Dan, do you ever cry at night when you find out about how teachers are misusing Desmos to teach their students “tricks” to get them through state testing?

This has come up a few times recently. Here is what is happening. The majority of state-level end-of-course exams in the United States include a button that opens up the Desmos Graphing Calculator, which is cool for lots of reasons, maybe the biggest of which is that students can use that calculator for free during the entire year.

Another reason that's cool is the Desmos Graphing Calculator will very quickly produce a graph when you type a function into it and it is very easy to type a function into the Desmos Graphing Calculator.

So if you're preparing students to answer this item from one particular state's EOC exam, you have two common options and then one that begs the question.

Teach students to

**factor trinomials**—for example 3+n-2n^2 = (1+n)(3-2n) in the numerator. Making a one from (1+n)/(1+n) gives us (3-2n) as an equivalent expression.Teach students to

**multiply binomials**—for example multiplying the denominator (1+n) by each of the four options until you get the numerator.Teach students that

**graphs are great.**A graph is a picture of all of the (x,y) pairs that make a statement about numbers true. So if you graph the question and then graph each of the four options, the correct option is the one that produces the same graph. (Mumbles about exceptions.)

Or here.

You can teach students solution methods that involve **substituting variables** or **eliminating variables **or **evaluating points **(toss all four points into the system to see which one is true in both equations) or, again, that **graphs are great**?

The person who DM'd me on Twitter is frustrated that teachers are teaching students that **graphs are great** here and I'm trying—honestly I'm trying—to get upset. But I don't think this is a situation where students are learning *tricks*—a series of steps which produce correct answers while making mathematical ideas less comprehensible.

Teachers are using technology here to make certain mathematical ideas *more* comprehensible to students. Big ideas, I'd argue. Ideas about equivalence and representation, for example. And, yes, they're minimizing others, but I'd argue those ideas are *smaller*. Ideas about symbolic manipulation, for example, even if they’re ideas that the people who wrote this assessment perhaps meant to assess.

You're welcome to share your perspective in the comments here, but this seems like yet another moment where advances in technology are inviting us to reconsider the kinds of mathematics students need for a rich intellectual life. Let's not turn down the invitation.

]]>Has anyone developed a checklist to review when choosing an online platform or app to recommend to teachers?

Over at the National Council of Teachers of Mathematics member forum, a teacher asks for help evaluating education technology.

"How do I know if this edtech is any good?" is the sort of question that was hard to find time and space to answer inside the fog of virtual teaching but seems newly fresh on everyone's mind.

My own rubric for evaluating edtech is very simple. One question tells me most of what I need to know.

**What happens to wrong answers?**

If I have one minute with your product, if you lure me over to your booth at a conference with candy, if someone sends me a link to your website, I'll do the same thing every time:

I'll read the question you're asking and I'll answer it incorrectly.

That's because **the vast majority of a student's time in a math class is spent producing wrong answers.** Which is to say it is spent producing thoughts that are under formalized, loosely gathered, under construction, and which, if pressed with even minimal force, would not produce a correct answer. I want to know what your product does with all of that time and thinking.

**Does edtech ****waste**** wrong answers or ****put them to work****?**

By this standard, you can put most education technology along an axis.

[Disclosure: I work at Desmos, a company that does *quite* well by this standard.]

Generally, a student writes an incorrect answer and the software tells the student they're incorrect. The student often gets another chance to answer. Sometimes they receive an encouraging message, a hint, or another resource. But the prime directive for most edtech is to let the student know they're wrong and get them to be not wrong as soon as possible.

I'm not saying we should protect students from knowing when they're wrong. I'm saying that **every wrong answer is a resource and *** we shouldn't waste it*.

Every wrong answer is the product of thinking that is correct and incorrect, formal and informal, developed and developing. The work of teaching is to help students understand and develop their thinking. Many of your most effective moments in a classroom, then, will be spent:

Scrutinizing wrong answers, not dashing past them.

Helping students identify the parts of their thinking that are correct, not writing it all off as incorrect.

Helping students backtrack through their thoughts to identify the moment they started to over- or under-generalize an idea, not pushing them forward to the next exercise.

Here is a concrete example. Your product is trying to help students learn to plot points using numeric ordered pairs. **Here is a student who is very wrong and very right.** All of the numbers are correct and none of them are in the correct place. The student has reversed the x- and y-coordinates.

When the student presses "Check," the software tells them "Not quite," which is the same message it sends if the student types in *anything—*the number 420, the entire shooting script of *Bee Movie*, a cry for help—*anything*. "Not quite."

The wrong answer is a very smart one and its brilliance is wasted on this edtech.

One approach that puts wrong answers to *work* is to simply *interpret* them and mirror them back to students in a form they can understand and learn from.

For example, in the activity *Sand Dollar Search* from the Desmos Math Curriculum, students try to locate sand dollars on a beach. If a student tries to locate the sand dollar at (3,4) and they type in (4,3), we don't tell them they're wrong. We don't even give them an encouraging message. We just take their thinking very seriously and make a crab pop up at (4,3).

That's all. The student experiences a moment of cognitive dissonance. "The crab did not pop up where I thought it would." Their mind works to resolve it. They try again.

On a later screen, we capture a huge range of student thinking by asking students to describe a new sand dollar in words, a sand dollar that's in a quadrant we have never seen before.

Three of my favorite student responses, all of them both wrong and right:

It is 3 on the y, and -4 on the x

It's 6 spaces away from the sand dollar

The sand dollar is in the up left box, and it's on (4, 2)

In the previous example, our technology interpreted a student's answer automatically. Here, we need the *teacher* to do that. We hope here that the teacher wrings as much value as possible out of these answers.

The teacher can do that by taking each one and interpreting them so literally that the teacher winds up looking for the sand dollar in the wrong place. (For example, by looking “6 spaces away” in the wrong direction.)

In doing so, they'll communicate the value of the student's answer and the need for more precise mathematics. Telling the student "Not quite" communicates none of that.

Whether the technology we're talking about is a curriculum, a computer, or (stretching the definition somewhat) a teacher, we should hope that it makes room for different, interesting right and wrong ideas and helps the entire class find value in those ideas. The best technology makes something easy out of something hard, which is why I'll always give a second look to edtech that takes a student's wrong answers and **puts them to work**.

A team of researchers at

**George Mason University**have developed a framework for thinking about technology centered in equity.**Audrey Watters**doesn’t play around with her “four key questions to ask of edtech.” (Scroll to the bottom.)If you’re tired of the usual professional learning offerings, tune into

**Elham Kazemi’s**ideas about job-embedded professional learning on the most recent Math Teacher Lounge podcast, hosted by**Bethany Lockhart Johnson**and myself.**Robert Talbert**has a great post inviting you to re-consider points-based scoring.Over on Twitter, I asked people “what is the most wasteful sport?” which was a fantastic experience in mathematical modeling.

The comments on my last post where I asked you to help me tune up a teaching moment were really tremendous. Just a high water mark for this newsletter. Can’t wait to get back into a class now.

If you're like me, every lesson is a mixture of success and failure, triumph and regret. I'm offering you here a short video of my teaching where I share one of those triumphs and also a regret, an area where I'd like your help. I'd like to tune it up. A Teacher Tune Up, if you will.

Have a look, please. Then come find me in the comments and help me figure out how to handle similar moments in the future.

**2022 Apr 27**

There’s at least a semester’s worth of knowledge in the comments here, over on YouTube, and in my inbox. This was exhilarating! And also nostalgic! I first started sharing my practice publicly in 2006 and will frequently claim in interviews that the combination of my transparency and the generosity and wisdom of this community helped me turn one year of teaching into two years of teaching growth. I was reminded of that feeling here as well. Thanks, everyone. Some highlights.

Lots of people have noted that I made the class an expensive offer. Decide now, in front of your peers, if you agree or disagree with an argument you maybe don’t really understand yet if you were even fully tuned into it in the first place. Lots of people have recommended a “turn and talk” or a “pair share” protocol to let students clarify their thinking and rehearse their argument first.

Thanks to **Shelly Herbert, Kristin Gray, Katie Derleth, Molly Vokey, Marilyn Burns**, and many others.

Other people here have offered valuable protocols for broadening the ways students can respond to the argument beyond agree / disagree.

**Johanna Langill** summarizes this well and offers a possible solution:

I like to start with “raise your hand if you thought of it in the same way/also used splitting for yours” or “fist to 5 how much sense does it make to you to split the shape like that” and then like Scott maybe a quick- “talk to your partner about why/how he split the shapes, and if you can do that” or maybe “about how his way of doing that connects to your way of figuring it out”

I think you got stuck because the question more about is he right or wrong (ambiguous about is the method right or wrong vs his final answer that the hangar is unbalanced) vs what made him think that/why his method does or doesn’t work.

**Janelle Schorg**

Focus your question around Error Analysis and/or a Misconception so students have something to think about instead of a a yes/no question: For example: Why might someone say B is balanced and identify the error? (Half of 3 is 1.5. Didn't take into account the square on the left side)

**Dave Kung**

What it if were a regular move in class, after seeing a student's work, to ask students to turn and talk with a more specific prompt. As a pair, they have to agree to respond to one of three possibilities:

1. Agree - why do you agree?

2. Disagree - why do you disagree?

3. Ask a specific question.

**Connor Wagner** makes sure everyone can give an answer to his next question.

… one of the moves I will sometimes make is take a poll of the class, "put a thumbs up if you think we are "allowed" to split a square in half; put a thumbs down if you think we can't do this; put a thumb to the side if you are unsure."

**Scott Farrand** calls this the *texture* of a question. Love that.

One thought: Beyond asking whether they think it balances or not, you could ask something with more texture. Three options: does it balance, or is the LHS heavier, or is the RHS heavier? Then send them to talk with their partners, seeking agreement and explanation. The ceiling is a bit higher on that question and the issue of which side is heavier might offer access to some students who have lost the thread.

Loads of people note that I fumbled the ball straight into the ground and all the way to the earth’s core when I ask the kid, “Who told you that you could do that?” only to recognize him freak out and then to freak out myself and quickly clarify that “it’s all good it’s all good little buddy.” Believe me when I say I will go to my grave replaying that moment ha ha.

@ddmeyer I'd love to see you comment or tweet or something a rundown of the infrastructure you used to do this.
from capture to presentation to feedback.

Video source #1 is an iPad on a Swivl. (That’s the far camera.)

Video source #2 is recorded from the webcam on my laptop. I use Screenflow here and a Logitech Brio webcam.

Video source #3 is a screen recording which is also handled by Screenflow.

Then I edited those three clips down to the segment of interest, cutting between them when it felt useful. I exported that as a video file.

Then for my commentary I used a program called Ecamm Live and a little image frame created by designer Kate Lam here at Desmos. Then I’m just yakking and scrubbing the video around in QuickTime Player.

Export then over to Youtube.

]]>I'm not saying my categorizations are perfect here or that any of the categories are *bad*. But as a generalization, it's relatively easy to find professional learning experiences that are big and that require adaptation to your own context. It's harder to find experiences that are brief and closely tied to your context, closely tied to the curriculum you teach, for one example.

I'm interested in "tiny teaching games." Or maybe "micro professional learning modules" if I'm looking for grant funding. Whatever we call them, they'd help prepare teachers to teach *their next lesson*, specifically by helping teachers anticipate and plan a response to the vast array of ideas students bring to class every day.

Okay, classroom video can be a useful “approximation of practice,” right? In our digital middle school math curriculum, we know *the lesson you just taught*. So we might send you two minutes of video of a teacher launching *your next lesson*. We might ask you what you could use or adapt from the video, and then share with you some responses from a larger group of participating educators.

Or check this out. We know how important it is for teachers to understand the objective of their lesson. We could invite you to preview your next lesson, then write what you think the objective of the lesson is in a) a sentence, b) a phrase, c) a word, and then see responses from other educators as well.

Or another. We have hundreds of thousands of student answers to every question in our curriculum. So the day before you start your next unit, you get a notification on your phone inviting you to predict the top five most common answers (which likely includes four *wrong* answers) to an upcoming question. Now you're less surprised by the diverse student thinking you'll encounter tomorrow.

It seems to me that key to all of these *tiny teaching games* are their small time commitment and the words "your next lesson." A lot of professional learning requires adaptation to your local context. These tiny teaching games would not. A lot of professional learning requires travel, time out of the classroom, or both. These would not.

Do these experiences already exist?

*Should* they exist?

Either way, I am having a very hard time not thinking about them.

I'm starting to venture out into the world. Let's hang out! (Responsibly!) Here is what's coming up.

March 24. California Math Framework Webinar. Stanford, CA. (Virtual)

March 26. Central Texas Council of Teachers of Mathematics Spring Conference. (Virtual)

April 1. Matematikbiennalen 2022. Växjö, Sweden.

April 13. Creativity, Collaboration & Communication in Math Class. Avenues School. (Virtual).

We completed our middle school math curriculum this month and are now rolling into Algebra 1. Check out some free lessons, including the artwork of Plenty of Parabolas, and sign up for an upcoming info session.

I love these Math Snacks from my colleagues in the United Kingdom—short videos that pose accessible and expansive mathematical challenges to little and big kids.

This activity from Daniel Wekselgreene had jaws dropping around Desmos. Draw some parentheses anywhere and learn some math.

A very wealthy local district is opening up 200 slots for students from (invariably poorer) neighboring districts, prompting Andrew Simmons to reflect on his commitment to his daughter, his commitment to public education, and the tension between the two.

I'm not sure I'll read anything in 2022 that's as powerful as Audrey Watters' essay, Hope for the Future.

In the last several weeks, a game called Wordle created by a single programmer in his spare time has captured national attention in a way that's pretty uncommon. Wordle has been written up in the New York Times and the Wall Street Journal. My friends and I are less likely right now to trade photos of our kids in our group messages than to discuss Wordle strategy.

It's a very enjoyable experience and I learn more about words by playing, so I've spent the last several weeks wondering, **"What can math class learn from Wordle?"**

Wordle has consumed a large and growing number of my Desmos colleagues as well, particularly the ones who develop the lessons in our middle school math curriculum. So I asked several of them which aspects of Wordle they find most professionally inspiring.

The games and stories that have endured longest in our imagination all have a question at their core that is very easy to explain and very hard to fully answer.

How do I hide?

How do I conquer all the territories?

How do I trap the king?

How do I get there first?

Asked to define a good math problem, Tom Sallee once said, "A good problem reveals its constraints quickly and clearly."

Wordle asks you quickly and clearly: *What is the five-letter mystery word?*

I really love that the rules of the game and the corresponding feedback are simple enough to explain to my 4 year old, but engaging enough to captivate adults of all ages.

In recent years, math teachers have worked very hard both to de-stigmatize wrong answers and to prioritize the journey toward mathematical understanding as much as the destination of a correct answer.

Wordle is wildly successful at both goals. "You're expected to fail on the first attempt," said John Rowe. Here, I found out that I got *none* of the letters correct on the first try, and this is expected.

There isn’t a huge consequence for being wrong and in fact you’re expected to be wrong at least the first few times you try.

Using all six tries is at least as valuable, if not more so, than using just two. Acclaim is given in the [internal Desmos] chat to the fewest tries each day, but sticking with it and pulling it out in the end might result in even more learning than getting it quickly.

Effective feedback attaches meaning to thinking—even and especially incorrect thinking. Expert teachers know that feedback messages like "you're right" and "you're wrong" are less effective than feedback that invites the student who got it right to deepen their thinking and helps the student who got it wrong to see the value in their incorrect answers.

Lots of my colleagues called out Wordle's feedback as particularly effective. Here, for example, I have learned that the mystery word:

contains I, R, and E in other locations,

contains S in exactly that place,

doesn’t contain D at all.

Wordle rewards every thought you bring to it with something new to think about.

Jay Chow appreciated that "the feedback helps guide you to the correct answer."

The feedback is specific to my response and shows me which parts of my response are useful in the solution and which parts of my response are not useful. The feedback leaves me with actionable next steps.

Shira Helft

Even if you get nothing correct, you still have information that helps you move forward.

If you look at games that have occupied humanity for decades and centuries, each of their central questions makes room for lots of different creative ways to succeed. Even questions that seem trivial—for example: *How do you put the basketball in the hoop?*—encourage infinite creative approaches.

The same is true with Wordle. Whenever my colleagues trade their answers, they all have the same mystery word, but no two people reached it through the same route of words.

Lisa Bejarano:

There's room for creativity and many different routes to each solution.

Shira Helft:

There’s something fun about recognizing that people take different paths to the answer using different strategies based on where their brain is at in the moment.

If you don't play Wordle, these tweets probably drive you nuts. Maybe you have muted "Wordle" on your timeline. I only feel sadness for you.

For people who play Wordle, these simple sets of colored emojis tell other players essential and interesting details about your gameplay, without revealing the specific words you attempted or the puzzle's actual answer.

Wordle puts the right amount of my business out there. Enough that it feels like I can contribute something personal, but without risking too much embarrassment. I can share my success, but I don’t have to share the messy details. I can even share my progress in a way that feels like it puts me in community with others, rather than feeling like I’m feeling evaluated.

I like the grid as a strong representation of my path to the correct answer, but stripped of specific info, because it can help me think about strategy and it also captures the emotional experience of solving the problem.

Wordle has created conditions for *social learning* even across time and space. Though my friends and I play Wordle in different cities and at different times, we feel a social connection through the game that is unlike any other digital learning experience I can remember, one that is stronger than even many in-person learning experiences. This is on account of one core and unusual decision the Wordle developer made.

Suzanne von Oy:

I think one of the most genius moves design-wise was to make it

only a single game per day. By doing this, a game that could become addictive and isolating instead brings people together. That right there is the reason it’s flooding Twitter. Everyone is working on the same word on any given day. People want to share results with each other, to talk strategy, and to congratulate each other. I have a Wordle chat with my parents and my sister, and after everyone has completed the day’s word, we tend to share screenshots and talk about the choices we made along the way. It’s very non-competitive and involves lots of encouragement from one another no matter the individual outcome. Quite lovely really.

I'm fascinated by the designer's decision to introduce scarcity (one puzzle per day), and how that decision seems to be impacting people's engagement. I have this sneaking suspicion that if I were allowed to complete as many puzzles as I want I probably would have done dozens in the first few days, and then walked away completely after a bit. There's something lovely about one-a-day pacing.

Sean Sweeney

One puzzle a day keeps people wanting more, but also allows people to chat about the day's puzzle which makes the experience much more social and brings people together, making it more memorable and more exciting to keep going.

If you're someone who designs learning experiences, I hope you'll take Wordle as a challenge.

Can you create a wealth of learning opportunities with only

**a simple prompt**?Can you design the activity and support so that

**everyone learns as much from failure as success**?Can you offer

**feedback that goes beyond "right" and "wrong,"**that helps learners identify everything right about their wrong answers?Can you

**make room for multiple paths to correctness**?Can you offer learners

**a representation of their learning**they can share with other people?

I'm not a disinterested bystander here, of course, but I love the way our middle school math curriculum answers each of these questions.

**Why else do you think Wordle works? What does it have in common with the lessons you've most enjoyed learning or teaching?**

Updates from Twitter people -

@party_shah @ddmeyer @Desmos Wordle is such a great example of how taking a break from a problem you're struggling with can be just what you need to break through and find the solution.

@ddmeyer @Desmos loved this. would also add that it speed isn't a valued factor.
people are given the time to think and challenge themselves.

**2022 Jan 24. **Something crucial IMO is that you can’t get behind in Wordle. Just like you can’t binge on Wordle games, you can’t develop a *backlog* of Wordle games either. “I should really catch up on Wordle,” is something no one has said, which may make it feel easier to pick it back up.

**2022 Feb 2. **Wordle shares a feature with great lessons in that it has a strong sense of core challenge but allows participants to modify it and extend it in interesting and personal ways. My friends and I are currently starting with the same first word … just to try something new and see if it’s fun for us. Great learning experiences offer the same opportunities.

**2022 Feb 13. **Maybe relevant also is Wordle’s required time commitment. It scales with your interest and can be very, very low.

**2022 Feb 15. **Imagine you type a word into Wordle and it shows you a red x if it’s correct and a green check if it’s wrong. That’s all the feedback you get. That’s what it’s like learning math on computers.

Please read and pass along this open letter from

**Steve Leinwand**and a bunch of math education leaders (plus yours truly) calling on our national teachers and supervisors of math education to develop new high school mathematics standards.**Robert Kaplinsky**straps on the Oculus Quest 2 virtual reality goggles and wonders "What will education look like in the virtual reality metaverse?" Y'all can roast me for being wrong about this when we're all bobbing up and down in vats of goo, doing all of life in the metaverse, but I'm pretty pessimistic about the opportunities for math education here. (VR's answers to my four questions about edtech aren't great so far IMO.)**Dr. Shelley Jones**, the president of the Benjamin Banneker Association, has a new book out with co-authors**Drs. Lou Matthews**and**Yolanda Parker**that I'm excited to check out: Engaging in Culturally Relevant Math Tasks.**Allison Hintz**and**Antony Smith**are Mathematizing Children's Literature. Elementary educators get to teach ELA*and*mathematics so I'm always excited about any and all efforts to connect the two disciplines.One more book: mine! Let me announce that I'm working on a book that researches the techniques of successful game designers and storytellers and steals all of them for math educators. It'll be edited by

**Tracy Zager**and published by Stenhouse in early [mumbles deliberately].

How can we resolve this crisis? Who represents the solution here and what is it? Some commentators suggest "teachers" and "better teaching."

For example, in his recent book *Dear Citizen Math*, Karim Ani writes that "[math teachers] may have more influence over the trajectory of democracy than anyone else in the country." In Education Week, mathematician and co-founder of the Institute for Mathematics and Democracy Ismar Volić writes that "the only substantive solution to this problem [an imperiled democracy] is more education aimed at cultivating political numeracy."

Both authors see the same problem—partisan division and political gridlock—and ask *math teachers* to solve it, specifically through better or different lesson plans.

If you're a teacher, it may be tempting to accept the responsibility for the fate of our democracy. Teaching is hard work for low pay, so teachers often accept extra compensation in the form of particular positioning in social and civic life. But this request from both Ani and Volić falls in a long line of requests people make of teachers to solve problems for which **teachers have little responsibility** and over which **teachers have little influence**. I hope teachers politely decline to try to save democracy through their lesson plans, both for their own sake and, ultimately, for the sake of democracy.

Teachers have tremendous influence. There is substantial evidence, for example, that teachers are the largest in-school factor affecting whether or not students learn. Beyond influencing student learning, teachers also influence a student's sense of themselves as capable learners, their sense of belonging in a class, a school, or a discipline, and a host of other socio-emotional outcomes. Anecdotally, adults I meet in my travels frequently communicate to me how a particular teacher many decades earlier communicated a particular message at a particular time that changed the course of their professional or even personal life.

Teachers should claim *those* areas of influence—their school, their classes, their lessons, their students—and reject the others.

They should reject the blame heaped on them by reactionary politicians from both parties whenever some national indicator declines, whenever other countries advance economically, technologically, militarily, or in this case, whenever our democratic institutions are imperiled.

There simply isn't sufficient evidence to suggest a teacher's influence extends to institutions as large as democracy itself. For example, educators are often asked to help close gaps between racial groups. But a Duke University analysis of consumer finance data found that "on average, a black household with a college-educated head has less wealth than a white family whose head did not even obtain a high school diploma." If the influence of education isn't sufficient to democratize wealth, teachers should hesitate before accepting responsibility for democracy itself.

Perhaps teaching hasn't yet had an impact on these large institutions because math teachers haven't yet, in unison, adopted the math lessons Ani and Volić prefer. We'll never know. But we *have* made large-scale nationwide changes to our teaching and curriculum in the recent and distant past, and none of them have corresponded to significant changes in any outcome or institution as large as our democracy.

My first claim is, pretty simply, that teachers should reserve their energy and obligation for areas of the world they *can* influence rather than areas they *can't*.

Both writers ask teachers to save democracy, an institution they obviously care a lot about, so it's surprising how little attention they pay to the people who are responsible for its diminished state, people who are right now stripping it for parts.

For example, Ani describes sitting in a contentious town hall meeting about the Affordable Care Act in 2009 and thinking, "If we could just have this conversation using seventh-grade probabilities, how much more productive would it be?" Volić similarly laments that we don't teach children about "mathematically superior ways to vote, such as ranked-choice methods that incorporate more information about voters’ preferences."

Both Ani and Volić regret the knowledge voters *don't* have, however neither commentator engages much with the competing knowledge voters *do* have or where it came from.

For example, from 2010 to 2014, the Kaiser Family Foundation found that 30–40% of voters believed the Affordable Care Act had a provision for so-called "death panels"—the false idea that the ACA created a government panel to make decisions about an individual's end-of-life care. That one false idea was directly responsible for a great deal of the anger Ani witnessed at his town hall.

But false ideas like death panels didn't emerge organically from the mist. They didn't float through the air and attach to an unused place in your uncle's brain that a teacher should have filled with a lesson on probabilities back in the seventh grade.

Instead, ideas of that sort were developed, disseminated, and coordinated by extremely well-funded organizations, including the health insurance industry and other pro-business groups.

This is what teachers and their lessons on probabilities and ranked-choice voting are up against. The health insurance industry makes billions of dollars of annual profits. The executives and shareholders of the private health insurance industry don't just take their profits and use them to buy yachts and mansions and other nice things. They pool those profits and invest them in lobbyists, media campaigns, political action committees, think tanks, research grants, political candidates, and other investments all designed to create the kind of status quo-preserving political gridlock Ani witnessed at the townhall in 2009.

The health insurance lobby has spent $498 million so far this year to influence the trajectory of our democracy, 30% more than the next largest lobbying sector. If you're a teacher and you're thinking about accepting responsibility for our democracy, you have to ask yourself, "What is my school's investment in my curriculum and instruction and how does it stack up next to $498 million?"

In spite of that massive asymmetry, Ani and Volić identify *teachers* as the people responsible for our coarse debates and frail democracy, rather than the ultra-rich, rather than an exploitative economic system that makes billionaires wealthy at the expense of their workers, rather than campaign finance laws that let them use their wealth to protect and strengthen their dominant position, rather than any of a number of more suitable targets.

In this, Ani and Volić actually find common cause with billionaires, who would love nothing more than for teachers to take responsibility for the problems caused by billionaires.

Teachers *should* teach interesting math lessons, lessons that examine the material, social, and mathematical worlds. They should adopt teaching practices that connect students to their own humanity and the humanity of their classmates. Teachers should recognize that every lesson is an opportunity perhaps not to *save* democracy, but certainly to *model* democracy, to model shared decision-making, shared knowledge construction, shared authority, and the equitable distribution of power and resources with their students. Teachers should do all of that because it is good for their students in the here and now, in the immediate and local, not because any of it will solve climate change or prevent the next pandemic or restore our democracy or reach into any of the areas teaching has never yet been able to reach.

If teachers can do anything to strengthen our democracy directly, I believe they'll do that work *outside* of their classrooms. That work will require an understanding of the ways our economic system concentrates wealth and power in the hands of the few, and the ways the wealthy and powerful work to protect that system. It will require the moral clarity to name an economic system that has allowed the richest 125 households to hoard more wealth than the 64 million poorest households *combined* as unjust and totally busted.

Our economic system distributes the vast wealth of our world to a very small number of people, even though all of us were born into it exactly the same way—naked and crying. Saving democracy will require our country to redistribute wealth from the few to the many and for teachers to realize that they are the single largest professional group within the many. It will be their numbers, their solidarity, and their moral clarity, far more than their lesson plans, that will amount to a teacher’s greatest influence over the trajectory of our democracy and the future of our country.

Where My Cynicism About Education Ends is a must-read from Michael Pershan. He reckons with the limits of his teaching's influence on students and society and emerges on the other side of that reckoning with *more* purpose and power rather than *less*.

The brightest spot of the day was when a student offered a wrong answer. I went from his desk to the front of the room and said to everyone, "Octavio has a really brilliant wrong answer I *have* to share with you."

I don’t know if it was the words “brilliant” and “wrong” right next to each other or if it was the urgency in my voice, but that line visibly scrambled the circuitry of several kids, including the student I named. But tell me this isn't a brilliant and wrong answer:

We were working on Stacking Cups. The student saw that 5 cups was 15 centimeters tall and calculated that 10 cups would be 30 centimeters tall. But it *isn’t*. It’s 22 centimeters tall.

From the front of the room, I trusted a bunch of strangers and some of them seemed to trust me to help all of us figure out why an answer that seemed very correct was actually incorrect. We figured out why, and afterwards, we talked about a version of the situation where, had we stacked cups differently, Octavio's answer would have been *correct*.

Kids are watching what we do and how we teach, more than the words we say or the posters we put on the wall. A Pathway to Equitable Math Instruction:

Though math teachers often tout the phrase “mistakes are expected, respected, inspected, and corrected,” their practices don’t always align. Teachers often treat mistakes as problems by equating them with wrongness, rather than treating them as opportunities for learning—which reinforces the ideas of perfectionism (that students shouldn’t make mistakes) and paternalism (teachers or other experts can and should correct mistakes).

As an exercise in developing a pedagogy of generosity towards student thinking, I highly recommend taking any math question and asking yourself (perhaps with colleagues) "What will be the most common wrong answer to this question? How is it brilliant? What can I do with it next?"

Start yourself with this one I posted to Twitter yesterday where 14% of students who answered it all had the same wrong and brilliant answer.

Several of the responses Twitter teachers proposed were so generous to students, so curious about their thinking, and so attentive to important mathematical ideas, they gave me a literal emotional reaction.

Give the exercise a try and let me know what it does for you.

3D printed chocolate Platonic solids. You could pick any three words from that sentence and I’d insta-click.

**Anna Weltman**is a fantastic author of interesting and accessible math books (I have a special place on my shelf for*This is Not a Maths Book*) and she has a new one out.Twitter teachers are going wild for this Desmos-ified homage to Green Globs from

**Rajeev Raizada**.**Libardo Valencia**describes how he used the Desmos Art Contest (on now!) with his students.I'm 100% sure you know people or students who would enjoy Mathigon's December Puzzle Calendar.

"How do we rid students of this misconception?" math teachers often ask. "Why does this mistake persist!?"

But it's important to first reject the framing of this *incorrect* idea as either a *misconception* or a *mistake*. The prefix "mis" connotes something deviant or poorly formed. But this idea is, in fact, very *sensibly* formed by the student in response to lots of features of their environment, most notably their curriculum and instruction!

Check out how the distributive property is introduced in two different curricula. How can you blame students?

In the human mind, every idea not explicitly forbidden is implicitly allowed. We can wish the process of learning were another way, but it’s this way instead.

In both excerpts above, students see that the distributive property means *you can apply a group operation to individual term*s**. **Both definitions tell you they're talking about *multiplication over addition*, but neither explicitly forbids other operations.

Learners take ideas that we only meant for *one* part of the world and—consciously or unconsciously—try them out in *other* areas. They over-generalize. They ride their bikes through the restaurant drive-thru. They take scissors to the cat’s whiskers. They use a vacuum to rake leaves. This is what humans do when they learn.

Rochelle Gutierrez invites us to stop naming this process as deviant:

… students don’t have misconceptions. They have conceptions. And those conceptions make sense for them,

until they encounter something that no longer works. They are only “misconceptions” when we begin with the expectation that others need to come to our way of thinking or viewing the world.

All of this is an introduction for a slide that captivated me and my Twitter mutuals the other week.

Notice how it offers learners a chance to *find the edges* of the distributive property through examples but also counterexamples. Over-generalization often results when students only experience positive examples. Too many negative counterexamples in sequence, however, and students might wonder, "Wait—does this idea apply *anywhere?*" This slide offers both opportunities.

Teachers develop people and their ideas. That's the work, and success requires us to take both ideas and people very seriously. It requires a default assumption that kids aren't broken and neither are their thoughts. Instead, both are in constant motion, constant change, and the work and pleasure of teaching for me is the chance to witness and support* *that change.

Area Man Who Talks a Lot About Teaching Teaches His First Full Day in >10 Years. A few reflections from yours truly on being a substitute teacher in my local district.

The team at Desmos just released feature improvements that makes it like 5000% easier to give students feedback and share student work.

New episode of my Math Teacher Lounge show with Bethany Lockhart Johnson! We chatted with Fawn Nguyen about problem solving.

“How can/should I teach dividing fractions if not with keep, change, flip?” an early-career teacher asks on Twitter. Loads of people respond, and I’ve got to say: if you’re a pre-service math teacher educator and you aren’t preparing teachers to interact with one another in online spaces like Twitter, those math teachers are going to miss out on some fantastic in-service learning opportunities.

I’ll be expanding on the ideas in this newsletter in a free session for the Make Math Moments Virtual Summit this weekend. You’re invited!

**1. The learning environment invites infinite expressions of thought.**

Consider basketball. There are functionally infinite ways to shoot a basketball.

Choose your spot on the court. Choose your velocity. Choose your angle. Choose your spin. Choose a bank shot or net. Choose choose choose.

**2. The learning environment returns useful information to the learner for ****each**** of those expressions.**

Whatever shot you take—literally any shot—the world around you *immediately* returns volumes and volumes of useful information. Whether or not the ball goes in is one of the most important and interesting pieces of information, sure, but there are others before and beyond that one.

Does the ball travel in line to the hoop? Does it peak where you wanted it to? Does it hit the backboard where you wanted? Does it hit the rim? Does it bounce away left or right? At what *speed* does it bounce away? Did your spin have its intended effect?

There are infinite ways to shoot a basketball, infinite ways to shoot it *successfully,* and even *unsuccessful *shots are interesting.

**3. The learning environment lets you try again.**

The fact that you can try your shot again, often from the same exact spot, adds enormous incentive to learn from your missed shots. You can put your new learning to immediate use.

**By contrast: a lot of math software.**

In a lot of math software, you’re allowed four possible thoughts. If none of them reflects *your *thoughts, you’re unlikely to feel the same agency someone feels when they play a sport.

And after you’ve constrained your thinking to one of the four options, what information does the environment return in response to your thinking? You’re either right or you’re wrong.

It doesn’t matter if we dress the “wrong” feedback up with a growth mindset message or the “right” feedback up with computer confetti. *It isn’t useful feedback.* It’s low information. It doesn’t help me understand “which *parts* of my thinking were right?” or “what should I try *next* time?”

**The work of teaching is to attach meaning to student thinking.**

I’m critiquing math software here because math software is what I think about every day. But it’s worth reflecting on curriculum and pedagogy as well.

If I’m evaluating any learning experience lately—math software, curriculum, or classrooms—I’m asking myself two questions:

How many different students get to express how many different kinds of thinking here?

How many of those students then get something interesting to think about in response to their thinking?

That’s it. The work of teaching is to attach meaning to as many kinds of student thinking as possible. That seems harder in some disciplines than in others and I can’t stop wondering why.

**2021 Oct 6. **Joe Matuch posts a video where he automates IXL exercises and asks, “Is our curriculum automatable?”

Check out the Bite-Sized Data Science Lesson Plan Competition. Submissions are due October 29 and I'm very excited to see the finalists here.

My favorite tweet thread of the last month dives deep into the One Laptop Per Child Project, an initiative that took the hype-per-unit-of-actual-impact ratio to heights previously unseen.

Berkeley's got shirts. I think I need convincing that the trapezoid debate is any more interesting than those deliberately ambiguous arithmetic problems that float around the internet every week. I'm open, but I don't know that debates over definitions are a facet of math I'm excited to foreground.

Math Teacher Lounge, my video series with Bethany Lockhart Johnson, is releasing new episodes—including some brief back-to-school interviews with some of our favorite educators.

Desmos cracked the EdTech Top 40. Encyclopedia Britannica is in our rear-view mirror. Watch out, Wikipedia.

I was homeschooled from kindergarten through eighth grade with a five-hour break in the fourth grade where I begged to be enrolled in the neighborhood public school and then begged to be unenrolled.

My mom and I visited my new teacher after school. Everything checked out. The classroom looked like a classroom, not a dining room, and I was kind of excited about that part maybe. The teacher gave me the day’s homework to take home and complete.

I got home and I got very stuck and then I didn’t go back to school.

I still remember the assignment, decades later. We were supposed to cut out plastic straws and tape them to a sheet in the shape of scalene, isosceles, and equilateral triangles.

These were names I had never seen before, through no one’s fault except the arbitrary ordering of mathematical ideas in two different curricula, but that was that.

**What is (mathematical) truth but a lie agreed upon?**

So much of mathematics rests on the foundation of social convention. Why is the horizontal axis labeled “x” instead of “m.” Why do we plot the horizontal coordinate first. Why do we evaluate arithmetic operations in the order we do.

You can’t take the square root of a negative number until you can. The angles of a triangle always add up to 180° until they don’t. Polygons always have an integer number of sides until they don’t.

It seems very easy, as a teacher, to ask questions that ensure only students who understand a social convention look mathematically smart—like wearing glasses that filter out every color but one.

It seems very easy, as a student, to try to answer those questions, run aground on the social convention, and assume that I am not mathematically smart.

Some #educatorgoals then:

Help students understand that their question “Why is it that way? Why couldn’t it have been another way” is an important and necessary one and has been, in fact, a primary engine for the advancement of the field of mathematics throughout history.

Help students understand other conventions were possible. “Yeah, we could have decided the first coordinate was the

*vertical*coordinate. That would have been just fine. We had to decide on*some*thing though and you and I weren’t in the room.”Ask the questions that gave

*rise*to the need for a social convention.

We developed a coordinate system to describe location precisely, so consider giving students the opportunity to try to describe location precisely *without one*.

We developed names for triangles because certain triangles exhibited certain interesting characteristics we wanted to group together. So consider asking students to look at a bunch of triangles and group them in ways *they* find interesting.

We created social conventions to serve our needs, not the other way around. We created social conventions—scalene, isosceles, etc.—to *facilitate* our brilliance. But social conventions aren’t a *substitute* for human brilliance, nor should we ever let social conventions undermine a student’s appreciation for human brilliance or their sense of *themselves *as brilliant.

Great recommendations from Mylène DiPenta.

... for any convention (i.e. co-ords given in order x,y) teach
- at least one system where the convention is different (i.e. lat, long)
- do the approaches have different pros/cons?
- who got credit?
- who invented a similar or competing system but didn't get credit?
- why?

Is a circle a regular polygon? Look at this beautiful mind out there picking fights with parents at the Minnesota State Fair.

Sarah Carter’s free start-of-the-year resources are getting downloaded left and right. Also—love this idea about creating emergency sub plans as Desmos activities.

Ontario’s Jason To is on a podcast talking about his board’s work de-tracking their math classes.

Rose Lam hosts a great conversation about how much emphasis to give your syllabus at the start of school.

If my adventures learning math through homeschooling interest you, here’s an old post where I describe learning from a proto Khan Academy in the eighth grade.

In my last newsletter, I asked you to share, “How did your journey with math and numbers begin? Where does your enthusiasm, your passion come from?” Some fantastic responses follow.

**Linda L.**

My HS Calculus teacher (1971-1972 school year) turned math class on its head for me. Each day he posed an interesting problem and put us to work in groups to grapple with it, followed by a lively whole-class discussion. I absolutely loved his class - first time I wasn't bored stiff in a math class. I decided to major in math and later to become a math teacher, because I fell in love with math that year and wanted to do for other kids what he did for me.

Almost 50 years later and newly retired, I hope I had some success at that.

50 years! The impact education has on macro-level factors like GDP and life expectancy is questionable, but it’s undeniable that a single teacher in a single year can have a massive impact on the course of a kid’s life.

**Sarah N.** loves math precisely *because of* the opportunities it gave her to challenge social convention.

As a young child I would "sneak" my mom's puzzle books, erase her answers, and then redo the puzzles. I have always enjoyed a challenging puzzle, and I think that just flowed over into math. I'm fascinated by the idea of math concepts growing and changing. Such as the idea that we "can't" divide by zero...

why not?It was once believed that we "can't" take the square root of a negative, yet we do! I like the creative and rebellious side of mathematics.

**Ashley T **describes #familygoals for me.

My mom was (and still is) a math teacher and so math was a natural part of my daily routine (we actually used to do mental math at the dinner table before eating together) so relationships with numbers began early. But as I grew, I actually began to struggle at random times in math throughout my high school and my college career as a math major. Along the way there were always specific teachers encouraging me, supporting and growing me. Realizing this, and how important persistence was to a math mindset, my passion for both teaching and social emotional learning grew. As a teacher that is what I founded our classroom culture on and now it has greatly impacted my work as a math specialist.

My mom, sister (also a math teacher), and I have a group text message that is almost like our family PLC where we chat math and teaching and I just love how we can all be so enthusiastic and bond together over what we do.

**Mark V’s **love for math and numbers evolved and hopped across careers.

]]>I remember in 4th grade crying in the classroom when I raced to take my answers to the teacher to get them checked and having to return, more than once, to redo them. So much for timed math tasks. She didn't do that much after that. I remember competing against John Wambaugh and Barry Moat to see who could do better at the work given us or on tests. We were good at it. But, competing to get some kind of prize, such as a grade, isn't what gave me a love for math. I think the first real eye opener came in college when in Elementary Analysis we derived the Fournier Series, and for the first time, I made the connection how the strain you could measure at an oil well at the surface related to what happens downhole at the pump. Basically, it is differential equations in action.

I was an engineer 20 years, then came to my passion, teaching.

In case you don’t have the same time or inclination to refresh twitter dot com as often as I do, here are some of those interesting back to school reflections and resources.

**Fawn Nguyen** wrote a note to new teachers.

You can measure the students’ enthusiasm for your class quantitatively: time how fast they arrive at your door the next day. :) What teachers do is really hard, and I hope you’ll reach out to people who have your back. Always make family and yourself a priority. The students and the lessons should bring you joy. If they don’t, I hope you’ve saved up to start a vineyard.

**Howie Hua** posted a thread of back-to-school activities.

It's Back-to-School season so here's a short thread of my beginning-of-the-course activities and activities I do consistently throughout the semester:

**Sarah Carter** is one of the craftiest math teachers I follow and was probably more bummed than most by the turn to remote learning last year. She’s clearly back in her element now, thinking about a yearly “Statistics Scrapbook” and back-to-school experiences that are tuned up for in-person learning.

The first week is one of my absolute favorite parts of the school year. I love crafting a fun and engaging experience for students that usually involves a little math and problem solving, too!

**Chanea Bond **wrote What I Wish My White Colleagues Knew, which includes lots of useful advice for White teachers working in multi-racial departments this next school year.

**Alex Shevrin Venet** invites us to reframe the common icebreaker question, “What do you wish your teacher knew?” to “What do you wish your teacher would *do?”*

When we ask students in the first days or weeks of school, “What do you wish your teacher knew?” we’re essentially asking for a disclosure. The question itself implies a secret. It acknowledges that there are things that are hidden between student and teacher, things that aren’t shared for some reason or another. The question asks students to vault over those barriers and share anyway.

**Sarah Strong** turned her Dear Math icebreaker into a Desmos activity as well.

Along similar lines as

**Sarah’s**activity above, this Getting to Know Each Other activity was the most-run Desmos activity last year by a long, long shot. Nothing else came close.Here’s the visualization of prime factors you didn’t know you needed.

**Howie Hua**created a gem of a tweet that helped zillions of people get more clarity on both the relationship between COVID vaccination and reinfection, but also Bayesian probabilities. Next best part was his follow-up tweet putting his viral explainer in the public domain.If I know you in person, I’ve already grabbed you by the arm and shared this Pre-K teacher explaining how to end the pandemic.

On my reading list:

**Steve Leinwand**and**Eric Milou’s**new book, Invigorating High School Math.

Author **Nihshanka Debroy** reached out to interview me about my path in math education and as I answered I found myself pretty curious how the thoughtful folks who subscribe to this newsletter would answer as well. Here’s one of his questions:

How did your journey with math and numbers begin? Where does your enthusiasm, your passion come from?

Hit reply, send me your answer, and I’ll send you mine.

Chanea Bond reached out on Twitter to mention that teachers in predominantly white institutions should also benefit from her post.

]]>The good news is that this next school year will be, in several crucial ways, *exactly* like any other year.

More good news is that while many of the most helpful resources for teachers are finite and nonrenewable, teachers have several resources at their disposal that are both infinite and infinitely renewable.

Time is a resource that is very helpful but finite and nonrenewable, for example. Many teachers had less instructional time with students last year and many of those teachers will try to teach grade-level standards this next year in spite of time lost in the previous grade.

I asked teachers in my last newsletter how they were planning for that task, and teachers *came through*, suggesting several **infinite and renewable resources**, each of which can make teaching easier next year.

Last year, student learning conditions shifted *categorically*—from in-person to remote, synchronous to asynchronous. That required categorically new solutions.

But if you’re all back in the classroom this next year, learning in the same room at the same time, it’s worth wondering if your challenges are in a different *category* from other years or just at a different point on a *continuum.*

Michael Pershan takes the latter perspective and it puts him more at ease:

When I taught at a school with fairly strict tracking for a place of its size, I would always end up with a few of the lowest track classes in my schedule. These were extraordinarily difficult classes to teach, for just a whole host of reasons: their expectation of failure, the challenging behavior, the unaddressed learning accommodations, and shaky understanding of the previous curriculum.

The case I'd make for why we shouldn't freak out about learning loss is because from the perspective of a classroom teacher,

we already are handling situations like it all the time.

Last year’s *categorical shift* asked teachers to devise new technological and pedagogical innovations. This year’s *continuous shift* asks you to do the same work of differentiating your instruction as always, just differentiating across a wider band of student experiences than you have in past years.

Teachers who believe that a positive and strong relationship with students is a medium for social, emotional, *and *mathematical learning will have an easier year than teachers who believe the work of teaching is only to support mathematical learning.

Here’s Jodi Donald in the comments:

What helps my students "catch up" is not so much about the math but my commitment to establish rapport between the students and me and create a safe learning space where kids can talk math, learn from each other, and use mistakes as learning opportunities.

Additionally, your belief that students have lots of mental images and early ideas that you can recruit to help them learn math is going to be even more helpful this year than it has been in previous years. It will require more effort to start with abstract and formal knowledge (like formulas and procedures that students didn’t experience last year) than to start with the sensory and intuitive ideas students were developing all throughout the last school year and then use those ideas to help them develop abstract and formal knowledge.

For example, starting with “Which is steeper?” offers students an entry point and offers you many more resources for instruction than starting with “Calculate the slope.”

A teacher who believes that math comprises a lot of small ideas—each one carrying the same mathematical load as any other, each one an essential prerequisite for some other idea—will experience a lot of unnecessary burden this next year. If math is a house of cards, then every individual card needs your full attention.

However, teachers who understand that math is about a small number of very large ideas will find it much easier to help students connect their previous learning to learning they may have missed last year. In particular, many teachers should be happy to have a) standards that treat math as a story, b) an understanding of the progressions of large ideas across grades, c) a curriculum that emphasizes different storylines in math proportional to their size and importance.

**When it comes to student learning this next year, imagine that students missed an episode of a good T.V. show.**

This hypothetical T.V. show is a good T.V. show, which means that the characters are well-developed. They behave in ways that are interesting, dynamic, sometimes surprising, but ultimately true to their nature.

This also means the story makes sense from one episode to the next. There is an “A” plotline that runs through the entire season. Then “B” plotlines that run across smaller arcs of episodes, and “C” plotlines that often start and end in a single episode. The T.V. show has spent much more time on “A” plotlines than “C” plotlines.

The characters and story are so well-developed that someone can miss an episode, jump into the next, and learn everything they need to know—especially for the “A” plot—from a short “Previously on Math Class” preface to the next episode.

For example, one of the big plotlines in our middle school math curriculum is the different ways we can represent equivalent ratios, which eventually gives rise to a new storyline with linear relationships.

In the Grade 6 episode of our curriculum, a student generates equivalent ratios in Pizza Maker.

In the Grade 7 episode, the student sees what those equivalent and nonequivalent ratios look like on a graph in DinoPops.

In the Grade 8 episode, the student sees graphs that are linear as well as proportional in Turtle Time Trials.

If a student missed *any* of those episodes, teachers can open up a screen from that lesson and offer a short “Previously in Math Class” preface that catches students up on the main characters and their “A” plotlines from the missed episode. That’s only possible if we understand math as a story and understand whether a particular plotline is an A, B, or C plotline.

I won’t trivialize the challenges of this next year. Teachers and students deserve many resources they may not have, time and political leadership perhaps chief among them. Those resources are finite and nonrenewable.

But beliefs are ideas you can cultivate without respect to time. I have developed some of my ideas about math, students, and learning much slower than my peers, and other ideas much faster. You can develop them on your own in a room while reflecting on your day, while reading something like this, while chatting with your colleagues.

Those beliefs are infinite and infinitely renewable, and each one will support your work during this next school year, a year that, in many important ways, will be just like any other.

For more along these lines, check out “The False Construct of Readiness” from Ashley Powell and Kristin Gray at Achieve the Core.

The New York Times is looking for stories about how educators are going to restart learning this next school year.

So is NCTM, with a new call for proposals [pdf]. They’re also publishing a guidance document in a few days.

Achievement for Good is a new project that seeks to create assessments that are culturally affirming, particularly of Black and Latino students. Their leadership team and advisory board make it a project to watch IMO.

RIP Robert (Bob) Moses.

While you’re here, I hope you’ll take a look at our commenters who are writing thoughtfully about their approaches to “opportunity loss.” Leeanne Branham describes a common theme of “weaving” old and new ideas together that we’ll return to.

]]>Having used the [Desmos] curriculum for the first time myself this summer with students who had struggled during the school year, the readiness checks were a great guide to what previous topics needed to be woven in. The curriculum is so inviting that there was 100% zero need to teach a unit of missed material. They could all find a way into that first lesson.

I posted a graph of Desmos usage in this newsletter that was incorrect. It expressed **curriculum usage** as a fraction of **lessons run** divided by **total possible lessons that could have been run**, and it’s come to my attention that my calculation of **total possible lessons** was too large by an amount I’m still sorting out. Please accept my apology for a careless error with numbers and graphs.

At this time last year, teachers were making plans for a *full year* of remote teaching rather than a *single emergency quarter*. Now, those same teachers are making plans for how they’ll teach in the wake of that year of remote teaching.

All kinds of debris bob around in that wake right now, all of it complicating the work of teaching. Students may have experienced the death of family or friends. They may have experienced health crises of their own. They may have experienced poverty in a new or deeper way.

Every teacher I talk to is very stressed about that debris from their year of remote teaching. They are particularly stressed about “learning loss,” something which they are frequently told (by people who I observe are often not classroom teachers) does not exist.

In one sense, the critics of “learning loss” are correct here. Learning is the sort of thing that can become inert and not easily used, but we can take deliberate measures to reconstitute that learning later. So “learning loss” isn’t a particularly accurate description of cognition. Neither does the term do justice to all the learning students *gained* while at home.

But my opinion is that those critics are working much harder to decry the term “learning loss” than they are working to name and address a real phenomenon that teachers *do *have every reason to worry about.

Students did lose *something* last year. We can argue over the value of that thing and its value relative to other things they gained. But every anecdotal source I have tells me that teachers were able to teach many fewer lessons this year than last year.

Yet many school districts are still asking teachers to teach next year as though the students they’re receiving had the same opportunity to learn as they would have in years that weren’t consumed by a world-historic pandemic.

Even if “learning loss” is a misnomer, teachers have every reason to be anxious about that “opportunity loss.” Students lost a lot of opportunities to learn math this last year and teachers would likely benefit from some concrete, specific, actionable suggestions for their preparation.

So let’s offer and trade some ideas together.

**How should teachers teach ideas this year knowing that many of their students will not have experienced related ideas from the previous year?**

Smash the reply button or leave a comment. In a future email, I’ll share some of your answers, some answers from my colleagues at Desmos, and some of my own.

I am obsessed with the question

**Jenna Laib**asked in a recent post, “Do students need to finish every problem?” One key to teaching with problems is knowing when the class has had sufficient experience with the problem to benefit from a conversation about it, which is generally earlier than when the entire class has*finished*the problem.If you’re a parent of small kids, check out this thread of children’s math books. I impulse bought a bunch on Amazon and my kids are loving Baby Goes to Market and Countablock, in particular.

What’s your favorite line from Math Person? High schooler

**Julia Schanen’s**ode to mathematics is by turns heartbreaking and actual-lol-level funny. Personally, “gnawing on a bone” put me on the floor.“Two lines are 2 apart.” Love an interesting tweet-sized problem.

Press the “Anonymize” button in Desmos and your students’ names will change to names of mathematicians. A small team at Desmos recently made large changes to the names of mathematicians that appear. Check out the what and the why.

Watching the Math Teacher Lounge thing and I want @ddmeyer (or really any of those @Desmos people) to come teach to my class

A few weeks ago, Atlanta math coach Lara Metcalf invited me to guest teach a lesson on prisms in a class of deaf geometry students, an experience that helped me understand better than any previous teaching experience the truth that:

What students know.

What students can communicate about what they know.

What I understand about what they're communicating about what they know.

… are very different sets that sometimes only barely overlap:

I had assumed the American Sign Language vocabulary was a subset of spoken English vocabulary. For example, I can speak the word *cat* or use my hands to sign the word *cat*—a direct translation. But ASL is, in fact, an entirely separate language from English and, as with other languages, some spoken English words don’t have a direct translation in ASL and vice versa. Deaf people might then combine signs for other known words, use grammatical structures that don’t exist in English to communicate a word’s meaning, or spell the word out letter by letter.

I had also assumed that ASL was communicated through hands alone—that the static pictures of signs you may have seen are the entire meaning of the word one communicates in ASL. Lara clarified, however, that ASL is communicated through hands moving multidimensionally through space, at different speeds, attached to a person with a face making expressions, the sum of which can endow communication in ASL *more* nuance and meaning than the same words in spoken English. When someone signs the word *cat*, they can communicate more than just the idea of a cat.

“I find English explanations of math concepts can actually be really clunky compared to ASL explanations,” Lara said. “I might argue that it’s a superior language for teaching math because of its visual nature.”

Mathematics is a discipline that *delights* in its nuances and distinctions. (It isn’t just a *triangle,* it’s an *obtuse* triangle. It isn’t just a *number*, it’s a *rational *number.) And my students and I were going to have very different languages for communicating nuance and distinction.

So in the days preceding my lesson on prisms and their volume, I sent my instructional materials to an interpreter Lara recruited so she’d have some familiarity with the nuances I might try to communicate.

I wanted to know the kinds of distinctions my students would make, so I grabbed a bunch of 3-D shapes and asked them to contrast the shapes as "Prism" or "Not Prism" and write about how they were separating the two. This didn't get me as close to understanding their thinking as I had hoped.

Lara told me that deaf kids often suffer from language deprivation since many deaf children are not exposed to an accessible language early in life. While hearing babies are constantly absorbing and learning language just by existing in a world full of spoken words, deaf babies and children often don’t receive the same resources, informally or even formally. With rare exceptions, hearing students receive 12 years of formal language education in the United States that build on the informal language education they received just from living and hearing in the world. Deaf students do not often receive 12 years of ASL language education because they often don’t get the opportunity to learn ASL until later in life.

One consequence of that deprivation is that my deaf students could more accurately express their thinking about prisms in ASL, a language I don’t understand, than they could in written English, a language I do understand. So the students would express their ideas in ASL to their interpreter who would express them verbally to me—a game of telephone with everyone using different languages and having different kinds of knowledge about the subject they're trying to discuss.

I tried to communicate several ideas to the students through that noise:

I loved their thoughts about prisms. Many of the ways they thought about prisms had never occurred to me.

A group of mathematicians have defined prisms in a particular way.

I still loved their thoughts about prisms.

An advantage of having a common definition of prisms is that prisms have particular properties we can use to get stuff done, like these goofballs who constructed the world's largest coffee cup (2,010 gallons, a record they set in 2010). That feat was only possible because the dimensions of prisms relate to their volume in a particular way.

I said throughout the lesson, as much to remind myself as to communicate it to students, "Other mathematicians have defined prisms differently from you folks, but your mind and your eyes are working together to produce some really interesting thinking."

Maybe it's easy to read this and think, "It must be hard to teach students who use a different language from their teacher." But what I'm suggesting is that those are the conditions under which *everyone* teaches.

We're always teaching students whose ideas about mathematics—the patterns they're noticing, their estimations about quantities and shapes, etc.—outstrip their ability to express them. It's perhaps more obvious when students haven’t received adequate instruction in written English or when their primary language is different from ours. But I'm grateful for the reminder that there is *always* more going on in a student's mind than meets our eyes or ears.

*Much gratitude to Lara Metcalf for her feedback before and after my substitute teaching experience, and for her feedback on drafts of this post. All errors are my own.*

Desmos released a free activity based on the classic Bridges of Konigsberg problem. The best part IMO is it has a Challenge Creator so students can create their own problems themselves. Feel free to use it as the year winds down or tuck it away for a warm-up when the new year begins.

Robert Banks IV continues to create some of the most fantastic and instructionally useful math visuals in the game. Head to this link and change the f(x) expression and watch the result. [via the Desmos Educators Facebook group]

Given the general horrific nature of this last school year, I appreciated the final exam Drew Lewis offered his students.

We saw huge growth in the use of our free activity creation tool this last year and people frequently asked us for resources for advanced usage. Better late than never: we’re now developing a multi-week, cohort-based online course called Activity Builder Academy. We’ll kick it off late summer / early fall. Sign up for more information.

Ben Orlin wrote a very good guide for math teachers joining Twitter.

Idil Abdulkadir in the comments of my last email:

]]>If pandemic schooling has taught us anything it's that schools + teachers are tasked with all manner of things that the state could do, should do, but actively decides not to do. I've been having this conversation about letting things fail & not saving broken systems on and off all year - and it's a hard sell for many teachers (including me, until recently). Teacher culture is making impossible, unreasonable things work and being called a hero if you're lucky.

Justin Reich and Rachel Slama offered me the opportunity to hang with some absolute heroes and also the opportunity to distill all my current ideas and questions about teacher learning into **four pictures** and **five minutes**.

Let me know if any of this connects to your own thinking about *supporting teachers*, or about *being supported as a teacher. *Let the record show that I often experience my largest leaps in understanding when strangers send me emails.

Here are my prepared remarks.

At Desmos, we’re convinced by a variety of research, personal experience, and trusted testimony that the most productive change a math teacher can make is to develop an asset orientation toward their students’ thinking.

From “What are this student’s *misconceptions* about an idea?” to “What are this student’s *conceptions* about the idea?”

From “Is this answer *correct*?” to “How is this answer *useful*?”

From “This student is broken—how can I fix them?” to “This student is brilliant—how can I invite, celebrate, and develop that brilliance?” borrowing from Dr. Danny Martin.

A lot of my colleagues and I spent years working with teachers on these changes, but we ran aground constantly on curriculum.

A curriculum will never expressly prohibit student brilliance, but lots of curricula only make room for expressions of grownup brilliance—the brilliance of the teacher or the textbook author, for example.

So supporting teachers meant building our own curriculum, a fork of the excellent curriculum from Illustrative Mathematics, optimized for our technology, design, and pedagogy.

Our curriculum is in a really useful position for supporting teacher development:

It’s rich—by which I mean it invites a broad array of student brilliance: noticings, wonderings, sketches, verbal arguments, equations, graphs, and text responses to interesting questions. Ideas that we couldn’t and wouldn’t want to label “correct” or “incorrect.”

It’s also digital. Students express a lot of that brilliance in a form that’s visible to teachers and the people who support teachers, including site-based coaches and coaches at Desmos. Teachers are able to enact some sophisticated routines through the platform like selecting student ideas, sequencing them, and presenting them. We have visibility into how teachers are taking up those practices.

Lots of curricula is one or the other. The IM curriculum is extremely rich but print-first. Lots of digital curricula only make room for student brilliance in the form of multiple choice responses and numbers—stuff that computers often grade as correct or incorrect.

The visibility we have into student ideas and teacher practices has expanded our options for teacher professional development.

We offer teachers a large kick-off session at the start of the school year to get us lined up on some of our hopes and dreams for the year and to get us excited for the work. This session is high intensity and one-time only, and we know the energy and learning dissipates quickly.

So we’re also developing and researching interventions that are lower intensity but continuous. Frequent booster shots if you’ll forgive the timing of that metaphor.

For example, teachers are always in their email. Let’s use that as a medium for professional learning. We know what lesson a teacher taught yesterday, so we send that teacher an email previewing the next lesson through the lens of one of the three themes, which we spiral throughout the year: inviting student brilliance, celebrating it, and developing it, with pedagogical techniques attached.

We offer those ideas to teachers not in a one-off session abstracted from their daily work, but rather as close to the teacher’s practice as we can get: “On this screen, here’s a great moment to tell students, ‘Look, you can’t break mathematics.’ ” These preview emails give teachers essential and generalizable teaching strategies and give us email open rates that email marketers can only dream about.

Big questions now:

What are the mid-range interventions?

*Medium frequency. Medium intensity.*How do we use this digital platform for rich student thinking to nourish professional learning community meetings or support site-based coaches?

*Don’t know.*

Also, we’re sitting on a *nice* pile of data. If you’re a researcher looking for interesting collaborators, let me know at dan@desmos.com.

The bad arithmetic problem that’s designed to go viral went viral again this week. I’m not going to fault your average internet user for passing along a bit of math that’s designed to excite the world’s passion for symbolic pedantry. But if you think math is about more than symbolic pedantry, well, please don’t share the bad arithmetic problem.

I’m obsessed with broken graphs. Graphs that reflect some kind of discontinuity or break in the social order itself. Over at Luke O’Neill’s Substack, I saw this grade A broken graph that indicates Americans are diagnosed with cancer at much greater rates at age 65. Over on Twitter, I asked people why.

Enjoy this time-lapse video of the introduction and disappearance of the Blockbuster Video franchises in the United States.

Education Doesn’t Work is the provocative title of a long essay that invites you to ask yourself, “What do I hope education will do? Is there any evidence it can do that?” If you hope education will bring about economic equality, for example, the evidence invites you to invest your efforts elsewhere.

Here is a quote I haven’t gone a week without thinking about since I first read it. “Everyone is mathematically smart as a result of living in the world.”

In the most recent episode of Math Teacher Lounge, Bethany Lockhart Johnson and I interviewed UCLA math education professor Megan Franke. She shared some thoughts about “mastery” that you should really read.

[“Mastery” often means that] if you master addition and subtraction you can move on and do other mathematical work, but you can’t move on until you’ve mastered that. What that does is sort students between who’s mastered and who hasn’t, and it keeps young people like this young person who may have said “twenty-ten” from being able to do more interesting mathematics. We say, “Well, we’ve got to practice the number sequence first. We can’t let you solve any problems that go into the twenties. We can’t let you . . . ” when actually letting you go farther is going to let you explore the idea of the number system better than if I stopped you. This idea of mastery gets us into some trouble because it keeps young people from exploring their mathematical ideas.

Check out the rest of the conversation, which is short and amazing, at mathteacherlounge.com. An idea that struck me by the end is that, for teachers, the most profound *professional* transformations require profound *personal* transformations as well. That raises the stakes on this work in ways I didn’t understand as a new teacher. As someone who has never successfully maintained any kind of separation between the personal and professional areas of my life, I wouldn’t have it any other way.

“Results show that essays have a stronger correlation to reported household income than SAT scores.” This is the study that everyone in education research is talking about right now.

“No tax exemptions for universities with selective admission processes,” is probably the take on the study I found most interesting.

Get a spot at the free Amplify STEM Forum next week. Lots of great speakers, and I’m excited to contribute a new talk called “Math Without Mistakes.”

If you like the Notice and Wonder math routine in

*principle*but struggle to structure that work in*practice*, check out this modification from Yorkville High School math teacher Kellie Stilson.Berkeley High School math student Veronika Price made a beautiful hummingbird out of hundreds of mathematical expressions (see the image at the top of this email!) and I made a fancam video breaking the whole graph down.

Our team at Desmos did something very similar this last month.

✅ Stole a time machine.

✅ Traveled decades into the future.

❌ Brought a sports almanac back to the present. (We are both too nerdy and not mercenary enough for this.)

✅ Instead, we brought back the best middle school math curriculum we could find in the future.

You can preview all of it right now, no hurdles to clear. Much of it is also available to try out for free with students. This curriculum builds on and enhances the skillful work of the Illustrative Mathematics curriculum team.

My current favorite sequence of lessons teaches eighth grade students about equations using the context of hangar diagrams. It starts by inviting students to play and experiment. (Try it!)

Students then learn that a point in the plane has *meaning*—that *one* point defines *two* pieces of information: the weight of the triangle and the weight of the circle. (Try it!)

Then students learn that a point's position on one line, two lines, or zero lines *also* has meaning. Every point in the plane represents a different pair of hangars. (Try it!)

I realize that interactive digital experiences in math class aren't all that rare. What's uncommon here is the room we've made for both students *and* teachers to be creative in their work.

For *students*, we ask questions that have more than one right answer and where even the *wrong* answers are interesting. We don't try to *fix* the wrong answers. They’re *interesting*, and worth talking about as a class.

So we pass all of those answers on to their teacher, who mixes the responses together in the giant mixing bowl we call the Teacher Dashboard, sprinkling in some of her *own* knowledge, and creating learning experiences that are far beyond the ability of computers to replicate both now and even in the future. (Take our word on this—we were there!)

For me, this is the team-est of team efforts and also the best embodiment of ideas that have excited me for close to 20 years now. It’s the best thing I’ve ever had a hand in making, and I hope you will check it out.

**What Else**

I got my substitute teaching certification here in Oakland, CA, so I could get back in the classroom more often and maybe give local teachers a dang break. Here I talk about how my first sub period went.

My favorite recent graph is this scatter plot of income perception vs. reality.

David Bowie called out the art establishment’s elevation of genius as a virtue, and the mathematics establishment starts shuffling its feet awkwardly, looking elsewhere.

I co-authored the latest draft of The Desmos Guide to Building Great (Digital) Math Activities with my colleagues Faith Moynihan and Lisa Bejarano. Five principles that inform everything the curriculum team makes.

Extremely good: Geoff Krall on what comes after the

*5 Practices for Orchestrating Productive Mathematics Discussions*.

Here is a question we might ask math students: what is this coordinate?

Let’s say a student types in (5, 4), a very thoughtful wrong answer. (“Wrong and brilliant,” one might say.) Here are several ways a computer might react to that wrong answer.

**1. “You’re wrong.”**

This is the most common way computers respond to a student’s idea. But (4, 5) receives the same feedback as answers like (1000, 1000) or “idk,” even though (4, 5) arguably involves a lot more thought from the student and a lot more of their sense of themselves as a mathematician.

This feedback says all of those ideas are the same kind of wrong.

**2. “You’re wrong, but it’s okay.”**

The shortcoming of evaluative feedback (these binary judgments of “right” and “wrong”) isn’t *just* that it isn’t *nice* enough or that it neglects a student’s emotional state. It’s that *it doesn’t attach enough meaning to the student’s thinking*. The prime directive of feedback is, per Dylan Wiliam, to “cause more thinking.” Evaluative feedback fails that directive because it doesn’t attach sufficient meaning to a student’s thought to cause more thinking.

**3. “You’re wrong, and here’s why.”**

It’s tempting to write down a list of all possible reasons a student might have given different wrong answers, and then respond to each one conditionally. For example here, we might program the computer to say, “Did you switch your coordinates?”

Certainly, this makes an attempt at attaching meaning to a student’s thinking that the other examples so far have not. But the meaning is often an *expert’s* meaning and attaches only loosely to the novice’s. The student may have to work as hard to *understand* the feedback (the word “coordinate” may be new, for example) as to *use* it.

Alternately, we can ask computers to clear their throats a bit and say, “Let me see if I understand you here. Is *this* what you meant?”

We make no assumption that the student understands what the problem is asking, or that we understand why the student gave their answer. We just attach as much meaning as we can to the student’s thinking in a world that’s familiar to them.

**“How can I attach ****more**** meaning to a student’s thought?”**

This animation, for example, attaches the fact that the relationship to the origin has horizontal and vertical components. We trust students to make sense of what they’re seeing. Then we give them an an opportunity to use that new sense to try again.

This “interpretive” feedback is the kind we use most frequently in our Desmos curriculum, and it’s often easier to build than the evaluative feedback, which requires images, conditionality, and more programming.

Honestly, “programming” isn’t even the right word to describe what we’re doing here.

We’re *building worlds*. I’m not overstating the matter. Educators build worlds in the same way that game developers and storytellers build worlds.

That world here is called “the coordinate plane,” a world we built in a computer. But even more often, the world we build is a physical or a video classroom, and the question, “How can I attach *more* meaning to a student’s thought?” is a great question in each of those worlds. Whenever you receive a student’s thought and tell them what interests you about it, or what it makes you wonder, or you ask the class if anyone has any questions about that thought, or you connect it to another student’s thought, *you are attaching meaning to that student’s thinking*.

Every time you work to attach meaning to student thinking, you help students learn more math and you help them learn about themselves as mathematical thinkers. You help them understand, implicitly, that their thoughts are *valuable*. And if students become *habituated* to that feeling, they might just come to understand that they are valuable *themselves*, as students, as thinkers, and as people.

**BTW**. If you’d like to learn how to make this kind of feedback, check out this segment on last week’s #DesmosLive. it took four lines of programming using Computation Layer in Desmos Activity Builder.

**BTW**. I posted this in the form a question on Twitter where it started a lot of discussion. Two people made very popular suggestions for *different* ways to attach meaning to student thought here.

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