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.

Screenshot 2017-08-23 at 6.58.36 AMScreenshot 2017-08-23 at 6.58.50 AM

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.

Screenshot 2017-08-23 at 6.59.03 AMScreenshot 2017-08-23 at 6.59.16 AMScreenshot 2017-08-23 at 7.00.19 AMScreenshot 2017-08-23 at 7.00.42 AM

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?

The Limitations of Improving Summative Assessment

A year ago my school started a project of trying to improve the way we do summative assessment. We had lots of different systems and not a lot of coherence, and decided to move in the general direction of standards-based grading.

A year in, it was a great learning experience for me. I had used elements of standards-based grading before, but working to assess students in consistent ways across classes and deciding what is worth being consistent with was a valuable exercise.

Here’s the thing. We put in a ton of effort over a year. Assessment was a huge focus of our meeting time. And I’m not sure we saw particularly strong results. I do think what we did helped, I’m just skeptical that fiddling with grading is the best way to improve a school. As Shawn Cornally says, it’s like putting lipstick on a pig. The pig is the collection of institutional obligations we have that mean we have to give grades. The lipstick is our fancy new assessment plan, which is really just tinkering around the edges of a broken system.

I’m not an expert on the literature around professional learning, but I’ve been diving into research on how teachers improve this summer. When research looks at the elements of teaching practice that help teachers improve, summative assessment is conspicuously absent. Improving the curriculum matters, teachers’ pedagogical content knowledge matters, purposeful formative assessment matters. I’m sure more deliberate summative assessment helps too, but of all the levers that help to improve schools, it doesn’t seem like it’s very high on the list, and we only have so much time to spend on new initiatives.

This is also consistent with conversations I’ve had with lots of teachers who jumped on the standards-based grading train five or ten years ago when it was the hot new idea in town. Most of them are still using some elements of standards-based grading, but they’re also not too starry-eyed about the transformation that standards-based grading has the power to bring.

None of this skepticism makes me regret the work we’ve put in. I’m a more thoughtful teacher now than a year ago when it comes to assessment. But it does make me want to put my effort elsewhere in the coming year.

I know there are folks out there who disagree with me. I’m curious for some pushback. What am I missing?

A Reminder to Be Bold at the Start of the Year

Ordinary, said Aunt Lydia, is what you are used to. This may not seem ordinary to you now, but after a time it will. It will become ordinary.

-Margaret Atwood, The Handmaid’s Tale

It’s a bit of a gloomy quote to start a blog post, but it’s also something I find to be a profound truth about humans: our capacity to adapt to new situations and a new normal. Often when I talk to teachers about a new idea or perspective on their teaching, they tell me, “well when I was a student I wouldn’t have liked that,” or, “I wouldn’t have learned if it was taught that way.”

Maybe that’s true, given your experiences and what you were used to. But you have a year to spend with your students. You have the opportunity to set new norms and new expectations, to create the classroom culture you want to teach in.

Assuming that students won’t learn a certain way because you wouldn’t have liked it is also a narcissistic approach to teaching. Human memories aren’t very good, and I’m skeptical that any teacher can look back on their school years and separate their likes, dislikes, and emotional attachments from what actually helped them learn.

This isn’t an argument for blind change. Instead, it’s an argument to be bold about how I think about what is possible in my classroom. These might be structural changes like grading less to spend more time on relationships and student thinking, shuffling groups more often, or asking students to move around the room. They might be about redefining what it means to collaborate on mathematics or asking students to approach unfamiliar problems in new ways. And there are more potential transformations down the road that I don’t know about yet. Many things seem impossible until they’re real.

In a digital, connected world, I have incredible opportunities to learn from other teachers about how they create powerful classrooms. This is a reminder to be bold enough to learn from and imagine possibilities beyond my own experiences.

Elements of Great Professional Learning

It is impossible to begin to learn that which one thinks one already knows.


I’ve been reflecting more and more the last few days on professional learning — how I have learned about my practice, how I want to continue to learn, and what that learning can look like when it is done well. I don’t claim to know very much about profession learning but here’s where my thinking is now, and how I plan to pursue new opportunities in the future.

A Rich Vision of Teaching 
The most powerful professional learning for me has always begun by pushing the boundaries of what I think great math teaching can look like, in ways small and large. It might simply present a new tool or approach I hadn’t thought of before. It might share a conception of what an impactful math class can look like that goes beyond my previous goals. It might share an engaging new approach to a challenging topic that I had thought was boring before. I can’t learn what I think I already know; learning needs to start by showing me what is possible and giving me new goals to work towards.

Focused Effort 
I often leave conferences with a dozen different ideas I want to implement, or come back from winter break with a list of bookmarked blog posts full of all the new lessons I want to teach. Neither of those translates very well to enduring learning. It’s really hard to pick one or two goals just outside my comfort zone and dive deep into making them happen in my classroom. At the same time, I think it can be empowering to put everything else on the shelf for a bit and focus on one or two small, concrete changes that will make a difference for students.

A Community of Support 
It’s hard to put focused effort into learning on a regular basis. It’s also hard to read blogs, attend presentations, and talk to teachers about the incredible things they’re doing in their classrooms without feeling a little bit inadequate. I learn more when I have people in my corner reminding me that getting better at teaching is hard, but it’s also worth it. Rather than comparing myself to other great teachers I should just be comparing myself to where I was months or years before. A community of support can come in lots of forms, but I need those reminders to stick with my goals.

One final truth for me is that it’s far easier to get better at talking about teaching than it is to actually get better at teaching. I’ve been writing on this blog for close to four years. I’ve become more thoughtful and articulate about teaching and learning. But too often I’m stuck at step one, talking about what I want my teaching to look like. The real work comes from focusing on a few specific ways to get better and building the community I need to sustain that work over time. That’s where the magic happens.

A Letter to a Teacher Who Refers to Students as “Ladies and Gentlemen”

Hi! I want to talk to you about something. I’ve heard you referring to groups of students as “ladies and gentlemen”, as in, “let’s quiet down ladies and gentlemen” or “alright ladies and gentlemen have a great day”. When you use those words, you make an implicit assumption that all of the individuals you are addressing identify as either ladies or gentlemen. Some may identify as transgender, gender nonconforming, genderqueer, or other identities that resonate with them more than male or female. Even if no students in the room use those identifiers, your assumption impacts how they see themselves and how those young people form their identities in the future.

Equity work is a process of becoming thoughtful and purposeful about things that were subconscious before. Language shapes how we think. It’s not easy to change, and it takes time. But it’s worth doing.

Have you ever taught a transgender student? It was hard to change the pronouns I instinctively wanted to use. I unintentionally misgendered them, often at first, but I got better over time. I noticed that an adult at my school who often referred to students as “ladies and gentlemen” was also the adult who had the most trouble referring to that student by their correct pronouns. It was tough for everyone, but he struggled far longer and often misgendered them publicly. I can’t say for sure that one was the cause of the other, but it struck me as a useful object lesson of what can often seem like an abstract idea.

While changing your language to words like “folks” and “y’all” is not likely to change the world, I see it as a microcosm of the change the world does need. For thousands of years, human social norms have changed on a time scale of generations. Change happened slowly. People grew up exposed to different perspectives and as older generations passed on new ideas took root.

Today we have a clearer view of the equitable and empowering world we want for everyone. We can also see how far away we are from that world. Generational change will not be enough; we need action that will change hearts and habits today. And while language is only one piece of the puzzle, a willingness to work consciously to change behavior and construct new social mores is the work that will make the world a better place, one step at a time.

I hope that this letter does not feel angry or resentful. That’s not how I feel. Change is hard, but it hope it can also feel empowering. When you become more thoughtful about your language, you are also influencing the way others think and speak, and embracing the potential of every student who enters your classroom. I hope you feel empowered about your role in creating a better world, one microcosm at a time.

Thanks to Grace Chen and Nik Doran who helped me find the words to write this.

Talking About Students

I’ve been thinking more about equity since Grace, Brette and Sammie’s great session at Twitter Math Camp. In particular, I’m working to pay attention to the elements of equity that I attend to in my reading and thinking about teaching this summer, and the elements that I want to get better at seeing, recognizing, and acting on.

One thing I’ve been paying attention to the last few days is how I talk about groups of students. Most teachers see a difference between “high” students and “low” students and find this distinction useful in their pedagogy. At the same time, teachers use all kinds of different euphemisms to communicate their perception of different students. My go-to descriptors in the past has been “struggling students” and “high-achieving students”. I’ve heard other teachers talk about low-ability and high-ability students, students who are behind, gifted-and-talented students, top and bottom students, and more.

I’m coming at this with the assumption that words matter, and that the way that I talk about students influences the way I think about students and make decisions about teaching.

I’m also coming at this thinking about what it means to have an asset-orientation toward students and their learning. I want to avoid deficit thinking — talking about students in ways that assume they are static, that their challenges define them, or that their prior experiences predict their future success. Lani Horn uses phrases like “lazy”, “a C student”, and “at-risk” as examples of this perspective. Whether or not these descriptors are true, they aren’t particularly helpful in moving forward and teaching that student. An asset-orientation replaces this language with a focus on students’ strengths and their potential to grow in the future. Importantly, an asset-orientation doesn’t ignore students’ prior challenges, but it is focused on the future and next steps moving forward rather than students’ past performance and experience.

Teaching toward equity and empowerment means acknowledging the differences in students’ prior achievement. It means paying particular attention and putting focused energy into supporting students who have been less successful or felt less included in the past. I don’t want to hide from those distinctions. At the same time, doing this work with an asset-orientation means approaching each day by acknowledging students’ prior experiences but leaving behind assumptions about what they can do that day and in the future.

When I’m talking about students who teachers might call “low students” or “strugglers”, I want to try to start using more language like “students who have struggled in the past”, “students who have been tracked into lower classes”, and “students with low prior achievement”. Those are more specific phrases, and they make no assumptions about what students are capable of in the future. I want to work on always accompanying that language with an explicit focus on what I am going to do to support students and what my goals are for them moving forward. My goal is not to erase students’ prior challenges in math class; my goal is to acknowledge those challenges but leave them in the past, and focus on what all students are capable of in the future.

I hope that this type of thinking will allow me to see one aspect of teaching and learning through a new perspective. I have no illusions that changing my language in small ways will make a large difference in my teaching. But as I work to see my classroom and my pedagogy through an equity lens, all I can do is add to my toolbox, one idea at a time.