why is it so hard to make effective teaching software?

For one, teaching is complex. A good human teacher does a lot of complicated things — gets to know their students, responds to the class’ moods and needs, asks “just right” questions, monitors progress, clarifies in real time as a look of confusion dawns on the class, etc., etc. — and it’s simply hard to get a computer to do that.

Maybe, theoretically, a piece of software could be designed that does these things. But in practice, many software designers don’t even try. It’s easier and cheaper to make pedagogical compromises, such as providing instruction entirely through videos. Yes, there are some thoughtful tools made by groups like those at Explorable Explanations, such as this lesson on the Prisoner’s Dilemma. But building high-quality tools well-adapted for a digital environment is difficult and time-consuming, and for prospective designers, destinations like Google or Blizzard tend to be more glamorous than working with schools. In practice, humans currently have a lot of advantages over computers in teaching.

Even if we overcame all the design issues, though, would students be motivated to stick with the program? Studies of online charters point to student engagement as the core challenge. When you put a kid in front of a computer screen, they jump to game websites, YouTube, SlateStarCodex, Google Images — anything other than their assigned learning. Many educational games that try to fix this resort to the “chocolate covered broccoli” tactic, trying to put gamelike mechanics that have nothing to do with learning around increasingly elaborate worksheets.

To be fair, student engagement is also the core challenge of conventional schools. But that’s precisely what the much-maligned structures of school are attempting to confront. The intensely social environment helps children identify as students and internalize a set of social expectations that are supportive of learning. The law compels school attendance, and schools compel class attendance. .And, once a child is in the classroom, their interactions with actual, live human instructors can set high academic expectations that a child will genuinely strive to meet.

The conventional story is that school is incredibly demotivating, but compared to their online counterparts schools are shockingly good at motivation. MOOCs like those on Coursera have an average completion rate of 15 percent — public schools do much better than this. Popular language app Duolingo’s self-reported numbers from 2013 would put their language completion rate at somewhere around 1%. If all a user has to rely on is their daily whim to continue a course, the most focused and conscientious may succeed, but those are the ones who already do well in schools. That’s a big part of why people lock themselves into multi-year commitments full of careful carrots and sticks to get through the learning process. Writers such as Caplan think that people are revealing their true interests when they skip learning to fart around on the web, but we might as well see a commitment to attend school as equally revealing. People need social institutions to help do things we’d truly like to do. As such, even as computers become better teachers, the motivational advantage of schools seems likely to persist.

How might tech-based learning tools address these factors, so they might stand a chance at holding students’ attention long enough to teach them? Art of Problem Solving, an organization promoting advanced math opportunities to children, makes a good case study. It’s found a balance worth examining. First, it provides accessible gamelike online tools that center on a careful sequence of thought-provoking problems. Second, it offers scheduled online classes with the promise of a fast pace, challenging content, and a peer group of similarly passionate students taught by subject matter experts. The online classes are more expensive offerings, but they preserve the human touch.

What does that balance mean for students? If they’re in the conscientious, self-motivated crowd that wants to learn everything yesterday, they can gorge themselves on software designed to be compelling. No barriers keep them from progressing. Software can always point to a next step, a harder problem. On the other hand, if they want to lock a motivational structure around themselves and keep the social benefits of school in a more challenging setting, they can.

Not every successful tool need look identical, but that core idea is worth repeating: software should enable the passion and self-pacing of eager kids, but should not rely on that to replace the power of social, human motivational structures. Yes, sometimes even the same structures used in “regular” schools.

Online learning, then, fits squarely within the history of attempts to automate teaching. Over and again we make the same mistakes and forget the lessons of history: that teaching is more complex than our machines have ever been, that motivation is largely social, and that schools will have a hard time distinguishing between altrustic designers and opportunistic profit-seekers.

For those in the market for online learning there are a lot of mediocre tools available, and many truly bad ones. Right now, there’s nothing that seems ready to serve as a full-on replacement for school without consistent, careful human guidance.

That said, depending on your passions, there are some excellent resources for learning out there. Especially if a student has a caring mentor or a passionate peer group, they can learn a lot online. As educators and designers create more tools that respect both the power and limitations of machines, that potential can grow. But it’s not quite science fiction.

Our algorithm has determined that you should watch the following two videos: here and here to balance realism and idealism

Education is complex and resists easy generalizations. That said, here are some generalizations.

On navigating school for your child:

• The brightest students do not thrive equally in every setting. Even the best students achieve more with teachers than on their own. Unless tutoring or some other private arrangement is possible, this means that a school is the best place to be for learning.
• But school right now doesn’t work for all kids. One fix: if a child wants to be accelerated and seems academically prepared for it, acceleration will usually help them.
• Most schools aren’t in the business of maximizing learning for every student, and in particular they tend to be skeptical of acceleration.
• Therefore: If your kid needs more than what school is offering, be prepared to be a nudge.
• But if you think your kid needs to be challenged more and your kid is perfectly happy in school, try really hard not to be a nudge.
• Don’t fight to move your child to a class that covers the exact same material at the exact same pace but has the word “Honors” next to it. That sort of ability grouping makes no educational difference.
• Prioritize free, open online tools. Don’t expect online tools to do the work for you or your child. Expect more distraction and less progress if online learning time is unstructured or unsupervised.

For educators:

• If you are an elementary teacher or administrator and your school is looking to try new things, consider cross-grade ability grouping by subject, especially in math and reading.
• Gifted kids are usually not equally talented in all fields. Consider options to accelerate to different levels in each subject based on demonstrated skill in that subject.
• A lot comes easily to smart kids, and sometimes they never get the chance to learn to struggle. Find something they think is hard, academic or not, so they are able to handle more important challenges later.
• If a child is bored in your class and knows the material, they probably shouldn’t be in your class.

For tech designers and users:

• If you’re making online tools, make the learning the most interesting part of them. Don’t rely on chocolate-covered broccoli or assume that just presenting the material is enough. Take the problem of motivation seriously.
• Look for passionate groups with robust communities, whether online or offline. Don’t overlook the social aspect of learning.

And for advocates of educational reform, in general:

• People almost only talk about educational efficacy. But don’t be fooled — educational debates are only sometimes about what works, and frequently about what we value.

One last thing: if you’re an educator or a parent or just somebody who spends time around children, take their feelings seriously, OK? If a kid is miserable, that’s absolutely a problem that has to be solved, no matter what district policy happens to be.

Acknowledgments: Thanks to /u/Reddit4Play from reddit, JohnBuridan from the SSC community, blogger Education Realist, and many others who read drafts and offered ideas along the way.

# Noticing Humans & Noticing Wonders – Guest Post by Benjamin Dickman

This is a guest post by friend of the blog Benjamin Dickman

• Noticing Humans refers to the importance of students being aware that they are seen by others – including, but not limited to, their math teachers – and that we, as teachers, would do well to interrogate how our external perspectives match or don’t match students’ internal perspectives.
• Noticing Wonders refers to a specific in-class activity related to Paul Lockhart’s essay, “A Mathematician’s Lament” (pdf), that I designed – based on suggestions from my Middle School English teaching colleagues – to gain insight into how my students were thinking about mathematics.

In noticing and wondering:

How can we follow up on our wonders about the ways in which students see mathematics?

Being Noticed

During a summer visit to my childhood home in Boston, Massachusetts, to see my family, I made a point of asking my mom[1] for a very specific suggestion: What can I say to students who voice anxiety or discomfort around an upcoming (math) assignment, especially when my prior knowledge of their quality of work and preparation suggests that they will do well? (For example, I would consider “Don’t worry about it!” or “I’m sure you’ll do fine” to be suboptimal responses.) Paraphrasing, her idea was that sharing evidence of strong past performance is fine, but that a helpful additional sentence would be one to the effect of: “As your teacher I have confidence in you, and while I know that you may not share in that confidence right now, I wish that you could see yourself through my eyes.” Less than a month later, I was reading a piece in Quanta Magazine about 2018 Fields Medal (an award sometimes called the Math Nobel) recipient Akshay Venkatesh, from which I excerpt (emphasis added):

Clearly, [Venkatesh’s] adviser must have written a glowing letter of recommendation for him — but why? Venkatesh took this question to Jordan Ellenberg, a friend and fellow mathematician. Ellenberg’s reply has stayed with Venkatesh over the years: “Sometimes, people see things in you that you don’t see.”

About a week after, I noticed on Twitter an open-request from Professor of Mathematics Education Ilana Horn about her son:

I happen to be a math teacher who had 5 minutes, so I watched the video and left a couple of comments. I thought that my first comment got to the heart of the matter, and left a second comment even though I deemed it less relevant. Here are the comments with their respective responses:

In retrospect, I conjecture that my first comment resonated primarily with other math educators, who can see what I see about habits of professional mathematicians, but that it would require more time and evidence to be fully believed by a child. My second comment had a different outcome – to “jump up and down with happiness” – because it comes from a different perspective: how one is seen (or even not seen) by their peers, and the ways in which one’s presence and absence are noticed, even from afar. The through line that I perceive in these three items – my mom’s suggestion to me, Ellenberg’s advice to Venkatesh, and the excerpted tweets – is that, although there is much talk in math education about noticing and wondering, I am concerned that we are (or: I am) not doing enough to ensure that I let my students know that I am noticing them for who they really are: as mathematical thinkers, as students, as humans.

I need to attend better to voicing what I notice in them, and thinking intentionally about how they share what they notice in one another. This is true for students who retain an anxiety around (math) assignments; it is true for future Fields Medalists; it is true for students who experience tracking systems in ways that are too often unjust; it is true for so many other students who cannot – or do not yet – see themselves through a caring adult’s eyes; and, I strongly suspect, it is true for so many teachers who long to be noticed and seen – knowingly or not – for who they really are: as mathematical thinkers, as teachers, as humans.

In noticing and wondering:

How can we follow up on our wonders about the ways in which students see mathematics?

Noticing Wonders

About once a month for each of the past two years, I have met with a group of teachers at my school – some Middle School, some Upper School, some both – to talk about “Writing to Learn” strategies. One of these is called a Focused Free Write, and although it was introduced as a way for Middle School students to get a foothold on Ovid, I decided to adapt it for use with an essay that is somewhat well-known in certain math education communities: Paul Lockhart’s “A Mathematician’s Lament,” which is often referred to simply as “Lockhart’s Lament” (PDF re-link). If you’ve never read it, or even if you have, you might try to complete the activity below; the students in our Problem Solving & Problem Posing course had no prior familiarity with any of the author’s writings. To summarize (over)simplistically, I feel that this essay, or Lament, speaks in a way that resonates with many mathematics teachers. I wanted to use it to gain insight into the specific mathematics learners with whom I was working. Here is the full prompt, for which I wrote the directions in red. No additional information was provided about the source text or its author:

[1] I happen to think “being my mom” is a sufficient criterion for giving good advice, but my mom also happens to be a child psychiatrist whose work includes advising for the PBS show Arthur, which created a therapist cartoon character named after her!

# I don’t focus my classroom on solving problems

This post is part of the Virtual Conference on Mathematical Flavors, and is part of a group thinking about different cultures within mathematics, and how those relate to teaching. Our group draws its initial inspiration from writing by mathematicians that describe different camps and cultures — from problem solvers and theoristsmusicians and artistsexplorers, alchemists and wrestlers, to “makers of patterns.” Are each of these cultures represented in the math curriculum? Do different teachers emphasize different aspects of mathematics? Are all of these ways of thinking about math useful when thinking about teaching, or are some of them harmful? These are the sorts of questions our group is asking.

I’ll update this post with links to all the pieces that are part of this group of posts. So far that includes: Anna, Dylan, CarmelJoshuaLara and Nadav, Dan, and David.

Here’s my entry.

I.

I’m not exactly sure how to say the thing that I want to say, because the thing I want to say isn’t quite true, and can easily be misunderstood. But the thing I want to say is that problem solving is overrated in math teaching.

A story may help explain what I mean. I’ve been planning a conference for math educators, and I work on recruiting people to lead sessions. In the middle of each day, we have a block of time for doing some math together. But the thing is, it can be a tricky session to lead. Some of our participants have advanced degrees in mathematics; others teach younger students and haven’t used algebra in years. How do you make sure you’re doing math that’s meaningful for everybody?

This gets especially hard if the session is framed as the attempt to solve a math problem. The problem is one of aim — will the problem be productive for everyone in the room? What if someone immediately sees the problem for what it is, and makes quick work of it? As Benjamin Dickman put it, “if the plan can be brought down with Oh I’ve Seen This Already…then to me it’s a risky move.”

(Sure, you can tell people what to do if they’ve seen it already, but doesn’t that feel like a consolation prize?)

It’s the same problem that I had to grapple with at math camp during the summer. One of my jobs at camp was to teach the counselor class. Counselors ranged from 16 to 20 years old. The class involved giving counselors time to grapple with some of the challenging math that the camp asked kids to do battle with.

Here is a typical problem from camp:

What is the sum of all the digits of the numbers from 1 to 1000, i.e. 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 1 + 0 + 1 + 1 + 1 + 2 + … + 9 + 9 + 9 + 1 + 0 + 0 + 0?

That’s tricky. But it was significantly trickier for some of the youngest counselors I taught, who were themselves high school students, and significantly less so for the seasoned math major who is headed straight toward a PhD program in math.

What to do?

In this case, I started class by emphasizing that the goal was not to solve the problem, but to find math to understand.

I put a representation — 10 through 15, with the digits all splayed out like 1 + 0 + 1 + 1 + 1 + 2 + 1 + 3 + 1 + 4 + 1 + 5 — up on the board, and asked the class what they saw and noticed.

I presented the original problem. Then I asked each counselor to come up with a question that was related, but not the same, as the original problem.

I asked everyone to pursue one of these questions, or maybe the original problem.

Explicitly framing class as a search for understanding, calling on students to notice whatever catches their attention, focusing attention on new representations, asking students to think about a problem as living in the context of a network of similar problems…these are all aspects of math that I emphasize, at the expense of solving a specific problem.

Getting a bunch of counselors who signed up to work at a math camp to work on math is not exactly teaching on the hardest difficulty setting. I mean, damn, if every group of kids that I worked with could spend a half hour essentially poking around mathematically after the set up I left these counselors…my job would be so much easier.

I’m not saying that my counselor class should be anyone’s model.

But I am trying to say that I felt like I was subverting problem solving. Because when you make problem solving the goal, you create a lot of challenges for yourself. What happens if someone finishes first? If the goal is solving the problem, why should the teacher ever help? or why should they refuse to help? How do you make sure everyone feels valued if they don’t solve the problem? How do you keep people from feeling that your extension is a consolation prize?

I’m still not quite sure what to say about problem solving, but maybe it’s like this: a lot of things get much harder once you tell your class that the goal is to solve a problem. So most of the time, I don’t.

II.

I think some people think of math as essentially focused on problems, but this is not how mathematicians necessarily describe their work.

For example, while Timothy Gowers (in Two Cultures of Mathematics) does describe himself as oriented towards problem solving, he sees this as the minority position within mathematics, where the dominant orientation is towards theoretical understanding. Michael Atiyah is his chosen representative of this dominant orientation. Atiyah describes himself as unmotivated by problems:

I’m interested in mathematics; I talk, I learn, I discuss and then interesting questions simply emerge. I have never started off with a particular goal, except the goal of understanding mathematics.

Talking, learning, discussing math: these are distinctly mathematical activities. It’s not all about problem solving — it’s not even primarily about problems.

There is a line of thought that argues that math education must be focused on problems and problem solving, because that’s what mathematicians do. But mathematicians do so many things beyond solving a problem.

More than anything else — Atiyah says this — mathematicians are interested in learning about math. They learn from colleagues, who explain things to them. They learn from classes and seminars. They ask questions. They teach, and learn from teaching. They learn new ways to represent ideas. They notice new things, learn to see new things as important. And, yes, they work on problems.

There’s another thing too. If you’re oriented towards problem solving, you’re oriented towards the achievement of the few. If mathematical success is about solving valuable problems, that’s going to be inherently inequitable. Your history of math becomes the history of individual achievement, mostly represented by men from privileged cultures. It’s the same inequities that show up in the classroom around problem solving — some have a head start, some finish first — but played out over an entire field.

And it’s the sort of thing that can get inside your head. Am I worthy of working on a problem? Will I ever contribute by solving something of value? (Or even: Has my culture ever contributed anything of value? Have people like me ever made important contributions?)

These worries make perfect sense in a world full of unsolved puzzles waiting to be solved. But, if we’re oriented towards understanding instead of problem solving, the whole situation gets turned on its head. Achievement is understanding, and there is more than one way to spread understanding. It’s not just about being first — in fact, it’s not about that at all.

These two ways of seeing achievement in math can clash. For example, when a graduate student worried online if he could meaningfully contribute to math, Bill Thurston answered that he was thinking of “contribution” too narrowly:

Mathematics only exists in a living community of mathematicians that spreads understanding and breathes life into ideas both old and new. The real satisfaction from mathematics is in learning from others and sharing with others. All of us have clear understanding of a few things and murky concepts of many more. There is no way to run out of ideas in need of clarification. The question of who is the first person to ever set foot on some square meter of land is really secondary. Revolutionary change does matter, but revolutions are few, and they are not self-sustaining — they depend very heavily on the community of mathematicians.

If we focus too much on solving a problem as a goal, then we end up focusing on the “solving.” And then we can end up paying too much attention on who did the solving, instead of looking at the way clarity was spread through some mathematical community.

(There are ways of combating this slide, but it’s hard work!)

If our culture and classrooms are focused on understanding, equity has a better shot.

III.

So much for teacher conferences, summer camp, and mathematical culture. But the last piece of this sort-of-critique of problem solving comes from my regular classroom work during the school year.

I have definitely found that, when students are starting off learning something new, solving a specific problem has limited use. The drive to answer the question can narrow attention in unhelpful ways, focusing students on whatever it is that will help them most solve the problem — even if that’s not a new and interesting feature.

So I try to hold off on problem solving, at first, and I very rarely frame that as the goal of a class session.

I remind students: the goal is spreading understanding.

I ask students to notice new things.

I ask students to study examples of student work, both correct and incorrect. Because learning from existing mathematical work is mathematical. It’s only second-rate if your goal is problem solving.

I ask students to write their own problems, and to see problems as existing within genres of problems. (If Monday was about finding the volume of prisms, very likely my first ask on Tuesday will be for students to come up with a prism to sketch and to find its volume. These sorts of tasks are surprisingly challenging! Remembering stuff is hard, but the work is deeply mathematical.)

I ask students to compare representations.

And if all these things aren’t exactly in conflict with problem solving, that’s also not our goal. We’re aiming for understanding, and problem solving happens along the way.

# Worked Examples and Loop-De-Loops

This is a Loop-de-Loop:

At camp, I taught a session on Loop-de-Loops, a mathematical object from Anna Weltman’s book. I had about 30 people in the room (15 counselors, 15 kids) looking for patterns and trying to figure out a bit about what Loop-de-Loops are all about.

I introduced Loop-de-Loops using Chris Lusto’s fantastic site, asking everyone what they noticed and were wondering after I hit “Show Me.” (Go try it; it’s a blast.)

This was perfect for us, as I wanted the focus of the session to be about asking questions. (Lord knows that the kids spend enough time solving problems in this camp. Asking questions is at least half of a full mathematical picture.) Lusto’s site makes it easy to quickly explore many different lists of numbers, generate theories and see if those theories hold up.

In fact, it’s too easy to generate these Loops with the software. For the sake of understanding, it’s always good to construct stuff by hand a couple times. Often we notice different things when we construct objects by hand than we do on the computer.

My first time running this session, though, I realized something: it’s hard to make a Loop-de-Loop!

Students and counselors struggled to draw these things — though, when they saw these objects on the computer, they were extremely confident that they understood how to make them. Surprise!

Today, I ran the session for another group of kids and counselors. This time, I came prepared. I wrote a worked example activity that aimed to help everyone better learn how to make these cool Loopy things:

I quickly made a handout right before class today. (Hence the marker.) I knew that I needed to include the three things I always include on one of these things:

1. The example, clearly distinguishing between the “task” and the “student work,” and trying to make sure to only include marks that contribute to understanding. (See those little arrow heads? I noticed that people had a hard time keeping track of directions while drawing these.) I tried to remove any distracting text or clutter — reading about the split attention effect helped me learn how to do this.
2. Prompts for noticing the most important stuff. Research suggests that students often don’t explain things to themselves, or do so superficially. (And experience totally confirms this.) Prompts, along with a clear call for students to spend a minute responding to the prompts, helps a lot with this I find.
3. A chance to try it out on your own, with the model nearby to help, if you get stuck.

I first read about this structure while reading about Cognitive Load Theory, but things didn’t click for me until I also saw the Algebra by Example project. Two other pieces have helped me better understand this bit of my teaching:

Here’s how I do these things. First, I put the “student work” up on the board and ask everybody to silently study it. (People need time to think before talking!) Then, I ask everyone to check in with a neighbor and to take turns making sure each can explain what’s happening. (This is usually where there are “ah!”s and “oh wait!”s.) Then, we talk about the prompts. After that, I tell everyone to try it on their own.

This part of the session went so much better today than my first pass. Everyone was still challenged by drawing these, and there were still a lot of mistakes. But the difference between these two sessions was precisely the difference between productive and unproductive struggle. Instead of flailing around when they got lost in the construction, everyone had something to go back to. Ah, OK, so this is how it’s done.

Part of my job is also to help the counselors support the students in their math work. A lot of the counselors tell me they struggle with knowing about how much to give away to a student when they are stuck. And while I totally know what they mean, I always tell them that their job isn’t to give away stuff or to avoid giving stuff away. Their job is to get the ball rolling for the student, get thinking happening, as quickly as possible — and then to step back and let that thinking happen.

Today I needed to get the ball rolling. It was a session about posing problems, and I started with asking everyone to notice and ask questions. But an important part of getting the ball rolling was a worked example.