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.

Technology I Use

I’m not particularly passionate about technology; I try to only use tools that meet a specific need and support learning more than they distract student attention. Here’s what I use on a regular basis on my classroom.

Paper & Pen
I use a ton of paper. Homework, classwork, and assessments. Nothing beats students doing math by hand for lots of the tasks I give.  I do all of my planning in one notebook — I have three preps, and I have a three-section notebook that I use to plan, follow my plan in class, and jot down notes for the future.

I have enough vertical non-permanent surfaces in my classroom this year to get all students working at the same time, as well as smaller whiteboards that students can use at their tables in small groups.

Desmos graphs, images, videos, spreadsheet for visible random groupings, and more.

Student Computers
I’ve been lucky to work at two schools where students have relatively easy access to computers. Mostly Desmos lessons and interactive demonstrations, but occasionally Google Docs, Sheets, and Forms. I’m pretty selective with computers; they can be an incredible tool but also a huge distraction. I’m a big fan of 2:1 computing to create some discourse and make activities feel less souless.

I have my Dropbox account set up so that, when I’m connected to wifi, photos I take with my phone sync automatically to my account. Lag time is typically around thirty seconds. Snap a photo and open up to quickly and easily share samples of student work with the class, with a decent degree of anonymity.

And that’s pretty much it. There’s lots of other great stuff out there that I’m not using. I like technology that’s free, practical, and doing as little as possible to get between my students and the math. I also like technology that I can use regularly and get good at working with; there are plenty of tools that would be useful on occasion, but I don’t want to use unless I can build a degree of consistency and routine with. I’m sure there’s plenty more I’ll adopt in the future, but these get the job done for me.

Why Students Don’t Learn

Memory is the residue of thought.

Dan Willingham

Lots of students struggle in math class. I would conjecture that, of the many students who I have been unsuccessful teaching, the principle reason they didn’t learn is that they spent much of their time in my class thinking about things besides mathematics. Here are some profiles of students who I’ve seen struggle.

Identity & Belonging
Some students come to my class each day feeling like they don’t belong. Maybe it’s because they perceive my actions as racist or sexist, maybe it’s because of societal messages about whether their identity is one that is supposed to be good at math. If students perceive that they don’t belong in my math class because of who they are, they are likely to quit thinking when things get hard, or otherwise find something else to think about because math isn’t for them.

Anxiety presents itself in a range of ways. I’ve seen students who physically shake, or put their head on their desk and shut down, or start crying. Others keep it inside. What they have in common is that, while they are in math class, their mind is consumed with fear and worry, and there’s no room for mathematical learning.

Students spend lots of hours in school during their lives. They are likely to learn some things during that time. For some students, they have learned that if they copy things from the board, write down answers figured out by others in their group, and generally stay hunched over a piece of paper looking studious, they are unlikely to be bothered by the teacher. They are doing some thinking during class, but that thinking is about how to look like a student, not how to reason about mathematics.

Some students have an intense dislike of mathematics; they would rather be anywhere in the world than my math class. Even if those feelings don’t trigger anxiety, they lead to avoidance mechanisms like doodling, refusing to answer questions, writing fake answers on the page, hiding misconceptions, and more. These students aren’t thinking about math; they’re thinking about how unhappy they are in that moment and everything else they’d rather be doing.

This is not an exhaustive list, but these are different behaviors I have seen with some frequency that prevent students from learning. I have one big takeaway here.

Never Blame Students 
These issues may not be my fault. They may be rooted in years of experience. They may be enormously challenging to work against. It might be really easy to just say “hey, Jimmy never paid attention, so he’s gonna fail this class”. That doesn’t matter. It’s my job to send anti-racist messages about how my classroom works. It’s my job to work with anxious students to create familiar routines, support systems, and a safe space for them to learn. It’s my job to teach students what effective effort and learning look like. It’s my job to create opportunities for engagement that teach students what it’s like to enjoy doing math.

I wrote last year about math ability, and made an argument that I didn’t see any evidence that math ability exists. It created quite a bit of argument on Twitter; many folks disagreed pretty strongly, for a variety of reasons.

I think my statement then was too strong. I’m mostly agnostic on math ability. I don’t know whether or not it exists. But I do know that the four phenomena above exist. And in every student I have taught who might be categorized as low ability, at least one, and often several, of the above issues are at play. In most cases they’ve been at play for years. I don’t know whether ability exists, but I know that these four issues are ones I can do something about. Talking about math ability seems unproductive in that context; I believe in teaching kids, I believe that every kid can learn, and I believe it’s my job to figure out what obstacles are in a student’s way and do something about them.

I don’t mean to stir up the math ability fight again. I don’t mind whether or not folks believe it. I do think that, when I’m confronted with a challenging student, I have lots and lots of tools that I can use to start moving them forward, and talking about ability is a dead end. There are no easy solutions to any of these problems, but they’re all problems I can do something about.

Mathematical Mindsets and Easy Fixes

I’m reading Jo Boaler’s most recent book, Mathematical Mindsets. She presents a compelling vision of what mathematics teaching can be, and I’m learning a great deal. At the same time, I wonder about the ways that Boaler talks about change in mathematics classrooms. One passage struck me.

In a chapter on tracking, Boaler describes a school that made two of the changes she recommends: de-tracking classes, and providing students with low floor, high ceiling tasks:

At the end of the first week of teaching the new classes, grouped for a growth mindset, one teacher exclaimed in amazement that after he gave out one task, astudent who “would have been in the bottom group” was the first to solve it. Over time the teachers continued to be surprised and pleased by the different creative methods shown by different students from across the achievement range. The teachers were thrilled with how well students responded to the de-tracking and with how disciplinary issues, which they had feared would increase, disappeared almost overnight. This was interesting to me, as the teachers had been quite worried about de-tracking and whether the students would work well together. They discovered that when they gave open tasks, all students were interested, challenged, and supported. Over time the stduents they thought of as low-achieving started working at higher levels, and the classroom was not divided into students who could and students who could not; it was a place full of excited students learning together and helping each other (p. 117).

I worked at a school that de-tracked students, and where I tried to implement low floor, high ceiling tasks. I didn’t feel particularly successful. I had some idea of my goals, and I had some tools to move in that direction, but I didn’t have the skill to use those tools effectively and actually make a difference for students.

The narrative that Boaler provides here is important to validate these two practices, which I really believe in. But I think the narrative also sends a message that they are practices which can change a culture of mathematics alone. That’s not my experience. De-tracking students requires concerted effort and systems to ensure the success of previously low-achieving students and make sure they get the support they need. Low floor, high ceiling tasks require deliberate pedagogy to create spaces where students are willing to take risks, ideas are shared productively, and tasks are used as opportunities for instruction.

I’m being tough on Boaler here. It’s an excellent book, and she does outline strategies for making these ideas work. She is particularly articulate about structures for group work and valuing multiple dimensions of mathematical thinking. But she also offers a narrative that I think oversimplifies what it means to transform a culture of mathematics in a school. I’m interested in structural changes that alter the way mathematics is practices and perceived. I’m much more interested in thinking about the pedagogy that supports those structural changes, and how teachers can develop that pedagogy in a deliberate way to make sure they are supporting all students.


A bit of wisdom from a grad school mentor on sarcasm, paraphrased:

Sarcasm is corrosive. It creates false intimacy. For every kid who might think it’s funny, there’s another who is terrified and will never say a word to you about it.

I’ve been thinking more about this recently. I can be pretty sarcastic at times, without much purpose. It’s easy to go for quick laughs. It’s much harder to think about whether there is a student in the corner who is terrified. And it’s especially important for students in math classrooms with deeply entrenched mindsets and dispositions toward math.

I wonder what my class would be like if I replaced my sarcasm with sincerity? Food for thought.

Preparing Students for Boring Teachers

But they need to listen to lectures in college; don’t we need to prepare them for that too?

I’ve heard the argument plenty of times. Sure, all that fun and engaging stuff you’re doing is awesome, learning by doing, etc, but we need to make sure they’re ready to get lectured at in college. (When I taught middle school some people said the same thing about getting lectured at in high school.) This seems to particularly resonate with folks who aren’t teachers or who are new to teaching. Not sure why.

I disagree, pretty strongly. A few reasons:

  1. I’m skeptical there’s some special skills called “listening to a lecture” that students need to practice for a few hundred or thousand hours (by getting lectured at) in order to develop. Sure, we should think about students’ attention spans, but I’m skeptical lecturing at kids is the most useful way to improve a kid’s attention.
  2. If students are going to be lectured at in the future, it’s probably most important that they have a broad base of knowledge and skills so that the lecture makes sense and connects with their prior knowledge. I’m going to build that base of knowledge the best way I know how.
  3. If college is some dystopia of endless lectures, it underscores the urgency and importance of building students’ curiosity and love of learning so that they are in a better position to pursue their passions, regardless of the quality of some subset of their education.
  4. Colleges are changing faster than many folks think. Not that they’re changing very fast, but they’re changing. We’re talking about aiming behind a moving target. No sense preparing students for the past.


I was asked recently if I lecture. I don’t really know how to answer that question. Do I explain things, model ideas and strategies on the board, point out connections, and teach by telling? Absolutely. All the time. But I don’t think that has to be the defining characteristic of my teaching. In my class, students do math. I use that math to elicit what they know and don’t know, and based on that knowledge I may choose to deliver some explicit instruction. Maybe for two minutes at a time, maybe for twenty.

I don’t see a lot of value in talking in absolutes here. “Lecture” seems to imply that the teacher talks for the whole period, maybe bold students are willing to ask questions, and that’s the lesson. Proponents of lecturing seem to view the alternative as hippy-dippy projects, or aimless discovery learning. My teaching hangs out in the middle. Sometimes I deliver information. Sometimes I ask students to figure something out. Sometimes I ask students to practice a skill. This can happen in any order based on what tools I think are most useful that day. There’s no magic bullet, no one right answer. That intellectual work of figuring out what is going to work tomorrow for my students is probably my favorite part of the job.

Two Types of Teachers

Below is an outburst I made at a meeting yesterday in response to a comment from the facilitator about the afternoon being a hard time for people to pay attention. Not the most tactful thing I’ve done recently.

There are two types of teachers in the world. There are teachers who love to tell you all of their excuses for why kids can’t learn last period, or first period, or after lunch, or Friday, or Monday, or the week before Christmas, or the month of December, or the month of June. Then there are teachers who say my students have brains and can use them and we are going to learn today and I don’t give a shit what anyone says.

(I don’t mean to claim that all times are equally conducive to learning. Just that excuses don’t help, and my experience is that, with a handful of rare exceptions, any student can learn at any time if they have teachers who care about them and create opportunities for meaningful learning.)