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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