Venn Diagram Puzzles

I really like Venn Diagram puzzles to get at what it means to solve a system of equations. Today I finally got around to making a digital/editable copy of these things (here). I feel like there are more ways to extend and play with this format beyond what I came up with.

Make your own, and enjoy!

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How to make worked examples work

I’ve written a few times about how I use worked-out examples (and mistake analysis) in my teaching:

The resource I’ve returned to again and again is Algebra By Example, which is really well-done. (And get excited for Math by Example, their expansion pack for 4th and 5th Grade!)

I realize that there are two things that I haven’t yet shared: some research on how to design these things well, and my routine for using these materials in my teaching.

The Research

Whenever you get a bunch of people talking about research in education, somebody eventually mentions how sad it is that research in education usually happens without the input of educators. Wouldn’t it be grand if researchers and educators were more collaborative?

Yes! This indeed would be grand, and it should happen more often. But sometimes it does happen, and those partnerships can produce really wonderful things. Algebra by Example is the result of one of these partnerships.

The project was led by Julie Booth of Temple University, and Booth has written a very nice piece in an NCTM journal about the design process her team uses and the research behind it:

McGinn, K. M., Lange, K. E., & Booth, J. L. (2015). A worked example for creating worked examplesMathematics Teaching in the Middle School21(1), 26-33.

Booth’s piece is very nice. It cites a more research-y piece that more carefully trawls and summarizes the worked example literature, and that is also an interesting (though more research-y) read:

Atkinson, R.K., et al. (2000). Learning from examples: Instructional principles from the worked examples research. Review of educational research, 70(2), 181-214.

Though it’s tough to summarize these pieces, here are a few important principles when designing example tasks:

  1. Try to put everything in one place…
  2. …but also try to visibly separate different steps.
  3. Switch up correct examples with incorrect work.
  4. Ask kids to explain what’s going on.
  5. Ask kids to try it on their own.

Try to put everything in one place. That’s a NO for splitting the problem and the solution with a not-so-useful line of text. Keep words at a minimum; ideally, you can see the whole example at once.


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It’s useful (according to Booth’s research and also me) to switch up correct with incorrect examples.

Can you spot the error?

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I like the way this incorrect work has some good thinking (I like the regrouping idea) and that it also clearly distinguishes each step.

From experience, I’ll say that people new to writing examples often try to do too much with the space they have. The goal really is to eliminate all but the most important thinking from the example, and it’s OK to use multiple examples to get at different aspects of the thinking. Overstuffing one of those little boxes can be a sign that it’s time to break this up into two smaller problems.

So much for the visual design of these materials, which is really a huge subject in of itself.

Some of the most interesting bits of those reviews are about self-explanation. Here’s a bit from the more research-y review:

Research on explanation effects suggests that self-explanations are an important learning activity during the study of worked examples. Unfortunately, the present research suggests that most learners self-explain in a passive or superficial way.

“Passive or superficial”: YES! All too often this is how students interact with an explanation (or with feedback).

So how do examples help at all, given the tendency of students to just passively say “OK fine yep I get it” when they see an example? I use a routine to really make example activities hum.

The Routine

Whenever I use an example activity with students, here is my routine.

  1. Get ready: I show the problem, covering up everything besides the problem itself. I usually use a document camera, so I mean I literally cover it up with my hand. Sometimes I ask students to begin thinking about how they’d approach the problem before I reveal the student work.
  2. Read: I ask everyone to read the student work to themselves. I ask students to offer a quiet thumb to let us know that they finished reading. I usually tell students to put a thumb up if they’ve read each line, even if they don’t understand everything fully yet.
  3. Discuss: I assign partners (really I’ve already done this) and ask students to discuss the example until each person understands every line of the student work. (I focus my class on spreading understanding, not on solving problems.)
  4. Explain: Every Algebra by Example problem has prompts for student explanation. When I make my own materials, I always include such prompts. It’s at this stage in the routine that I make sure everyone tries to answer the prompts. Then I usually call on a student to explain.
  5. You Try: This is where we help students try it on their own. There’s a whole art to picking good problems for students to try here — they should change the surface-details in some way, while keeping the underlying ideas constant. This is where we try to keep kids from making false generalizations, and it’s another level of protection against superficial understanding.

Using the Algebra by Example materials, the routine flows from left to right:

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It’s not so easy to design these prompts, either! Here are some suggestions, from Booth’s piece:


If you’re in the mood for a bit of practice, you might think about how you’d fill in the rest of this example task:

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Example analysis works especially well as a feedback routine. There was a problem that kids especially had trouble with? Write an example that focuses on how to improve, and then follow the routine with some time to revise and improve the quizzes (or a re-quiz, or etc.).

All of this — the examples, the prompts, etc. — is worth your time, I think. It’s a lot of fun for kids to puzzle out how someone else is thinking, and the format allows us to really focus on a whole idea. It’s a bit of a myth, I think, that these types of problems are boring.

It’s also a myth that this kind of math is boring. It’s not, because there’s something to be understood here. And fundamentally, understanding is interesting.

Multiplication Resources for 3rd/4th Grade

As with all topics, the stronger your main curricular resource is the better. There have been years when I used a heavily modified version of Investigations for 3rd/4th Grade, but now I don’t find it as useful.

I assume that most teachers do have some main curricular resource for math, so here are the things that I use to supplement:

There are a lot of people with strong disagreements about how to teach multiplication. As in a lot of areas of my teaching, I see myself as being one of those annoying people that accepts each side’s critique of the other. I aim to teach mental strategies while also teaching facts for memory and skills for automaticity. I will very clearly explain both mental strategies and written procedures, though not as early as the traditional math crowd would prefer.

I don’t know. Part of this certainly has to do with the sort of place where I teach. The kids are for the most part not poor, I’m not under state test pressure, and the kids don’t have grades. My goals are for kids to know their stuff and be prepared for future coursework, but I also want them to have fun, and I don’t feel as if those goals are in tension in my current situation. (At other places that I’ve taught, though, I have felt those sorts of tensions.)

I do think that it can be a big mistake to focus only on strategies, as I’ve seen kids left behind on their fact knowledge when I taught that way. I wrote at length about this in Teaching Rachel.

I’m sure other resources will come to mind and I’ll try to add them to this post. Feel free to drop your own favorite resources in the comments.

Isn’t “roots of unity” a nice bit of mathematical phrasing? I think so. Nice job, mathematicians.

[Mathematical content warning: roots of unity]

Over the weekend I was reading and practicing with problems from Learning Modern Algebra. I came across these:

Let n be a positive integer and let \zeta = e^{\frac{2\pi i}{n}}.

1. Establish the identity

x^n - 1 = (x - 1)(x - \zeta)(x-\zeta^2) \cdot \cdot \cdot (x-\zeta^{n-1})

2. If x and y are integers, show that

x^n - y^n = (x - y)(x - \zeta y)(x-\zeta^2 y) \cdot \cdot \cdot (x-\zeta^{n-1} y)

3. If x and y are integers, show that

x^n + y^n = (x + y)(x + \zeta y)(x+\zeta^2 y) \cdot \cdot \cdot (x+\zeta^{n-1} y)

I enjoyed thinking about these from a geometric perspective.

Most of the work of the first identity is being done by the definition of \zeta and the very notion of a root of unity. What are the solutions to x^2 = 1? to x^3 = 1? x^9 = 1? x^{anything} = 1?

Some quick facts, in case you’re bravely reading this extremely sketchy blog post without already being familiar with roots of unity:

  • x^n = 1 always has n solutions, no matter what n is.
  • 1 is always a solution.
  • -1 is often a solution.
  • Besides for 1 and -1, all the solutions are non-real complex numbers.
  • If you graph them in the complex plane, all the solutions are on the radius 1 circle away from (0,0).
  • The points are always equally spaced from each other on that circle (which is all that $latex.
  • Another way of saying the previous point is that \zeta = e^{\frac{2\pi i}{n}} provides one solution, and then the rest of the solutions are powers of \zeta.
  • This guarantees that the solutions of x^n = 1 will form a regular n-gon.

A picture is worth a lot of words, via Wolfram MathWorld, and possibly violating their reuse permissions:


That first identity really is just summarizing the roots of unity situation. If all these powers of \zeta are the solutions of x^n = 1, then these \zetas are the roots of x^n - 1 = 0 and so each of those powers of \zeta is a factor of some polynomial that is equivalent to P(x) = x^n - 1. Multiply all those factors together, and you get back the original polynomial:

x^n - 1 = (x - 1)(x - \zeta)(x-\zeta^2) \cdot \cdot \cdot (x-\zeta^{n-1})

The second identity is the one that I enjoyed thinking about as the result of scaling the entire unit circle by a factor of y. So being on a circle of radius 1, the circle is now of radius y. Which means that each of the roots has been multiplied by y, so that the roots around the circle are now y, \zeta y, \zeta ^2 y, ..., \zeta ^ {n-1} y.


But if y is now a root of the equation, instead of x^n - 1 = 0 it must be x^n - y^n = 0, and likewise each the roots arranged around the circle will make y^n when raised to the nth power. And this is just what the second identity is saying:

x^n - y^n = (x - y)(x - \zeta y)(x-\zeta^2 y) \cdot \cdot \cdot (x-\zeta^{n-1} y)

The third identity is just what happens when you replace y with -y, though this replacement is only relevant when the power of n is odd and we get a negative (and so we end up swapping from subtraction to addition):

x^n + y^n = (x + y)(x + \zeta y)(x+\zeta^2 y) \cdot \cdot \cdot (x+\zeta^{n-1} y)

My textbook reports that the above factorization represents “an early attack on Fermat’s Last Theorem.” Feel free to link me to a readable explanation of what this early attack consisted in!


I find all of this so interesting, both mathematically and sort of meta-mathematically. I love the historical transition of algebra from “hey let’s solve some equations” to “hey here’s this entire metaphysics of structures whose names will give you absolutely no sense as to what they represent.” (I’m thinking of rings, ideals, modules, fields, domains, etc.) How did this happen? Where are the motivations?

Anyway, the above is a small part of that story.

Teach at BEAM this summer in NYC, LA or at one of their residential programs

I’ve taught at BEAM NYC twice. I’ve had so much fun being part of their work. Here’s how they describe what they do:

This summer, change the lives of underserved students with exceptional potential in math. We are looking for people who love math and working with youth to create a community of deep thinking and mathematical joy.

They’re hiring for this summer, and you should strongly consider making BEAM part of your summer plans.

Here is some info about teaching at their non-residential programs in NYC (where I’ve taught) and LA:

Staff responsibilities run weekdays June 26th – July 24th in Los Angeles, and weekdays July 2-August 12 in New York City…Both part-time (teaching an hour and helping for an hour of study hall) and full-time (teaching two courses per day) positions are available. The salary ranges from $2000-$5800 depending on course load and experience.

The non-residential programs are for 6th Graders, going into 7th Grade. If you teach full-time, it’s a full day of work that includes helping kids work on their problem sets after each class.

The residential program is for 7th Graders, going into 8th. Maybe you’re interested in teaching there:

The program runs at two college campuses in upstate New York with dates July 4-July 31 and at one college campus in Southern California with dates to be determined, starting in mid-June and ending in mid-July. You will have some flexibility in deciding which site you prefer. The salary is $5000 for faculty, and $3300 for junior faculty.  In addition, food, housing, and transportation are all provided.

There are a lot more details here. I am enthusiastic about my experiences teaching at BEAM, and this is really a situation where classroom teachers are needed and can make a difference. So, please, spread the word and share this opportunity with good teachers everywhere.

I’m happy to answer questions about my experiences and working at BEAM in the comments.

Update: I forgot that you can read about BEAM in the NY Times, here.

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.

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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.

Homework software is ready

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


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.


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.
  • 2016-2018: People talk about all the problems with Twitter.

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.


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.


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 #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:

Screen Shot 2018-10-24 at 2.53.33 PM.png

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 A: I am an excellent Radiohead album. Also, 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:ums-copier@saintannsny.org_20181004_101230-page-001


  • 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.)