Meaning Making and Structure Building

Of all I saw and learned this past half year, one thing stands out. What goes on in the class is not what teachers think — certainly not what I had always thought. For years now I have worked with a picture in mind of what my class was like. This reality, which I felt I knew, was partly physical, partly mental or spiritual. In other words, I thought I knew, in general, what the students were doing, and also what they were thinking and feeling. I see now that my picture of reality was almost wholly false. Why didn’t I see this before?

-John Holt, How Children Fail

I was fooling around on Youtube and ended up watching Beyonce’s Irreplaceable. It’s a song I’ve liked for a while, though I’m only a casual fan. Watching it last night I realized I’d been both hearing and singing it wrong for years. I had been singing, “everything I own in a box to the left” when the actual lyric is “everything you own in a box to the left”. And that’s a pretty significant distinction. Here’s a context clue from the video to help:

Screenshot 2017-04-24 at 10.20.48 PM.png

I had been listening to and enjoying the song for a long time — but in all that time, I had managed not to change a significant misconception or probe beyond the surface of my understanding of what was happening.

I’m curious how many of my students experience my teaching in this way, spending their time in class thinking about surface features of the mathematics we are studying without putting significant cognitive work into the underlying meaning of the content.

At NCTM a few weeks ago, I attended a talk by Skip Fennell, Beth Kobett, and Jon Wray on formative assessment. You can check out their slides here. One strategy that stuck with me was using an interview to explore student thinking after a task, asking the student how they solved the problem, why they solved it that way, and what else they can tell me about their thinking. It’s obviously impractical to do this every day or with every student. But it’s also a strategy I’ve never used to explore student thinking in depth, and with the premise that students often know less than I think they do I’m sure I would get some great insights out of it. One more point the presenters made was that the interview doesn’t have to be with a student who is struggling; talking with a student who is effectively using certain strategies could be useful in figuring out what moved their thinking forward and how to help other students with that thinking.

Here’s a final thought:

There do appear to be cognitive differences in how we learn. … One of these differences is the idea … that psychologists call structure building: the act, as we encounter new material, of extracting the salient ideas and constructing a coherent mental framework out of them. These frameworks are sometimes called mental models or mental maps. High structure-builders learn new material better than low structure-builders. The latter have difficulty setting aside irrelevant or competing information, and as a result they tend to hang on to too many concepts to be condensed into a workable model (or overall structure) that can serve as a foundation for future learning.

-Brown, Roediger & McDaniel, Make It Stick

I see structure building as the biggest difference between successful students and students who struggle. The most important piece of the research that the authors present on structure building is that guidance toward the key elements of a problem that makes explicit the essential relationships can support all students in structure building and making sense of the mathematics.

I’m not sure how well these ideas are connected — Holt, Beyonce, formative assessment interviews, and structure building. But it’s been some good food for thought in probing more meaningfully into student thinking, and constantly asking myself whether students are actually doing the thinking that I hope they are doing.

21st Century Skills

Many educators advocate for schools to focus on 21st century skills. They argue that we need to teach students to be creative, critical thinkers, collaborators, and effective communicators. At the same time, many argue that other skills have become obsolete. Computers can do math for us, the internet has made all of the world’s knowledge available at our fingertips, and procedures are increasingly done by robots rather than humans.

I often feel like the old boring traditionalist in the room when these arguments are made. Research in cognitive science suggests that critical thinking is not a general skill, but needs to be taught in context. A student’s ability to think critically depends more on the depth and breadth of content knowledge than on experience learning generic critical thinking skills. I’m not familiar with research on creativity, collaboration or communication, but I would conjecture that these must also be taught in context.

I’m also skeptical that the 21st century has made very many skills obsolete. Sure, calculators can multiply for us. But a fluency with multiplication and familiarity with its structure builds essential knowledge that students need to engage in more challenging problem solving. It’s easy for those with knowledge to underestimate the extent to which that knowledge makes higher-order reasoning possible, called the “curse of knowledge” by psychologists. I’m a big believer in content. The more people know, the better they are able to reason about new situations in the future. This isn’t an indiscriminate argument for a rote curriculum. How students know things is just as important as what they know, and students need to move beyond shallow knowledge and have opportunities to probe for deeper structure, apply what they know, and transfer their knowledge to new contexts. Elon Musk presents a fascinating case study, explored in this article, which to my reading reinforces the need for both a broad base of content knowledge and learning that content in a way that facilitates transfer. This type of knowledge is how I want to incorporate critical thinking, creativity, collaboration and communication into my classroom.

At the same time, the learning landscape has changed in the last several decades. To be successful citizens, students absolutely will need new skills. Digital literacy, statistics, spreadsheets, research skills, and more. I’m excited about expanding my pedagogy to a broader view of what students need to know and be able to do, and to incorporate these skills into my class.

This leaves me with a few questions:

  • In an ideal world (without pressures from standards and tradition), what are the skills that should be cut from the math curriculum?
  • What strategies will help students learn content in meaningful ways?
  • What are the essential new, teachable skills that students need?
  • How can I find the time to balance these competing demands?

Rethinking “Formative Assessment”

I got to see Dylan Wiliam speak at NCTM last week. I’m a big fan of his work and I think Embedding Formative Assessment may be the book I have learned the most from in my teaching.


I’ve also increasingly realized that formative assessment is misunderstood. I had one conversation with a teacher who told me that their school’s formative assessments are the exams students take each quarter. In another conversation, a teacher described formative assessment as a diagnostic assessment that happens at the start of a unit to see what students know and don’t know.

Both of these can be valuable formative assessment opportunities. At the same time, one big lesson I have learned from Wiliam’s work is that formative assessment is something that happens at multiple levels, using multiple strategies, to constantly measure the goals of teaching against the learning that is actually happening and attempt to do a little better the next day. In his book, Wiliam identifies five strategies of formative assessment, each of which consists of a chapter full of techniques to try in the classroom:

  • Clarifying, sharing, and understanding learning intentions and success criteria
  • Engineering effective discussions, tasks, and activities that elicit evidence of learning
  • Providing feedback that moves learning forward
  • Activating students as learning resources for one another
  • Activating students as owners of their own learning

I think each of these strategies is an important element of effective teaching. At the same time, Wiliam’s book could easily be criticized for trying to capture too much of what happens in classrooms under the umbrella of formative assessment, as well as presenting a framework that is complicated and difficult to get one’s head around.

In Wiliam’s talk, he spoke about two core ideas that underlie much of his thinking on formative assessment: that teachers need to have purposeful pedagogies of engagement and pedagogies of responsiveness.

I really like these phrases. They are focused on pedagogy — what I’m doing in my classroom, and the actions I’m taking to further student learning. They are broad, yet at the same time capture what I think are key ideas of impactful teaching. And they offer a useful synergy: if students are fully engaged but teaching is not responsive to their needs, or if teaching responds brilliantly to where students are but does not engage them in meaningful thinking, learning seems unlikely. I’m going to go ahead and define them more precisely for my purposes. Just drafts here, still working out my thinking.

Pedagogies of engagement: a variety of strategies that cause students to think deeply about content, practice essential skills, and invest themselves in their learning.

Pedagogies of responsiveness: a variety of strategies designed to elicit information about where students are and where to go next so that teaching can be responsive to the needs of students and classes.

These seem like useful questions to ask myself while either planning for or reflecting after a lesson. Am I using effective pedagogies of engagement? Am I using effective pedagogies of responsiveness?

Maybe the difference between these pedagogies and formative assessment is just semantics. At the same time, more than a week after NCTM, these are the ideas I have reflected the most on, and these are the ideas that seem to me to have the most potential to impact my teaching.

Mathematical Anthropology

Geoff Krall spoke at NCTM’s ShadowCon last week about “The Art of Mathematical Anthropology”. You can watch his talk here starting at about 52:30.

Geoff made a compelling argument for the value of reflections to help students see their own growth. His call to action was to:

  • Assign complex tasks
  • Invite written reflection
  • Have a conversation

By giving students complex tasks and producing complex work, we can ask them where the work was difficult, where they struggled, and how they got better at it. I can learn a great deal about my students from these reflections and conversations. More importantly, students can authentically see their own growth, rather than focusing on shallow external indicators like grades.

I’m interested. I’m hoping to learn more from Geoff’s online course this fall. But in the meantime, I’ve been thinking about how to apply the lens of mathematical anthropology to other areas of my teaching. I have two ideas.

Mathematics is…

I wrote a while back about Nat Banting’s idea of asking students what they think mathematics is, and how much I learned from reading my students’ reflections. This year, I set up a progression of thinking about the purpose of mathematics from the beginning of the year. I began in the first week of class by asking students whether math is worth learning and making my elevator speech for the value of learning mathematics. I then returned to these ideas with content we are working with as the year goes on, and we talk about why these topics may or may not be worth learning and how students may or may not use that type of thinking again. At the end of this sequence, I ask students to finish the sentence or paragraph “Mathematics is…” however they’d like. I find it gives me useful feedback about the extent to which I have changed students’ beliefs about mathematics, as well as entertaining me with students’ reflections.


Sam Shah wrote recently about group work. In short, Sam argues for having the class generate a list of what they value in group work. Then, several weeks later, each student answers questions about how they are doing relative to these student-generated values. Then, students have a facilitated conversation in groups, discussing what they see going well and what they want to improve on. I’m excited to try this out, and I think individual student reflections on their own collaboration after these conversations could be both useful for students and illuminating for me.

Mathematical Anthropology 

I’m excited about the idea of having conversations with students about rich artifacts that can help to illuminate their growth. Geoff convinced me that it is worth having these conversations about student work on complex tasks. In addition to students’ knowledge and mathematical skill, I also want them to think about the way they work together and their relationship with mathematics. As I think about collecting student work on complex tasks, I’d like to add to that portfolio student thinking about their collaboration and what they see as the role of mathematics in their lives. These ideas aren’t fully formed, but I’d like to keep thinking about what a portfolio looks like that effectively captures the range of goals I have for students. At the same time, I want to keep that portfolio small enough that it is manageable to look through the artifacts and have a conversation about what they mean. Definitely some figuring out left to do.

On NCTM and the #MTBoS

I just finished NCTM and am reflecting on the experience in the airport. I have three specific observations, an anecdote, and a possible connection on my mind.


  • At the Desmos trivia night, one trivia category asked what different acronyms stand for. One clue was MTBoS. A number of tables had a blank line for the answer to that question.
  • One woman told me, very insistently, that MTBoS is a Twitter handle that five people have control of
  • A search for #mtbos and math twitter blogosphere, and #mtbos and mathtwitterblogosphere turns up only about a tweet per month explaining the meaning of the hashtag


I wrote this post after doing some data collection two years ago suggesting there are vastly more lurkers — teachers aware of people and ideas in the MTBoS — than active participants in the community. I believe that’s still true. It seems like some ideas, like Estimation 180, have reached a pretty significant level of saturation among attendees of the conference.

At the same time, and this is completely subjective, the MTBoS community felt less visible at NCTM than my last time at the annual meeting two years ago. It felt like fewer presentations were mentioning the community, there were fewer people wearing the MTBoS ribbon on their name tags, and in general less buzz around the MTBoS and more buzz about specific speakers or events that I would formerly have associated more closely with the community, like the Mathalicious happy hour and Shadowcon.


I very often teach a lesson and realize at some point that, despite my efforts, students don’t actually know the purpose of the lesson or how it fits with the broader trajectory of our class. I wonder to what extent a similar thing is happening with the way new people learn, or don’t learn, about the MTBoS. It seems like the community is being explained to newcomers at a far slower rate than would be necessary for teachers arriving to Twitter or other sites to adequately understand what the MTBoS is or wants to be.

I notice that the popularity of the specific projects that the MTBoS has created seem to only increase in visibility and popularity. I wonder what impact that will have on the future of the community that calls itself the MTBoS.

More Than Resources: The Internet and Deliberate Practice

This is the short version of my presentation at NCTM (Thursday 12:30). Slides are linked here. The substance of this talk is very similar to my talk at TMC last summer, though geared toward the audience at NCTM.

Quick plug — the MTBoS tab on this blog is meant to be a “starter kit”. If you’re unfamiliar with the community, that page could start you down the rabbit hole.

The goals of my talk are first, to sell folks on the #MTBoS, an online community that creates tons of free, awesome resources you can use in your classroom, and second, to think about using those resources purposefully to get better at teaching.

When I started teaching, I struggled. I started using lessons from blogs and resources on the internet in an effort to be a more engaging and effective teacher. Some of those lessons were pretty engaging, but my students weren’t learning very much. Strong students had the background to make sense of my clever new lessons, but weaker students floundered and my curriculum became disjointed and messy.

What I learned from all this is that clever ideas don’t add up to coherent curriculum. Great curriculum focuses student thinking on the essential mathematics, builds on previous understandings, and is responsive to what students know and don’t know. I put this veneer of student engagement on top of the curriculum and expected huge changes.

There’s lots of great stuff on the Internet. I don’t meant to bash all of the resources out there. I want to argue the opposite. But that stuff doesn’t instantly make me a better teacher. Instead, getting better at teaching takes practice. And more specifically, it takes deliberate practice — practice where I push my comfort zone, work toward specific goals, focus intently on practice, use feedback, and develop a mental model of what expertise should look like.

While throwing new resources into my class didn’t work very well for me, I do want to highlight all of the amazing stuff out there. Open Middle, Three-Act Tasks, Visual Patterns, Estimation 180, Emergent Math’s Problem-Based Curriculum Maps, Nix the Tricks, Which One Doesn’t Belong. And there are tons more, that’s just a sampling.

So here’s the tension. There are awesome resources on the internet, but those resources don’t instantly make me a better teacher. I want to explore that tension by looking at two different resources that I’ve used. What has made these impactful on my teaching is that, instead of one-off lessons, they are structures I can use on a regular basis. I’m not going to be great at using them right away, but I can put effort into these specific structures, learn how to use them well, and add a new tool to my teaching toolbox.

One resource is Connecting Representations, an instructional routine developed by Grace Kelemanik and Amy Lucenta. We’ll do the routine, both to learn about the routine, and to unpack the type of thinking I try to use to get better at using this routine, and use those lessons to become a better teacher.

[this is the part you’d really have to be there for]

At the same time as Connecting Representations can be a powerful routine, there is also an incredible community out there supporting it. When I tweeted out this task with the hashtag #connectingreps, I got a ton of responses helping me to refine my thinking and figure out how this fit into my broader curriculum. And the best part is, there are dozens more Connecting Representations tasks out there.

A second resource is Visual Patterns, collected by Fawn Nguyen. Similarly, let’s do a visual pattern together and think about how this tool can be used effectively, and how it can inform other areas of teaching.

The purpose of all of this is both to sell folks on the power of online resources to supplement curriculum and impact your classroom, and also to think critically about how to use those resources purposefully to improve teaching practice. I’m not trying to sell either Connecting Representations or Visual Patterns specifically — there’s a ton of awesome stuff out there, and I don’t know what’s right for you. I do want to wrap up by framing these ideas in the broader context of teaching and offer a challenge going forward.

Dylan Wiliam wrote that “like so much else in education, ‘what works’ is the wrong question because everything works somewhere and nothing works everywhere”. One of the things about the #MTBoS that I find so powerful is that you can use these resources however you like, choose what you want to use, and adapt them to your specific classroom. There are no methods that work everywhere. Instead, I think of great teaching as having a broad toolbox, and having deep knowledge of how to use all of those tools effectively. People often talk about 21st century learning, how what students need to learn is changing. I think of this as 21st century teaching. Today there are more resources than ever before to improve teaching practice. I think we have an imperative to broaden our toolbox, and to use those tools purposefully to get better in the classroom and better serve our students.

Steve Leinwand has said that it’s unprofessional to ask teachers to change more than 10% a year. It’s unrealistic. I’m not asking for a huge shift, or hours spent online poring over internet resources. But he also says that it’s unprofessional for teachers to change less than 10% a year. My 10% is focused on developing new tools through easy to use resources, and figuring out how to use those tools effectively. My challenge to you is to figure out what your 10% will be.

Deliberate Practice and Mountain Biking

I’ve argued in a few places recently for the value of deliberate practice (see here, and here, and here). My argument in short is that five principles of deliberate practice can help teachers to maximize their improvement, and the synergy of the five principles together is particularly powerful. The five principles are:

  • Push beyond one’s comfort zone
  • Work toward well-defined, specific goals
  • Focus intently on practice activities
  • Receive and respond to high-quality feedback
  • Develop a mental model of expertise

I recently encountered an interesting example of deliberate practice outside of teaching that reminded me why I think these principles are worth sharing.

In December, I registered for a 100 mile mountain bike race this summer. I started mountain biking a little less than two years ago when I moved to Colorado, and I ride regularly when there’s no snow on the ground (which is only from May/June to October around here) and when I have the chance to get down to the desert during the winter. The race is in my town and starts just a few blocks from my front door. Several friends have ridden it, so there are plenty of people around who know some things about endurance racing.

If you had asked me in December how I was going to train for the race, I probably would have said that I need to go on a lot of long rides starting as soon as I can, and that I should do rides with long climbs as there are five long climbs in the race. I would characterize that perspective as naive practice — it lacks focus, and while I would definitely improve, I wouldn’t improve as quickly as with more effective practice techniques. Having done more research into training over the last few months, I have some more purposeful ideas about how I want to get better.


The goals of “go on long rides” and lots of climbing” could become much more specific. After more research, here’s a list of some things I’m working on:

  • Leg speed
  • Power while climbing
  • Aerobic capacity
  • Descending
  • Pushing my bike (everyone except Lance Armstrong ends up pushing their bike up at least one or two really tough sections)
  • Being in the saddle for 5+ hours
  • Starting in a crowd of 2000 riders
  • Climbing in a crowd of riders
  • Intervals work to improve my climbing
  • Technical climbing
  • Nutrition for a 10-11 hour race

I’m sure I could get even more specific with many of these goals, and the more specific goals I have, the more effective my practice will be.

Comfort Zone 

Connected with specific goals, I need to be willing to work on my areas of weakness and try things I haven’t before. Relative to my other goals, I have pretty decent power while climbing, but my leg speed isn’t very good. I don’t enjoy spending 15 minutes of a ride working on sustained leg speed, but that work is likely to lead to more improvement than focusing on goals that I’m more comfortable with.


Naive practice might be going for a fun 3 hour ride rambling through some of my favorite trails every day. Instead, I could do a shorter ride focused on leg speed through a few flat sections, do intervals while climbing, and work on efficient descents going downhill. The next day I can rest with a fun casual ride. Focus means quality over quantity.


I’m still a beginning rider in many ways. One area I know I need feedback on is my bike setup — I bought my bike used from a friend, and haven’t changed much of anything since then. While I don’t know enough to do this myself, with feedback from a more experienced rider I can optimize my seat, handlebars, brakes and more to my body size and riding needs. That’s just one area I can use expert feedback — I’m sure there are plenty more opportunities for feedback that I don’t know about, but that I will learn a great deal from as I seek out feedback this summer.

A Mental Model 

A mental model means having the knowledge to link deliberate practice to the goals I’m working toward. I need an accurate mental model based in exercise science of how my body responds to different workouts as I plan my training. I need to understand how my goals fit together and synergize to get me ready for race day. Without that understanding, the elements of deliberate practice can start to fall apart.

Training for a mountain bike race has some other interesting things in common with practice in teaching. Different riders will focus on different goals, just as teachers have different elements of teaching that they choose to prioritize. Feedback is tough because I can’t see how much I improve from a particular workout. In the same way, I may think students learned from a particular lesson until a few days later when I realize nothing I tried to teach has made it into students’ long term memory. It’s hard to make effective use of the principles of deliberate practice and there’s no perfect training technique.

My biggest lesson from thinking about my training has been that it is both easy and natural to fall into naive practice. For most things in life, that’s totally fine. But using the principles of deliberate practice offers an opportunity to increase the quality of practice and maximize improvement, and those techniques are often lying unused right under my nose.