Category: tech enthusiasm

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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:

  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?

Larry Cuban’s Eight Features of Highly-Effective Tech-Using Schools

Larry Cuban has spent the last year observing and documenting the practices of schools that are known for successful technology implementation.

Here are eight different yet interacting moving parts that I believe has to go into any reform aimed at creating a high-achieving school using technology to prepare children and youth to enter a career or complete college (or both).

Notably, none of them are explicitly about technology.

Who Wore It Best: Maximizing Area

I have a recurring happy dream that I’m on Jeopardy. It’s the final round. The Trebekbot 2000 reads the final clue:

“These are the dimensions of the rectangle that has the largest area given a fixed perimeter.”

“WHAT IS A SQUARE!” I yell out while my competitors are still thinking quietly. I have disqualified myself and ruined the round, but I don’t care. I start high-kicking around the set while security tries to wrangle me away and I still don’t care because I finally found some use for this fact that takes up a significant chunk of my brain’s random access memory.

It’s a question you’ll find in every quadratics unit, every textbook, everywhere. I could have selected this week’s Who Wore It Best contestants from any print textbook, but instead I’d like to compare digital curricula. I have included links and attachments below to versions of the same task from GeoGebra, Desmos, and Texas Instruments, three thoughtful companies all doing interesting work in math edtech. (Disclosure: I work for Desmos, but don’t let that fact sweeten your remarks about the Desmos version or sour your remarks about the others. Just be thoughtful.)

So: who wore it best?

Click each image for the full version.

Version #1 – GeoGebra


Version #2 – Desmos


Version #3 – Texas Instruments


Steve Phelps suspects I stacked the deck in favor of Desmos here, taking full advantage of our platform while taking only partial advantage of GeoGebra and the Nspire. John Golden concurs, hypothesizing that “there would be a worksheet to go with the GeoGebra sketch.”

So a note on sampling: the GeoGebra example is the most viewed lesson on the subject I could find at their Materials site. The Texas Instruments lesson is the only lesson on the subject I could find at their Activities site. I told Steve, and I’ll tell you, that if anybody can come up with a better lesson on either platform. I’ll be happy to feature it. This isn’t much fun for me (or useful to Desmos) if I stack the deck.

Both Lisa Bejarano and John Golden call out the Desmos lesson as “too helpful” – they know how to make it sting – in the transition from screen 5 (“Collecting data!”) to screen 6 (“Here! We’ll represent the data as a graph for you.”).


I’l grant that it seems abrupt. I don’t think this kind of help is necessarily counterproductive, but it doesn’t seem as though we’ve developed the question well enough that the answer – “graph the data!” – is sensible. The Texas Instruments version has a solution to that problem I’ll attend to in a moment.

My concern with the GeoGebra applet is that the person who made the applet has done the most interesting mathematical thinking. I love creating Geogebra applets. I generally don’t have a good story for what students do with those applets, though. In this example, I suspect the student will drag the slider backwards and forwards, watching for when the numbers go from small to big and then small again, and then notice that the rectangle at that point is a square. The person who made the applet did much more interesting work.

Let me close with one item I prefer about the Desmos treatment and one item I prefer about the Texas Instruments treatment.

First, my understanding of Lisa Kasmer’s research into estimation and Paul Silvia’s research into interest led me to create this screen where I ask students “Which of these three fields has the biggest perimeter?” knowing full well they all have the same perimeter:


Still later, I ask students to estimate a rectangle they think will have the greatest area. That kind of informal cognitive work is largely absent from the TI version, which starts much more formally by comparison.


TI does have a technological advantage when they allow students to sample lots of rectangles and quickly capture data about those rectangles in a table.


Desmos is working on its own solution there, but for now, we punt and include prefabricated data, which I think both companies would agree is less interesting, less useful, and more abrupt, as I mentioned above.

That’s my analysis of these three computer-based approaches to the same problem. What’s your analysis? And it’s also worth asking, “Would a non-computer-based approach be even better?” Is the technology just getting in the way of student learning?

You can also pitch your thoughts in on next week’s installment: Pool Table Math.

2016 Jul 8. Steve Phelps has created a different GeoGebra applet, as has Scott Farrar.

2016 Jul 9. Harry O’Malley uploads another GeoGebra interpretation, one that strikes a very interesting balance between print and digital media.

“The Cup Is the Y-Intercept”

Are your students overgeneralizing their models? After working exclusively with proportional relationships for the last month, are they describing every new relationship as proportional?

This isn’t a task, or a lesson, or anything of that scope. It’s a resource, a provocation, one that gives students the chance to check their assumptions about what’s going on.

Play this video and pause it periodically, asking students to decide for themselves, and then tell a neighbor, what’s coming next.


10 marbles weigh 350 grams. So 20 marbles should weigh how much? I’m curious which students will say the answer is less than, exactly, or more than 700 grams. I’m curious which students will say it’s impossible to know.

Reveal the answer.


That will be surprising for some. Now invite them to speculate about 30 marbles. 40 marbles. And 0 marbles.

Let me end with three notes.

First, my thanks to Kevin Hall who had the fine idea for the video and encouraged me to make it. I’ve never met Kevin. That’s the kind of internet collaboration that makes my week.

Second, the stacking cups lesson offers a similar moment of dissonance. Can you find it?

Third, here’s Hans Freudenthal on technology in 1981:

What I seek is neither calculators and computers as educational technology nor as technological education but as a powerful to arouse and increase mathematical understanding.

Featured Tweet

Michael Jacobs:

I always like creating a proportional reasoning speed bump by giving these types of questions.



Featured Comment

Kate Nowak:

Hey! Nice idea for helping kids make the turn from proportional to linear relationships. There were two things I wanted to change:

• the discrete nature of the domain
• the way it’s not clear in the still images whether we are being shown the mass of just the marbles or the mass of the marbles + the glass together (the brief shot of the balance scale with the glass on it at the beginning of the video wasn’t doing it for me).

So I made a video! Here! It was shot on my phone using a jar of cumin to stabilize, so it could certainly be professionalized.

The Difference Between Sketching and Graphing

Here is what I mean. Ask a student to:

Give an algebraic function whose graph has one positive root, a negative y-intercept, and an asymptote at x = -5, if that’s possible. If it’s impossible, explain why you can’t.

Maybe the student can determine the function. At some point, an advanced algebra student should determine the function. But what do I learn from a student who can’t determine the function? What does a blank graph tell me?


The student might understand what roots, intercepts, and asymptotes are. She might understand every part of the task except how to form the function algebraically. I won’t know because I’m asking a very formal task.

This is why a lot of secondary math teachers ask a less formal question first. They ask for a sketch.

Sketch a function whose graph has one positive root, a negative y-intercept, and an asymptote at x = -5, if that’s possible. If it’s impossible, explain why you can’t.


Think of what I know about the student that I didn’t know before. Think of the feedback that’s available to me now that wasn’t before.

Desmos just added sketching into its Activity Builder. That was the result of months of collaboration between our design, engineering, and teaching teams. That was also the result of our conviction that informal mathematical understanding is underrepresented in math classes and massively underrepresented in computer-based mathematics classes. We want to help students express their mathematical ideas and get feedback on those ideas, especially the ones that are informal and under development. That’s why we built sketch before multiple choice, for example. I’m stating this commitment publicly, hoping that one or more of you will help us live up to it.