Collective Effervescence Is the Cost of Personalized Learning

Cross-posted from the Desmos blog. I’m happy enough with this post to re-broadcast it here. The Desmos blog doesn’t have comments, also, which makes this a better forum for you to tell me if I’m wrong.


We’re proud to debut our free Classroom Conversation Toolset, which has been the labor of our last three months. You can pause your students’ work. You can anonymize your students’ names. You can restrict the pace of your students through the activity. We believe there are productive and counterproductive ways to use these tools, so let us explain why we built them.

First, the edtech community is extremely excited about personalized learning – students learning at their own pace, uninhibited by their teacher or classmates. Our Activity Builder shares some of that enthusiasm but not all. Until last week, students could click through an activity from the first screen to the last, inhibited by nothing and nobody.

But the cost of personalized learning is often a silent classroom. In the worst-case scenario, you’ll walk into a classroom and see students wearing headphones, plugged into computers, watching videos or clicking multiple choice questions with just enough interest to keep their eyes open. But even when the activities are more interesting and cognitively demanding than video-watching and multiple choice question-clicking, there is still an important cost. You lose collective effervescence.

Collective effervescence is a term that calls to mind the bubbles in fizzy liquid. It’s a term from Émile Durkheim used to describe a particular force that knits social groups together. Collective effervescence explains why you still attend church even though the sermons are online, why you still attend sporting events even though they’re broadcast in much higher quality with much more comfortable seats from your living room. Collective effervescence explains why we still go to movie theaters; laughing, crying, or screaming in a room full of people is more satisfying than laughing, crying, or screaming alone.

An illustrative anecdote. We were testing these features in classes last week. We watched a teacher – Lieva Whitbeck in San Francisco – elicit a manic cheer from a class of ninth-graders simply by revealing the graph of a line. She brought her class together and asked them to predict what they’d see when she turned on the graph. They buzzed for a moment together, predicted a line, and then she gave the crowd what they came for.

She brought them together. She brought back the kids who were a bit ahead and she brought forward the kids who were a bit behind. She de-personalized the learning so she could socialize it. Because arguments are best with other people. Because the negotiation of ideas is most effective when you’re negotiating with somebody. And because collective effervescence is impossible to experience alone.

So these tools could very easily have been called our Classroom Management Toolset. They are useful for managing a class, for pausing the work so you can issue a new prompt or so you can redirect your class. But we didn’t build them for those purposes. We built them to restore what we feel the personalized-learning moment has missed. We built them for conversation and collective effervescence.

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What Should Math Teachers Do When They Don’t Know the Math?

In a comment on my last post, Tracy Zager wrote about a childhood math teacher who responded to one of her questions with, essentially, “Just go with it, Tracy, okay? That’s how math works.”

How do we handle the moment when it becomes clear, in front of the class, that we don’t understand some math we thought we understood? Do we handle it like Mr. Davis did, and view kids’ questions as challenges to our expertise and authority? Or do we say, “You know, your question is making me realize I don’t understand this as deeply as I thought I did. That’s awesome, because now I get to learn something. Let’s figure it out together.”

You don’t transition from a novice teacher to an expert in a day. The transition isn’t obvious and it isn’t stable. You become an expert at certain aspects of teaching before others and some days you regress. But one day you wake up and you realize that certain problems of practice just aren’t consistent problems anymore.

One strong indicator for me that I had changed as a teacher in at least one aspect was when I no longer felt threatened by students who caught me in an error at the board or who asked me a question which I couldn’t quickly answer. I knew some of my Twitter followers would feel the same way, so I asked them a version of Tracy’s question above:

What do we do when it becomes clear, in front of a class, that we don’t understand the math like we thought?

Here are my ten favorite responses. If you have a response that isn’t represented here, please add it to the comments.

BTW. David Coffey has answered the same question about college mathematics, where students are sometimes very unforgiving of mathematical errors and lapses.

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Raymond Johnson:

The most lasting memory from my Modern Algebra I class: It was a Monday, and the instructor was about 20 minutes into his lecture when he got stuck in the middle of a proof. He stopped and stared at the board, then down at his notes, then back at the board, then back at his notes. The class paused their notetaking as the instructor (who was well-respected and always prepared) mumbled and tried to sort things out. After an awkward few moments, he said, “I know there’s something wrong here and I can’t figure it out, and my notes aren’t helping. We really can’t go on before we’ve proven this, so you are all dismissed and we’ll start here again on Wednesday.” We left, returned two days later, and the instructor enthusiastically explained what had caused the problem, how he worked past it, and we moved on. The episode might not have represented great pedagogy, but it was a refreshing example of humility.

Ethan Weker:

I have a space on my whiteboard for questions that come up that I don’t have answers for at the moment. So far this year, I have a couple of favorites:
“What do you call quadrants in 3d?”
“Why do we use p and q in logic? Is it the same origin as ‘Mind your p’s and q’s’?”
Students can find answers or I find answers but either way, it reminds students (and me) that I don’t know it all and I don’t have to.

Elizabeth Raskin:

I hated making mistakes in front of students when i first started teaching. I became conformable with not appearing perfect when my classroom culture transitioned from being myself as the expert and students as the learners to all of us learning from each other.

Mr K:

I’m teaching a small, highly gifted class this year. One of the things we’ve started doing is solving 538’s Riddler each Friday.

For the first couple of weeks, they always looked to me for an answer. It took them a while to realize that I didn’t know it either. Today, while we were doing it, they treated me more as a colleague than as an authority. They’d propose ideas, I’d ask them to justify the ideas, we’d try them out, and decide whether it got us closer to the final answer or not.

It’s really fun modeling my thinking process, and narrating it at the same time. I’ve started identifying when I have interesting things to look at, aha experiences, and most importantly how I test out my suppositions rather than just assuming that they’re correct.


Students who catch my mistakes at the board receive a prize: a mechanical pencil. They become sought-after tokens by the end of the year, and keep students following my reasoning as we work through complex problems!

Diane Way:

In my middle school classroom, I also use 24, WODB, and Set as daily warm ups. Because I don’t “automatically” know the solutions, when students don’t find them, we are able to reason them out together. They observe me trying things out and persevering, and are often inspired to “beat” me which keeps the engagement level high. They all become more comfortable risk takers over time.

Maria Rose offers similar thoughts to Diane’s, right down to the activities they use.

Corey Null:

There is a certain amount of excitement in not knowing. I try to translate that to the students. We wouldn’t be in this game if we didn’t want to know an answer to a question but had no idea where to begin! That’s the beauty of both mathematics and of teaching. Share that enthusiasm for the chase with them. Some questions are unknown to the teacher but easily answered. Others are not. Try your best to answer them, but more than that, try to engage them with your excitement for discovering the unknown.

2016 Oct 18. An excellent companion post from Dan Teague: Demonstrating Competence by Making Mistakes.

Teaching for Tricks or Sensemaking

Here are two approaches to teaching zero exponents that are worth comparing and contrasting:

First, a Virtual Nerd video:

Once you have your non-zero number or variable picked out, put it to the zero power. Now no matter what number or variable you picked, once you put it to the zero power, I know what the answer is. One. How do I know this? Well in math, if we put any non-zero number or variable to the zero power it always equals one. No matter what.

Second, Cathy Yenca’s activity, which has students completing the following table:


One approach will lead students to understand that math is a fragile set of rules that have to be transmitted and validated by adults. The other will help students realize that these rules are strong and flexible, and exist to make math internally coherent, with or without any adults around.

BTW. This is as good a time as any to re-mention Nix the Tricks, the MTBOS’s collection of meaningless math tricks and great strategies for teaching those concepts with meaning instead.

BTW. Check out Cathy Yenca’s own post on the comparison.

BTW. Check out the comments on that YouTube video. Interesting, right? What do we do with that?

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Tracy Zager:

I wrote a story about this exact moment in my book. Cliffs notes version: 8th grade. Mr. Davis told us the rule a^0=1. I questioned why. “Because that’s the rule.” I said, “But why?” Stern voice now. “Because that’s what’s I just told you.” The first boy I’d ever kissed said, “Give it up T! It’s in the book, that’s why!” Everybody laughed. Me too, on the outside. Not on the inside. On the inside I was angry and frustrated and humiliated. If I write my math autobiography, this moment goes in it. It’s one of the moments math lost me.

By the way, my favorite way to see the pattern Cathy shows here is with Cuisinaire rods. You literally go from cubes to squares to rods to single unit squares. It’s this amazing moment to see one as the fundamental unit.

But I didn’t get to see that in 8th grade. I didn’t see that until my late 30s.

The saddest part? Most kids don’t try again after they’re burned. They never come back. The not-quite-saddest-part-but-still-sad-part? I doubt Mr. Davis ever learned why either.

That’s the biggest legacy of that story to me, as a teacher. How do we handle the moment when it becomes clear, in front of the class, that we don’t understand some math we thought we understood? Do we handle it like Mr. Davis did, and view kids’ questions as challenges to our expertise and authority? Or do we say, “You know, your question is making me realize I don’t understand this as deeply as I thought I did. That’s awesome, because now I get to learn something. Let’s figure it out together.”

Kent Haines:

Something that has been effective for me (that I mention in the video above) is to really emphasize that 1 is the origin and invisible starting point of all multiplication problems.

Scott Farrand offers another helpful way for students to make sense of zero exponents:

There’s a truly great old lesson on exponentiation that I believe comes from Project SEED, that has been used with amazing success in hundreds of elementary classrooms.

education realist brings up a Ben Orlin post that includes a) a beautiful technique for teaching lots of rules of exponentiation and b) this beautiful paragraph:

Math’s saving grace, though, is that it can make us feel smart for another reason: because we’ve mastered an ancient, powerful craft. Because we’ve laid down rails of logic, and guided a train of thought smoothly to its destination. Because we’re masters—not over our peers, but over the deep patterns of the universe itself.

The Desmos Guide to Building Great (Digital) Math Activities

[cross-posted to the Desblog]

We wrote an activity building code for two reasons:

  1. People have asked us what Desmos pedagogy looks like. They’ve asked about our values.
  2. We spend a lot of our work time debating the merits and demerits of different activities and we needed some kind of guide for those conversations beyond our individual intuitions and prejudices.

So the Desmos Faculty – Shelley Carranza, Christopher Danielson, Michael Fenton, me – wrote this guide. It has already improved our conversations internally. We hope it will improve our conversations externally as well, with the broader community of math educators we’re proud to serve.

NB. We work with digital media but we think these recommendations apply pretty well to print media also.

Incorporate a variety of verbs and nouns. An activity becomes tedious if students do the same kind of verb over and over again (calculating, let’s say) and that verb results in the same kind of noun over and over again (a multiple choice response, let’s say). So attend to the verbs you’re assigning to students. Is there a variety? Are they calculating, but also arguing, predicting, validating, comparing, etc? And attend to the kinds of nouns those verbs produce. Are students producing numbers, but also representing those numbers on a number line and writing sentences about those numbers?

Ask for informal analysis before formal analysis. Computer math tends to emphasize the most formal, abstract, and precise mathematics possible. We know that kind of math is powerful, accurate, and efficient. It’s also the kind of math that computers are well-equipped to assess. But we need to access and promote a student’s informal understanding of mathematics also, both as a means to interest the student in more formal mathematics and to prepare her to learn that formal mathematics. So ask for estimations before calculations. Conjectures before proofs. Sketches before graphs. Verbal rules before algebraic rules. Home language before school language.

Create an intellectual need for new mathematical skills. Ask yourself, “Why did a mathematician invent the skill I’m trying to help students learn? What problem were they trying to solve? How did this skill make their intellectual life easier?” Then ask yourself, “How can I help students experience that need?” We calculate because calculations offer more certainty than estimations. We use variables so we don’t have to run the same calculation over and over again. We prove because we want to settle some doubt. Before we offer the aspirin, we need to make sure students are experiencing a headache.

Create problematic activities. A problematic activity feels focused while a problem-free activity meanders. A problem-free activity picks at a piece of mathematics and asks lots of small questions about it, but the larger frame for those smaller questions isn’t apparent. A problem-free task gives students a parabola and then asks questions about its vertex, about its line of symmetry, about its intercepts, simply because it can ask those questions, not because it must. Don’t create an activity with lots of small pieces of analysis at the start that are only clarified by some larger problem later. Help us understand why we’re here. Give us the larger problem now.

Give students opportunities to be right and wrong in different, interesting ways. Ask students to sketch the graph of a linear equation, but also ask them to sketch any linear equation that has a positive slope and a negative y-intercept. Thirty correct answers to that second question will illuminate mathematical ideas that thirty correct answers to the first question will not. Likewise, the number of interesting ways a student can answer a question incorrectly signals the value of the question as formative assessment.

Delay feedback for reflection, especially during concept development activities. A student manipulates one part of the graph and another part changes. If we ask students to change the first part of the graph so the second reaches a particular target value or coordinate, it’s possible – even likely – the student will complete the task through guess-and-check, without thinking mathematically at all. Instead, delay that feedback briefly. Ask the student to reflect on where the first part of the graph should be so the second will hit the target. Then ask the student to check her prediction on a subsequent screen. That interference in the feedback loop may restore reflection and meta-cognition to the task.

Connect representations. Understanding the connections between representations of a situation – tables, equations, graphs, and contexts – helps students understand the representations themselves. In a typical word problem, the student converts the context into a table, equation, or graph, and then translates between those three formats, leaving the context behind. (Thanks, context! Bye!) The digital medium allows us to re-connect the math to the context. You can see how changing your equation changes the parking lines. You can see how changing your graph changes the path of the Cannon Man. “And in any case joy in being a cause is well-nigh universal.”

Create objects that promote mathematical conversations between teachers and students. Create perplexing situations that put teachers in a position to ask students questions like, “What if we changed this? What would happen?” Ask questions that will generate arguments and conversations that the teacher can help students settle. Maximize the ratio of conversation time per screen, particularly in concept development activities. All other things being equal, fewer screens and inputs are better than more. If one screen is extensible and interesting enough to support ten minutes of conversation, ring the gong.

Create cognitive conflict. Ask students for a prediction – perhaps about the trajectory of a data set. If they feel confident about that prediction and it turns out to be wrong, that alerts their brain that it’s time to shrink the gap between their prediction and reality, which is “learning,” by another name. Likewise, aggregate student thinking on a graph. If students were convinced the answer is obvious and shared by all, the fact that there is widespread disagreement may provoke the same readiness.

Keep expository screens short, focused, and connected to existing student thinking. Students tend to ignore screens with paragraphs and paragraphs of expository text. Those screens may connect poorly, also, to what a student already knows, making them ineffective even if students pay attention. Instead, add that exposition to a teacher note. A good teacher has the skill a computer lacks to determine what subtle connections she can make between a student’s existing conceptions to the formal mathematics. Or, try to use computation layer to refer back to what students already think, incorporating and responding to those thoughts in the exposition. (eg. “On screen 6, you thought the blue line would have the greater slope. Actually, it’s the red line. Here’s how you can know for sure next time.”)

Integrate strategy and practice. Rather than just asking students to solve a practice set, also ask those students to decide in advance which problem in the set will be hardest and why. Ask them to decide before solving the set which problem will produce the largest answer and how they know. Ask them to create a problem that will have a larger answer than any of the problems given. This technique raises the ceiling on our definition of “mastery” and it adds more dimensions to a task – practice – that often feels unidimensional.

Create activities that are easy to start and difficult to finish. Bad activities are too difficult to start and too easy to finish. They ask students to operate at a level that’s too formal too soon and then they grant “mastery” status after the student has operated at that level after some small amount of repetition. Instead, start the activity by inviting students’ informal ideas and then make mastery hard to achieve. Give advanced students challenging tasks so teachers can help students who are struggling.

Ask proxy questions. Would I use this with my own students? Would I recommend this if someone asked if we had an activity for that mathematical concept? Would I check out the laptop cart and drag it across campus for this activity? Would I want to put my work from this activity on a refrigerator? Does this activity generate delight? How much better is this activity than the same activity on paper?

The Explanation Difference

Brett Gilland coined the term “mathematical zombies” in a comment on this blog:

Students who can reproduce all the steps of a problem while failing to evidence any understanding of why or how their procedures work.

When I think about mathematical zombies, I think about z-scores – how easy it is to calculate them relative to how difficult it is to explain those calculations.

Check it out. Here is the formula for a z-score:


In words:

1. You subtract the mean from your sample.
2. You divide that by the standard deviation.

Subtraction and division. Operations simple enough for a elementary schooler. But the explanation of those operations – why they result in a z-score, what a z-score is, and when you should use a z-score – is so challenging it eludes many graduates of high school statistics. Think about how easily you could solve these exercises without knowing what you’re doing.

That difference brings this chart to mind and helps me understand all of the times I’m tempted to just tell students, here’s how you do it already so now just do it. That’s where the operational shortcuts are most tempting.


All of this is preface to a lesson plan on hypothesis testing by Jeremy Strayer and Amber Matuszewski, which is one of the best I’ve read all year.

Hypothesis testing is, again, one of those skills that’s far easier to do than to understand. As you read the lesson plan, please keep in mind that difference. Also notice how capably the teachers develop the question, disclosing the mathematics progressively, and resisting the temptation to shortcut their way to operational fluency.


It’s spectacular. I’m struck every time by a moment where Strayer and Matuszewski ask students to model an experiment with playing cards, only to model the exact same experiment with a computer later. They didn’t just jump straight to the computer simulation!

Here is a video of an airline pilot landing an Airbus A380 in a crosswind. This is that for teachers.

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I always think of z-scores as a set of transformations from one plain-vanilla normal curve to the hot-fudge-sundae Standard Normal curve. Maybe once you see it this way, you can’t unsee it. To me, that helps make sense of the “why” you would bother standardizing and the “how” it’s done.

David Griswold:

I’m not sure I agree that z-score is so conceptually difficult as to be worth the shortcut. Though I suppose it requires understanding of standard deviation, which is kind of hard. But if you think of standard deviation as “typical weirdness distance” then z-score as the idea of “how many times the typical weirdness is this point” becomes pretty straightforward. A z-score magnitude of 1 becomes average weirdness, less than 1 becomes less weird than average, etc. The bigger the magnitude of the z-score, the weirder the point.

Bob Lochel:

In introductory stats courses, much of what we do simply comes down to separating “Is it possible?” from “Is it plausible?”. We have seen a wonderful growth in the number of free, online applets which allow teachers and students to perform simulations designed to assess these subtly different questions.