Here is what it takes to make effective education technology:

**The technologists on your team need to understand the nature of learning in schools and the educators need to understand how software gets made.**

Neither group will become an expert in the other’s domain. But once they cross a certain threshold of empathy and understanding, they can start to make truly wild stuff together.

The technologists who once assumed learning is something like performing write operations on a database now understand that those write operations are contingent on circumstances like teacher skill, the social environment of the classroom, the school calendar, etc. The teachers who at one point might have made new feature requests for every new day of the year now understand that constraints are liberating, that the constraints of a platform’s architecture and user interface free its users to start *using* a platform rather than *learning* it.

At that point of empathy and understanding, the educators and technologists start to make products that are *multiplicative* of their skills, not just *additive*.

A major conceptual leap for the technologists on an edtech team is to understand Ed Begle’s maxim, that “... education is much more complicated than you expected, even though you expected it to be more complicated than you expected.”

Technologists need to understand that the experiences they have learning in front of their laptops during their workdays—typing questions into Stack Overflow; watching explanatory YouTube videos; perhaps even riffing with some AI chatbot—while powerful and legitimate have very little in common with the work of learning in schools.

Here is one of ∞ examples of what I mean:

This is Jalah Bryant of Evergreen Public Schools in Vancouver, WA, shared with permission. She makes maybe a dozen key instructional decisions in the span of 44 seconds, some of which include:

Offering some direct instruction about one student’s work.

Noticing another student’s head is down

*while she’s concluding that first instructional moment*.Speculating on why that student’s head is down.

Making a plan for intervention.

Asking the student about their situation.

Suggesting a solution.

Reiterating high expectations for mathematical work.

Realizing the implications of

*that*move on the student’s deskmate.Suggesting a solution for

*that*student.

The way she’s still wrapping up one interaction while planning her next one reminds me of the professional basketball players who have a complete and current inventory of all ten players on the court at all times, looking one direction while whipping a pass through traffic in another. It’s incredible.

That interaction between the teacher and student both *precedes* and *determines* any learning about the day’s subject! Their interaction isn’t *about* math per se. But the teacher’s conduct within it—is she inquisitive or presumptuous? is she empathetic or punitive? does she get what’s *actually* going on?–is *highly* predictive of how much math the student will learn.

Students are learning constantly in classrooms. It’s just that what they’re learning isn’t always explicitly about the subject matter. They’re learning about the world, about grownups, about power, and about themselves. When education technologists misunderstand or are oblivious to that complexity, they lower the odds dramatically that they’ll successfully build a product that makes a difference in a teacher’s teaching or a kid’s learning. When education technologists appreciate and learn to work *within* the complexity of classroom teaching, when educators and technologists enlist one another in *partnership*, well, okay, the odds are still very long but I like them a lot better.

**BTW**

Here’s a recently released study about our curriculum: “The Effect of Desmos Math Curriculum on Middle School Mathematics Achievement in Nine States”

FWIW I think Bryant offers a useful illustration of Deborah Ball’s concept of “discretionary spaces,” which you can watch more about in this keynote.

Here’s a recording of a webinar I gave last week called “What Amazing Math Looks Like.”

Would love to hear what #iteachmath / #mtbos thinks about these responses. Students were SUPPOSED to use the distance formula, but look at the response in the pic. How do you grade?????

Nick posed a teaching dilemma which elicited ~100 replies on Twitter, a couple of responses *off* of Twitter (including the one you’re reading), and saw several of the replies spin off lengthy conversations of their own. (Like this one about students who are reluctant to show their work.)

So first thing: I wanted to marvel with you at that professional learning experience.

Second, I wanted to offer a response to Nick’s question, which is that, if you have to ask the question, “should I give full credit?” the answer is almost always “yes.”

This is simply an application of Blackstone’s 18th century legal doctrine to math class: “it is better that ten students get full credit who don’t deserve it, than one student who deserves full credit doesn’t.”

Look at this count of Google results and tell me we should do otherwise.

Loads more students feel bad at math than other disciplines—or at least they’re more inclined to write about it on the internet.

I’ll wager that if we asked students which of their classes feels most concerned with rightness and wrongness, most concerned with precision, and least concerned with their personal subjectivity, the majority of them would name “math.”

It’ll be the project of multiple lifetimes to undo those perceptions. And by giving Nick’s student less than full credit, we’d be communicating that, “Not only must you find the correct answer, you must find it in the correct *way*.” It’s too much, and I encourage us to risk letting the guilty go free in these situations.

**BTW**

As an exercise for the reader, what would an assessment item look like that more explicitly looked into a student’s understanding of the distance formula?

If you’re interested in hanging with me and learning more about the kinds of curriculum and instruction we’ve been developing at Amplify, I and some of my buddies are offering a webinar series that kicks off this next week.

There is a fundamental question in edtech, one that’s rarely asked out loud even though every edtech company has an answer. **What are the humans good for and what is the tech good for?** And by humans I mean both the teachers and students.

One answer to that question has been dominant for decades and is very well represented by Khan Academy. That answer says that what the computers are good for is for instantaneously evaluating certain forms of student thinking; for instantaneously serving up a learning resource if the student is evaluated as wrong, like a video, a worked example, feedback, a nudge; and, if the student is evaluated as right, for instantaneously serving up a star or some other kind of positive feedback.

What are teachers good for? The teachers are good for backup. They come along to fill in whatever needs the computer can’t meet.

What are the students good for? The answer, almost explicitly in these models, is “not much.” This edtech model sees other students as inhibitors of your learning, which is why they call it “personalized learning” rather than “socialized learning.”

The costs of this model are becoming more broadly understood. For example, the Gates Foundation has been studying personalized learning nationwide since 2015 and has found that students in personalized learning schools feel diminished feelings of belonging and safety than students in the comparison group.

My team and I at Amplify are working on a different set of answers to the question of what humans and computers are good for. In short:

Students are useful because they have lots of early mathematical ideas, ideas that are not very well captured by multiple choice items, ideas they need to offer for the sake of their learning and the learning of their classmates.

Teachers are useful because they can recognize the value of those early ideas and build on them in effective ways.

And computers are good for supporting that interaction. They aren’t backup. They aren’t primary. They’re partners. They can elicit that early thinking. They can collect it for the teacher. They can connect student thinking and teacher thinking together. This is a different kind of answer to the fundamental edtech question—a centaur model. And a little abstract. So here’s what I mean.

In this lesson, which is pretty typical of our curriculum, the computer shows the student something that doesn’t feel particularly “mathy” at first—a person getting shot out of a cannon at a carnival—which is what helps elicit that early, concrete thinking from students

.We ask students to sketch the graph of height over time here and students graph lots of different things. Straight up. Straight down. Not a lot of precision yet.

What do we do with this thinking? We don’t evaluate it as right or wrong because **early student thinking is always a mix of both**. Instead, we *interpret* student thinking back into the context. Here is the cannon person you meant to graph and here is the cannon person you graphed. Students can revise and resubmit.

This is nice but I want to suggest that where this edtech model hits a new gear is when we get the teacher involved. We give teachers a view of this early student thinking in real time. Here the teacher has lots of options.

They can say, “Everyone stop what you’re doing. Check this out. This is so interesting.” And pull one student graph up. “What are two things you like about this graph? What are two things you’d change? Talk to your neighbor and then I’ll share some of my own thoughts.”

Or they can pull up *two* student graphs and say, “How are these two graphs the same? How are they different? Talk to your neighbor and then I’ll share some of my thoughts.”

I was on this screen not long ago in a class in San Jose, CA, where I saw one particular student response that surprised me.

Who is this kid?

I want to offer the possibility that this is a protest. This is a student who maybe feels a low sense of belonging or engagement in math class, a student who maybe feels stupid and is going to make me, the teacher, feel stupid as well.

So what does it do for that student to hear the teacher say, “Everyone stop what you’re doing. Check this out. This is so interesting. **What do you think will happen when we press play on this graph?** Talk to your neighbor and then I’ll share some of my thoughts.”

What did that student learn about mathematics here? What did the *class* learn about mathematics here, and about their classmates?

These two answers to the fundamental question of edtech can complement one another to a limited degree but they have many more differences than similarities. And those differences result in students learning very different ideas about mathematics and about themselves and their classmates as math learners.

**2023 Mar 16. **I corrected the RAND study link.

One group of educators has a very clear answer here which is, roughly, “Just tell ‘em! How is this even a question?” Some will emphasize Barak Rosenshine’s Principles of Instruction, which recommends you a) break the stuff you know into small steps, b) model those steps, and then c) ask students to practice those steps.

What’s great about this answer is that it is quite clear and also quite accessible to all the teachers who are wondering what to do with the stuff they know.

Another group of educators got together, formed an organization called the National Council of Teachers of Mathematics and occasionally publishes documents that have a lot of influence on the field of math education in the United States. For example, NCTM published *Principles to Actions* in 2014, a document with eight recommendations for teaching, all of which have a lot of wisdom. However, *not one of them* offers a clear or accessible answer to the question, “When do I get to tell students the stuff I know?” Check them out.

Groups that speak clearly and accessibly about the value of *student *knowledge need a perspective that is similarly clear and accessible about the value of *teacher* knowledge.

Here’s Schwartz and Bransford writing about the limitations of the “just tell ‘em!” model and offering a perspective on how teachers can create “a time for telling” (1998, p. 511):

Distinctions can seem obvious to experts who, therefore, do not bother to illuminate them. Moreover, the expert's ability to discern is often tacit and, hence, goes unrecognized. As a result, neither instructor nor students may recognize that students have missed important distinctions.

Even when there is an attempt to help students differentiate, the task can be difficult when novices are simply told about distinctions they should make. It is relatively easy to tell a distinction to someone, if that person shares the same set of experiences. However, with respect to the content of instruction,

shared experiences are exactly what novices and experts are missing.

For explanations to be effective, teachers and students need a shared set of experiences to talk about.

These conversations are often too abstract so I want to offer a short excerpt of classroom video, shared with permission.

In this video, Liz Clark-Garvey shares her own knowledge of unit rates but does so only after having used an interactive activity to create a shared set of experiences with her students.

Have a look:

It’s worth noting what the teacher does with her explanation:

She reminds students of previous learning. (A number talk.)

She models several example rows from the table.

She responds to a student’s hypothesis by interpreting it through ideas offered by two students, Gael & Zoli. (This is an absolutely breathtaking moment IMO.)

She summarizes the learning of the day.

She invites students to try another example on their own.

Schwartz and Bransford’s research suggests the teacher’s students have learned more because of their shared experiences than they would have if the teacher had shared her knowledge straight away.

Moreover, **students are always learning more than math in math class**. I invite you to wonder what the teacher’s students have learned about their capacity to think valuable mathematical thoughts, about their relationship to power, about their value in that class, to have heard their teacher speak about *the* *experiences they share*, rather than just about *the stuff she herself knows*.

A durable finding dating back zillions (sixty) years and replicated in different forms in different studies is that **a student’s ability to learn new stuff depends on the stuff they know now**. It winds up being an area of rare agreement among educational partisans who otherwise disagree about most things.

One of the biggest challenges here is **convincing teachers who are very aware of the math ****they know**** that the math ****their students know**** is more interesting and more important**.

At a session last week in Santa Cruz County, CA, I tried to develop that interest in a systematic way.

We split into groups.

We looked at pre-assessment questions for an upcoming unit.

We predicted the most common answers to several questions among a national sample of Desmos Math students—awarding ourselves zero points for knowing the correct answer and one point for predicting each of the most common incorrect answers.

We revealed the answers and tallied the scores.

The energy was strong and the conversation was tightly focused on student thinking and all the interesting detours students take en route to a correct answer. One teacher would suggest a possible wrong answer that another teacher hadn’t considered and, in explaining the origin of that wrong answer, the first teacher essentially had to make a case that the student was doing something sensible.

When I revealed that 4% of students incorrectly answered “1 cm” to the question above, a teacher first said, “*What??*” followed almost immediately by *“Ohhh .…” *as he realized that students had correctly solved for the *perimeter *of 10 rather than an *area*.

I don’t want to make *too much* of that moment but we should make* something* out of it. In a moment, just a blink, a teacher went from imagining students and their thinking to be alien and nonsensical to imagining them sensible and clever even when they’re wrong.

A common feature of engaging teachers is that they’re *engaged*. Engaged with their discipline, yes, but also *engaged* with you, the student. Engaged* by* you and your thinking. It’s easy to imagine those teachers are born, rather than made. But I’m wondering lately whether exercises like the one above, repeated over time, can help *make* more engaging and more effective teachers.

While I have no insight into how the Rosenthal committee makes their decisions, I do think it’s a very good problem that gives us some very useful insights into how to design very good problems for math students. Here are three.

The lesson starts with a billiard ball sailing across a table at a 45° degree angle and then spends three days asking the question “What’s going to happen?” in different forms.

“What’s going to happen?” is a question that every middle school student asks themselves hundreds of times a day. Every middle school student has experienced objects in motion and ricocheting off surfaces, and every middle school student understands on an intuitive level the physical forces at work here.

In her introduction to *Rehumanizing Mathematics for Black, Indigenous, and Latinx Students*, Rochelle Gutierrez urges teachers to draw on a student’s “voice, vision, touch, and intuition.” Every student—whatever their experience with mathematical abstractions like angles and proportions—has an intuition for where that billiard ball might go next. That’s a great launch.

Is this a “real world” math problem? Please do not attempt to answer that question. It has ensnared and defanged countless promising curriculum designers who proceed to drape math on top of contexts that clearly have no use for it, designers who miss the fact that math *itself—*a world of numbers and shapes and patterns—is a world that is deeply *real* to students.

More to the point, contexts aren’t real or fake in any kind of Platonic sense. They *become* real as we inhabit and explore them, an observation made by Hans Freudenthal and many others.

In Day 2 of Finkel’s lesson, he senses that our work on the “What happens?” question has made the billiard context more real to us. We understand the trajectory of the bounce better. We understand the range of possibilities better. Finkel sees all those cognitive resources, and invites us to play with them like toys. “Can you predict which corner the billiard ball will end in?” he now asks, a question that wouldn’t have felt as interesting or accessible on Day 1.

On Day 3, Finkel encourages students to *prove* some of their conjectures. But proof logic requires an ability to manipulate abstractions that eludes many adults, to say nothing of kids.

Finkel is operating with a full palette, though, and understands there are colors between a) deductive proof and b) verification through drawing. And that’s to ask students a question that makes verification through drawing unappealing, one that requires a leap with logic.

If full arguments prove too difficult for students, ask students to write down a solution for how to use the patterns and conjectures to

make a prediction for what corner a billiard ball would end up in on a 36 by 98 table. This is too big to do by hand, so they’ll have to apply the conjectures from Day 2.

Congrats to Dan on a very nice lesson. Check the lesson out yourself and let us know any features you like about it.

**2023 Feb 23**

Check out Dan Finkel’s comment here.

**Dylan Kane**describes “What I Mean When I Say ‘Blogging is Dead’” which is a lovely time capsule and ode to a time of teacher development that was very meaningful for a bunch of us.**The Desmos curriculum team**(my colleagues) just completed their Algebra 1 curriculum and are clearly operating at the height of their powers. I recommend you check out Lizard Lines, Finding Desmos, Solutions and Sheep, Shelley the Snail, and then get yourself a free trial of the whole 6-8 & A1 curriculum.

I don’t mean that I was late to *start* the class. I mean that students had figured out a mathematical trick before I got there which meant they didn’t have the same need to understand the math which, unfortunately, was my goal for the class.

Floating in Groups is a lesson designed to help students understand one of the most counter-intuitive truths of mathematics: that multiplying two negative numbers together results in a positive number. It does that with a helpful metaphor and helpful feedback.

The metaphor is that multiplying two negative numbers is like removing several groups of anchors from a submarine, which makes the submarine go up. (“Removed” is the first negative number. “Anchors” is the second.) That’s a nice metaphor. Then we pair that metaphor with feedback designs that let students *play* with the metaphor, subtracting and adding groups of floats and anchors dynamically.

We spent the lesson developing that context and its connection to an important mathematical idea. I was feeling great. Then the lesson asked students a question:

Use the floats and anchors scenario to explain why it makes sense that (-2)(-4) is positive.

And I felt less great. Here’s video of the moment.

Students resorted to a mnemonic that was far removed from the context we’d been developing the whole class and also really, really fragile.

because 2 negatives make a positive

because a negative times a negative = positive

a negative plus a negative is a positive

because negatives make a positive

When a student says “because a negative times a negative = positive,” it’s *possible* their understanding of operations on numbers is robust but it’s very likely they’ll see a question like, “What is -2 - 6?” and answer +8 because they think that two negative numbers in any operation produce a positive number no matter what. (One student actually says that more or less explicitly: “a negative plus a negative is a positive.”)

In case it bears mentioning: none of the adults in the room had offered up any of those mnemonics.

My response (which you can see in the video above) felt clumsy enough that I wanted to bring it back to y’all here. **What do you do when a trick or mnemonic has already taken hold of a class?** How do you help students strengthen a fragile understanding they believe is very strong?

**2023 Feb 17**

The commenters come through every time! One suggestion was to put a little extra pressure on the mnemonic by either asking “is it always, sometimes, or never true?” about statements like “Two negatives make a positive” and “A calculation involving adding always gives you a positive answer.” (Thanks, anne!)

Also other commenters suggesting keeping two numbers fixed, changing operations and signs around them, and asking, not for the result, but for the sign of the result. Like:

-2 -4 (-2)(-4) -2 +4

Thanks, Leandra, Marty, and DK!

**2023 Feb 21**

Dylan Kane has written up a useful response here, including this paragraph:

So instead of telling students “tricks are bad,” I propose we show students the limits of that trick, and help them understand why the trick stops working where it does. This might seem obvious. But there’s a huge difference between telling a student “don’t use FOIL, tricks are bad” and showing them its limits with a thoughtful set of problems.

Some technologists have argued that innovations in artificial intelligence like ChatGPT are increasing fast enough in quality, speed, and scale that they may soon replace teachers, or at least transform the work of teaching.

Though I work in education technology, I’m skeptical. The more classrooms I observe and teach, the more I understand the process of teaching to be too complex across too many facets to reduce to a transfer of questions and information between a teacher (human or AI) and a learner.

I know I’m not alone in this skepticism. And it’s to my fellow skeptics that I want to pose the question:

**What hurdles would AI have to clear in order for you to believe that it had, if not eliminated a teacher’s job, at least transformed it so as to be unrecognizable? **

I have three proposals here.

In mathematics, ChatGPT can currently answer with 100% accuracy only a very narrow selection of questions from a given curriculum: the kind with highly structured inputs and outputs and no surprises.

Ask ChatGPT, “What is the solution to 2x - 3 = 6?” and it offers an answer that is correct and well justified:

Ask ChatGPT a question that is more difficult by a single term, “What is the solution to 2x - 3 = 5x + 6?” (a question that middle school students routinely answer in the United States), and it falls apart:

It combines the proof diction of a PhD (“therefore!”) with an approach to algebraic operations that’s less thoughtful than a kitchen blender. ChatGPT transforms math teaching here only insofar as it gives math teachers a new source of interesting errors for their students to analyze and learn from.

Once we broaden our scope beyond highly structured problems like these ones toward problems where students have to answer interpretive questions about multiple connected representations of data, it’s clear that ChatGPT doesn’t yet deserve the trust and authority that even a novice teacher enjoys.

It’s one task for an AI chatbot to offer an explanation for finding the solution to the equation 2x - 3 = 6. But it is another task entirely, and a much harder one, to understand the different routes students might take toward understanding that question and its answer, including the many partially correct answers. This is called pedagogical content knowledge, as defined by Schulman, Grossman, and many others, a special kind of knowledge that AI hasn’t yet mapped.

For example, if you ask a teacher of even moderate skill, “Why is the solution to 2x - 3 = 6 not x = 1.5? What did I do wrong?” they’ll recognize that the student may have *subtracted* 3 from 6 and then divided by 2, rather than undoing the subtraction by *adding* a 3 to both sides of the equation. That pedagogical content knowledge helps the teacher connect the student’s path to other paths, including one that leads to a correct answer. When I asked that same question of ChatGPT, though, I received the following response:

Essentially, ChatGPT has said, “You were wrong because you weren’t careful. Here is how I would do it.” No part of ChatGPT’s response refers to the ways I might have been thinking about the equation. No part indicates the sense behind my thinking (I performed the same operation to both sides of the equation), leaving me to believe I am 100% incorrect.

All of this means that if I am to learn from ChatGPT’s response, I need to draw connections from my thinking to expert thinking on my own, as a novice. This is a cognitive tax, one that skilled teachers and tutors generally pay on behalf of learners.

An AI has passed the “Turing Test” if it can have a conversation with a human that the human finds indistinguishable from another human being. An AI has passed the “Caring Test” (let’s call it) if the human wants to *continue* that conversation another day.

This kind of social affinity is important for learning. Researchers have found that a student’s belief that they are taking a socially relevant action creates conditions for greater learning and engagement. If you think you are interacting with a human, someone with whom you feel kinship, you’re more thoughtful about your explanations and put more effort into their preparation.

Even if I know the AI will give me consistently correct answers to questions I ask, how much do I care that it thinks my *own* answers are correct? A whole raft of academic objectives essentially ask, “Can you persuade other people?” Do students care that they have persuaded ChatGPT? Before we can say AI has transformed teaching or tutoring, it needs to convince students that their interactions are socially meaningful.

Obviously AI hasn’t yet cleared any of these hurdles in education to say nothing of clearing them all at the same time. It’s possible that teaching is so irreducibly complex that AI may simply hop into a long line of technologies—including television, the radio, the graphing calculator, and even the printing press–that contemporaries believed meant the end of education as they knew it then.

Rather than wonder “Will AI replace teachers?” educators and technologists might instead wonder how teachers and artificial intelligence can partner in productive ways, for example with AI offering recommendations to teachers that grow more relevant and more accurate over time. If past technologies are any indication, this hybrid model of human and machine is likely to retain the best of teaching’s past while also pointing towards a happier, more productive future.

]]>“Previously on [TV Show],” says the announcer in a flashback at the start of the show, followed by excerpts from previous episodes.

Those flashbacks are, psychologically speaking, an attempt to take our “inert knowledge” of the show and *activate *it, making that knowledge available for new stories. The flashbacks remind us of a character’s nature because we are about to learn something *new and different* about that nature. We are reminded of events that happened in a particular place because we are about to see *new and different* events in that place.

Paul Silvia’s research found that interest is generated by situations that are both novel *and* familiar*. *The flashback generates familiarity and the new episode generates novelty.

It is common in many classes for teachers to write an objective on the board and say it out loud.

Today we will learn how to multiply positive and negative integers together.

Today we will learn how to solve one variable equations using hangar diagrams.

This style of lesson opener introduces what is *novel* about the day but not what is *familiar*. It emphasizes what is *new, *but not how it is *different* from what came before. It lacks a flashback. Drawing those connections, highlighting the limitations of *the old* in order to create a need for and a connection to *the new, *is essential work for teachers.

Here is a short video of two teachers starting their lessons like storytellers. One teacher is me, teaching *Floating in Groups* to seventh graders in Oakland, CA; the other, by permission, is San Diego educator Gen Esmende, teaching *Balancing Equations* to seventh graders in San Diego, CA.

Both of us are teaching from Desmos Math. Without coordinating, both of us spend thirty seconds at the start of the lesson in exactly the same way: trying to turn our lessons into stories. In both classes, you’ll then see students talking and drawing connections, activating inert knowledge that will be useful later.

Cognitive psychologists have called stories a “psychologically privileged” form of information–the square peg for our brain’s square hole. So I’m always excited to steal the lessons of effective storytellers and apply them to our classes. “Previously on math class,” is one of those lessons I think about every day I’m in the classroom.

I remember exactly who I was and where I was when Mr. Bishop put this one on the board.

If you teach secondary students who think math class is about how quickly and accurately you can perform operations, this problem will help them understand math as conceptual and elegant. I think I had expanded all the way to, like, (x-f), resulting in a messy sixth degree polynomial, before I decided the problem would have to crack with careful thought rather than brute force.

]]>My late policy is whatever their late policy is for their students. So any time a teacher—one of my students—comes to me and says, “Well I’m not going to finish this project on time,” I say, “Alright just show me your late policy, and that’s going to be applied to you.” I changed 100% of my students’ late policies.

I appreciated the reminder here that no matter what a teacher says, no matter what’s between the covers of the textbook, some of the most enduring lessons a student will learn in a classroom are taught through its policies.

Grading policies—especially participation grades—teach students how the teacher defines success and competence.

Seating charts teach students about the kinds of classroom interactions the teacher values.

Late policies, and a host of related policies, teach students what the teacher thinks about the

*packaging*of the thinking relative to the thinking itself.Bathroom policies teach students what the teacher thinks about their bodily autonomy.

And on and on. Thurbee is obviously conscious of this “implicit curriculum” but it’s another more difficult matter to help new teachers (and *veteran* teachers!) become conscious of it and its effects on students. I thought his meta late policy was a particularly graceful lesson.

**Thurbee**offers several other gems in that podcast, including (a) how an effective principal plans staff meetings and (b) how an effective math teacher educator gently helps new teachers understand the limits of their*content knowledge*and their need to develop*pedagogical*content knowledge.If you ever find yourself wondering, “What does effective technology use look like in a math class?” please help yourself to this video from

**Berwyn South School District 100**. You’ll see technology used to support student thinking and conversation. You’ll see multiple students looking at a single device, talking about what they’re seeing. You’ll see students move seamlessly from the device to paper to whiteboards and back again. It’s very easy to find math classes where digital technology is either absent or omnipresent, deployed sparingly or overbearingly. BSD is setting a standard here IMO.I went shopping for a toilet last week, which led to a mathematical investigation of weighted averages that is, in one sense, perfect for middle school and, in another, more important sense, completely inappropriate for middle school.

**Robert Kaplinsky**attended the*Get Your Teach On*conference and brought back a fun piece of ethnography looking at its differences with conferences like the National Council of Teachers of Mathematics annual event.I just finished teaching a middle school class here in Oakland, CA, and I’ll share with you a screen from the activity that drove the class into an absolute problem solving frenzy. Please enjoy.

Whenever you find yourself learning and enjoying learning, ask yourself, “

What can schooling borrow here?”How can we sneak in here, stuff all these really good ideas about learning in a duffel bag, and bring them back to our schools and classrooms and edtech startups?

Lately, my kids and I are playing *Toad Treasure Tracker* on Nintendo Switch. It’s a joy. I have convinced myself it basically doesn’t count towards any screen time tally because my kids are obviously learning collaboration, fine motor skills, problem solving, etc. When we’re stuck, my youngest kid will murmur, “Something I’m wondering is ….” reflecting my own problem solving mannerisms back to me in a kind of funhouse mirror.

I’m increasingly jealous of one aspect of *Toad Treasure Tracker* that simultaneously accounts for tons of its appeal but also seems really challenging to import into school learning.

**Replayability**

In *Toad Treasure Tracker*, the main goal is the same in every level. Capture the star.

It feels awesome. We love to capture the star, don’t we, kids?

We do. I think it’s worth noting that this moment itself is sometimes rare in school learning, a moment where you can sit back and say, “No matter what I haven’t yet done, one thing is certain, and that is that I just did a thing.”

And what seems *really *rare in school learning is our eagerness to **replay the same level again**. We’re eager because each level contains *four* goals of which capturing the star is only one, and also the easiest to accomplish.

Capture the star.

Find all three gems hidden somewhere in the level.

Accomplish the challenge goal, which varies.

Find a tiny pixelated toad character hiding somewhere in the level.

All four of those goals deepen our understanding of the levels in some pretty key ways—the level map, how to move efficiently and precisely through it, how to avoid hazards, etc. If your learning objective was, “Deeply understand every square inch of this particular level,” then you couldn’t do much better than *Toad Treasure Tracker* has in setting up those four goals.

From the perspective of school learning, **how weird is this?**

Yes, I realize the pacing calendar of schooling feels like it only speeds *up*, that it only *subtracts* room for discretionary experiences of any kind. But let’s pretend for a second that a student had an extra 30 minutes at the end of your week. Something I’m wondering is:

What incentives exist for that student to

*return*to a unit, a lesson, a book, a project after it’s completed, to understand it more deeply?What secondary goals could we establish for each lesson that would encourage replay and deeper, more flexible learning? (While capturing the star occasionally requires speed, interestingly, none of the side goals in

*Toad Treasure Tracker*relate to time or speed.)Is there something about school learning that makes this idea of replayability a poor fit? Game theorists will cite “alternate endings” and “unlockable characters” as contributing factors to replayability, neither of which seem like they have an obvious home in school learning. But other researchers have found that collaboration increases a desire to replay, which obviously

*does*have a home in school education.

All hypotheses welcome in the comments.

]]>Here is a system called Individually Prescribed Instruction from the late 1960s:

Students are provided with self-study aids, such as pre-recorded cassettes and videos, library references, computer-assisted instruction, sample tests, or programmed learning modules. [..] When a prescription is completed, the work is checked by the teacher, and if it is satisfactory, the student proceeds with the unit.

Sound familiar? You might recognize the same kind of premise with Khan Academy and the “flipped classroom” movement in the early 2010s:

She's now on her way to "flipping" the way her class works. This involves replacing some of her lectures with Khan's videos, which students can watch at home. Then, in class, they focus on working problem sets. The idea is to invert the normal rhythms of school, so that lectures are viewed on the kids' own time and homework is done at school.

Or the Modern Classrooms project here in the 2020s:

In a Modern Classroom, educators forego [sic] the traditional front-of-class lecture to provide shorter forms of direct instruction that students can access whenever and wherever they might be. In this way, educators duplicate themselves digitally, allowing them to more freely respond to the various needs in their classrooms.

In a Modern Classroom, students go at their own pace. They don’t move onto the next lesson simply because it’s time or because they completed the requisite work. They move on when they have mastered the concept and are ready to build on that skill.

If you get close to the surface of classrooms, it can seem like we all have drastically different ideas about teaching and learning. Everyone is using a different curriculum, different strategies, different stuff!

But what’s interesting to me about this same technology returning from cold storage every few years is that it reveals a *secret*. It reveals what a huge group of people—including large subsets of the national media, venture capitalists, startup founders, parents, teachers, administrators, etc—believe about education but don’t often say out loud or even necessarily think about consciously.

The most prevalent of those beliefs seems to be that:

Teaching involves the transfer of extended amounts of information from expert to novice, enough information to make pre-recording it worthwhile.

We can measure teaching quality by the quality of the pre-recorded information transfer—its clarity, its engagement, its effect on learners—and we would ideally invite the people who are the best of the best at that kind of information transfer to pre-record the information.

Learning is the act of an individual and the presence of other learners with differing traits (existing knowledge, interest, participation styles, etc.) inhibits that learning. Therefore, each learner should learn from the pre-recorded information that is best for them.

You can use, love, and probably even invent these technologies without holding onto any of those beliefs exactly. I’m not describing any one person’s beliefs about education. I’m trying to describe why these particular technologies, all of which share a bunch of common DNA, become very popular every few years. That popularization is the work of a *group*, not an *individual*.

Part of my work right now is communicating the value of a very different kind of educational technology than the one we keep re-animating. Perhaps you’re in a similar position in your school, district, or board. From that vantage point, I think it’s extremely helpful to understand these enduring beliefs about education, the ones shared by lots of people you’ll need to persuade. Change requires us to speak *with* those ideas, rather than *past* them.

[s/o to basically everyone in this Twitter thread sharing their ideas on this same question.]

]]>At the end of January, Wordle was acquired by the New York Times, and the #1 most shared link was a math-themed adaptation called Ooodle. Relatedly, the most shared article was my post analyzing Why Wordle Works.

*Most Active Conversation:* “I don't mean to start any drama but...do you prefer doing math in pen or in pencil?” [link]

In February, we were still sharing Wordle knock-offs, now including Quordle (#1 link in February), Worldle (#3), and Sedecordle (#4). The Olympics were live in Beijing and Ralph Pantozzi had the most shared photo with this beautiful skiing parabola.

*Most Active Conversation:* “How many years have you been teaching and, at this point, what has most influenced how you teach?” [link]

The top three links in March were all Wordle knock-offs, now including Heardle at #3.

A bunch of math teachers in the UK got together and wrote “If I Could Tell You One Thing,” which was most shared in the non Wordle-related division.

*Most Active Conversation:* “In honor of pi day, replace a word in a movie title with the word ‘pi.’” [link]

Okay, I am no longer reporting Wordle-related rankings here. The most shared link here was for Julia Aguirre’s webinar, “Grades and Test Scores Do Not Define Us as Math Learners: Cultivating Transformative Spaces for Anti-Racist Mathematics Education.”

Zak Champagne had the most shared article: “The Teaching and Learning of Mathematics is NOT About “Right Answers”

*Most Active Conversation:* “What's the best teaching advice you can give in 3 words?” [link]

Desmos had the top two links in May. Sean Sweeney’s Desmos Escape Room took top honors. Then Desmos Classroom announced its merger with Amplify, taking the silver medal overall, though a strong gold in the “press release” division.

The most shared photo was the gorgeous little number sense problem above.

*Most Active Conversation:* “Suppose that your pupils are having difficulty with this question. [If a = 10 and b = -2 find the value of a-b.] How do you go about getting them over this hurdle?” [link]

Simon Singh’s “home for curious mathematical minds” Parallel was the most shared link in June.

In the most shared photo, Jonathan Hall asked, “Where do we lose the 2?”

*Most Active Conversation:* “Make a math teacher cry with 4 words.” [link]

The most shared link in July was Brian Bushart’s Numberless Word Problems website.

Aleda Klassen’s sketchnotes for *Building Thinking Classrooms* were also a hit.

The most shared image included these Mathematical Points of Power from Margie Pearse.

*Most Active Conversation:* Something about high school mascots and it isn’t hard to conclude that much of math education Twitter took a break this month. Good!

Desmos Classroom had the top three links as the school year as the US started ramping up, including:

Polypad integrating into Activity Builder

A new Twitter handle for @DesmosClassroom.

Fawn Nguyen starting work at Amplify, the parent company of Desmos Classroom.

Robert Kaplinsky wrote the most shared article in August, an ode to Peter Liljedahl’s book *Building Thinking Classrooms*.

Most active conversation: “It's Mental Math Monday. How would you mentally calculate 80% of 150?” [link]

Sound on!!! 🎼🥁🎸🎼
Polygons, number bars, and fraction bars on Polypad @MathigonOrg now can play audio!
What music can you make? How about your students? How can we use these to explore and discover mathematics?
#iteachmath #mtbos

In the most shared video, David Poras showed us how Polypad can turn mathematical shapes into musical notes.

In the most shared photo, Emily Rae turned a complaint about an Amazon purchase into a promising lesson about surface area.

*Most Active Conversation:* A roll-call of attendees of NCTM’s annual conference. [link]

A Desmos Classroom upgrade took the #2 spot but otherwise this month belonged to the United Kingdom, including:

A Black Heroes of Maths poster from the UK’s International Centre for Mathematical Sciences.

A resource called “Developing Your Use of Manipulatives in Maths Teaching” from the National Centre for Excellence in the Teaching of Mathematics.

This resource called “Worked Examples” which is exactly what you think it might be.

At this point in the year, the Wordle adaptations have largely faded from the Top 20. They’re now surpassed by a new contender for daily-puzzle-you-complete-while-brushing-your-teeth called Fiddlebrix.

*Most Active Conversation:* Howie got real about the haters and hopefully felt the love in return. [link]

The call for proposals for next year’s national NCSM conference was most shared in November. NCSM has some juice!

Ayelian had the most shared video with a construction of 1/7 of a circle that is the only way that *this* household will cut a pizza into seven slices from now on.

400+ educators signed a petition urging The College Board to reconsider its syllabus for AP Precalculus.

*Most Active Conversation:* Tracy Zager, stellar human and editor, announces she is leaving Stenhouse Publishers under circumstances not of her choosing. There are other great editors at Stenhouse but working with Tracy was the biggest reason I signed a contract for a book there, one which I have since withdrawn. [link]

Nothing was shared even half as much in December as Mathigon’s annual December puzzle calendar. I’d invite you to bookmark it for next year but I’ll wager it’ll get shared just as much.

All Ten from Beast Academy is another puzzle that placed pretty high as well.

No one kept the math education Twitter chatting in 2022 like Howie Hua so it’s fitting that he had the most shared video, turning math into an infomercial, and conversation this month. In fact, he had the most active conversation of all of 2022.

*Most Active Conversation:* Make a math teacher cry in 4 words. [link]

My Twitter use has changed quite a bit since I first joined in 2007, but true then as now, there are forms of knowledge sharing, professional development, and community you can only find on Twitter, hints of which you’ll find all throughout this rundown of 2022. Let’s keep it going in 2023.

]]>… 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.

I saw an illustration of Gutierrez’s quote in a lesson where a bunch of students had been successfully graphing inequalities all morning until they crashed on a new one:

Over the last ten years, math education has done a top-shelf job destigmatizing making mistakes and revealing misconceptions. But when teachers fixate on the idea of a student having a misconception, even if to say *that’s okay!*, they are fixated on what *they themselves* know.

Psychologist David Ausubel said, “The most important single factor influencing learning is what a learner already knows.” So Gutierrez’s critique isn’t *purely* social here; it’s also cognitive and pedagogical. Our teaching is *less* effective when we’re focused on what *we* know and *more* effective when we focus on what *students* know.

**What do these students likely know?**

There is a connection between the inequality and the graph.

They both describe a bunch of solutions.

There is a critical point on the graph where solutions turn into not-solutions.

**What I did next:**

Gutierrez:

And those conceptions make sense for them,

until they encounter something that no longer works.

These students don’t have a *misconception*. They have a *conception*, an idea, that whatever number is by itself on one side of an inequality, that’s the critical point. They need to encounter something that helps them find the limits of that idea. It works sometimes but they’ve overgeneralized it.

So as a class we evaluated x = 1 and convinced ourselves that it definitely works in the inequality *and* the graph. I used my authority to herald the fact that *this is a very good thing, yea verily, these should match!*

Then we evaluated x = 20 and convinced ourselves that it definitely works in the graph but definitely *not* in the inequality where it produces the false statement 34 ≤ 25. Then I asked students to discuss what they saw at their table, try again, and people found a new path.

**PD is PD**

Something exciting and a little unnerving about Gutierrez’s quote up there is that it applies equally to our classroom and our homes, our neighborhoods, our communities, our entire social lives. Everyone whose behavior befuddles us in some way—spouses, partners, friends, neighbors, etc—offers us the opportunity to either focus on what we know and how we would have done it or to wonder why what someone else did made sense to them and what we’ll need to do to develop a different shared understanding.

What I’m saying is that PD is also PD–*professional* development in teaching is *personal* development in life. This is not true for many, many jobs on offer. But because our work is social and very intensely personal, personal transformation results in professional transformation and vice versa. I’d like this work to transform me into a more loving, curious individual and my life to do the same for my teaching, and starting with the conviction that “students don’t have misconceptions,” is one such transformation.

Math class might take note here! Here’s Howie Hua doing this week something we might call “Infomercial Pedagogy.”

This is the most ridiculous math video I've made but I hope it's still good 😅
Tired of "borrowing" when subtracting? Never "borrow" when subtracting with this one method.

Don’t you hate it when your standard algorithm requires you to make exchanges all the way across the minuend? Well you should try thinking about subtraction as the distance between two numbers instead!

OMG I want to show this to my students and have them make their own math infomercials!!!

Howie Hua @howie_hua

Joe Herbert suggests this process of identifying the *need* for new math ideas might be a useful project for students. Me, I feel pretty convinced that, because we’re forcing students to spend 10-12 years of their only childhood learning mathematics, the obligation to identify the need for new math ideas rests with the teacher, and the curriculum, instead.

For example, here is one of my favorite examples from our curriculum. Don’t you hate it when you try to sketch a scale drawing and it doesn’t quite work out as you intended?

Well you should try a grid and numerical precision instead!

By starting with the knowledge students already have and helping them encounter the *limits* of that knowledge, we’re making learning math feel more *purposeful*. But by connecting old and new knowledge, we’re strengthening both and making learning math more *effective *as well.

**BTW**

If you find these ideas interesting and want something more theoretically grounded than infomercials, definitely check out Guershon Harel’s paper on the necessity principle.

You might also enjoy the Directory of Mathematical Headaches which leaps from a different metaphor than infomercials.

Little book help, Twitter? I need to hear from some young (6 y.o. and under?)mathematicians about this shape.
Is it a circle? How do you know?

This extraordinary bit of tweetcraft from Christopher Danielson reminded me this week of one of the most important principles in the whole of teaching:

**Everyone knows something about everything.**

All learning interacts with earlier knowledge. So whatever you’re trying to teach an old kid right now is interacting with knowledge they developed as a *young* kid which is interacting with knowledge they developed as a *very, very young* kid and so on.

We might wish learning worked a different way, a way where I say stuff the way I understand it and then you immediately understand that stuff the same way I do no matter what you knew before I started saying stuff. But we go to the classroom with the principles of cognition we *have* not the principles of cognition we might *want*.

That early knowledge is a *responsibility *for teachers. We need to let students know that their knowledge and experience are as essential to their learning as their *teacher’s* knowledge and experience.

But that early knowledge is a *resource *for teachers as well. It’s a gift. It makes teaching *easier*. When a student says, “it’s not only round, it has little bumps,” we have resources that can help us all move in the direction of *new* ideas.

Okay you’re saying this one is bumpy. How do you know this one

isn’tbumpy? Which of these shapes is most bumpy and least bumpy? Can we come up with a rule to figure out if a shape is bumpy or not?

Jere Confrey said, “Students are our most underutilized resource in school” and I believe she was referring to resources like “bumpy.”

**The failing of most math curricula is that they waste the gift of early student thinking.**

For example, this is the *first* paragraph in a unit from a major math curriculum called “Circles in Math.”

In Maths or Geometry, a circle is a special kind of ellipse in which the eccentricity is zero and the two foci are coincident. A circle is also termed as the locus of the points drawn at an equidistant from the centre. The distance from the centre of the circle to the outer line is its radius. Diameter is the line which divides the circle into two equal parts and is also equal to twice of the radius.

The vast majority of math textbooks make no *use *of early student thinking. They make no *room* for it. Those textbooks share the formal, developed thinking of an adult and then ask students to develop their own ideas on *those *terms.

Those textbooks have no use for “bumps.”

**This is why teachers modify their curricula, but they shouldn’t have to.**

When teachers modify their curriculum, they’ll often describe their efforts to make it more “engaging.” But another name for what they’re doing is j”creating room,” creating room for early ideas from students and finding ways to treat those ideas like the resource, the gift, that they are. This makes teaching easier for the teacher. And, for the student, there are few experiences more engaging than experiencing some knowledgeable other person as deeply engaged in *you*.

You do a bunch of stuff you

*want*to do—watch movies, watch trailers, etc.Netflix’s computers think about the stuff you’ve been doing.

Netflix recommends stuff you might want to do next.

Here is how Netflix would work if it was built by an edtech company:

You would

*stop*doing stuff you want to do and take a quiz about it.

Netflix would display some graphs of what you did in your quizzes.

Netflix would require

*you*to decide what to do with that data or even how to make sense of it.

Something I don’t like admitting because I feel like I’m Someone Who is Supposed To Know Stuff is that I have no idea what I would do with most edtech data displays.

The typical approach in edtech to data display is a rare combination of **overwhelming and passive**. It offers you *loads* of data and doesn’t help you figure out what to do with it.

By contrast, Netflix’s approach is **invisible and active**, meaning Netflix doesn’t show you *any* data but it makes some active recommendations that I, personally, find pretty accurate. (Why yes, I *would* like to watch another paranoid political action-thriller from the late 1990s.)

I feel pretty dumb looking at graphs in edtech because, absent any recommendations from the software, I feel like it’s supposed to be obvious what I’m supposed to do with them. But it isn’t obvious to me.

I don’t think I’m alone in feeling more confusion than clarity with edtech data displays. Heather Hill reviewed studies of how those displays get used and reported that, “Studying student data seems to not at all improve student outcomes in most of the evaluations I’ve seen.”

Why doesn’t data analysis work? All three researchers explained that while data is helpful in pinpointing students’ weaknesses, mistakes and gaps, it doesn’t tell teachers what to do about them. Most commonly, teachers review or re-teach the topic the way they did the first time or they give a student a worksheet for more practice drills.

So here is one question I’m thinking about with some collaborators at Amplify:

A teacher gives a digital assessment. How can we help that teacher make the most out of a

15-minute reviewof that assessment afterwards?

The 15-minute review of an assessment is already a part of the routine of many classes. We are not asking for *extra* time here.

What if we threw away the pie charts, 3D scatter plots, and all the other over-determined representations of student learning and, like Netflix, made our most confident recommendation for those 15 minutes?

“Here’s a movie you might like” could then become “Here’s a question you might ask.” Or, perhaps more specifically, “Here is the most common wrong answer from your last assessment. Ask your students to write a question where it is the right answer.”

One of the best outcomes of Desmos Classroom merging with Amplify has been the pressure to think about supporting a *huge* group of teachers with very different capacities instead of a *smaller* group of very high-capacity early adopters. That change, more than any other, accounts for my recent heel-turn away from depictions of teaching as martyrdom, and over-determined frameworks for teaching.

There are too many teachers working too hard right now and those of us who consider ourselves teacher supporters should ask ourselves at every moment, “Are we making life easier or harder for teachers? Are we taking weight off their shoulders or adding to it?”

The way Netflix uses data takes weight *off* my shoulders. The majority of edtech data displays only *adds* to it.

I put the question out on Twitter, “How do you use these things?” and got some great responses. Some people described a lack of faith in the visualizations themselves.

Math ANEX is doing interesting work in this space and replied with a relevant blog post asking, “What does 2 grades below grade level really mean?”

**Scott Davidson**, a math coach in Clovis, CA, offered a visualization of assessment data that struck me as unambiguously helpful, a visualization that takes weight off the teacher’s shoulders, offering the teacher recommendations for what to do next depending on the problems students struggled with on a pre-assessment.

]]>Basically, every presenter in every session I attended was very concerned that **only certain ways of thinking mathematically are welcomed in math classrooms**—specifically the formal, correct, and precise kinds, leaving out students who have lots of *other* kinds of valuable mathematical thinking to offer.

For example,** Annie Fetter** likened math to art in her opening keynote, by way of this Tim Madigan quote, and argued that, similar to art education, we need to offer experiences before explanations.

Art is something natural to all human beings, which should not be overly explained or theorized about before being experienced.

Some of the experiences Fetter recommended were Numberless Word Problems, Contemplate Then Calculate and Which One Doesn’t Belong routines.

**Howie Hua** argued that “math needs a better marketing team” and shared his methods for marketing math more effectively, including transforming word problems to “would you rather”-style problems which he argued would welcome more students into the math.

My Math Teacher Lounge co-host **Bethany Lockhart Johnson** helped her participants identify “Opportunities for Joyful Sense-Making Within Assigned Curriculum.” With one particular strategy, Bethany would remove structures embedded in a curriculum (instructions for solving a math problem using a particular tool like a number bond or number line, for example) letting students reveal their own tools and adding the original tools back into the problem as needed.

**Megan Franke **recommended we welcome what she called “partial ideas.” Franke argued that the point of offering a student a problem to think about isn’t *just* to engage and welcome them but also to help the student “understand what they already know,” a/k/a those partial ideas.

I share these concerns. Too many students feel like math class isn’t for them, like math class only has room for certain kinds of ideas.

And I also wonder, as a field, **are we over-theorizing the welcome into math class and under-theorizing what we do with students ****after**** they feel welcome?**

Let’s assume kids are through the door. Let’s assume they feel like they are in a place that welcomes them and their ideas. How do we make sure that once they *leave* the room we have helped them make *more* of their ideas than they could have on their own?

**Jessica Balli** and **Jeremy Thiesen’s **session fascinated me, then, because they’re developing digital assessments and scoring processes that are asset-based, that assign value to the ideas students already have, even if those ideas stop somewhere short of precise and formal correctness.

But they also proposed some interesting routines for helping students *develop* those partial ideas. (Check out their slides.) For example, they’d take some common answers from the class to a particular question and ask students to ask a question that would have each number as its answer.

I see lots of value in students sharing the ideas they brought to class and have little doubt that those students are learning more mathematics (and also learning *more than* mathematics) in those conversations. But *how* that happens seems, frankly, a bit mysterious to me sometimes. Like I should place ingredients next to one another on a counter, leave the room, return, and find that they have assembled themselves into a meal.

I think a key challenge for teacher supporters is to demystify the mysterious. So I‘m excited to spend more time thinking about, reading about, and perhaps even developing routines of the sort described by Balli and Thiesen—routines that ask “**how can we ****systematically**** help students make more out of their own partial ideas?” **in addition to “how can we help students feel *welcome* to offer their partial ideas?”

If you have any favorite strategies, or anything to add to a syllabus, please leave it in the comments. 👇

As a presenter, it’s impossible not to learn about the craft of presentation when you’re watching other presenters. Here are a few moves I wanted to write down and remember.

**Howie Hua**had tons of humility, expressing gratitude that we were there and giving us explicit permission to bail if the session didn’t feel like it was meeting our needs. Nobody bailed and many probably felt like Howie was watching out for them.**Annie Fetter**used Google Slides because they’ll automatically generate closed captions for deaf participants.**Megan Franke**started her session by asking us to stand up, meet someone new, and share what we’d learned at the conference so far. "The whole point of this conference is to meet people,” she said.

In the unlikely case that he is *not* at this second worrying about our community, in the unlikely case that he is instead considering a raft of changes that will make Twitter less democratic, more oligarchic, and more crowded with junk, I want to ask a couple of questions:

What has made Twitter work so well for teachers to date?

If you’re a teacher who is leaving Twitter, then where are you off to next? TikTok? Mastodon? Club Penguin? Blogs? Message boards? Real life?? Why?

My take on why Twitter has worked:

Twitter features low barriers on both consumption and production and lots of ways to participate. On the consumption side, you can read tweets without an account. You can get an account and follow a hashtag or a set of users. You can then start *producing* in simple ways—liking or retweeting a tweet for example. You can dial up your production by replying, quote tweeting, and then sending out your own tweets. Your tweets can also include a variety of media (images and videos, for example) that have helped us represent our teaching practices in interesting and useful ways.

Twitter also features a delicate and important balance between producers and consumers. Quote tweets and replies often receive more engagement than the tweets getting quoted (d/b/a “getting ratio’d”). A new conversation might spin up in the replies of a tweet that the original tweeter didn’t intend or even want. People with huge followings put ideas out into the ether and they are often taken in entirely new directions by the crowd. Twitter is the closest we have had to a digital commons.

Every day on Twitter, you can watch that community produce knowledge in real time.

Content knowledge:

How do people teach finding the nth term of quadratics? I’ll reply to this with a few of my own thoughts shortly but wanted to sound out how others do this…

Pedagogical knowledge:

What are your favorite things for students to do after a test?
We have 82 min blocks. One of my classes has assessments like every other week (it’s a very fast paced class). I don’t want to give them homework and some students need more time than others. #mtbos #iteachmath

Math education research:

Next week there is a workshop for mathematics education professors exploring doctoral programs across the country, funded by NSF.
What should be the goals in doctoral education in mathematics education? How should the programs be improved?
Anyone have thoughts on this matter?

etc.

I am interested here as someone who realizes how much a) community, b) professional learning, and especially c) knowledge production arose from the particular context (including technological, political, personal, epidemiological, generational factors, etc) of 2005 to 2015. And I’m wondering where teachers–especially *new* teachers–will get that next.

Heck if I know. I mean I *know* that’s where you’ll find young teachers these days.

@ddmeyer Each semester in my undergrad class we talk about building a PLN and how Twitter can help. The younger generation clearly goes to Pinterest, IG, TikTok over Twitter. Most don't even have a twitter account. I would be curious to see the ages of those who use #mtbos and #iteachmath

What I *don’t* know is how that platform supports community, professional learning, or knowledge production.

I asked a couple of people who are active in both communities to describe how life would change for them if Twitter were to disappear.

Howie Hua (48,000 followers on TikTok) said, “I would miss the deep conversations. There can be conversations in the comments section on TikTok but it doesn't compare to what it's like on Twitter. I would also miss the Twitter community.”

Tim Ricchuiti (144,000 followers on TikTok) said, “I’d feel fine as a content creator,” he said, “but much less fine as a math educator.”

Ricchuiti is the only math teacher at his school in his grade and has found community on Twitter in ways he hasn’t on TikTok. “On Twitter, I feel like I have the world's greatest professional development,” he said. “I can see other teachers posing questions, citing research, developing lesson plans, sharing warm-ups, and, perhaps most importantly, talking about what hasn't worked. There may be the equivalent to that on TikTok, but if so, I haven't found my way to that part of TikTok yet.”

The generation of teachers immediately *before* mine experienced community, professional learning, and idea generation, in large part, because of a few key federal grants into curriculum and professional learning. Without those grants, their story changes entirely—maybe for the better but also maybe for the worse!

The current teaching generation is writing their story right now, including how they learn about teaching, form community, and create new ideas. I’d love to know that it ends as well for them as it has for many of us from earlier generations.

]]>It’s a beautiful bit of craftsmanship that I think is bad for math students. In my usual spirit of comity, I encouraged its inventors to invest themselves in a project with greater returns to society.

This prompted a bunch of people to rush to the defense of *fluency practice *in general. Fluency practice doesn’t need defenders, least of all from me.

I mean, check out the TIMSS study which surveyed teachers on their instructional activities in 2011. They found the US leading the world in students working problems.

Fluency is doing fine.

People used different metaphors in their defenses of fluency practice, though, including basketball, carpentry, music, and others. Fluency is every bit as important in math as it is in each of those disciplines but those disciplines develop fluency in ways that are distinct from math drill sheets, and those distinctions are worth a quick look.

@KochNina @JonTorsch @ddmeyer The carpentry analogy is limited. The fact remains every year I get 10th and 11th graders who don't know basic arithmetic.
On the other hand professional athletes become the best at their sport by doing the same thing over and over, hundreds even thousands of times

@KochNina @ECW3Math @ddmeyer I mean, carpentry is so much more than a hammer and drill, but it's pretty hard to do if you don't know how to use the tools in the first place. The movement to enrich & broaden mathematics is good, but binary this-or-that thinking is harmful on both ends of this debate imo.

**The math drill sheet offers worse feedback.**

When you whack a nail into a board or take a jump shot or press a key, the world returns to you a bounty of information about your efforts. Is the nail crooked? How many swings did you need? Does the ball go in, go long, go short, go sideways? Did the note sound flat or sharp?

If carpentry or basketball or music were like the drill sheet, you’d shoot a shot and the ball would disappear as it left your hand. The nail would evaporate on contact with the hammer. You’d press the piano keys and hear nothing. You’d move to the next shot or swing or song having grown no wiser from the last.

**The math drill sheet is lonely.**

There are *digital* math drills that offer students certain meager amounts of feedback but even here they tend to isolate students in ways that are distinct from the disciplines people namecheck when they defend math drill sheets.

It’s possible to work on a carpentry project by yourself or shoot drills alone in a gym. But those are less common than working *with *someone on a project, participating in group drills, or, at least, observing other people doing *their *drills, gaining inspiration and knowledge from others.

**The math drill sheet conceals the ****performance**** of mathematics.**

When people practice sports or music, they generally do so with a strong understanding of the performance that practice is meant to support. They’ve watched other people play the games or listened to other people play music. They have very likely messed around clumsily on a court or at a piano. They understand its point and like it well enough that the practice feels welcome and necessary.

Define the performance of math however you want and ask yourself, first, how well do your students understand it? Have they watched other people play math? Have they messed around clumsily with math themselves? Do they understand the point of math and like it well enough that the practice feels welcome and necessary?

**We need skill fluency that blends practice and performance.**

When you look at the ways professionals (and even highly engaged non-professionals) develop fluency, they do so in *a blend of practice and performance*.

Read up on some of Steph Curry’s shooting drills. He isn’t standing stock still from different positions in the court, shooting over and over again, assuming (as many do in math) that fluency in those discrete areas will lead to expert performance in a game.

Rather, he’s moving between positions as he would in the performance of basketball. He’s trying to do it quickly because the performance of basketball (unlike the performance of math) is timed and requires speed.

What I appreciate about games like Garbage, which I’m playing with my own kids right now, task structures like Open Middle, ideas like variation theory, task fields, etc, is that they take practice seriously while also insisting that *the practice should testify to the performance*. They realize that even while students are learning lessons about fluency, they are learning other lessons as well, including lessons about the nature of performance.

Math drill sheets—whatever they teach students about operational fluency—also teach students a distorted view of the performance of mathematics.

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