Category: tech enthusiasm

Total 103 Posts

[Desmos Design] Why We’re Suspicious of Immediate Feedback

One of our design principles at Desmos is to “delay feedback for reflection, especially during concept development activities.” This makes us weird, frankly, in Silicon Valley where no one ever got fired for promising “immediate feedback” in their math edtech.

We get it. Computers have an enormous advantage over humans in their ability to quickly give students feedback on certain kinds of work. But just because computers can deliver immediate feedback doesn’t mean they always should.

For example, Simmons and Cope (1993) found that students were more likely to use procedural strategies like trial-and-error in a condition of immediate feedback than a condition of delayed feedback.

I think I can illustrate that for you with this activity, which has two tasks. You get immediate feedback on one and delayed feedback on the other.

I’ll ask you what I asked 500 Twitter users:

How was your brain working differently in the “Circle” challenge [delayed feedback] than the “Parabola” challenge [immediate feedback]?

Exhibit A:

The circle one was both more challenging and fun. I found myself squinting on the circle to visualize it in my head while with the parabola I mindlessly did trial and error.

Exhibit B:

With the circle, the need to submit before seeing the effect made me really think about what each part of the equation would effect the graph in each way. This resulted in a more strategic first guess rather than a guess and check approach.

Exhibit C:

I could guess & check the parabola challenge. In the circle challenge I had to concentrate more about the center of the circle and the radius. Much more in fact.

Exhibit D:

I couldn’t use trial and error. I had to visualize and estimate and then make decisions. My brain was more satisfied after the circle.

Exhibit E:

I probably worked harder on [the circle] because my answer was not shown until I submitted my answer. It was more frustrating than the parabola problem – but I probably learned more.

This wasn’t unanimous, of course, but it was the prevailing sentiment. For most people, the feedback delay provoked thoughtfulness where the immediate feedback provoked trial-and-error.

We realize that the opposite of “immediate feedback” for many students is “feedback when my teacher returns my paper after a week.” Between those two options, we side with Silicon Valley’s preference for immediate feedback. But if computers can deliver feedback immediately, they can also deliver feedback almost immediately, after a short, productive delay. That’s the kind of feedback we design into our concept development activities.

BTW. For a longer version of that activity, check out Building Conic Sections, created by Dylan Kane and edited with love by our Teaching Faculty.

[Desmos Design] Algebra Is Power, Not Punishment

This is the first of several posts where I’ll use the activities Desmos created last quarter to illustrate our design principles.

One of those principles is:

Create an intellectual need for new mathematical skills.

Nowhere is that principle more necessary, in our view, than in the instruction of algebraic expressions. Three of my least favorite words in the English language are “write an expression” because they so often mean we’re asking the student to do the difficult work of variable manipulation without experiencing any of the fruit of that work.

In both of the questions below, students are likely to experience the work of writing an expression as punishment, not power.

Given the width of the lawnmower (W) and the length of the rope (L), write an expression for the pole radius (R) that will make the lawnmower cut the lawn in a perfect spiral.

Given the width of the pool in tiles (n), write an expression for the number of tiles that will fit around the pool border.

We recognize that one reason variables give us power is that they let us complete lots of versions of the same task quickly and reliably. So in our version of both of the above problems, we asked students first to work numerically, both to acclimate them to the task, but especially to establish the feeling that, “Okay, doing a lot of these could get tedious.”

And then we use their expression to power ten pool borders.

And ten lawnmowers.

Those activities are Pool Border Problem and Lawnmower Math. In Picture Perfect, for another example of this principle, we give students the option of either a) filling in a table with 24 rows, or b) writing an algebraic expression once.

In each case, students are more likely to see algebra as power than punishment.

Here Are Ten New Desmos Activities

The Desmos quarter just ended and this was a huge one for the team of teachers I support.

First, we made substantial upgrades to our entire activity pool. Second, we released ten new activities in the same amount of time it took us to release one activity two years ago. This is all due to major improvements to our technology and our pedagogy.

Technologically, our engineers created a powerful scripting language that hums beneath our activities, enabling us to set up more meaningful interactions between teachers, students, and mathematics.

Pedagogically, my teaching team has spent the last year refining our digital mathematics pedagogy through daily conversations, lesson pitches, lesson critiques, summary blog posts, occasional lunch chats with guests like the Khan Academy research team, and frequent consultation with our Desmos Fellows.

The result: we cut an activity pool that once comprised 300 pretty good activities down to 127 great ones, and we gave each one of those 127 a serious upgrade, making sure they took advantage of our best technology and pedagogy. Then we added ten more.

I don’t think I’ve learned as much or worked as hard in a three-month span since grad school, and I owe a debt of gratitude to my team – Shelley Carranza, Christopher Danielson, and Michael Fenton – for committing the same energy. Also, it goes without saying that none of our activity ideas would have been possible without support from our engineers and designers.

In future posts, I’ll excerpt those lessons to illustrate our digital pedagogy. For today, I’ll just introduce the activities themselves.

Picture Perfect

Hang loads of pictures precisely and quickly using arithmetic sequences.

Game, Set, Flat

Your shipment of tennis balls has been contaminated. Use exponential functions to find the bad ones.

Graphing Stories

Graphing Stories comes to Desmos just in time for its tenth birthday.

Pool Border Problem

One of the oldest and best problems for exploring algebraic equivalence. We wouldn’t have touched it if we didn’t think we had something to add.

Laser Challenge

Use your intuition for angle measure to bounce lasers off mirrors and through targets.

Lawnmower Math

Use Algebra and the properties of circles to help you mow ten lawns automatically and quickly.

Land the Plane

Use linear equations to land airplanes safely and precisely.

Circle Patterns

Practice circle equations by completing artistic patterns.

Constructing Polynomials

Develop your understanding of the behavior of polynomial graphs by creating them piece by piece, factor by factor.

What’s My Transformation?

This is my favorite introduction to the concept of a transformation. “Actually, there’s really only one parabola in the world – we just move it around to make new ones.”

We are still testing these activities. They are complete, but not complete complete, if you know what I mean. You won’t find all of them in our search index yet. We welcome your feedback.

New Activity: Marcellus the Giant


In Marcellus the Giant, the new activity from my team at Desmos, students learn what it means for one image to be a “scale” replica of another. They learn how to use scale to solve for missing dimensions in a proportional relationship. They also learn how scale relationships are represented on a graph.

There are three reasons I wanted to bring this activity to your attention today.


Marcellus the Giant is the kind of activity that would have taken us months to build a year ago. Our new Computation Layer technology let Eli Luberoff and me build it in a couple of weeks. We’re learning how to make better activities faster!


When we offer students explicit instruction, our building code recommends: “Keep expository screens short, focused, and connected to existing student thinking.”

It’s hard for print curricula to connect to existing student thinking. Those pages may have been printed miles away from the student’s thinking and years earlier. They’re static.

In our case, we ask students to pick their own scale factor.


Then we ask them to click and drag and try to create a scale giant on intuition alone. (“Ask for informal analysis before formal analysis.”)


Then we teach students about proportional relationships by referring to the difference between their scale factor and the giant they created.

You made Marcellus 3.4 times as tall as Dan but you dragged Marcellus’s mouth to be 6 times wider than Dan’s mouth. A proportional giant would have the same multiple for both.

Our hypothesis is that students will find this instruction more educational and interesting than the kind of instruction that starts explaining without any kind of reference to what the student has done or already knows.

That’s possible in a digital environment like our Activity Builder. I don’t know how we’d do this on paper.


Marcellus the Giant allows us to connect math back to the world in a way that print curricula can’t.

Typically, math textbooks offers students some glimpse of the world – two trains traveling towards each other, for example – and then asks them to represent that world mathematically. The curriculum asks students to turn that mathematical representation into other mathematical representations – for instance a table into a graph, or a graph into an equation – but it rarely lets students turn that math back into the world.


If students change their equation, the world doesn’t then change to match. If the student changes the slope of the graph, the world doesn’t change with it. It’s really, really difficult for print curriculum to offer that kind of dynamic representation.

But we can. When students change the graph, we change their giant.


There is lots of evidence that connecting representations helps students understand the representations themselves. Everyone tries to connect the mathematical representations to each other. Desmos is trying to connect those representations back to the world.

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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