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## [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.

First

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!

Second

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.

Third

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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## The Desmos Guide to Building Great (Digital) Math Activities

[cross-posted to the Desblog]

We wrote an activity building code for two reasons:

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?

• Match My Line is an activity for practicing graphing but we also ask students to sketch, settle a dispute, and analyze.

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.

• In Lego Prices, we eventually ask students to do formal, precise work like calculating and graphing. But before that, we ask students to estimate an answer and to sketch a relationship.

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.

• In Picture Perfect, students can either calculate answers numerically across dozens of problems, or solve the problem once algebraically.

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.

• In Land the Plane, the very first screen asks students to “land the plane.” We try to keep that central problem consistent and clear throughout the activity.

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 the graph of 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.

• In Graphing Stories, we ask students to sketch the relationship between a variable and time. Their sketches often reflect features of the context that other students missed, and vice versa.

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.

• Our Marbleslides series offers students lots of opportunities for dynamic trial-and-error, for manipulating slopes and intercepts one tenth of a unit at a time until they collect all of the stars. But we also offer several static reflection questions where students can’t manipulate the graph before answering. We give them the chance to check their work only after they commit to an answer.

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.”

• In addition to the examples above, Marcellus the Giant invites students to alter the graph of a proportional relationship. Then students see the effect of that altered graph on the giant the graph was describing. We connect the graph and the giant.

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.

• Our Card Sort activities feature only a handful of screens but offer students and teachers an abundance of opportunities to discuss both early and mature ideas about mathematics.

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.

• Charge! presents a relationship, a cell phone charging over time, that seems quite linear. As the cell phone completes its charge, the rate of charge slows down considerably, confounding student expectations and preparing them to learn.

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.”)

• In Game, Set, Flat, we ask students to create a bad tennis ball, one that doesn’t bounce right at all. Then we tailor our explanation of exponential models to their tennis ball, explaining how it violates assumptions of exponential models and how they can fix it.

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.

• In Smallest Solution, we don’t ask students to solve a long list of linear equations. Instead we ask them to create an equation that has a solution that’s as close to zero as they can make it.

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.

• In the conclusion of Water Line, after graphing the rising water line in several glasses we provide, we ask students to create their own glass. Then that glass goes into a shared classroom cupboard, giving students many more challenges to complete.

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?

2016 May 11. Updated to add references to activities.