I have thought about this tweet from David Coffey at least once per week for the last five months.

I'm really enjoying @HrishiHirway's @SongExploder. His episode with @Lin_Manuel was especially informative.

I wish #MTBoS would do some Lesson Exploders – breaking down your process of making lessons.

— David Coffey (@delta_dc) November 26, 2017

The Song Exploder podcast interviews artists about the craft of songwriting. The artists describe their motivations for creating their songs, what they were trying to accomplish, and how they tried to accomplish it, all while the Song Exploder team teases out key elements of the song for illustration. I feel smarter about the craft of songwriting whenever I listen to it. Maybe not as smart as if I had spent a year at the Oberlin Conservatory of Music, but for how much smarter I’m feeling, it’s hard to argue with Song Exploder’s cost (free) and scale (internet-sized).

Now swap “teacher” for “artist” and “lesson” for “song.” I *know* what we can swap in for “Oberlin Conservatory of Music.” Classroom visits. Lesson studies. Problem solving cycles. Professional learning communities. Those are all very effective and also very expensive. I *don’t* know what to swap in for “Song Exploder,” though – an option that is less effective but basically free and scales with the internet.

What kind of digital media could we use to a) highlight something significant and useful about the craft of teaching b) as quickly as possible c) distributed as widely as possible d) in a form that’s replicable and episodic? (Song Exploder is up to 133 episodes right now.)

What current examples can we find? Teaching Channel videos? Blog posts? Lesson plans? Unedited classroom video? Marilyn Burns distills classroom anecdotes into really popular tweets.

What inspiration can we take from other fields? Delish videos? NFL Red Zone / Mic’d Up? Mystery Science Theater 3000? Twitch streaming?

I can’t figure out the tolls on the New Jersey Turnpike.

If you don’t come from turnpike territory, how it works is you enter the turnpike somewhere and you exit the turnpike somewhere else. You pay depending on where you entered and exited.

My assumption is that the pricing would look pretty linear as a function of the miles traveled. Like this:

But it doesn’t. It looks like *two* linear functions with the second piece starting maybe at the Garden State Parkway. (Why?) And the Pennsylvania Turnpike exit is also *way* more expensive than a linear function would predict. (Why again?)

Here is the website that tells you the cost of different trips on the turnpike. Eric Berger, our CTO at Desmos, helped me type code into my browser’s Javascript console that returned all the data. Feel free to dig in. I’m looking for answers to my questions about pricing and I’m also interested in possible classroom applications of the data.

Cape Town has a water crisis and a website that until recently calculated a “Zero Day” for their water reserves, a day when faucets will run dry and people will collect a daily allotment of water from central locations throughout the city.

That’s either terrifying or mathematically interesting, depending on which part of my brain I subdue while I’m thinking about it. How do they calculate that zero day? How can we put students in a position to appreciate, replicate, and even adapt those calculations for their own contexts?

]]>Surprise occurs when the world reveals itself as more orderly or disorderly than we expected. When we’re surprised, we relax assumptions about the world we previously held tightly. When we’re surprised, we’re interested in resolving the difference between our expectations and reality.

In short, when we’re surprised we’re ready to learn.

We can *design* for surprise too, increasing the likelihood students experience that readiness for learning. But the Intermediate Value Theorem does not, at first glance, look like a likely site for mathematical surprise. I mean read it:

If a continuous function, f, with an interval, [a, b], as its domain, takes values f(a) and f(b) at each end of the interval, then it also takes any value between f(a) and f(b) at some point within the interval.

[I slam several nails through the door and the floor so you’re stuck here with me for a second.]

Nitsa Movshovits-Hadar argues in a fantastic essay that “every mathematics theorem is surprising.” She continues, “If the claim stated in the theorem were trivial it would be of no interest to establish it.”

What surprised Cauchy so much that he figured he should take a minute to write the Intermediate Value Theorem *down*? How can we excavate that moment of surprise from the antiseptic language of the theorem? Check out our activity and watch how it takes that formal mathematical language and converts it to a moment of surprise.

We ask students, which of these circles *must* cross the horizontal axis? Which of them *might* cross the horizontal axis? Which of them must *not* cross the horizontal axis?

They formulate and defend their conjectures and then we invite them to inspect the graph.

In the next round, we throw them their first surprise: functions are fickle. *Do not trust them.*

And then finally we throw them the surprise that led Cauchy to establish the theorem:

But you can’t expect me to spoil it. Check it out, and then let us know in the comments how you’ve integrated surprise into your own classrooms.

**Related**: Recipes for Surprising Mathematics

This wasn’t a dream. The MidSchoolMath conference organizers had proposed the idea months ago. “Why don’t you do some actual teaching instead of just talking about teaching?” basically. They’d find the kids. I was game.

But what kind of math should we *do* together? I needed math with two properties:

**The math should involve the real world in some way**, by request of the organizers.**The math should ask students to think at different levels of formality**, in concrete and abstract ways. Because these students would be working in front of*hundreds*of math teachers, I wanted to increase the likelihood they’d all find a comfortable access point*somewhere*in the math.

So we worked through a Graphing Stories vignette. We watched Adam Poetzel climb a playground structure and slide down it.

I asked the students to tell each other, and then me, some quantities in the video that were *changing* and some that were *unchanging*. I asked them to describe in *words* Adam’s height above the ground over time. Then I asked them to trace that relationship with their finger in the air. Only then did I ask them to graph it.

I asked the students to “take a couple of minutes and create a first draft.” The rest of this post is about that teaching move.

I want to report that asking students for a “first draft” had a number of really positive effects on me, and I think on us.

First, for me, I became less evaluative. I wasn’t looking for a correct graph. That isn’t the point of a rough draft. I was trying to interpret the sense students were making of the situation *at an early stage*.

Second, I wasn’t worried about finding a really *precise* graph so we (meaning the class, the audience, and I) could feel successful. I wanted to find a really *interesting* graph so we could enjoy a conversation about mathematics. I could feel a lot of my usual preoccupations melt away.

After a few minutes, I asked a pair of students if I could share their graph with everybody. I’m hesitant to speculate about students I don’t know, but my guess is that they were *more* willing to share their work because we had explicitly labeled it “a first draft.”

I asked other students to tell that pair “three aspects of their graph that you appreciate” and later to offer them “three questions or three pieces of advice for their next draft.”

- I like how they show he took longer to go up than come down.
- I like how they show he reached the bottom of the slide a little before the video ended.
- I think they should show that he sped up on the slide.
- Etc.

If you’ve ever participated in a writing workshop, you know that **workshopping one author’s rough draft benefits everyone’s rough draft**. We offered advice to two students, but

And then I gave everybody time for a second and final draft. Our pair of students produced this:

Notice here that **correctness is a continuous variable, not a discrete one**. It wasn’t as though some students had *correct* graphs and others had *incorrect* ones. (A discrete variable.) Rather, our goal was to become *more* correct, which is to say *more* observant and *more* precise through our drafting. (A continuous variable.)

And then the question hit me pretty hard:

**Why should I limit “rough-draft talk” (as Amanda Jansen calls it – paywalled article; free video) to experiences where students are learning in front of hundreds of math teachers?**

My students were likely anxious doing math in front of that audience. Naming their work a first draft, and then a second draft, seemed to ease that anxiety. But students feel anxious in math class *all the time!* That’s reason enough to find ways to explicitly name student work a rough draft.

That question now cascades onto my curriculum and my instruction.

**How should I transform my instruction to see the benefits of “rough-draft talk”?**

If I ask for a first draft but don’t make time for a second draft, students will know I really wanted a *final* draft.

If I ask for a first draft, I need to make sure I’m looking for work that is interesting, that will advance *all* of our work, rather than work that is formally correct.

**How should I transform my curriculum to see the benefits of “rough-draft talk”?**

“Create a first draft!” isn’t some kind of spell I can cast over just any kind of mathematical work and see student anxiety diminish and find students workshopping their thinking in productive ways.

Summative exams? Exercises? Problems with a single, correct numerical answer? I don’t think so.

What kind of mathematical work lends itself to creating and sharing rough drafts? My reflex answer is, “Well, it’s gotta be rich, low-floor-high-ceiling tasks,” the sprawling kind of experience you have time for only once every few weeks. However I suspect it’s possible to convert much more concise classroom experiences into opportunities for rough-draft talk.

To fully wrestle my question to the ground, how would you convert each of these questions to an opportunity for rough-draft talk, to a situation where you could plausibly say, “take a couple of minutes for a first draft,” then center a conversation on one of those drafts, then use that conversation to advance *all* of our drafts.

I think the questions each have to change.

**Geometry**

**Arithmetic**

**Algebra**

[photo by Devin Rossiter]

]]>We also don’t have the assessments to place kids with any precision on the map. The existing measures are not high enough resolution to detect the thing that a kid should learn tomorrow. Our current precision would be like Google Maps trying to steer you home tonight using a GPS system that knows only that your location correlates highly with either Maryland or Virginia.

If you’re anywhere adjacent to digital personalized learning – working at an edtech company, teaching in a personalized learning school, *in a romantic relationship* with anyone in those two categories – you should read this piece.

Berger closes with an excellent question to guide the next generation of personalized learning:

What did your best teachers and coaches do for you—without the benefit of maps, algorithms, or data—to personalize your learning?

My best teachers *knew what I knew*. They understood what I understood about whatever I was learning in a way that algorithms in 2018 cannot touch. And they used their knowledge not to suggest the next “learning object” in a sequence but to challenge me in whatever I was learning then.

“Okay you think you know this pretty well. Let me ask you this.”

What’s *your* answer to Berger’s question?

**BTW**. It’s always the right time to quote Begle’s Second Law:

Mathematics education is much more complicated than you expected even though you expected it to be more complicated than you expected.

**Featured Comment**

SueH:

I have come to believe that all learning is personalized not because of what the teacher does but because of what’s happening inside the learner’s brain. Whatever pedagogical choices a teacher makes, it’s the student’s work that causes new neural networks to be created and pre-existing ones to be augmented or strengthened or broken or pruned.

]]>I’ll accept the risk of stating the obvious: my best teachers cared about me, and I felt that. Teaching is an act of love. A teacher who cares about each student is much more likely to, in that instant after a student responds to a question, find the positive value in the response and communicate encouragement to the student, verbally and nonverbally. And students who feel cared for are more likely to have good things going on in their brains, as described by SueH.

I love the processes in the middle – like seasoning and sautéing. I can use that process in lots of different recipes, extending it in lots of different ways. It’s the right level of technical challenge for me right now.

In the same way, I’m enamored lately of *instructional routines*. These routines are sized somewhere between the routine administrative work of taking attendance and the non-routine instructional work of facilitating an investigation or novel problem. Just like seasoning and sautéing, they’re broadly useful techniques, so every minute I spend learning them is a minute very well spent.

For example, Estimation 180 is an instructional routine that helps students develop their number sense in the world. Contemplate then Calculate helps students understand the structure of a pattern before calculating its quantities. Which One Doesn’t Belong helps students understand how to name and argue about the names of mathematical objects.

(Aside: it’s been one of greatest professional pleasures of my life to watch so many of these routines begin and develop *online*, in our weirdo tweeting and blogging communities, before leaping to more mainstream practice.)

I first encountered the routine “Two Truths and a Lie” in college when new, nervous freshmen would share two truths about themselves and one lie, and other freshmen would try to guess the lie.

Marian Small and Amy Lin adapted that icebreaker into an instructional routine in their book *More Good Questions*. I heard about it from Jon Orr and yesterday we adapted that routine into our Challenge Creator technology at Desmos.

We invite each student to create their own object – a circle graph design in primary; a parabola in secondary.

We ask the student to write three statements about their object – two that are true, and one that is a lie. They describe why it’s a lie.

Here are three interesting statements from David Petro’s circle graph design. Which is the lie?

- The shaded part is the same area as the non shaded part.
- If these were pizzas, there is a way for three people to get the same amount when divided.
- If you double the image you could make a total of 5 shaded circles.

And three from Sharee Herbert’s interesting parabola. Which is the lie?

- The axis of symmetry is y=-2.
- The y-intercept is negative.
- The roots are real.

Then we put that thinking in a box, tie a bow around it, and slide it into your class gallery.

The teacher encourages the students to use the rest of their time to check out their classmates’ parabolas and circle graphs, separate lies from truth, and see if everybody agrees.

Our experience with Challenge Creator is that the class gets noisy, that students react to one another’s challenges verbally, starting and settling mathematical arguments at will. It’s beautiful.

So feel free to create a class and use these with your own students:

**2018 Feb 6**. I added eight more Two Truths & a Lie activities on suggestions from y’all!

- Circle Graphs
- Parabolas
- Logarithms
- Trig Graphs
- Systems of Linear Equations
- Ellipses
- Rational Functions
- Quadrilaterals
- Right Triangles
- Box Plots
**2018 Mar 22**. Line Segments**2018 Mar 23**. Linears**2018 Mar 23**. Exponentials**2018 Mar 27**. TWO Box Plots

**BTW**. Unfortunately, Challenge Creator doesn’t have enough polish for us to release it publicly yet. But I’d be happy to make a few more TTL activities if y’all wanted to propose some in the comments.

Taylor registered her Twitter account this month. She’s *brand* new. She’s posted this one tweet alone. In this tweet, she’s basically tapping the Math Teacher Twitter microphone asking, “Is this thing on?” and so far the answer is “Nope.” She’s lonely. That’s bad for her and bad for us.

**It’s bad for her because we could be great for her.** For the right teacher, Twitter is the best ambient, low-intensity professional development and community you’ll find. Maybe Twitter isn’t as good for development or community as a high-intensity, three-year program located at your school site. But if you want to get your brain spinning on an interesting problem of practice in the amount of time it takes you to tap an app, *Twitter is the only game in town*. And Taylor is missing out on it.

**It’s bad for us because she could be great for us.** Our online communities on Twitter are as susceptible to groupthink as any other. No one knows how many interesting ways Taylor could challenge and provoke us, how many interesting ideas she has for teaching place value. We would have lost some of your favorite math teachers on Twitter if they hadn’t pushed through lengthy stretches of loneliness. Presumably, others didn’t persevere.

Interesting looking at my early #MTBoS tweets. Most of the time I got no response at all. I wonder why I kept at it.

— David Butler (@DavidKButlerUoA) July 29, 2017

So we’d love to see fewer lonely math teachers on Twitter, for our sake and for theirs.

Last year, Matt ~~Stoodle~~ Baker invited people to volunteer every day of the month to check the #mtbos hashtag (one route into this community) and make sure people weren’t lonely there. Great idea. I’m signed up for the 13th day of every month, but ideally, we could distribute the work across more people and across time. Ideally, we could easily distinguish the *lonely* math teachers from the ones who already experience community and development on Twitter, and welcome them.

I’m not the first person to want this.

wondering: is it possible to make a bot that auto RTs any #MTBoS tweet that's gotten no response in x hours, so ppl are more likely to see?

— grace a chen (@graceachen) July 29, 2017

So here is a website I spent a little time designing that can help you identify and welcome lonely math teachers on Twitter: lonelymathteachers.com.

It does three things:

- It searches several math teaching hashtags for tweets that a) haven’t yet received any replies, b) aren’t replies themselves, and c) aren’t retweets. Those people are lonely! Reply to them!
- It puts an icon next to teachers who have fewer than 100 tweets or who registered their account in the last month. These people are
*especially*lonely. - It creates a weekly tally of the five “best” welcomers on Math Teacher Twitter, where “best” is defined kind of murkily.

That’s it! As with everything else I’m up to in my life, I have no idea if this idea will work. But I love this place and the idea was actually going to bore a hole right out of my dang head if I didn’t do something with it.

**BTW**. Thanks to Sam Shah, Grace Chen, Matt Stoodle, and Jackie Stone for test driving the page and offering their feedback. Thanks to Denis Lantsman for help with the code.

**Related**

**18 Jan 22**:

Site got slammed with more traffic than I knew what to do with it. I'll figure something out soon. Fun to see all the interest in welcoming lonely math teachers. https://t.co/UbJyaOHYTn

— Dan Meyer (@ddmeyer) January 22, 2018

It’s the muscle that connects my capacity for *noticing the world* to my capacity for *creating mathematical experiences for children*. (I should also take some time in 2018 to learn how muscles work.)

By way of illustration, this was my favorite tweet of 2017.

Gathering some data re: perfect banana ripeness. Have 5 seconds? Fill out a one-question survey! https://t.co/r9UQA1RlC6 #mathchat #MTBoS pic.twitter.com/BrYtUpfa7h

— Ilona Vashchyshyn (@vaslona) September 29, 2017

Right there you have an image created by Brittany Wright, a chef, and shared with the 200,000 people who follow her on Instagram. Loads of people before Ilona had *noticed* it, but she *connected* that noticing to her capacity for

That skill – taking an interesting thing and turning it into a challenging thing – is one of teaching’s “unnatural acts.” Who *does* that? Not civilians. Teachers do. And I want to get awesome at it.

But Ilona ran a marathon and I want to run some wind sprints. I need *quick* exercises for strengthening that muscle. So here are my exercises for 2018:

I’m going to **pause when I notice mathematical structures in the world**. Like flying out of the United terminal in San Antonio at last year’s NCTM where I (and I’m sure a bunch of other math teachers) noticed this “Suitcase Circle.”

Then I’ll **capture my question** in a picture or a video. Kind of like the one above, except pictures like that one exist in abundance online.

Civilians capture scenes in order to *preserve as much information as possible*. That’s natural. But I’ll *excerpt* the scene, *removing* some information in order to provoke curiosity. Perhaps *this* photo, which makes me wonder, “How many suitcases are there?”

In order to gauge the curiosity potential of the image, I’ll **share the media I captured with my community**. Maybe with my question attached, like Ilona did. Maybe without a question so I can see the interesting questions other people wonder. You may find my photos on Twitter. You may find them at my pet website, 101questions.

I want to get to a place where that muscle is so strong that I’m *hyper*-observant of math in the world around me, and turning those observations into curious mathematical experiences for children is like a reflexive twitch.

(Plus, that muscle will be more fun to strengthen in 2018 than literally any other muscle in my body.)

**BTW**. Check out the 3-Act Task I created for the Suitcase Circle. It includes the following reveal, which I’m pretty proud of.

**BTW**. The suitcase circle later turned into Complete the Arch, a Desmos activity, which has some really nice math going on.

[Suitcase Circle photo by Scott Ball]

**Featured Comment**

]]>I would just add that we shouldn’t forget that the classroom is a world within a world for us to notice, and that while many great, unforgettable tasks are based on interesting phenomena that we’ve observed or collected outside of school, on a day-to-day basis, high-impact tasks are probably more likely to be rooted in our observations and interactions with our students (in fact, even the banana tweet and post were sparked by a conversation with a student who was eating what was, to me, an exceptionally green banana). They tend not to be as flashy, but can have just as much impact because they’re tailored to the kids, norms, relationships, and histories in our classrooms.

Top Chef. Project Runway. The Voice. Live competition shows have introduced audiences to the worlds of cooking, fashion, and singing — and opened a window into the intricate craftsmanship that these industries demand. Now it’s time for one of America’s most under-recognized professions to get the same treatment. Hi, teachers!!

Two teams of math teachers will teach a lesson to a live audience and receive judgment from a panel of “teacher celebrities.”

I linked to that description on Twitter and people were unsparing in their criticism:

There is so much more to teaching than what is viewable in a short lesson presented to a group of individuals. Connections with students matter.

— Lisa Bejarano (@lisabej_manitou) December 31, 2017

I fear this will perpetuate false ideas of what teaching entails. https://t.co/5QUB6yo80M

My mind is spinning with all the reasons why I think this is an awful idea. https://t.co/99X9owPAPk

— Marilyn Burns (@mburnsmath) December 31, 2017

Seems to me about as wise as a televised, competitive Parent Off. [Which is to say: Unwise.]

— Benjamin Dickman (@benjamindickman) December 31, 2017

Goodness. This is describably horrific.

— Ben Wichser (@BenWichser) December 31, 2017

Singing and “chef”ing (different from cooking) are performative arts centered on the entertainer.

Teaching must always be about helping the students grow. Not about judging the entertainer aspects of the instructor.

Teaching is and should always be a collaborative endeavor. Competition is what causes rifts among staff, encourages teaching in silos, and prevents us from growing together.

— Mike Flynn (@MikeFlynn55) December 31, 2017

I agree with the spirit of those criticisms, and David Coffey’s in particular:

Also, much of @Chalkbeat’s “craft of teaching” occurs under the surface. #TeachingLearningCycle pic.twitter.com/wKi3Feqne5

— David Coffey (@delta_dc) December 31, 2017

Good teaching requires complicated decision-making based on a teacher’s long-range knowledge of a student and of mathematics. We should reach for any opportunity to make those decisions transparent to the public, who will always benefit from more education about good education. But a live event with an audience you don’t know and can’t interact with individually will necessarily flatten “teaching” down to its most presentational aspects, down to teachers dressing up in costumes, down to Robin Williams standing on desks in *Dead Poets Society*.

I asked teachers what kind of TV show *would* do justice to the complexity of teaching, if *The Voice* and *Top Chef* were the wrong models. Jamie Garner and James Cleveland both suggested *The Real World*, which seems dead on to me.

Maybe more like The Real World - stick a bunch of teachers in the same house, and working at the same school.

— James Cleveland (@jacehan) December 31, 2017

Rather than a game show, I envision that it would be more like The Real World: Math Class. This would allow for a development over time of understanding of the work of a classroom, not in one hour segments focused on competition. 1/2

— Jamie Garner (@mavenofmath) December 31, 2017

*The Real World* a) isn’t a competition, b) allows for characters to develop over time, and crucially, c) isn’t a live event. It is edited. You don’t watch the cast members do anything mundane. In the case of teaching, we’d love for the public to understand that good teachers assess what students know and adjust their instruction in response. But no one wants to watch a class work quietly on a five-minute exit ticket in real time. So the show would edit quickly past students *completing* the assessment and straight to the teacher trying to make sense of a student’s thinking, involving the audience in that process.

The challenge I’d like to see the folks at Chalkbeat take up is how to make those invisible aspects of teaching – the work that takes place after the bell – visible to the public. The work of *presenting* is already teaching’s most visible aspect.

**BTW**. Jamie Garner expands on The Real World: Math Class.

**2018 Jan 1**. Chalkbeat’s Editor-in-Chief, Elizabeth Green, clarifies her rationale for launching the competition and responds to some concerns raised here and on Twitter. She describes lesson study as the touchstone for her *Teach Off* and how she’s had to alter that format to fit SXSW.

It’s a really interesting article, full of references to the education scholars who have inspired her work for a decade. But I still tend to think she and the members of her design team have underestimated the magnitude of those compromises and how they’ll distort the approximation of good instruction her audience will encounter.

**2018 Jan 8**. In a revised contest page the organizers have eliminated the competition and clarified other aspects.

**Featured Comment**

Organizer Elizabeth Green weighs in:

]]>I’m weighing in late here, but in response to one of the above threads, we never intended to have the whole audience serve as the students. As we’ve clarified in our revised page, which has more specific language, we’ll have 7-10 adult audience volunteers serve as students. Imperfect as a representation, for sure, but we still think everyone will get something important out of the 20-minute instructional activity + the followup discussion — that “something important” being better understanding about the nature of teaching and math teaching in particular. And for the record, Dan, at the 1,000-person “Iron Chef”-style teach off in Japan that Akihiko described, the students were the teacher’s actual students, and they all sat onstage.

But *Arthur* nails the nuance in “Sue Ellen Adds It Up,” and reports three important truths about math in ten minutes.

**We are all math people. (And art people!)**

Sue Ellen is convinced she isn’t a math person while her friend Prunella is convinced there’s no such thing as “math people.” You may have this poster on your wall already, but it’s nice to see it on children’s television. Meanwhile, Prunella is convinced that, while she and her friend are both “math people,” only Sue Ellen is an “art person.” Kudos to the show for challenging that idea also.

**Informal mathematical skills complement and support formal mathematical skills.**

Sue Ellen says that she and her family get along fine without math everywhere “except in math class.” They rely on estimating, eyeballing, and guessing-and-checking when they’re cooking, driving, shopping, and hanging pictures. Prunella tells Sue Ellen, accurately, that when Sue Ellen estimates, eyeballs, and guess-and-checks, she *is* doing math. Sue Ellen is unconvinced, possibly because the only math we see her do in math class involves formal calculation. (Math teachers: emphasize informal mathematical thinking!)

**We need to create a need for formal mathematical skills.**

Sue Ellen resents her math class. She has to learn formal mathematics (like calculation) while she and her family get along great with informal mathematics (like estimation). Then she encounters a scenario that reveals the limits of her informal skills and creates the need for the formal ones.

She’s made a painting for *one* area of a wall and then she’s assigned a *smaller* area than she anticipated. She encounters the need for computation, measurement, and calculation, as she attempts to crop her painting for the given area while preserving its most important elements.

Nice! Our work as teachers and curriculum designers is to bottle those scenarios and offer them to students in ways that support their development of formal mathematical ideas and skills.

[h/t Jacob Mehr]

]]>This is my best attempt to tie together and illustrate terms like “intellectual need” and expressions like “if math is aspirin, how do we create the headache.” If you’re looking for an elaboration on those ideas, or for illustrations you haven’t seen on this blog, check out the video.

**The Directory of Mathematical Headaches**

This approach to instruction seriously taxes me. That’s because answering the question, “Why did mathematicians invent this skill or idea?” requires a depth of content knowledge that, on my best days, I only have in algebra and geometry. So I’ve been *very* grateful these last few years to work with so many groups of teachers whose content knowledge supplements and exceeds my own, particularly at primary and tertiary levels. Together we created the Directory of Mathematical Headaches, a collaborative document that adapts the ideas in this talk from primary grades up through calculus.

It isn’t close to complete, so feel free to add your own contributions in the comments here, by email, or in the contact form.

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