Author Archives: dkane47

Task Propensity: Pause

Task propensity refers to situations where students are so focused on the features of a specific task that they don’t generalize their thinking in a way that is useful to solve different problems in the future. In short, they lose the forest for the trees. I’m exploring how task propensity relates to Desmos activities and how this thinking could help me teach more thoughtfully with Desmos tools. I first learned about task propensity through this paper, and you can read the rest of my series on the topic here.

I’ve been having a ton of fun with this series on task propensity. It’s fun because I love Desmos activities. I find them a joy to teach, I find them useful for student learning, and they let me do things that aren’t possible through other means. I also love thinking critically about where these activities are the right tools and where something else would work better. I probably spend about 15% of class time on Desmos activities, and that feels about right to me.

In my last two posts, I was critical of how Desmos activities fall into the trap of task propensity, and how keeping activities humble and designing deliberate follow-up tasks can help to avoid this challenge. In this post, I want to talk about the pause button — something that the folks at Desmos have baked into their interface and designed for student learning.

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When the folks at Desmos talk about the pause button, they talk about classroom conversations and 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.

There’s a ton of collective effervescence in Desmos activities. Take an activity I did recently, where students practice graphing sine and cosine functions by writing equations for different curves. There’s not much to it; it’s just meant to allow some low-floor practice early in the unit as we begin to formalize an understanding of how these functions work. Students like it a lot. It’s easy to experiment and change different parts of the function, and the challenges feel just within their reach.

Early in the activity, I noticed that most groups were using guess and check strategies to figure out the phase shifts. Take this screen:

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One solution is to use a negative amplitude and then shift the function down. However, many groups end up trying to use a phase shift instead. This is a great opportunity for students to make connections through a quick discussion. Even better, some students write their phase shift in terms of pi, while others guess and check their way using decimals. Lots to draw on here.

Of course, at that moment there’s a ton of collective effervescence in the room. Students are engaged in the activity and in working collaboratively. It’s hard to tear them away. That’s where the pause button comes in.

There’s something really unique about the energy in the room when an activity is paused. There are often groans, and students wish they could keep working. It takes a moment to let that work its way out of their systems. The collective effervescence remains, but the task propensity — the feeling that students are “in it”, focused on meeting the challenges and nothing else — is put on pause as well. That excitement for math is great, but as students try to match the functions they see, they aren’t thinking about connections between different representations, the equivalence of multiple functions, or other big ideas of trigonometric functions I want them to learn. They’re stuck on the task in front of them.

The pause button is my chance to change their focus, to start a conversation with all the energy and engagement in the room that takes a step back, thinks more broadly about the math in front of students, and thinks about how some of the thinking that they’re doing could be applied in new contexts in the future.

When the pause button was added to the activity dashboard, I rarely used it. I felt like I didn’t want to interrupt student thinking and tear them away from what they wanted to do — from their goal of completing an activity they found enjoyable and challenging. I’ve shifted since then to see opportunities to redirect student thinking as the heart of my role as a teacher in that moment. My goal for a Desmos activity isn’t for students to get really good at Desmos. It’s for students to get really good at math. That means I need students to step out of the task, and to think metacognitively about what they’re doing and how it connects with what they already know and what they’ll do in the future. The pause button is by far the best tool I have for doing that.

Global Math Week

Next week my students and I will be participating in Global Math Week. The goal of the project is for one million students around the world to have a common, joyous, and uplifting experience with math. Over 750,000 students from over 100 countries are already registered. You should too! The topic for this first year is exploding dots, which you can learn more about on James Tanton’s GDay Math website. I want to describe roughly what I plan to do, to help any teachers out there who may be on the fence or unsure how to proceed. I teach upper high school so take that as useful context, but I think the heart of the math works down to upper elementary and could absolutely be adapted for lower grades.

First, I plan to share the video at the bottom of this page to introduce the idea of exploding dots, and have students try to figure out the question at the end of the video.


Watch it if you haven’t seen it already! It’s a great, perplexing introduction to the principles of exploding dots as students work together to figure out what the number is. Depending on how that goes, I may give them a few more or just move on if they get it quickly. Then, we’ll discuss the connections between the two sets of numbers, talk about the “1-2 machine” demonstrated in the video, and expand to a “1-10 machine” — base ten rather than base two.

I’ll have students add a few numbers using the 1-10 machine, and then introduce multiplication. I’ll do a few examples with small numbers, then offer the puzzles of 24617×10 and 24617×11 to see what they can figure out.

Next up is subtraction with dots and antidots, and division. I’m not sure how far we will get but I hope to give students a few division problems to attempt, and to ask them whether they like long division or dividing using dots better. Finally, I hope to introduce the “1-x machine” and do a bit of polynomial division. Not sure if we’ll get to this.

The heart of the lesson is a bit of perplexity and the chance to play with math in a new way. I’ll let different parts of the lesson go longer or shorter as students are engaged with those goals. I would recommend spending time with the examples on the Exploding Dots page and getting to know the content to have some flexibility with what students are engaged with.

I’ll close by saying that I hope students enjoyed the experience, and that this was helpful for them to look at arithmetic from a new perspective.

Global Math Week is not meant to restrict what teachers can do. If you have fifteen minutes, maybe just share the video and connect it to base ten addition. If you have three days, explore polynomial operations and tease students with some of the fascinating unsolved problems that exploding dots reveals. If you’re somewhere in the middle, find some middle ground. If you’re interested in some more technology, check out the web app. Watch the videos on James’s site. Find a way to make this work for you. And happy Global Math Week!

Catalyzing Change: Feedback

NCTM recently released the document Catalyzing Change in High School Mathematics for public review and feedback. The purpose of the document is “to identify and address some of the challenges to making high school mathematics and statistics work for each and every student”. You can find the document and give feedback on the NCTM site here. This post is adapted from my feedback.


I just finished reading Catalyzing Change in High School Mathematics, and I’d like to offer feedback.

Positive Feedback

I love the content of the document. Your choice of what to include — the purpose of school mathematics, tracking, equitable instruction, essential concepts for focus, and pathways through high school math — paints a broad and compelling vision of what an effective high school mathematics program should look like.

A chapter devoted specifically to tracking is an incredibly important message. Eliminating tracking, both of students and teachers, is the most important change that could happen in high school math. Dodging this question in such a document would be a shame, and Catalyzing Change takes it on admirably, presenting a research-informed argument for eliminating tracking while also distinguishing tracking from acceleration.

The essential concepts for focus are both deliberately chosen and framed effectively. The content is framed through the lens of proof and modeling, two big ideas that should be present throughout all topics. The topics in Algebra, Functions, Statistics, Probability, and Geometry will form a practical and manageable curriculum that will set students up for future success while avoiding the bloat of many current high school curricula. Based on my read, the curriculum would be similar to a year of Algebra I, significant portions of typical Geometry courses, and a semester or more of statistics and probability, though with a focus on essential ideas that would allow significantly more flexibility than many current curricula and standards. For those who don’t want to explore the document in detail, the specific essential concepts are:

  • Algebra
  • Connecting Algebra to Functions
  • Functions
  • Data Analysis
  • Statistical Inference
  • Probability
  • Making Decisions and Quantitative Literacy
  • Transformations
  • Measurement
  • Geometric Arguments, Reasoning, and Proof
  • Modeling in Geometry

I love that Catalyzing Change is not trying to be a standards document. Each essential concept for focus begins with a narrative explaining its place in the broader curriculum, the relevance of those topics to students’ mathematics experiences, and the habits of mind students should exercise through those concepts, including the intersections with proof and modeling. Where the list of topics above would be insufficient to guide a high school math program and a standards document is likely to get lost in the weeds, this paper balances the two. It offers a manageable number of topics, names 2-6 big ideas that should guide each topic, and makes clear how the topics are connected to students’ broader mathematics progression. Any math educator who reads the essential concepts will come away with a deeper understanding of the essential ideas in high school mathematics.

I think the increased flexibility of 2 1/2 years of common experiences before the possibility of multiple pathways will be great for students. Whether they go on to calculus, statistics, discrete math, modeling, or other courses, this plan will create a trajectory that minimizes the race to calculus and creates opportunities for more students to feel engaged and connected to their math courses, while also ensuring every student engages with essential core content.

Finally, I love the focus on equity. A chapter on creating equitable structures and a chapter on implementing equitable instruction zoom in specifically on what equity looks like in high school mathematics. At the same time, equity is not sidelined to its own chapter; it is also evident throughout the document in language that constantly asks how the guidance in Catalyzing Change supports positive experiences and outcomes for all students.

Constructive feedback

While I think the core content of this document is important and necessary for high school math programs, I worry that it will not have the intended impact. The front matter is vague and avoids making the goals of the document explicit, with statements of purpose like, “Creating equitable structures in mathematics–confronting the impact of student and teacher tracking and support systems”. The end tries to appeal to too many different groups, with separate recommendations for teachers, district and building leaders, K-8 teachers and leaders, policymakers, curriculum developers, and mathematics educators. There are four key recommendations at the end of the document, but it is easy to lose the forest for the trees and miss essential ideas in Catalyzing Change because there is no single place where it makes a compelling case for what high school mathematics should look like. The document as a whole feels a bit like it is trying to appease too many groups at once, like there are competing ideas vying for attention instead of a coherent message. I don’t get this sense on a page-by-page level; instead, I get this sense because after reading I wasn’t sure how I would capture the essence of Catalyzing Change in an elevator speech. Compare this with Principles to Actions — a similarly detailed, research-based document — which derived much of its success from the concise guiding principles and eight mathematical teaching practices. Catalyzing Change will be read by many, but more of the audience will learn about it by word of mouth, summaries, and excerpts. What is the positive, ambitious vision that NCTM has for high school mathematics? What is it that mathematics educations should do to realize this vision? And how can this be communicated clearly and effectively to the broadest audience possible?

One might add to the front matter and the recommendations at the end of Catalyzing Change a clear and affirming statement of what effective high school mathematics programs must do. For instance:

An effective and equitable high school mathematics program must:

  1. Emphasize the multiple purposes of school mathematics: expanding professional opportunities for all students, using math as a lens to understand and critique the world, and experiencing the wonder, joy, and beauty of the discipline of mathematics.
  2. Eliminate student and teacher tracking systems that perpetuate inequitable experiences for students in qualitatively different, dead-end pathways, and work to provide common experiences and necessary supports so that all students engage with essential mathematical concepts.
  3. Implement equitable instruction that promotes effective mathematics teaching practices with attention to how those practices foster positive mathematics identity and create opportunity for all students.
  4. Focus on essential concepts in algebra, functions, statistics, probability, and geometry that will best prepare students for higher education and the workforce while engaging with big ideas of high school mathematics.
  5. Develop an equitable and common 2 1/2 year pathway for all students that includes the five essential concepts and allows for student choice in a final 1 1/2 years of study that best suits their individual goals.

These are welcome suggestions for many in math education, but they are also radical in the sense that they recommend a serious departure from math programs in the vast majority of high schools. A radical set of recommendations deserves a prominent place in the document that is easily shared and easy to remember.

I am an early career teacher, and I hope to be in this profession for several decades to come. I know that change will not happen quickly, but I hope that this is the start of a transition that I will get to see through in my time in the classroom. I hope that this document is able to catalyze that change. Thank you for your hard work, and I look forward to reading the final version.

Task Propensity: The Follow-Up Task

Task propensity refers to situations where students are so focused on the features of a specific task that they don’t generalize their thinking in a way that is useful to solve different problems in the future. In short, they lose the forest for the trees. I’m exploring how task propensity relates to Desmos activities and how this thinking could help me teach more thoughtfully with Desmos tools. I first learned about task propensity through this paper, and you can read the rest of my series on the topic here.


One example where I see task propensity degrade learning in engaging, well-intentioned activities is Polygraph. For context, Polygraph is a Guess Who-like game, where one student chooses a rational function among 16, and their partner asks yes or no questions to figure out which function they chose.

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This is aided by technology to make the setup and transitions effortless and let me watch through the Desmos teacher dashboard. For instance, check out these student responses in Polygraph: Rational Functions:

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We pause the activity and talk about vocabulary, looking at questions that worked well and making explicit the language that students could use to talk more precisely about the graphs they see. The activity is well-designed to create an intellectual need for vocabulary, and it’s effective for many students. At the same time, I still see responses like this through the end of the activity:

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There are some missed opportunities here, and these types of exchanges persist for some students every time I run Polygraph.

The Follow-Up Task 

We finish with Polygraph, and the game and the computers go away. We maybe even wait a day for ideas to bounce around.  Then I give students this:

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I don’t think this is anything brilliant or revolutionary. But I do think that spending 15 minutes working on this task in class and completing it for homework, perhaps over a few days, is worth far more than extending Polygraph for those additional minutes. The practice is more focused and less haphazard. I can have students trade with each other and try to find counterexamples, and suddenly there is even more need for precision in language, and more opportunity for students to revise their language and feel a need to add to their vocabulary for talking about rational functions. There’s even a more rigorous logic in writing questions with a much more challenging goal, and some classes students ask questions of the form, “say yes if the graph has two vertical asymptotes, say no if the graph has one vertical asymptote, and say I don’t know if the graph has no vertical asymptotes”, which I think is awesome. This task doesn’t work without Polygraph, but I think Polygraph is incomplete without some type of follow-up task to consolidate and extend student thinking.


I love Marbleslides. Students love Marbleslides. They get to experiment with different functions, watch marbles roll around, and feel like they’re playing a game. Yet, despite the best intentions of Marbleslides’ design, students are guessing and checking far more than I would like. I love sequences of screens like these:

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But too often students rush past these opportunities for reflection and consolidation of their learning to play with the marbles and the stars.

The Follow-Up Task 

Enter the follow-up task. I have students do Marbleslides, and a day or two later I give them this:

Screenshot 2017-09-26 at 7.38.54 AM.png

I get a lot of bang for my buck with this task. Students need to articulate differences between sine and cosine functions, and then write a function when they can’t see a full period of what they’re trying to model. Finally, they have to do all of that thinking without being able to guess and check their way to an answer. While this type of task is possible using the Pause and Teacher Pacing tools in Desmos activities, separating this task from the activity lets me take my time formatively assessing students strategies, selecting different approaches, sequencing them for a discussion, and unpacking different choices students made in writing their functions. And all of this happens without feeling like I’m interrupting students from playing a fun game in math class.

In short, I’ve found that Marbleslides is insufficient. It needs some bridges to connect work on Desmos with the thinking I need students to be able to do without the support of technology. There’s no substitute for Marbleslides in the scope and sequence of my curriculum. There’s also no substitute for a carefully designed follow-up task to reinforce and consolidate the big ideas, to make sure students don’t lose the forest for the trees.

More broadly, I find Desmos activities enormously valuable, but I also find them incomplete. They are incredibly engaging opportunities to work at formal and informal levels, experiment with a low floor, challenge students by raising the ceiling, and make connections through rich multimedia experiences. However, Desmos activities are only one element of a coherent curriculum, and there need to be bridges between those two — bridges that explicitly return to the thinking students do during a Desmos activity, consolidate it, and build on it toward larger goals.


Starting Conversations

(Thanks to Anna for this tweet.)

While I’m often intrigued by research-oriented claims about education, I don’t like the way these statements are framed. They paint in broad strokes when the topics are subtle and dependent on context. They are dogmatic where they could examine both sides. They rely on research happening outside classrooms to tell us what to do in our lessons tomorrow. Most importantly, they masquerade as statements of fact rather than asking provocative questions.

That’s not to say I disagree with the assertions made above. Just that I think there’s an important difference between a statement that is true and a question that is worth discussing. I don’t want to present ideas like “There is no such thing as developing a general skill” to a group of teachers as a “truth” that “they” figured out. Instead, I think of it as a question for discussion — an idea that teachers should entertain and debate about, to question their own beliefs and improve their practice. Here are a few questions that I don’t think I know the answers to, but are worth discussing to see what a group of teachers might learn together:

  • Do students need to be motivated to higher levels of achievement, or will higher levels of achievement motivate students?
  • What is engagement, and how is it related to learning?
  • What is the difference between grading and feedback?
  • What should students do with feedback, and how can teachers encourage it?
  • Is it possible to teach problem solving? Is it possible to teach critical thinking?
  • Is it possible to teach someone to have a growth mindset?
  • How much time should a teacher spend grading each day?
  • What should grades mean?
  • Is learning the student’s responsibility or the teacher’s?
  • Are some students more intelligent than others?
  • Should we personalize learning? What does personalized learning mean? How much of students’ learning should be personalized?
  • What does effective practice look like?
  • Are the principles of effective teaching for students with special needs the same as those for mainstream students?
  • Why is math worth learning?
  • What does effective collaboration look like?
  • What are the goals of math class?
  • What does it mean to teach toward equity and empowerment? Is it important to do so?

I’m skeptical of broad generalizations about teaching and learning. I can find someone who disagrees with almost any statement I read or hear. More importantly, the vast majority of teachers are turned off by statements that go against their everyday practice.

What I do think can change teachers’ practice are questions — well-posed and well-timed questions where teachers can share their experiences, learn from other perspectives, and move their pedagogy forward.

Scaffold Success

I had a great time presenting with Lisa Bejarano at the Colorado Council of Teachers of Mathematics Conference on Friday. We did a session called “Beyond Rhetoric: Everyday Growth Mindset Interventions”. It’s been fun to watch my thinking involve from my first attempts to write about mindset two years ago, to a presentation at NCTM in Phoenix last fall, to now.

In thinking about mindset, I’ve focused on three strategies:

  • Carefully define what success looks like in math class
  • Build relationships so that students are willing to take risks
  • Scaffold success for struggling students

That last piece is the one that has influenced my teaching the most this year, and has changed the way I think about supporting my students who struggle.

Scaffold Success

I used to think of scaffolding in terms of the goals I had for students. I take a problem I want students to be able to learn from and build in some scaffolds to help students access the math. I start with the goal, and I scaffold down to make that goal more accessible.

When I say “scaffold success”, I mean something different. Instead of starting with the goal, I start with students. I start with what they can do, and build from their strengths. I take what they already know and feel successful with, and scaffold up from there to more and more ambitious thinking.

This is how the warmup routines that Lisa shared in our presentation work. They don’t tell students to have a growth mindset, they create moments where students can experience success in math class. For a little bit of every class, they put away content goals. They are routines that every student can access, that allow multiple strategies and entry points for different students, and value student voice and student ideas. It’s impossible to hide from content standards, curricular goals and pacing guides. But I can carve out a piece of class where, instead of starting from the content and thinking goals and scaffolding down, I start from where students are and scaffold up, with a focus on how students experience success in my classroom.

(slides from our presentation here)

Task Propensity: Stay Humble

Task propensity refers to situations where students are so focused on the features of a specific task that they don’t generalize their thinking in a way that is useful to solve different problems in the future. In short, they lose the forest for the trees. I’m exploring how task propensity relates to Desmos activities and how this thinking could help me teach more thoughtfully with those tools. I first learned about task propensity through this paper, and you can read the rest of my thinking on the topic here.

I think the best example of task propensity is Marbleslides.


Students solve challenges like the one above by rewriting the function so that when the balls drop, they capture all of the stars. I love Marbleslides and I use variations on it often. At the same time, I find that a subset of students — usually the students who are already struggling — learn less than I would hope through these tasks. They are likely to solve Marbleslides challenges through trial and error without paying attention to the structure of the mathematical objects they’re working with, or they get frustrated and use functions outside of the family I want them to learn about.

Marbleslides offers one paradigm for what Desmos activities can look like. These activities are incredibly engaging — students love them, and are often asking for more. They let students see math as a dynamic process, learning about objects that make sense and follow certain rules — and learning those rules is what learning math is all about. They are valuable activities and I’m glad I am able to use them.

But in this post I want to offer an alternate perspective that tries to avoid the challenge of task propensity. I spent a bunch of time this summer thinking about polynomials. My polynomial units often feel flat and uninspired and I wanted to add a wider variety of activities. I’ve previously used this Desmos activity.


Students solve challenges where they need to build functions that meet certain conditions. It can be great for certain features of polynomials, but can also suffer from some elements of task propensity. Students just end up fiddling with different functions until they find something that works, and in the process they may or may not learn what I want them to learn about polynomials.

I wanted to design a new activity, one that falls after Polygraph at the start of the unit but before students start doing more formal algebraic work. My goal was to bridge some of the gaps between using vocabulary to describe polynomial functions and writing polynomial functions that meet certain conditions. I also wanted to write something humble. There’s nothing very flashy about this activity, no high-engagement tasks that students will want to keep coming back to. I want this activity to provoke useful thinking, and to do so using tools like sketching and interactive Desmos graphs that are impossible with a pencil and paper or whiteboard and marker. And I want it to stay laser focused on a few key ideas that I want to get across. I’m having trouble clearly articulating what I like about this activity that I don’t find in some others, but I’ll try to lay out what I was going for below. The activity is linked here if you’d like to play along.

Below are the first two screens. They are meant to give me a rough idea of how my students conceptualize polynomials.

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I would use teacher pacing here, so that students can only work on these two screens and can’t go further ahead. Then, I would pause and project a few examples anonymously to discuss why they are or are not polynomials, and show students a few different ideas of what the function could look like. Nothing crazy, just trying to see where student thinking is and help them do some informal work sketching and seeing sketches of polynomials.

The next four screens are also formative, and are more focused on multiplicity, where I find many students get tripped up when working with polynomials. I would use teacher pacing on these screens as well so that students can’t go ahead.

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The goal here is to explore how even and odd multiplicity influence a function, and see how well students can sketch a function that has specific characteristics while connecting multiplicity to other properties of that function. Nothing too crazy, but also something that I can only let students experiment with informally through a Desmos activity. They’re also meant to be really carefully focused on an informal understanding of multiplicity, making sure students are doing the right thinking on these screens. There’s a great opportunity to share different students’ graphs with the whole group and discuss both the specific properties and how they come together to create the larger function.

Next they would likely finish the activity at their own pace. I don’t want work through the potential management challenges of continuing teacher pacing. I’m watching the dashboard and looking for two things: interesting disagreements or misconceptions to surface and discuss at the end of the activity, and where student thinking is more broadly as I figure out what to do after this activity.

I think this is far from perfect and some of its rough edges could be smoothed out. But this lesson sticks with some values I want to try to use more often with Desmos activities. It isn’t trying to tour through an entire concept in 45 minutes. It isn’t supposed to be my most engaging lesson. Instead, the goal is to be laser focused on an important development in student thinking — reasoning flexibly about polynomial graphs and the vocabulary we use to describe them, without getting into algebraic notation. By staying really focused and living in that specific place, I’m trying to avoid some of the challenges of task propensity. There are no fancy challenges that students have to work through. The focus is on sketching and explaining their thinking. There is less emphasis on guess-and-check than many other Desmos activities. And I built this activity thinking specifically about how I want to use teacher pacing and other Desmos conversation tools in order to create useful moments of formative assessment and class discussion.

I don’t mean this to be a criticism of other Desmos activities, just a change in emphasis for me on what is missing in some of my pedagogy. It’s also meant to be something that complements what I’m already doing, rather than replacing other activities. There’s a time for engagement and excitement, and there’s a time for humble activities that zoom in on specific goals and focus on getting all students to meet those goals.

I would love feedback. Is this a distinction worth making? Is this activity really just a mess? Where else might this type of thinking be useful? Where could I go further?