## 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 â€“Â Shelley Carranza, Christopher Danielson, Michael Fenton, me â€“Â 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, letâ€™s say) and that verb results in the same kind of noun over and over again (a multiple choice response, letâ€™s say). So attend to the verbs youâ€™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. Itâ€™s also the kind of math that computers are well-equipped to assess. But we need to access and promote a studentâ€™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, â€œWhy did a mathematician invent the skill Iâ€™m trying to help students learn? What problem were they trying to solve? How did this skill make their intellectual life easier?â€ Then ask yourself, â€œHow can I help students experience that need?â€ We calculate because calculations offer more certainty than estimations. We use variables so we donâ€™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 â€œland the plane.â€ 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, itâ€™s possible â€“ even likely â€“ 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 canâ€™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 â€“ tables, equations, graphs, and contexts â€“ 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. â€œAnd in any case joy in being a cause is well-nigh universal.â€

• 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, â€œWhat if we changed this? What would happen?â€ 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 â€“ 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 itâ€™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 studentâ€™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. â€œOn screen 6, you thought the blue line would have the greater slope. Actually, itâ€™s the red line. Hereâ€™s how you can know for sure next time.â€)

• In Game, Set, Flat, we ask students to create a bad tennis ball, one that doesnâ€™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 â€œmasteryâ€ and it adds more dimensions to a task â€“ practice â€“ that often feels unidimensional.

• In Smallest Solution, we donâ€™t ask students to solve a long list of linear equations. Instead we ask them to create an equation that has a solution thatâ€™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.

I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. He / him. More here.

1. #### David Wees

September 15, 2016 - 10:16 am -

1. I love that you folks at Desmos are working on getting this right. Norming around this kind of idea in our own project’s work has been gigantic for our team. It’s super important work that is, unfortunately, not done often enough.

2. I think you might want to check out the Mathematical Quality of Instruction rubric, not because the rubric itself is especially helpful in your work directly, but because it does a decent job of naming a bunch of different elements of good teaching some of which you have included in your work above, and others which you have not. Maybe the elements you have not included don’t make sense in the context of a Desmos activity, maybe they do?

3. I notice from reading your description above that if I were to draw a Venn diagram representing students, mathematics, and teachers, that once a task is designed all of the work above looks like it resides in the students and mathematics circles and none in the teacher circles. So I wonder; while an activity is running, what should a teacher do based on your description above?

2. #### Dan Meyer

September 15, 2016 - 3:48 pm -

Thanks for the passing along the resources, David.

I notice from reading your description above that if I were to draw a Venn diagram representing students, mathematics, and teachers, that once a task is designed all of the work above looks like it resides in the students and mathematics circles and none in the teacher circles. So I wonder; while an activity is running, what should a teacher do based on your description above?

Two notes there. First, if you run back and search for “teacher” in the document, I think you’ll find some significant notes on how we hope teachers will use our technology. Second, our conversations on how teachers should use our activities tend to converge much faster than our conversations about whether an activity is any good or not. Meaning we haven’t felt a lot of internal need for the document you propose, though that doesn’t mean it wouldn’t be useful externally. We’ll kick the idea around.

3. #### Mark Hansen

September 15, 2016 - 4:47 pm -

Thank you Dan. At our primary school in Australia we have an awareness of these guidelines, but always good to have a written version to assist with discussions and verify how we are travelling with respect to them. Particularly like the focus on mathematical conversation you have spoken of. Regarding David’s query, whilst our students are tackling these tasks, our role moves to that of facilitator, and questions we expect to hear include:
“Was your method the most efficient?
“How do you know you’re right?”
“When would your strategy not work?
“What is the same/different?”
“What if…”
We would also expect teachers to differentiate for students when required, which is always easier if there is that low floor/high ceiling Dan has spoken of. Rules for our students during this problem solving time include:
1. No wrong thinking
2. You must have a go
3. You can copy/change your mind but must tell us why
4. You can work with who you like (or on your own)
5. You may use anything to assist you (just ask a teacher)
6. Donâ€™t rub anything out- let us see your thinking

4. #### Dan Meyer

September 15, 2016 - 8:40 pm -

@Mark, thanks for passing along your school’s own guidelines. I could read sets of these all night.

@Scott, that’d be a worthwhile task. Let me get my team on it.

5. #### Robert

September 16, 2016 - 12:16 pm -

Desmos’ user interface is generally excellent for doing all kinds of computation stuffs that can hardly be done on the top of the head. In my opinion the minimalistic design makes it more flexible for designing activities that are more straight to the point.

Interesting that you talk about activities that are easy to start and hard to finish. One can imagine how Desmos Art probably falls into this category.

The informal analysis over formality is actually a great point too, as people from the higher maths background tend to disregard it a bit and let the rigor takes over.

6. #### Cathy

September 16, 2016 - 1:53 pm -

Thanks so much for sharing these! When “Team Desmos” edits an activity “with love” these guidelines help clarify the team’s “lens”, which helps teachers see why an edited activity is better… and ideally, helps teachers create better activities for our students.

One thought – I hope teachers feel as though they have liberties to create mini-experiences with Desmos as well as the beefy ones that follow many/most/all of these guidelines. Sometimes a little warm-up can go a long way.

Appreciate all that Team Desmos is doing to improve math education globally!

7. #### Roberto Catanuto

September 17, 2016 - 6:23 am -

Thank you for these advises, they look great.
Isn’t it time to start a mooc about “Math Education through Digital Media” from the Desmos faculty?
It’d be a success…

8. #### Ayanna Ramsey

September 17, 2016 - 5:12 pm -

This is a great resource and definitely makes me REALLY think about the purpose of each and every screen: “what is it I really want students to be able to do or gain from each part of a task”. This is a must share with colleagues!

9. #### Levi Patrick

September 19, 2016 - 4:20 am -

I am a firm believer that creating (and reiterating) a well-framed pedagogical perspective is critical to our pursuit of excellent teaching. I love what you’ve done here, certainly because it adds new wrinkles into my thinking, but mostly because the conversation for educators should often return to focus on how we believe students come to a deep understanding of mathematics. We have to grasp that and then work to both confirm our thinking and test the limits of our hypotheses every time we select, design, enact, or reflect on a instructional moment.

As always, you and the Desmos team have done an excellent job moving the conversation forward and focusing it on the essential components of what it takes to intentionally be an excellent mathematics educator–thanks.

10. #### Xavier B.

September 19, 2016 - 9:38 am -

Can someone give us some (physical) activities conforming this guides?

@Dan: good brand new wordpress theme ;-)

11. #### Jen Silverman

September 24, 2016 - 6:36 am -

Thanks for this, team Desmos. It’s great to have all of these points of reflection in one place. I’m bookmarking it for sure.