**Daniel Willingham**argued in*the Los Angeles Times*that math specialists should teach children math in elementary school instead of elementary teachers. (A thread of unhappy math specialists and elementary teachers on Twitter.)**Ben Orlin**interviews**Matt Enlow**about his lined paper from another dimension, an art and math project that may transport you into a skewed version of your childhood in school.**Gallup**surveyed teachers and students on Creativity in Learning, revealing several disagreements about how often they think students memorize facts, apply their learning to the real world, and discuss topics with no right or wrong answer.**Maya Kosoff**writes Big Calculator: How Texas Instruments Monopolized Math Class, another critical examination of TI’s profit margins and their effect on working class families. (My thread about the article.) Very related: Desmos is now permitted on the Texas end-of-course exam.

Any tips around a young lad with ASD who cannot get his head around estimation? He just cannot see that it would be ‘nearly’ or ‘around’ something when he can clearly work the answer out. My gut says give him something else to do, but if anyone’s come across this in the past….

— Ashley Booth (@MrBoothY6) September 11, 2019

I think it’s possible we should cut the student some slack here.

If the student has all the tools, information, and resources necessary to *calculate* an answer, we should be *excited* to see the student calculate it. Asking students to do anything less than calculate in that situation is to ask them to switch off parts of their brain, to use less than their full capacity as a thinker.

If we treated skills in other disciplines the way we often treat estimation in math …

… we’d ask students to spell words incorrectly before spelling them correctly.

… we’d ask students to recall historical facts incorrectly before recalling them correctly.

Estimation shouldn’t ask students to switch off parts of their brains or use less than their full capacity as thinkers. It should ask them to switch on *new* parts of their brains and *expand* their capacities as thinkers. Estimation tasks should broaden a student’s sense of what counts as math and who counts as a mathematician.

Estimation and calculation should also be mutually supportive in the same way that …

… knowing *roughly* the balance of yeast and sugar in bread supports you when you pour those ingredients *exactly*.

… knowing the *general direction* of your destination supports you when you drive with *turn-by-turn directions*.

… knowing *the general order* of your weekend schedule supports you when you carry out *your precise itinerary*.

Engaging in one aspect of mathematics makes the other easier and more interesting. That’s what Kasmer & Kim (2012) found was true about estimation. When students had a chance to first *predict* the relationship between two quantities it made their later *precise operation* on that relationship easier.

If we want students to develop their ability to estimate, we need to design experiences that don’t just ask them to calculate badly on purpose.

**Create tasks where estimation is the most efficient possible method.**

Take that worksheet above. Give students the same sums but ask them to *order* the sums from least to greatest.

Students may still calculate precisely but there is now a reward for students who estimate using place value as a guide.

**Create tasks where estimation is the only possible method.**

This is the foundation of my 3-Act Task design, where students experience the world in concrete form, without the information that word problems typically provide, without sufficient resources to calculate.

“Estimate the number of coins.” Estimation feels natural here because there isn’t enough information for calculation. Indeed, estimation is the only tool a student can use in this presentation of the context.

Meanwhile, in this presentation of the same task, there is enough information to calculate, which makes estimation feel like calculating badly on purpose.

Estimation isn’t a second-class intellectual citizen. It doesn’t need charity from calculation. It needs teachers who appreciate its value, who can create tasks that help students experience its benefits.

**BTW**

- Here’s a beautiful children’s book on exactly this topic. [via Julia McNamara]
- Contemplate then Calculate is an instructional routine that cleverly blocks calculation by only showing a mathematical structure for a limited time only.

**Featured Comment**

One thing I love about calculus is is proceeds from estimation to exact calculation, and there’s no way to justify the exact calculations without working through the estimation first. We often think of mathematics as a discipline that proceeds deductively from perfect truth to perfect truth, but there are whole swaths of mathematics where the best way forward is to work from an answer whose incorrectness we understand towards an answer whose correctness we don’t yet understand.

I agree with you, but I think it’s interesting to turn your non-math examples into better activities that reflect what we’re trying to do with “good” math estimation tasks.

Mr. K references Fermi problems, which fall really nicely in the category of “tasks where estimation is the *only* possible method.”

]]>At the beginning of the year, I fill four jars around the room. One with M&M’s, one with eraser caps, one with cotton balls, and one with paper clips. They are all allowed a guess for how many in each jar. They enter their answer and their name on a slip of paper and place it in a collection jar. Whenever we come to a question where I want them to estimate first, I remind them of what they did when they first looked at the jar. I don’t tell them how many in each until the winter break – the suspense is awesome. Then in January I start with four new jars.

**CMC-South**. Palm Springs, CA. November 15-16. I’m going to describe how “rich tasks” and “bland tasks” both fail our students. And I’m going to do it with 15 students on a stage in a live lesson demonstration. Let’s gooo! [register]

**CMC-North**. Pacific Grove, CA. December 6-8. I’ll share some of the ways my colleagues and I at Desmos are designing for belonging in math class, specifically how we try to expand the list of who counts as a mathematician and what counts as mathematics. [register]

- One of Idil Abdulkadir’s students asked her “Are there any other people of colour?” about a summer program she was attending. She describes what she thought before she responded, illustrating how much of the work of great, relational teaching takes place invisibly and nearly instantaneously.
- Sarah Schwartz writes in Education Week about the most interesting controversy in math curriculum right now: the publisher suing a parent for allegedly lying about their curriculum.
- Bob Janes describes his implementation of the five practices for orchestrating productive mathematics discussions with technology and without.
- Chatbot company Hubert recommends teachers regularly ask their students three questions to promote the development of their teaching practice.
- Jessica Wynne photographed mathematicians’ blackboards and the results do not disappoint.

Every new school year, Twitter lights up with caregivers who can’t believe they have to buy their students a calculator that’s wildly underpowered and wildly overpriced relative to other consumer electronics.

The Hustle describes Texas Instruments as having “a near-monopoly on graphing calculators for nearly three decades.” That means that some of the students who purchased TI calculators as college students are now purchasing calculators for their *own* kids that look, feel, act and (crucially) cost largely the same. Imagine they were purchasing their kid’s first car and the available cars all looked, felt, acted, and cost largely the same as *their* first car. This cannot go on forever.

As the chief academic officer at Desmos, a competitor of Texas Instruments calculators, I was already familiar with many of The Hustle’s findings. Even still, they illuminated two surprising elements of the Texas Instruments business model.

First, the profit margins.

One analyst placed the cost to produce a TI-84 Plus at around $15-20, meaning TI sells it for a profit margin of nearly 50% — far above the electronics industry’s average margin of 6.7%.

Second, the lobbying.

According to Open Secrets and ProPublica data, Texas Instruments paid lobbyists to hound the Department of Education every year from 2005 to 2009 — right around the time when mobile technology and apps were becoming more of a threat.

Obviously the profits and lobbying are interdependent. Rent-seeking occurs when companies invest profits not into product development but into manipulating regulatory environments to protect market share.

I’m not mad for the sake of Desmos here. What Texas Instruments is doing isn’t sustainable. Consumer tech is getting so good and cheap and our free alternative is getting used so widely that regulations and consumer demand are changing quickly.

Another source told The Hustle that graphing calculator sales have seen a 15% YoY decline in recent years — a trend that free alternatives like Desmos may be at least partially responsible for.

You’ll find our calculators embedded in over half of state-level end-of-course exams in the United States, along with the International Baccalaureate MYP exam, the digital SAT and the digital ACT.

I *am* mad for the sake of kids and families like this, though.

“It basically sucks,” says Marcus Grant, an 11th grader currently taking a pre-calculus course. “It was really expensive for my family. There are cheaper alternatives available, but my teacher makes [the TI calculator] mandatory and there’s no other option.”

Teachers: it was one thing to require plastic graphing calculators calculators when better and cheaper alternatives weren’t available. But it should offend your conscience to see a private company suck 50% profit margins out of the pockets of struggling families for a product that is, by objective measurements, inferior to and more expensive than its competitors.

**BTW**. This is a Twitter-thread-turned-blog-post. If you want to know how teachers justified recommending plastic graphing calculators, you can read my mentions.

Twitterverse, I need your help! I am going to be teaching math next year for the first time. 7th grade! I’ve previously taught ELA & SS. Any teachers/gurus I should follow? Advice/tips for me? Resources you recommend? I know @Desmos is a good resource! #iteachmath #WEAreLakota

— Emily Ladrigan (@EmilyLadrigan) June 28, 2019

I’ll be teaching 7th grade math next year. Switching from Social Studies. Any tips or helpful websites?? #iteachmath

— Ms. Nesmith (@ms_nesmith) July 24, 2019

I saw those tweets and had to sit back and collect myself.

That’s because I know how well I’m served by my knowledge of mathematics, how that knowledge helps me find value in early student thinking, how that knowledge helps me connect and build on thoughts from different students that, *without that knowledge*, might seem totally unrelated.

This isn’t a critique of those two newly drafted math teachers at all. Most of my horror here results from the thought of being drafted to teach *history* after a career teaching *math*. So what can they do?

You’ll find lots of people in those threads recommending resources and curricula. But resources and curricula are only as good as the teacher using them. A developing teacher can make a good resource bad and an expert teacher can make a bad resource good. (This is why John Mason prefers to talk about “rich teaching” instead of “rich tasks.”)

So my *own* advice is for these teachers trained in the humanities to focus on their *teaching*, not the resources or curricula.

Specifically, I hope they’ll resist the idea that math should be taught any differently than the humanities. I hope they’ll resist the idea that only the humanities deal in subjectivity, argumentation, and personal interpretation, while math represents objective, inarguable, abstract truth.

Math is only objective, inarguable, and abstract for questions defined so narrowly they’re almost useless to students, teachers, and the world itself.

In social studies, an analogous question might ask students to recall the date of the Louisiana Purchase or the name of the king who signed the Magna Carta – questions that are so abstracted from their context, so narrowly defined, and so objective that they make no contribution to a student’s ability to think historically.

The National Council for the Social Studies describes what’s necessary for students of social studies:

Students learn to assess the merits of competing arguments, and make reasoned decisions that include consideration of the values within alternative policy recommendations. [..] Through discussions, debates, the use of authentic documents, simulations, research, and other occasions for critical thinking and decision making, students learn to apply value-based reasoning when addressing problems and issues.

All of which rhymes perfectly with recommendations from the National Council of Teachers of Mathematics:

Teaching mathematics with high expectations for all students in mathematical reasoning, sense making, and problem solving invites students to learn to identify assumptions, develop arguments, and make connections within mathematical topics and to other contexts and disciplines.

Teaching math like the humanities asks us to:

**Broaden the scope of the problems we assign**. We can always*narrow*the scope in collaboration with students but the opposite isn’t true. Students don’t have the opportunity to “identify assumptions,” for example, if we pre-assume every detail in the problem.**Focus on mathematical ideas that are big enough to be understood in different ways**. Ask students to make claims that demand to be*argued and interpreted*rather than*evaluated by an authority for correctness*.**Celebrate novel student contributions to mathematics**. History is made every day and so is mathematics. If our students leave our classes this year without understanding that they have had made unique and original contributions to how humans think mathematically, we have defined “mathematics” too narrowly. (For example, someone just decided to call this shape a “golygon.” If*that person*has the right to notice and name things, then so do*your students*.)

Instead of the worksheet above, show your students this video of a pallet of bricks and then immediately hide it.

“Does anybody have a guess about how many bricks we saw up there?”

“Did anybody notice any features about the bricks that might help us figure out *exactly* how many bricks we saw there?”

“Let’s look at the video again. Okay, what’s the most efficient way you can think to figure out the number of bricks.”

“How were *you* thinking about the number of bricks you figured out? What assumptions did you make?”

“Someone else got a different answer from you. How do you think *they* thinking about the number of bricks?”

“Here’s the number of bricks. What’s another question we could ask now?”

These questions rhyme with the kinds of questions you’d hear in a productive, engaging humanities classroom, questions which are no less possible in mathematics!

Humanizing math class means teaching like the humanities. And if you’re joining us from the humanities, please be generous with your pedagogy. We need all of it.

**BTW**: This is my contribution to the Virtual Conference on Humanizing Mathematics, a fantastic learning opportunity hosted by Hema Khodai and Sam Shah through the month of August 2019.

Most recently, they let me think about *time* with some second graders – the youngest kids I’ve ever taught and probably ever met, I can’t be sure – and especially how to tell time on an analog clock.

My goal in these experiences is always to find *areas of agreement* between the teaching of different age groups and different areas of math. Whether I’m learning about time with second graders or about polynomial operations with high schoolers or about teaching with math teachers, I’m asking myself, what’s going on here that crosses all of those boundaries, not one of which is ever drawn as sharply as I first think.

The odds of me stepping on a child go way up, for one.

For another, these students were *inexhaustible*. Their default orientation towards me and my ideas was *rapt engagement* and an earnest, selfless desire to improve my ideas with stories about their friends, their pets, and their families.

My tools for curriculum and instruction were forged by students who communicated to me that “none of this matters” and “I can’t do it even if it did.” Those tools seemed less necessary here. Instead, I needed tools for *harnessing* their energy and I learned lots of them from my friends at Beach Elementary – popsicle sticks for group formation, procedures for dismissing students *gradually* instead of *simultaneously*, *silent* signals for agreement instead of *loud* ones, etc.

Even still, with these second graders, I tried to *problematize* conventions for telling time, just as I would with high school students. I asked students to tell me what bad thing might happen if we didn’t know how to tell time, and they told me about being late, about missing important events, about not knowing when they should fall asleep and accidentally staying awake through the night!

I tried to elicit and build on their early language around time by playing a game of Polygraph: Clocks together. I told them I had picked a secret clock from that array and told them I would answer “yes” or “no” to any question they asked me. Then they played the game with each other on their computers.

One student asked if she could play the game at home, a question which my years of teaching high school students had not prepared me to hear.

I saw in second grade the students I would eventually teach in high school. Students who were anxious, who shrunk from my questions, either wishing to be invisible or having been invisibilized. Other students stretched their hands up on instinct at the end of every question, having decided already that the world is their friend.

Those students weren’t handed those identities in their ninth grade orientation packets. They and their teachers have been cultivating them for *years*!

Rochelle Gutierrez calls teachers “identity workers,” a role I understood better after just an hour teaching young students.

All mathematics teachers are identity workers, regardless of whether they consider themselves as such or not. They contribute to the identities students construct as well as constantly reproduce what mathematics is and how people might relate to it (or not).

I have wanted not to be an identity worker, to just be a *math* worker, because the stakes of identity work are *so high*. (Far better to step on a child’s foot than to step on their sense of their own value.) We wield that power so *poorly*, communicating to students with certain identities at astonishingly early ages – especially our students who identify as Latinx, Black, and Indigenous – that we didn’t construct school and math class for their success.

I have wanted to give up that power over student identities and just teach math, but as Gutierrez points out, students are always learning more than math in math class.

My team and I at Desmos are forging new tools for curriculum and instruction and we’re starting to evaluate our work not just by what those tools teach students about *mathematics* but also by what they teach students about *themselves*.

It isn’t enough for students to use our tools to discover the value of *mathematics*. We want them to discover and feel affirmed in their *own* value, the value of their *peers*, and the value of their *culture*.

We’ve enlisted consultants to support us in that work. We’re developing strategic collaborations with groups who are thoughtful about the intersection of race, identity, and mathematics. A subset of the company currently participates in a book club around Zaretta Hammond’s Culturally Responsive Teaching and the Brain.

Before undertaking that work, I’d tell you that my favorite part of teaching Polygraph with second graders is how deftly it reveals the power of *mathematical* language. Now I’ll tell you my favorite part is how it helps students understand the power of their *own* language.

“Is your clock a new hour?” a second-grade student asked me about my secret clock and before answering “yes” I made sure the class heard me tell that student that they had created something very special there, a very interesting question using language that was uniquely theirs, that was uniquely valuable.

]]>Why did you become a math teacher?

Perhaps you loved math. Perhaps you were good at math, good, at least at the thing you called math then. Friends and family would come to you for help with their homework or studying and you prided yourself not just on explaining the *how* of math’s operations but also the *why* and the *when*, helping others see the purpose and application behind the math.

Helping other people understand and love the math *you* understood and loved – perhaps that sounded like a good way to spend a few decades.

Or perhaps you loved kids. Perhaps even at a young age you were an effective caregiver, and you knew how to care for more than just another person’s tangible needs. You listened, and you made people feel *listened to*. You had an eye for a person’s value and power. You understood where people were in their lives and you understood how the right kind of question or observation could propel them to where they were going to *be*.

Spending a few decades helping people feel heard, helping them unleash and use their tremendous capacity – perhaps you thought that was a worthwhile way to spend what you thought would be the hours between 7AM and 4PM every day.

Or perhaps you loved both math and kids. It’s possible of course that neither of the two previous exemplar teachers will speak fully to the path that brought you to math teaching, although one of them speaks fully to mine. Yet, in my work with math teachers, I find they often draw their professional energy from one source or the other, from math’s ideas or its people.

It took me several frustrated years of math teaching – and years of work with other teachers – to realize that *each* of those energy sources is vital. Neither source is renewable without the other.

If you draw your energy only from mathematics, your students can become abstractions, and interchangeable. You can convince yourself it’s possible to influence *what they know* without care for *who they are*, that it’s possible to treat their *knowledge* as deficient and in need of fixing without risking negative consequences for their *identity*. But students know better. Most of them know what it feels like when the adult in the room positions herself as all-knowing and the students in the room as all-unknowing. A teacher’s love and understanding of mathematics won’t help when students have decided their teacher cares less about them than about numbers and variables, bar models and graphs, precise definitions and deductive arguments.

If you draw your energy only from students, then the day’s mathematics can become interchangeable with any other day’s. Some days it may feel like an act of care to skip students past mathematics they find frustrating, or to skip mathematics altogether some days. But the math you skip one day is foundational for the math another day or another year. Students will have to pay down their frustration later, only then with compound interest. Your love and care for students cannot protect them from the frustration that is often fundamental to learning.

I could tell you that the only solution to this problem of practice is to develop a love of students *and* a love of mathematics. I could relate any number of maxims and slogans that testify to that truth. I could perhaps convince some of you to believe me.

But the maxim I hold most closely right now is that we act ourselves into belief more often than we believe our way into action. So I encourage you more than anything right now to adopt a series of productive *actions* that can reshape your *beliefs*.

Here are five such actions: anticipate, monitor, select, sequence, and connect.

Those actions, initially proposed by Smith and Stein in 2011 and ably illustrated here with classroom videos, teacher testimony, and student work samples, can convert a teacher’s love for math into a love for students and vice versa, to act her way into a belief that math and students both matter.

For teachers who are motivated by a love of students, those five practices invite the teacher to learn more mathematics. The more math teachers know, the easier it is for them to find value in the ways their students think. Their mathematical knowledge enables them to monitor that thinking less for *correctness* and more for *interest*. Would presenting this student’s thinking provoke an *interesting* conversation with the class, whether the circled answer is correct or not? A teacher’s mathematical knowledge enables her to connect one student’s interesting idea to another’s. Her math knowledge helps her connect student thinking together and illustrate for the students the enormous value in their ideas.

For the teachers like me who are motivated by a love of mathematics, teachers who want students to love mathematics as well, those five practices give them a rationale for understanding their students as people. Students are not a blank screen onto which teachers can project and trace out their own knowledge. Meaning is made by the student. It isn’t *transferred* by the teacher. The more teachers love and want to protect interesting mathematical ideas, the more they should want to know the meaning students are making of those ideas. Those five practices have helped me connect student ideas to canonical mathematical ideas, helping students see the value of both.

Neither a love of students nor a love of mathematics can sustain the work of math education on its own. We work with “math students,” a composite of their mathematical ideas and their identities as people. The five practices for orchestrating productive mathematical discussions, and these ideas for putting those practices into practice, offer the actions that can develop and sustain the belief that both math and students matter.

You might think your path into teaching emanated from a love of mathematics, or from a love of students. But it’s the same path. It’s a wider path than you might have thought, one that offers passage to more people and more ideas than you originally thought possible. This book will help you and your students learn to walk it.

]]>It’s my annual tradition to scrape all the handouts and slides from the NCTM annual conference website and put them in a more usable form and location, including a single, massive download link [2.8 GB].

But this year I also found *recordings* from every session.

Not video, nope.

Twitter!

People were recording the sessions on Twitter. People were in sessions the entire conference tweeting key findings, memorable phrases, and impactful slides.

Those tweets were enjoyed in the moment by people who were near their computer during the session. But every session’s tweets were mixed up with every *other* session’s tweets from that same time period. Then they were lost immediately afterwards to a timeline that *doesn’t stop moving*.

**So I re-captured those tweets and attached them to every talk.**

I wrote a script that copied every speaker’s Twitter handle and the time of their session from the program book. Then the script searched Twitter for their handle and “#NCTMSD2019” and – here’s the thing! – captured the results *only* from the time period of their session. Not during their flight to NCTM. Not during the fantastic happy hour after their session either.

So at this site you’ll see the usual listing of speakers and sessions – but you’ll *also* see columns that let you know a) how many attachments those speakers made available and b) how much Twitter coverage their sessions received.

Here’s why this is a blast. Lauren Baucom gave a dynamite talk describing her work supporting the de-tracking of her high school. No handouts or video, but here are 34 tweets to help you connect with her ideas. The same is true for loads of other presenters.

I think a lot about Jere Confrey’s statement that “students are the most underutilized resource in our schools.” Students bring value to our classes – their identities, their aspirations, their early and developing understandings of mathematical ideas – that too often goes uncelebrated, unexplored, and underutilized.

Similarly, thousands of teachers traveled at great length and great expense to San Diego last week. Thousands of brains, bodies, and souls all together in community. What did we *make* of that experience? What do we have to *show* for it? Here’s one thing and I’d like to know more.

If you have been successful in math, by public consensus, you must be smart. If you have been successful in the humanities, you *may* also be smart but we cannot really be sure about that now can we, says public consensus.

In a world where our finest mathematical minds ruined the global economy and perpetuate unequal social outcomes, outcomes most ably critiqued by people trained in the humanities, public consensus is wrong.

This worksheet is worse.

This worksheet associates smartness with a certain *way* of doing math, diminishing *other* ways your students might develop to do the same math. Because there are lots of possible ways to tell time – some new, some old, and some not-yet-invented!

Worse, this worksheet associates smartness with a certain way of doing math that is *culturally defined*, diminishing entire cultures. For example, depending on your location in the world, “2/5/19” and “5/2/19” can refer to the same calendar date. Neither of those ways are “smart” or “dumb.” They work for communication or they don’t.

If I’d like students to learn a certain way of doing math – whether that’s adding numbers a certain way or solving equations a certain way – I need to understand the reasons why we invented those ways of doing math and put students in a position to *experience* those reasons. I also need to be excited – thrilled even! – if students create or adapt their own ways of doing math when they’re having those experiences. Anything less is to diminish their creativity.

If I want students to learn how to communicate mathematically, I need to ask them to *communicate*.

So in this Desmos activity, one student will choose a clock and another student will ask questions to narrow 16 clocks down to 1.

I have no idea what ways students will use, create, or adapt in order to tell time. I will be excited about all of them.

I will also be excited to share with them the ways that lots of cultures use to tell time. When I share those ways, I will be honest that those ways aren’t “smart” any more than they are “moral.” They are merely what one group of people agreed upon to help them get through their day.

So I’d also offer students *this* Desmos activity, which tells students the time using several different cultural conventions, including the one the worksheet calls “smart” above.

Students set the clock and then they see how easy or hard it was for the class to come to consensus using that convention.

Later, we invite students to set the clock themselves and name the time using three different conventions. They make two of them true, one of them a lie, and submit the whole package to the Class Gallery where their classmates try to determine the lie.

The words we use matter. “Real world” matters. “Mistakes” matter. “Smart” matters. Those words have the power to shape student experiences, to extend or withdraw opportunities to learn, to denigrate or elevate students, their cultures, and the ideas they bring to our classes.

Defining smartness narrowly is to define “dumbness” broadly. Instead, we should seek to find smartness as often as possible in as many students as possible.

shoot. i say five fifteen and five forty five routinely. i guess i'm not "smart"

— Ms.B (@MathIsNotScary) March 18, 2019

Write each time "my way."

— corey andreasen (@coreyandreasen) March 19, 2019

There's nothing wrong with familiarizing students with these phrases. How about "Write each time in words in at least two different ways. Tell which way is your favorite."

Re: time as culturally bound

— Idil A (@Idil_A_) March 30, 2019

growing up in Mogadishu, Somalia my mom said they used a 12 hour am/pm system, but it ran 6am-6pm.

Makes a lot of sense living on the equator, sunrise was 0 and sunset was 12.

@ddmeyer more cultural time context. I always have to ask what’s “half seven” and there’s more than one answer. https://t.co/daR81RCAUC

— Calley Connelly (@CalleyMath) March 30, 2019