How many linear equations did the student use to create it? You might start counting lines and assume it required dozens. For some students, you’d be right. They typed 40 linear equations and corrected a handful of typos along the way.

But other students created it using only *four* linear equations and many fewer errors!

The seventh mathematical practice in the Common Core State Standards asks students to “look for and make use of structure.” The second half of that standard is a heavier lift than the first by several hundred pounds.

Because it’s easy enough for me to ask students, “What structures do you notice?” It’s much more difficult for me to put them in a situation where *noticing* a mathematical structure is more useful than *not noticing* that structure.

Enter Match My Picture, my favorite activity for illustrating my favorite feature in the entire Desmos Graphing Calculator and for helping students see the *use* in mathematical structures.

First, we ask students to write the linear equations for a couple of parallel lines.

Then four lines. Then nine lines.

It’s getting boring, but also easy, which are *perfect* conditions for this particular work. A boring, easy task gives students lots of mental room to notice structure.

Next we ask students, “If you could write them all at once – as *one* equation, in a form you made up – what would that look like?” Check out their mathematical invention!

Next we show students how *Desmos* uses lists to write those equations all at once, and then students put those lists to work, creating patterns much faster and with many fewer errors than they did before. With lists, you can create nine lines just as fast as *ninety* lines.

What are the four equations that created this graph? Personally, I find it almost impossible to discern by just looking at the graph. I have to write the equation of *one* of the lines. Then another. Then another. Then another, until that task becomes boring and easy. Only then am I able to notice and make use of the structure.

**Mathematical Play**

*Kassia Omohundro Wedekind*

Kassia’s ShadowCon talk was such a blast, integrating several different bodies of scholarship all arguing for the mathematical and social value of *play*. Her course has insightful readings, illustrative classroom video, and Zak Champagne and Mike Flynn as teaching assistants.

**The Art of Mathematical Anthropology**

*Geoff Krall*

Geoff has loads of experience with innovative assessments as a coach in the New Tech network of schools. In his course, he’ll help you understand what *portfolio assessments* offer students and how to *develop* them. You’ll find me in Geoff’s course as a teaching assistant.

I’ll pick a number between 1 and 100. I’ll give you ten guesses to figure out my number. And every time you guess, I’ll tell you if my number is higher or lower.

I always wagered whatever cash I had in my pocket – generally between $2 to $20. The math teachers, meanwhile, owed me nothing if they lost. I had no trouble finding people to take the other side of that wager.

Watch one of the wagers below.

I pick my number.

She first guesses 61. I’m higher.

Then 71. I’m higher.

Then 81. I’m higher.

Then 91. I’m lower. She’s got me trapped. Six guesses left.

Then 86. I’m lower. Five guesses left. I’m an injured gazelle.

Then 83. I’m lower. Between 81 and 83. Four guesses left, but she only needs one. The crowd smells blood.

Then, with a trace of sympathy in her voice, 82. The crowd thinks it’s over.

But I’m *higher*.

*Aaaand the chase is back on, y’all!*.

Tentatively now: 82.5. I’m still higher. One by one, members of the crowd are wise to my scam.

Then 82.75. I’m lower. She has one guess left.

Then 82.7. I’m higher, at 82.72.

I asked her what I’d ask any crowd of sixth graders at this point:

If I offered you the same wager again, what follow-up questions would you have for me?

“What kind of number are you picking?” she said.

My point in all of this is that math teachers have *names* for their numbers, much in the same way that ornithologists have *names* for their birds. And much in the same way that ornithologists haven’t given me a reason to care about the difference between a Woodlark and a Skylark, math teachers often fail to motivate the difference between rational numbers and integers and whole numbers and imaginary numbers and supernatural numbers.

The difference is that ornithology isn’t a course that’s required for high school graduation and university enrollment and labor market participation. Kids aren’t forced to study ornithology for twelve years of their childhood.

So I’m inviting us to ask ourselves: “Why did we invent these categories of numbers?” And if we agree that it was to more effectively *communicate* about numbers, we need to put students in a place where their communication *suffers* without those categories. If we can’t, then we should confess those categories are vanity.

Before we give students the graphic organizers and Venn diagrams and foldables designed to help them *learn* those categories, let’s help them understand that they were invented for a *reason*. Not because we have to.

There are always ways to make kids memorize disconnected, purposeless stuff.

But because we *should*.

**Featured Comment**

Via email:

Did you ever lose?

I never once lost. I was never once asked to specify the *kind* of number I was picking.

NCTM went about that goal in the second half of *Catalyzing Change*, enumerating a set of “Essential Concepts” along with two pathways students can take to learn them. I’ll comment on those concepts and pathways in a moment. But it’s worth mentioning first what I *didn’t* anticipate: a document full of moral ambition, the first half of which is a reimagination of the *purpose* of a math education along with a high-decibel endorsement of equity in that education.

You should read the latter half of the document if you have any stake in *high school* math education. But you should read the *first* half of the document if you have any stake in math education at all, at any level.

While the Obama administration proposed college and career-readiness as the purpose of schooling, NCTM broadens that purpose here to include “Understanding and Critiquing the World,” addressing the question, “When will I ever use this?”, and also “Experiencing Wonder, Joy, and Beauty,” acknowledging the millions and millions of people who love studying math even apart from its immediate application to the world outside the classroom.

NCTM reinvokes its call for equitable math instruction, citing Gutiérrez’s perspective that until it is no longer possible “to predict mathematics achievement and participation based solely on student characteristics such as race, class, ethnicity, sex, beliefs, and proficiency in the dominant language,” we haven’t finished the work. To advance the cause of equity, NCTM pulls precisely zero punches in its condemnation not just of student tracking (which allocates students inequitably to the best classes) but *teacher* tracking (which allocates teachers inequitably to the most underserved students), also double-year math courses, and other less overt ways in which students are tracked even in elementary school.

This is what I mean by “moral ambition.” NCTM hasn’t merely underlined its existing statements on equity or de-tracking. Rather it lets those statements stand and then opens up several new fronts and runs at them. *Catalyzing Change* doesn’t arrive pre-compromised.

So again: everyone should read the first half of *Catalyzing Change*, which addresses much of the “why?” and “who?” of mathematics education. The second half of the document makes several clear and ambitious claims about the “what?”

NCTM proposes that all students take four years of math in high school. 2.5 of those years will comprise “essential concepts,” taken by every student regardless of career or college aspiration. Students may then take one of two paths through their remaining 1.5 years, one towards calculus, the other towards statistics and other electives.

40 essential concepts cluster under five conceptual categories:

- Algebra
- Functions
- Statistics
- Probability
- Geometry

If we only examine the *number* of concepts and not yet their content, this proposal compares very favorably with the Common Core State Standards’ *over 100 required standards* for high school. Under NCTM’s proposal, students may come to understand a proof of the similarity of circles (Common Core State Standard G-C.1) or a derivation of the equation of a parabola from its directrix and focus (G-GPE.2) but only as an *incidental* outcome of high school math, not an *essential* outcome.

Then, as I read the *content* of the concepts, I asked myself, “Do I really believe *every* student should spend 2.5 years of their limited childhood learning this?” In nearly every case, I could answer “yes.” In nearly every case, I could see the concept’s applicability to college and career readiness, and even more often, I could see how the concept would help students understand their world and nurture their joy and wonder. (I wouldn’t say that about the derivation of a parabola’s equation, by contrast.)

That’s such an accomplishment. The writing team has created a “Director’s Cut” of high school mathematics – only the most essential parts, arranged with a coherence that comes from experience.

If I’m concerned about any category, it’s “Algebra” and, particularly, essential concepts like this one:

Multi-term or complex expressions can represent a single quantity and can be substituted for that quantity in another expression, equation, or inequality; doing so can be useful when

rewriting expressionsand solving equations, inequalities, or systems of equations or inequalities.[emphasis mine]

Without any evidence, I’m going to claim that one of the top three reasons students leave high school hating mathematics is because their algebra courses required weeks and weeks of transcribing expressions from one form into another for no greater purpose than passing the class. I’m talking about conjugating denominators, converting quartic equations into quadratic equations through some clever substitution, factoring very special polynomials, completing the square, and all other manner of cryptic symbology, none of which deserves the label “essential.”

NCTM has done much more work here defining what is “essential” than what is “inessential,” which means their definitions need to be air tight. Some of their definitions in “Algebra” and “Functions” leave room for some very inessential mathematics to slip through.

My other concern with *Catalyzing Change* is the bet NCTM makes on technology, modeling, and proof, weaving that medium and those habits of mind through every category, and claiming that they have the greatest potential to enable equitable instruction.

I don’t disagree with that selection or NCTM’s rationale. But add up the bill with me here. NCTM proposes a high school course of study premised on:

**modeling**, which students most often experience as pseudocontextual word problems,**proof**, which students most often experience by filling in blanks in a two-column template,**technology**, which students most often experience as a medium for mealy, auto-graded exercises,- to say nothing of
**joy and wonder**, which most students typically experience as boredom and dread.

This is a multi-decade project! One that will require the best of teachers, teacher educators, coaches, administrators, edtech companies, assessment consortia, policymakers, publishers, and parents. It will require new models of curriculum, assessment, and professional development, all supporting modeling and proof and eliciting joy and wonder from students. It will require a constant articulation and re-articulation of values to people who aren’t NCTM members. That is, changes to the K-8 curriculum required articulation to high school teachers. Changes to the high school curriculum will require articulation to college and university educators! Does anybody even *know* any college or university educators?

I’m not finding fault. I’m identifying challenges, and I find them all energizing. *Catalyzing Change* is an invigorating document that makes a clear case for NCTM’s existence at a time when NCTM has struggled to articulate its value to members and non-members.

If you haven’t heard that case, let me try to write it out:

]]>Hi. We’re NCTM. We want to restore purpose, joy, and wonder to your high school math classrooms. We know that goal sounds ambitious, and maybe even impossible, but we have a lot of experience, a lot of ideas, a lot of resources, and a lot of ways to help you grow into it. We’re here for you, and we also can’t do any of this without you. Let’s do this!

We’re excited to release our latest activity into the world: Transformation Golf.

Transformation Golf is the result of a year’s worth of a) interviews with teachers and mathematicians, b) research into existing transformation work, c) ongoing collaboration between Desmos’s teaching, product, and engineering teams, d) classroom demos with students.

It’s pretty simple. There is a purple golf ball (a/k/a the pre-image) and the gray golf hole (a/k/a the image). Use transformations to get the golf ball in the hole. Avoid the obstacles.

Here’s why we’re excited to offer it to you and your students.

**Teachers told us they need it**. We interviewed a group of eighth grade teachers last year about their biggest challenges with their curriculum. Every single teacher mentioned independently the difficulty of teaching transformations – what they are, how some of them are equivalent, how they relate to congruency. Lots of digital transformation tools exist. None of them quite worked for this group.

**It builds from informal language to formal transformation notation.** As often as we ask students to define translation vectors and lines of reflection, we ask them just to describe those transformations using informal, personal language. For example, before we ask students to complete this challenge using our transformation tools, we ask them to describe how they’d complete the challenge using words and sketches.

**The entire plane moves**. When students reach high school, they learn that transformations don’t just act on a single object in the plane, they act on the entire plane. We set students up for later success by demonstrating, for example, that a translation vector can be anywhere in a plane and it transforms the entire plane.

**Students receive delayed feedback on their transformations**. Lots of applets exist that allow students to see immediately the effect of a transformation as they modify it. But that kind of immediate feedback often overwhelms a student and inhibits her ability to create a mental concept of the transformation. Here students create a transformation, conjecture about its effect, and

**Students manipulate the transformations directly.** Even in some very strong transformation applets, we noticed that students had to program their transformations using notation that wasn’t particularly intuitive or transparent. In this activity, students directly manipulate the transformation, setting translation vectors, reflection lines, and rotation angles using intuitive control points.

**It’s an incredibly effective conversation starter.** We have used this activity internally with a bunch of very experienced university math graduates as well as externally with a bunch of very inexperienced eighth grade math students. In both groups, we observed an unusual amount of conversation and participation. On every screen, we could point to our dashboard and ask questions like, “Do you think this is possible in *fewer* transformations? With just *rotations*? If not, why not?”

Those questions and conversations fell naturally out of the activity for us. Now we’re excited to offer the same opportunity to you and your students. Try it out!

]]>**First**, Cathy Yenca will help you “Seek Students Who Hide,” describing her rationale for formative assessment and then the tools (both low-tech and high-tech) she uses to draw the best out of all of her students.

Check out her talk below for an introduction and then register for the free class.

**Second**, Anurupa Ganguly will examine the participation gap in STEM majors for historically disenfranchised students, asking, “Why don’t the humanities see the same attrition rate, and what can we do about it?”

Check out her talk below for an introduction and then register for the free class.

Both courses will feature lots of interaction with the speakers (including a live webinar halfway through), helpful readings, provocative discussions, and plenty of support from the teaching assistants, Mike, Zak, and I.

**Third**, in response to recent evidence that personalized learning is struggling to see positive results on important issues of belonging and engagement, I asked on Twitter for an invitation to the meeting where we question the core assumptions of the model.

Can you send me a calendar invite to the meeting where we question the core assumptions of personalized learning? https://t.co/ar2WrwW0tW pic.twitter.com/DnkIZZXTxU

— Dan Meyer (@ddmeyer) August 16, 2017

One thing led to another and on Tuesday, September 5, 2017, at 4:00 PM Pacific, I’ll be having exactly that meeting with Mary Jo Madda of EdSurge, livestreamed on YouTube. I’m excited to examine those failures and, wherever the failures aren’t *endemic* to the model, brainstorm some solutions.

**2017 Sep 5**: Update – link changed for the EdSurge chat.

Any tips for first day of first year of teaching?!?!? #MTboS

— Nancy Pendleton (@PendleNA) August 4, 2017

If you feel anywhere close to how Nancy feels, click through for some great advice from your friends on Math Teacher Twitter. You’ll see very few people encouraging her not to smile until December and very many people encouraging her to *do some math with your students on day one*. Great advice. We crowdsourced loads of ideas for those math tasks last year. Please add more there.

As much as I’m curious what happens *within* the four walls of your classroom on day one, I’m also curious what happens *on* the four walls of your classroom.

This tweet caught my eye for a couple of reasons:

Leave some white space on your walls for your students to fill with their creations-don't let it look like Pinterest threw up in your room

— Lynn (@LMGirolamo) August 5, 2017

First, “… like Pinterest threw up in your room” is going to be a hard image to shake.

Second, I love the thought that our students would walk into rooms that aren’t fully authored by their teachers, that the space would be shared and awaiting their co-authorship.

Yes! I used to put up huge blocks of colored bulletin board paper so room looked cheerful, but open and ready for S work.

— Lorri Sapp (@LorriSapp) August 5, 2017

If you have experience or ideas here, please add them in the comments. I’ll add the Feltron Project as my own contribution to this planning potluck, and I’d love to learn more.

**Featured Tweets**

I purposefully leave the biggest wall mostly blank, with a "Teamwork!" poster at the top. Day 1- Ss work on posters hung at end of class.

— Susan Glassburn (@MsGTeachesMath) August 8, 2017

I keep a Poloroid camera and let students take pics at epic moments.

— x-tina marie (@keepingmypma) August 7, 2017

We make a histogram with their birth dates on day one and a scatter plot with their bedtime and awake time #CPM 6thgrade #iteachmath #charts

— Jennifer Profumo (@JennProfumo) August 8, 2017

I use a board with tacks glued to the back of clothes pins to easily swap out student work. Ss can easily choose and swap out.

— Mrs. Beauchemin (@MrsBeauchemin) August 8, 2017

]]>Let MS Ss dictate when they have "Wall worthy" work-- let them share stuff they're proud of!

— Ashley Nesbit (@AGoTeach) August 7, 2017

This community has also been built and nurtured by hundreds of people in thousands of big and small ways – from huge initiatives like Twitter Math Camp, ExploreMTBoS, and Global Math Department, down to folks who watch out for new Twitter users and say “howdy.” This post wasn’t and isn’t meant to critique any of those efforts, but I realize that it came across that way, and that was wrong of me. Precisely because there are thousands of those efforts, I can’t reach out and apologize to each of you individually for dismissing them, so please accept my apology here. Keep on making this place awesome.

Whatever else you think of this post, the people who have commented on it and whose tweets I’ve excerpted below are real people who have found *our name* alienating. (Not the community. The name itself.) That’s a problem that countless people in the last few days have told me isn’t worth tackling, or one that pales in comparison to other problems. I respect that opinion. I’d like to work on it anyway, and also work on the other problems. But rather than use my platform here to set a unilateral course, I should have found out who is already doing that work and found out how I could help. I’m generally skeptical of leaders and I’ve never been particularly eager to be one, but that isn’t any excuse for setting a bad example. If you’re doing that work, and if I can help or collaborate, please let me know in the comments or at dan@mrmeyer.com.

**2017 Jul 28**. Thanks to everyone who helped me think this through, especially the ones who did so in spite of being annoyed and hurt. Much love to you all, and to this place. My current plan is to introduce teachers to Math Teacher Twitter by inviting them to attach “#iteachmath” to a tweet, a tag that is intuitive, pronounceable, and importantly, a declarative statement. Meanwhile, “#MTBoS” has less certain pronunciation and, *for newcomers*, it has been unintuitive and felt a bit like you’re inviting yourself into a secret club. (Seriously, don’t trust me on this. Read the dozens of tweets and comments I’ve excerpted below.) I hope that the thousands of people who find community around “#MTBoS” will continue to enjoy it! But I’m hopeful that “#iteachmath” will be a better invitation for the hundreds of thousands of math teachers who don’t yet know how great we have it.

The original post follows.

——

I’m not asking us to retire *the* #MTBoS (unabbreviated: the Math Twitterblogosphere) the collection of people, ideas, and relationships that has provided the most satisfying professional development and community of my life.

I’m asking us to stop referring to it as “the MTBoS” and to stop using the hashtag “#MTBoS” in online conversations.

That’s because this community is only as good as the people we invite into it. We currently represent only the tiniest fraction of the math teachers in the world, which means we (and I’d like to believe *they* also) are missing out.

That fraction will stay tiny so long as our name alienates people. And it alienates people.

Umm, MTBoS? What is that please?

— Prof Rick Fletcher (@TRFletcher) July 26, 2017

People don’t know how to pronounce our name. Whenever I use it, I get tweets back asking me what I’m talking about. Whenever I invite new teachers to get on Twitter and search for “#MTBoS,” their confusion is plain at that seemingly random assortment of vowels and consonants, capitalized in seemingly random ways.

The more used hashtag for physics is #iteachphysics. I'm curious: how are you quantifying "good" in comparison to #MTBoS ?

— Andrew Morrison (@achmorrison) July 27, 2017

This morning I read a tweet from a science teacher named Andrew Morrison. I learned from Andrew that the physics teaching community hashtags their work “#iteachphysics.” I felt such a sense of *invitation* when I read that hashtag – “This is who we are and what we do. You should join us.” And then I felt envy.

*We* should be so inviting.

This community of ours has no leader. It has no high council. Each one of us has to be the change we want to see in it. I want to see a more inviting community, a community that doesn’t shroud its entrance behind a hedge or protect its door with a password.

**So I’m going to stop referring to my participation in “the MTBoS” and instead talk about how much I love “Math Teacher Twitter.” I’m going to stop tweeting using “#MTBoS” and instead tweet using “#iteachmath.”**

No one has to join me, and I absolutely won’t be offended if you don’t, but I hope you will, and I hope you at least understand why I’m doing this. I think this change is necessary for our growth and this is how I’ll try to be that change.

**Reservations That I Had About This Proposal That I Don’t Anymore**

*“#iteachmath” is five more characters than “#MTBoS. That’s five fewer characters for my tweets!”*

I accept that those five characters are the cost of a more inviting community.

*Twitter users outside the United States will want to use “#iteachmaths.”*

The MTBoS has a very, very tiny handful of community members outside the United States as it is. I think we can only improve from here. Me, I’m going to add both “#iteachmaths” and “#iteachmath” to the same column in Tweetdeck.

*“MTBoS” includes blogs (the “B”) but “Math Teacher Twitter” just refers to Twitter.*

“MTBoS” also fails to refer to Slack, Voxer, or any of the other ways teachers collaborate online. “Math Teacher Twitter” *hints* at all those ways. It doesn’t try to *catalog* them.

*But I’m a coach / consultant / curriculum author / administrator. I don’t teach math so I’ll feel weird using “#iteachmath”.*

Let’s not treat this hashtag like it’s a sworn statement in a court of law. It’s an invitation. It’s how we’ll gather community around a conversation. It doesn’t need to serve any higher purpose than that, and I think it’ll serve that purpose better than anything we have right now.

By default, never felt a part of it because I don't blog--felt like I couldn't take part. I'm all for #iteachmath. Thx for initiating Dan.

— Maria Riverso (@RiversoMaria) July 28, 2017

Exactly. I used it to search and find awesome content but never felt "cool" enough to use it, partially bc I don't blog. #iteachmath is *me*

— Kelly Hagan (@haganmath) July 28, 2017

I agree with your point, I've never used the #MTBoS hashtag because I could never remember what it meant AND didn't know if I belonged. 🤔

— MrsLyonsLibrary (@mrslyonslibrary) July 28, 2017

Agreed. I've never even known what #MTBoS meant, but didn't think it meant me. However, I can definitely say #iteachmaths https://t.co/ZAHosO5oQi

— Emmaface (@emmaemma53) July 28, 2017

Ah thank you! I actually thought #MTBoS was an elite group of math teachers. I favor #iteachmath

— Elizabeth Smith (@lizmaismith) July 28, 2017

Honestly, I'd feel less like an imposter tweeting #iteachmath than #MTBoS. It's insecurity, but it's likely shared by many.

— Sara Philly (@PanglossCritic) July 28, 2017

Finally!! Cause I could never figure out how to be a part. I want to be a part. #iteachmath

— Natalie Moon (@themathgirl) July 27, 2017

Totally agree. I THINK I'm a "member;" it's intimidating. #MTBos, #iteachmath

— ジョアン (@TeacherKaizen) July 27, 2017

I agree. It 'sounds' like an organization you have to apply to. And existing members are better than you, and always know more than you

— Justin Yantho (@iYantho) July 27, 2017

Justin’s tweet seems really, really important to me. Consider the *perceived* requirements for membership in the #MTBoS vs. #iteachmath.

#MTBoS: who knows, but a blood sample and credit verification is probably part of it.

#iteachmath: it’s right there in the hashtag. That’s it. No guessing. You’re invited.

Thank You! I have followed for mult yrs but always felt like until some1 "in" knew who you were, you weren't actually in the club...

— Jason Kissel (@MrKisselMATH) July 27, 2017

This is awesome. I've been on the very fringes for many these reasons. It might just be semantics, but definitely #iteachmath

— Angela Cooper (@angecooper) July 28, 2017

Even at NCTM, I never wore the MTBoS because I thought it was an exclusive group. Thank you, Dan! #iteachmath #iteachmaths #inclusivemath

— Maggie Lee McHugh (@mlmchugh) July 28, 2017

Via direct message:

I always felt a little worried or unsure about joining the community and when it was ok to tweet #mtbos.

Also via direct message:

I actually had to look up the #MTBoS. I am not a member and not sure I am a blogger. I do have a question for the group. May I ask a question with the hashtag without the membership? Thank you!

And on and on and on and on and on and on and on and on and on and on and on and on and on and on and on and on and on and on and on and on and on and on and on and on and on and on and on and on and on and on and on and on and on.

**Featured Comments**

I joined the community of online teachers this last year and attended the national conference. MTBoS felt like a secret society that I wished to be a part of but didn’t know how to get in.

… my honest-to-goodness first thought about being invited was, “Am I ‘in’ the #MTBoS ‘enough’ to speak about it with these other mathies who seem to be ‘in’ it ‘more’?

This makes me happy. For

monthswhen I first discovered #MTBoS, I hadnoidea what it stood for and felt so left out! And then I had no idea how to talk about it to others. (And usually resorted to “it’s basically math teacher twitter.”)

100% agree…I (found) find #MTBoS “clickish…and therefore offputting…even if/though that isn’t (wasn’t) the intent, it has (had) a mysterious and exclusive feel which made me, a 30 year teacher, feel “out of the loop”

Thank you, I’m on board! #MTBoS confuses me and I even know what it stands for.

This proposal made the rounds among the veterans of, let me try this out, Math Teacher Twitter, and they largely aren’t buying it. No hard feelings on my end. This project has become sharper with feedback from the community.

Here are the four most common responses.

**We are inviting, in particular at Twitter Math Camp.**

So, I'm here, at #tmc17, did speed dating w newbies. They all told me how happy they are that we all are so INCLUSIVE. #mtbos

— Julie (@jreulbach) July 28, 2017

I have no doubt that everyone at Twitter Math Camp who comes within forty feet of Julie or the other organizers will feel warm and welcomed. But TMC hosts only a few hundred math teachers out of millions. What is the best way to invite people into this community who have never sent a tweet? Or who have only watched *other* people tweet? Too many people find our current approach alienating. Check the featured tweets and featured comments above for a sample. If they bother you, what solutions are you thinking about?

**This is their problem, not ours.**

Yeah, it's a pretty secret code that is tough to crack. #MTBoS pic.twitter.com/jHbfOpMH6T

— Meg Craig (@mathymeg07) July 28, 2017

If the alienated people in the featured tweets and comments above don’t burden you, or if you think their lack of comprehension at our hashtag and how to use it is their own problem, don’t let this proposal weigh on you for a second more. And don’t feel any guilt from me about it. This is my project, which doesn’t mean it has to be yours.

**This won’t fix everything.**

Do you want #Mtbos to be called #iteachmath, or do you want MTBoS to BE something else? Just notation, or norms, or this whole thing here?

— Carl Oliver (@carloliwitter) July 28, 2017

Using a different hashtag won’t make everything great. Totally true. I think it’s a necessary step, and an important one since it’s our figurative front door, but it’s insufficient. How can we sufficiently welcome teachers to professional community online? I don’t know, but I’m enjoying that conversation also.

**I won’t use #iteachmath because I don’t teach math.**

I don't teach math, I teach teachers. Can I still tweet with the hashtag? It certainly doesn't include me.

— Glenn Waddell, Jr. (@gwaddellnvhs) July 28, 2017

I’ve already addressed this above, but it’s possible that #iteachmath isn’t ever going to feel right for folks who *aren’t* practicing classroom teachers. That makes a lot of sense to me. I have may have chosen the wrong hashtag for these efforts, but that doesn’t change the reality of all the alienated teachers in the featured tweets and featured comments above. If they weigh on you as they do on me, let me know the solutions you’re thinking about.

I’m not sure if it’ll surprise you to find out that the people most enthusiastic about this proposal have been a) classroom teachers, and b) total strangers to me online. Very few people whose names I recognized. These are people whose ideas may nourish us, people who may need our nourishment also.

So here’s a new proposal: let’s treat “#iteachmath” as the welcoming lobby for new Twitter teachers. When I meet new teachers at conferences or in professional development, I want to recommend they post an idea or a question to a hashtag they’ll find intuitive and inviting. From there, perhaps a bit more emboldened, I hope they’ll venture out towards any number of our other hashtags and communities.

**2017 Jul 29**. Harry O’Malley has written up a really interesting proposal extending these ideas.

**2017 Aug 7**. Interesting to see medium-sized groups of educators with fewer than 30 combined tweets and followers popping up on the #iteachmath hashtag. See: Algebra for All; #NCLargeMath

That single comic has put thousands of students in a position to encounter the power and delight of the coordinate plane. But I’ve never been happier with those experiences than I was when my team at Desmos partnered with the team at CPM to create a lesson we call Pomegraphit.

Here is how Pomegraphit reflects some of the core design principles of the teaching team at Desmos.

**Ask for informal analysis before formal analysis.**

Flip open your textbook to the chapter that introduces the coordinate plane. I’ll wager $5 that the *first* coordinate plane students see includes a grid. Here’s the top Google result for “coordinate plane explanation” for example.

A *gridded* plane is the formal sibling of the *gridless* plane. The gridded plane allows for more power and precision, but a student’s earliest experience plotting two dimensions simultaneously shouldn’t involve precision or even numerical measurement. That can come later. Students should first ask themselves what it means when a point moves up, down, left, right, and, especially, diagonally.

So there isn’t a single numerical coordinate or gridline in Pomegraphit.

**Delay feedback for reflection, especially during concept development activities.**

It seemed impossible for us to offer students any automatic feedback here. After a student graphs her fruit, we have no way of telling her, “Your understanding of the coordinate plane is incomplete.” This is because there is no *right* way to place a fruit. Every answer could be correct. Maybe this student *really* thinks grapes are gross and difficult to eat. We can’t assume here.

So watch this! We *first* asked students to signal tastiness and difficulty using *checkboxes*, a more familiar representation.

*Now* we know the quadrants where we should find each student’s fruit. So when the student then *graphs* her fruit, on the next screen we don’t say, “Your opinions are *wrong*.” We say, “Your graph and your checkboxes *disagree*.”

Then it’s up to *students* to bring those two representations into alignment, bringing their understanding of both representations up to the same level.

**Create objects that promote mathematical conversations between teachers and students.**

Until now, it’s been impossible for me to have one particular conversation about the tasty-easy graph. It’s been impossible for me to ask one particular question about everyone’s graphs, because the answer has been scattered in pieces across everyone’s papers. But when all of your students are using networked devices using some of the best math edtech available, we can collect all of those answers and ask the question I’ve wanted to ask for years:

“What’s the most controversial fruit in the room? How can we find out?”

Is it the lemon?

Or is it the strawberry?

What will it be in your classes? Find out and let us know.

**2017 Jun 16**. Ben Orlin adds several different graphs of his own. Delete his objects and ask your students to choose and graph their own. Then show Ben’s.

Enter Tony Riehl’s cell phone policy, which I love for many reasons, not least of which because it isn’t exclusively a cell phone policy. It’s a *distractions* policy.

What Tony’s “distraction box” does very well:

**It makes the positive statement that “we’re in class to work with as few distractions as possible.”**It isn’t a negative statement about any*particular*distraction. Great mission statement.- Specifically,
**it doesn’t single out cell phones**. The reality is that cell phones are only*one*kind of technology students will bring to school, and digital technology is only*one*distractor out of many. Tony notes that “these items have changed over time, but include fast food toys, bouncy balls, Rubik’s cubes, bobble heads, magic cards, and the hot items now are the fidget cubes and fidget spinners.” **It acknowledges differences between students**. What distracts you might not distract me. My cell phone distracts my learning so it goes in the box. Your cell phone helps you learn so it stays on your desk.**It builds rather than erodes the relationship between teachers and students**. Cell phone policies often encourage teachers to become detectives and students to learn to evade them. None of this does any good for the working relationship between teachers and students. Meanwhile, Tony describes a policy that has “changed the atmosphere of my room,” a policy in which students and teachers are mutually respected and mutually invested.

Read his post. Great, right? How would you build on his work?

**2017 May 26**. Okay, okay, we have a bunch of font critics in the comments thread!

**Featured Tweet**

This is a different approach. The cell phones are in jail. But I admire the incentive for parking your phone.

]]>I saw this in a colleague's class the other day and thought it was pretty clever. pic.twitter.com/F6fEkcyE6H

— Sarah Newton (@sarahnewtonmath) May 25, 2017