Growth Mindset: From Rhetoric to Action

I presented yesterday at NCTM Phoenix. It was titled Growth Mindset: From Rhetoric to Action. The ideas come largely from this post from last year and the thoughtful responses in the comments. My slides are here, and here is some fascinating further reading on academic tenacity.

My thesis is that promoting a growth mindset is hard, and is particularly hard for a subset of students who are most disaffected and have had the most negative experiences with mathematics. In my experience, most of the interventions that are commonly talked about — praising effort rather than ability, encouraging students to try new strategies when they are struggling, creating space for collaborative work — are ineffective for these students.

I see a student’s mindset as a function of two variables:

If I am telling a student to have a growth mindset, but those words don’t match the experiences that student is having in my class, they’re unlikely to think of themselves differently.

I do think there are a few things we have control over that can influence this function. We can

  • Carefully define what success looks like in math class
  • Build relationships so that students are willing to take risks
  • Pay particular attention to students who have a history of failure
  • Have scaffolds and supports ready to move struggling students toward success

None of these ideas are groundbreaking. I asked the group at the start of the presentation to think about and discuss some possible negative consequences of growth mindset, and folks in the room named pretty much everything I shared. If there’s one idea I hope people took away, it’s that this work is both extremely hard and absolutely worth doing.

I got to spend some time with Henri Picciotto, and he said something smart at dinner last night. “When you grade, you help one child at a time. When you plan, you help all kids. Spend your time accordingly.” I’d love to add a corollary — in the same way that our time planning lessons is valuable, we can spend extraordinarily valuable time planning for how we create a classroom that makes students feel like they belong and that they can be mathematical thinkers. It’s relatively easy to spend some time puzzling over how I want to introduce polynomial division next week. It’s much harder to spend some time trying to figure out how I can define success in my class so that every student is able to feel like they can be a math person.


Student Feedback

I think #observeme is a really cool thing. I’ve enjoyed following the hashtag and seeing great things happening in a ton of classrooms. My school is working to increase professional collaboration and observation, and I’ve had several colleagues visit my class so far this year. Thing is, I haven’t received much helpful feedback. I know I’m not alone with this challenge. Other teachers often say they weren’t there for long enough to give helpful feedback, or just never follow up.

I don’t mean to criticize #observeme. It’s powerful just to have other teachers in my classroom and get these conversations started. I’m excited that this is a priority for my school and we are working on common language and goals to focus these observations. I’m sure that over time I’ll receive more useful feedback from my peers. The goal of #observeme isn’t just to get me some useful feedback; it’s also to create a professional community that values observation and continual learning.

But if my goal is to get some useful feedback on my teaching that I can use to get a little better tomorrow, I think I’ve found a different way to do that. Here’s a survey I asked students to fill out via a Google Form:


It’s adapted from the ideas in Dylan Wiliam’s book Embedding Formative Assessment. I tried to make the language more accessible for students, and briefly framed what these ideas mean to clarify what I was looking for from students. The survey was anonymous, and only took students a few minutes. Here are a few of the responses I received:

Even though we have been given a sheet that tells us what we are being assessed on for synthesis tasks it can still be a bit confusing what precisely we are being assessed on for a specific synthesis task. Maybe this is a misunderstanding that I have but it would be helpful just to revisit that topic quickly again.

I think that it is really hard to give peer to peer feedback. Coming up with a strategy to help facilitate that would be something to look at. I want to know how these problem solving skills can help me in the future. So every one in a while just vocalizing what that looks like would be an area to improve on. I think also when giving feed back on synthesis tasks, it can be nice to have a very brief verbal explanation as well as what is written on the page.

I feel that even when we do collaborative work it still is very independent. I think finding a better way of holding students accountable for their group work is an area to improve on.

These are pretty insightful comments into my classroom, from the people who are likely to best understand what is happening. There were plenty more, affirming things I do well, reminding me of weaknesses I need to work on, and providing insights into areas of my teaching I hadn’t been thinking about. For a few minutes of my time mocking up this survey and a few minutes of their time filling it out, I’ve got a lot of great feedback to work with.

How My Teaching Has Changed

I think I’m a better teacher than I was a few years ago. I look pretty different on the surface. I’m more relaxed in the classroom, I speak more clearly and confidently, I can plan classes more quickly, and I’m generally less stressed about the day to day responsibilities of teaching.

But most of that is incremental and only tangentially connected to student learning. Here are some more substantive ways my teaching has changed that I think have actually made a difference.

I ask myself, pretty incessantly, what my goals are for a lesson and how an activity I have planned meets those goals. I don’t do activities that don’t meet my goals, and I’m likely to think in terms of these goals when I decide to allow more time for an activity or cut it short.

If students don’t know something, my old response would be to jump in, offer an explanation of what I thought they were missing, and try to push my knowledge into their brains on the spot. Now, I’m much more likely to step back and realize that the best response is often to circle back to that topic later in class, the next day, or the next week, once I’ve had time to think through student misconceptions and figure out next steps that are more likely to make a difference.

Student Thinking 
When I first started teaching, a lot of my goals were around getting students to say right answers. I spent class time asking questions that were implicitly seeking validation of my teaching by trying to lead students to say clever things. I’m much less interested in that now, in comparison with students thinking smart things. I’ve become much more comfortable with wait time, and I’m less concerned with that perfect series of leading questions to get kids to say some right answers than with a smaller number of questions that kids think about for more time, talk about in partners or groups, then share with the class. Maybe they don’t share that perfect answer. That’s fine. It’s about the thinking.

When I give a task, I almost always give students time, individually, in partners, or in small groups, to work through a problem or task. Then, I try to start any full-class sharing with a few students chosen in advance to move discourse toward my goals for the task. I still take hands, but far less of the time, and I always have a voice in the back of my head reminding me that when I take hands, I’m hearing from an unrepresentative subset of the class.

Discovery, Follow-Up

Lots of really thoughtful people got me thinking in their responses to yesterday’s post. Check the comments and Twitter for a sample. Three are really sticking with me.

Anna Blinstein:

I think that my only addition or caveat to your post, Dylan, is to push back a bit on the goals of math class. If the primary or only goal is remembering/applying mathematical content knowledge, then your post makes complete and total sense – we should probably use discovery sparingly; it is helpful as a motivator (basically, the intellectual need and wonder categories you listed) and maybe helps some students remember some ideas some of the time. But, if one’s goal is to teach students to think like mathematicians, then I don’t know of a better way than having them engage in the process of doing math consistently and frequently while also seeing models of what this might look like and getting feedback on their efforts and ideas. I don’t think that anyone would argue that a discovery approach is the most efficient method of transferring knowledge, but for me at least, that’s not the primary goal.

Dan Anderson:


(my answer)


Avery Pickford:


These all hit me pretty hard, and I’m questioning a bunch of what I wrote yesterday. I’m going to try and reframe my argument and see where this goes. Call me out if I’m totally off track here. There are two issues I see. Does discovery lead to better learning of content, and does discovery learning teach students to think like mathematicians and acquire future knowledge for themselves? The second question is the one I want to tackle.

I think the answer is yes. First, an example.

Introducing the Unit Circle

I recently introduced the unit circle. Students started by constructing right isosceles and equilateral triangles to derive the values of sine and cosine at a few angles. I chose a discovery approach for this lesson because it had students practicing relevant triangle geometry, seemed manageable, and the much more didactic alternative I was imagining sounded boring.  The activity felt successful; every group found the value they were looking for, we followed up with a brief discussion, and then groups derived the rest of the unit circle.

I was surprised when, three days later, many students failed to produce a similar proof on an assessment. I don’t mean to argue that discovery never works, just that in this instance it came up short, and that doesn’t feel like an isolated example to me.

Contrast: after that assessment, I gave students this image:

I told them that this was a new unit circle, with angles at 15, 75, 105 and etc degrees. I gave them the values of sine and cosine at 15 degrees and asked them to figure out as much additional information as they could.

I would argue that this activity also asks kids to think like mathematicians. It’s much more subtle, in that there is no grand reveal of the unit circle at the end. But it’s still valuable thinking. And, more importantly for me, I find that more students have a chance to access a task like this, because they all have similar background knowledge at this point. This task also involves acquiring knowledge, and can be metacognitive in thinking about strategies that are useful for acquiring knowledge. And if I have to pick between a discovery activity introducing the unit circle and a demanding task asking students to reason with what they’ve learned, I prefer this one. Not by a ton, but I do.

Discovery often meets ambitious goals of mathematics learning. I think that, for many students, my unit circle exploration did that. But it didn’t land for everyone. And, based on my experience with that activity and similar activities, it’s too often the same few kids who are left behind.

I don’t believe that discovery has a monopoly on teaching students to think like mathematicians, and I don’t believe that every lessons has to involve discovery in order to teach kids how to acquire knowledge themselves. I’m at a 3.5 on Dan’s scale, which I take to mean somewhere around a third of my lessons involve some element of discovery. I would like to nudge that number upward a bit — but only if it means I’m doing discovery well, for every student. I would rather focus a small amount of energy on these goals but do that well than try to do it every day and possibly undermine those same, important messages.

I hear the argument that it’s important to teach students to think like mathematicians and acquire knowledge for themselves. But I want to avoid being dogmatic about using discovery as a means to do so. Instead, I want to focus on doing discovery well when I use it. Maybe I’m just coming up short in my knowledge and skill with that particular pedagogy; I’d love to hear folks sound off in the comments with the practices that have proven useful for them with these types of lessons. But in the meantime, I’m going to try and meet all of the above goals with as many different means as I can.


Here’s a statement that I expect some folks will disagree with.

I don’t believe that students necessarily learn better when they figure things out for themselves.

To clarify, I’m not trying to argue that kids shouldn’t have a chance to figure things out, or that it’s inherently bad. Just that, in and of itself, student discovery is not a particular priority for me. Some reasons:

Cognitive Load Theory
Cognitive Load Theory is fascinating to me. It’s also interpreted lots of different ways. Fundamentally, it says that while trying to figure something out, working memory resources are consumed in such a way that it is difficult to move information to long term memory. I’ve definitely seen this happen. Kids spend a great deal of effort searching for a possible solution to a problem and meet various dead ends along the way. By the time they reach a viable strategy, much of their thinking has been spent at cross purposes with the goal of the lesson. That realization they make at the end is just one synapse firing over a long lesson and isn’t particularly durable.

I’m skeptical this is the case in every instance, and I think there’s a lot more subtlety to things than “increased cognitive load of discovery = bad”. But it’s something important to think about — in the words of Ben Blum-Smith, “any thoughtful teacher with any experience has seen students get overwhelmed by the demands of a problem and lose the forest for the trees”.

How much does one activity matter?
I tried hard to create powerful discovery experiences early in my career. An implicit belief embedded in that instruction was the idea that, if I could just find the perfect way to introduce students to an idea, they would remember it and be able to apply it in the future. At best I had mixed success. One activity, no matter how clever it is, never makes as much of a difference as I might think. The sum of student experiences — from the introduction of an idea, through practice activities, to opportunities to transfer that understanding to new contexts — are what make a difference for learning. I’m much more interested in focusing my energy on a range of activities that allow students to practice and extend the ideas we’re learning about than putting all my eggs in the “they discovered it so they’ll remember it” basket. Here’s my core value: it matters less how a student figures our something new than what they do with that knowledge in the future.

I do still use discovery-oriented activities. Here are two goals I have that this type of lesson can effectively support.

Intellectual Need
One idea that pushes back against an aspect of Cognitive Load Theory is the generation effect. In short, trying to figure something out before learning it leads to more durable learning. I want learning to be active for students whenever possible. I want to avoid overwhelming them, but I prefer active tasks whenever possible, and if the task is within students’ reach, I’m likely to use it. In addition, I want to create some intellectual need for what students are going to learn. If struggling with a problem makes students aware of what they don’t know and what that knowledge might be useful for, they are well-positioned for future learning.

Neither of these strategies requires that students actually reach the big ideas of a lesson on their own or struggle for a great deal of time; they just argue that attempting to do so to begin a lesson can lead to more durable learning. And, in both cases, the sequence ends with an opportunity to make the learning explicit and consolidate understanding. That explicit instruction doesn’t have to come from me. One way I often do this is to use group work to provide opportunities for students to share strategies, and lead into a full-class discussion sharing those strategies and consolidating the big ideas that students need to move forward. That discussion can be largely student driven. But the consolidation is essential; it’s incredibly rare for me to see every student figure something out in a discovery lesson. And, honoring the principles of Cognitive Load Theory, even if many students figure something out on their own, further clarification and exposure beyond a discovery activity is essential to further cement their understanding.

I love doing math because of the joy of learning something new through discovery. This doesn’t mean that students necessarily feel the same way or learn best that way. But it does mean I want students to have the opportunity to experience the wonder and joy of mathematics. Even if creating these experiences is an inefficient use of class time or leads to less learning, it’s a priority for me. Not every day and often not every week, either. Being judicious with how often I use these activities allows me to focus on the ones I do and do them right. And watching a student light up when they have that moment of insight is a special thing to see and a special thing to experience.

Conditions for Discovery 

  • A discovery activity should either focus on an incremental, manageable step forward or something so mathematically spectacular that it is worth significant effort
  • I must be willing to cut an activity short if it’s clear we’re hitting a dead end
  • An activity has to end with time spent consolidating understanding, either student driven or teacher driven
  • I won’t choose a discovery activity at the expense of time spent on practice activities that allow students to deepen their understanding, and allow me to see who understands and who doesn’t and adjust future instruction appropriately

I don’t mean to get too down on discovery. Instead, I want to clarify more specific and concrete goals than “students figure things out themselves”. For me, the goals of intellectual need and wonder are worth working toward, and are much more connected to experiences and outcomes I care about than discovery for its own sake.

Letter to a Future Teacher, or Why I Teach

For anyone thinking of becoming a teacher, or wondering what the work is like. Also for me, to remind myself what’s important after hard days, hard weeks, and hard months.

Learning teaching came slowly to me, and too many of those hours were spent formatting worksheets, learning the ins and outs of Microsoft Word’s equation editor, and sitting at my desk terrified, dreading fourth period. Then, all I wanted was for students to be silent when I wanted, to dazzle with my deep understanding of eighth grade math, and to find a perfect challenge question that prompted students to figure out the Pythagorean Theorem on their own. I thought teaching was about saying clever things, getting students to give lots of right answers, and passing on a love of mathematics mostly through force of will. That all didn’t go so well, and it also didn’t leave me very happy at the end of the day.

Growing to love my content and my time in class has been one part of the work, but it’s easy to oversimplify teaching as the process of moving knowledge from my brain into students’ brains. That’s one part of my job, and a challenging and important part of it. But we have computers now that can do much of that work, and talking heads who think they should do more of it for us. That’s not what teaching is about. For me, it’s much less flashy, without theatrics or instant gratification, and with a great deal more concern for the individuality of every student I teach.

Teaching is about convincing that girl who thinks girls can’t be mathematicians that they can, and she can, bit by bit, day by day. It’s about staying after school to hang out with two kids who love to play chess and can’t find anything else about school to enjoy. It’s about knowing that one kid is anxious and terrified every time they walk into math class, and with every direction or transition keeping an eye on their eyes and their shaking knees under the desk and figuring out just what to say to help them feel a little bit more at home. It’s about having the courage to throw out a lesson plan after Walter Scott was killed to teach students something relevant that day. It’s also about screwing up all of the above, and having the humility to know it and learn from it. It’s about being wrong. Realizing you don’t understand fraction division nearly as well as you thought you did. Realizing that you’ve been making assumptions about who kids are and where they come from that are slowly crippling them. Fundamentally, it’s about being a human who can create space for children to share their ideas and truly, truly listen to them.

It’s often thankless work. Kids aren’t often very good at gratitude. Much of the difference you will make won’t manifest for months or years; for many students you’ll never know. There will be far fewer days than you would like where class goes well, where that discussion reaches the conclusion you’d planned for, where that great problem you’d prepared is just right to pique your students curiosity.

That’s the terrifying, humbling, wonderful work of teaching. It is also the part of teaching that will never be replaced by a computer program, that we need more humans to appreciate and love and work for. If you are considering becoming a teacher, know that teaching is a job with an enormous amount of humanity behind it. It’s exactly that humanity which makes it so challenging and so rewarding, and keeps me coming back.

Curriculum Coherence

Matt Larson wrote a great article last month as the NCTM President’s Message titled “Curricular Coherence in the Age of Open Educational Resources“. He argues that, without serious effort and collaborative work in professional learning communities, using online tasks and activities is likely to lead to an inconsistent experience for students that undercuts curricular coherence.

I’ve never had access to a coherent, high-quality curriculum. I have always had to supplement with resources that I’ve either found on the internet or created myself. It’s easy to be defensive and argue that Matt is wrong — that there are lots of high-quality resources on the internet and that many teachers have the expertise to put them together into an effective curriculum.

Case Study
I wish I had time to do that on a regular basis. Too often I don’t. I am teaching expected value in Precalc right now. So I went to my usual haunts for resources. A few tasks from Illustrative Mathematics, some questions I threw together based on the Wheels of Fish, Chips, and Peas that Darryl and Bowen used at PCMI, a Yummy Math lesson I just found on two point conversions, something cool from Dan Meyer, something else cool from Dan Meyer, and some Mathalicious lessons in their probability unit.

It’s a bit of a mess. There’s high-quality stuff in there, but I’m having trouble sequencing it, figuring out the best places to deliver explicit instruction, and using consistent representations across tasks. I think kids will have some good opportunities for learning — there are benefits to seeing a concept in a wide range of contexts. But this is far from an ideal unit.

This is my first time teaching expected value; I am particularly in need of high-quality curriculum for that unit. That’s less the case for quadratics. I’m teaching graphing quadratics in Algebra II right now; I’ve taught this unit before, as well as introduced quadratics in Algebra I. I have a much better idea what a learning progression looks like, and instead of scrambling for anything to work with, I’m trying to supplement what I’ve done before with some more high quality activities. In this case, I’m plugging in the great Desmos lesson Build a Bigger Field, and then on the following day teaching the Mathalicious lesson Prescripted. These two lessons get at a challenging idea — writing a quadratic model for a particular situation of the form y=x(a-x), and analyzing that function to learn something about the situation.

This feels like a much stronger curricular choice. I’m definitely lacking in meaningful application tasks for quadratics. The activities complement each other well, fill a gap in my prior curriculum, and I have the chance to put some effort into figuring out how I will teach them and link them together.

I love all of the resources that exist through online communities; I’m sure that, no matter what curricula I have access to in the future, I will keep using many of them. But Matt’s right — it’s easy to divorce the potential of great online activities from the reality, and to mistake a lesson that’s cool and fun for one that leads to meaningful student learning. I have trouble doing this. But no single lesson, no matter how amazing it is, is likely to make or break a student’s understanding of some topic in math. That happens with lots of effort, over time, as ideas build on previous knowledge and connect to the broader curriculum.