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NCTM’s Bold and Invigorating Plan for High School Mathematics

NCTM released Catalyzing Change in High School Mathematics last week. I anticipated it would address the gap between the K-8 Common Core State Standards, which feel tightly designed both within and across grades, and the high school standards, which feel loose and shaggy by comparison.

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 expressions and 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!

Free New Desmos Activity: Transformation Golf

[cross-posted to the Desmos blog]

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 then press a button to verify those conjectures. Elsewhere in the activity we remove the play button entirely so students are only able to verify their conjectures through argument and consensus.

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!

Three Opportunities for Free Online Professional Development That Start Today

Mike Flynn, Zak Champagne, and I are very excited to bring you two month-long ShadowCon courses that each begin Tuesday, September 5, 2017. They’re both free.

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.

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.

Co-Authoring Your Classroom

Welcome back to school! Do you remember how this felt?

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:

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.

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

Let’s Retire #MTBoS [as an INTRODUCTION to Math Teacher Twitter]

2017 Jul 31. I have apologized directly to a number of people for aspects of this post. Among others, I apologized to the organizers of Twitter Math Camp (including Lisa Henry, Mary Bourassa, Tina Cardone, James Cleveland, Daniel Forrester, Megan Hayes-Golding, Cortni Muir, Jami Packer, Sam Shah, Glenn Waddell) for posting it during their camp weekend and distracting even a bit from their efforts. Loads of other people stepped up in unofficial, totally voluntary ways to make TMC an awesome, inviting time, and I regret however much I spoiled their efforts.

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

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.

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.

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.

Featured Tweets

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.

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

Angel Martinez:

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.

Cathy Yenca:

… 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 months when I first discovered #MTBoS, I had no idea 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”

Beth Baker:

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

2017 Jul 28.

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.

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.

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.

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’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