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Fave Five

Five of my favorite articles from the last month.

Humanizing Math Class Means Teaching Math Like The Humanities

Here are a couple of terrifying tweets from my summer.

I saw those tweets and had to sit back and collect myself.

That’s because I know how well I’m served by my knowledge of mathematics, how that knowledge helps me find value in early student thinking, how that knowledge helps me connect and build on thoughts from different students that, without that knowledge, might seem totally unrelated.

This isn’t a critique of those two newly drafted math teachers at all. Most of my horror here results from the thought of being drafted to teach history after a career teaching math. So what can they do?

You’ll find lots of people in those threads recommending resources and curricula. But resources and curricula are only as good as the teacher using them. A developing teacher can make a good resource bad and an expert teacher can make a bad resource good. (This is why John Mason prefers to talk about “rich teaching” instead of “rich tasks.”)

So my own advice is for these teachers trained in the humanities to focus on their teaching, not the resources or curricula.

Specifically, I hope they’ll resist the idea that math should be taught any differently than the humanities. I hope they’ll resist the idea that only the humanities deal in subjectivity, argumentation, and personal interpretation, while math represents objective, inarguable, abstract truth.

Math is only objective, inarguable, and abstract for questions defined so narrowly they’re almost useless to students, teachers, and the world itself.

find the volume of an abstract compound shape where all side lengths are known

In social studies, an analogous question might ask students to recall the date of the Louisiana Purchase or the name of the king who signed the Magna Carta – questions that are so abstracted from their context, so narrowly defined, and so objective that they make no contribution to a student’s ability to think historically.

The National Council for the Social Studies describes what’s necessary for students of social studies:

Students learn to assess the merits of competing arguments, and make reasoned decisions that include consideration of the values within alternative policy recommendations. [..] Through discussions, debates, the use of authentic documents, simulations, research, and other occasions for critical thinking and decision making, students learn to apply value-based reasoning when addressing problems and issues.

All of which rhymes perfectly with recommendations from the National Council of Teachers of Mathematics:

Teaching mathematics with high expectations for all students in mathematical reasoning, sense making, and problem solving invites students to learn to identify assumptions, develop arguments, and make connections within mathematical topics and to other contexts and disciplines.

Teaching math like the humanities asks us to:

  • Broaden the scope of the problems we assign. We can always narrow the scope in collaboration with students but the opposite isn’t true. Students don’t have the opportunity to “identify assumptions,” for example, if we pre-assume every detail in the problem.
  • Focus on mathematical ideas that are big enough to be understood in different ways. Ask students to make claims that demand to be argued and interpreted rather than evaluated by an authority for correctness.
  • Celebrate novel student contributions to mathematics. History is made every day and so is mathematics. If our students leave our classes this year without understanding that they have had made unique and original contributions to how humans think mathematically, we have defined “mathematics” too narrowly. (For example, someone just decided to call this shape a “golygon.” If that person has the right to notice and name things, then so do your students.)

Instead of the worksheet above, show your students this video of a pallet of bricks and then immediately hide it.

bricks stacked in an interesting way on a pallet

“Does anybody have a guess about how many bricks we saw up there?”

“Did anybody notice any features about the bricks that might help us figure out exactly how many bricks we saw there?”

“Let’s look at the video again. Okay, what’s the most efficient way you can think to figure out the number of bricks.”

“How were you thinking about the number of bricks you figured out? What assumptions did you make?”

“Someone else got a different answer from you. How do you think they thinking about the number of bricks?”

“Here’s the number of bricks. What’s another question we could ask now?”

These questions rhyme with the kinds of questions you’d hear in a productive, engaging humanities classroom, questions which are no less possible in mathematics!

Humanizing math class means teaching like the humanities. And if you’re joining us from the humanities, please be generous with your pedagogy. We need all of it.

BTW: This is my contribution to the Virtual Conference on Humanizing Mathematics, a fantastic learning opportunity hosted by Hema Khodai and Sam Shah through the month of August 2019.

It Isn’t Enough to Love Kids or Math: My Foreword to “The Five Practices In Practice”

NB. I was honored to write the foreword to Peg Smith and Miriam Sherin’s fantastic new book The Five Practices in Practice, reprinted here with permission. Smith & Stein’s original book, 5 Practices for Orchestrating Productive Mathematics Discussion, was transformative for me professionally, but also personally, as I narrate in kind of oblique second-person fashion below. (Suffice to say: I am very much one of the two teacher types I describe.) Smith and Sherin’s follow-up book contextualizes those five practices in some extremely useful ways.

Why did you become a math teacher?

Perhaps you loved math. Perhaps you were good at math, good, at least at the thing you called math then. Friends and family would come to you for help with their homework or studying and you prided yourself not just on explaining the how of math’s operations but also the why and the when, helping others see the purpose and application behind the math.

Helping other people understand and love the math you understood and loved – perhaps that sounded like a good way to spend a few decades.

Or perhaps you loved kids. Perhaps even at a young age you were an effective caregiver, and you knew how to care for more than just another person’s tangible needs. You listened, and you made people feel listened to. You had an eye for a person’s value and power. You understood where people were in their lives and you understood how the right kind of question or observation could propel them to where they were going to be.

Spending a few decades helping people feel heard, helping them unleash and use their tremendous capacity – perhaps you thought that was a worthwhile way to spend what you thought would be the hours between 7AM and 4PM every day.

Or perhaps you loved both math and kids. It’s possible of course that neither of the two previous exemplar teachers will speak fully to the path that brought you to math teaching, although one of them speaks fully to mine. Yet, in my work with math teachers, I find they often draw their professional energy from one source or the other, from math’s ideas or its people.

It took me several frustrated years of math teaching – and years of work with other teachers – to realize that each of those energy sources is vital. Neither source is renewable without the other.

If you draw your energy only from mathematics, your students can become abstractions, and interchangeable. You can convince yourself it’s possible to influence what they know without care for who they are, that it’s possible to treat their knowledge as deficient and in need of fixing without risking negative consequences for their identity. But students know better. Most of them know what it feels like when the adult in the room positions herself as all-knowing and the students in the room as all-unknowing. A teacher’s love and understanding of mathematics won’t help when students have decided their teacher cares less about them than about numbers and variables, bar models and graphs, precise definitions and deductive arguments.

If you draw your energy only from students, then the day’s mathematics can become interchangeable with any other day’s. Some days it may feel like an act of care to skip students past mathematics they find frustrating, or to skip mathematics altogether some days. But the math you skip one day is foundational for the math another day or another year. Students will have to pay down their frustration later, only then with compound interest. Your love and care for students cannot protect them from the frustration that is often fundamental to learning.

I could tell you that the only solution to this problem of practice is to develop a love of students and a love of mathematics. I could relate any number of maxims and slogans that testify to that truth. I could perhaps convince some of you to believe me.

But the maxim I hold most closely right now is that we act ourselves into belief more often than we believe our way into action. So I encourage you more than anything right now to adopt a series of productive actions that can reshape your beliefs.

Here are five such actions: anticipate, monitor, select, sequence, and connect.

Those actions, initially proposed by Smith and Stein in 2011 and ably illustrated here with classroom videos, teacher testimony, and student work samples, can convert a teacher’s love for math into a love for students and vice versa, to act her way into a belief that math and students both matter.

For teachers who are motivated by a love of students, those five practices invite the teacher to learn more mathematics. The more math teachers know, the easier it is for them to find value in the ways their students think. Their mathematical knowledge enables them to monitor that thinking less for correctness and more for interest. Would presenting this student’s thinking provoke an interesting conversation with the class, whether the circled answer is correct or not? A teacher’s mathematical knowledge enables her to connect one student’s interesting idea to another’s. Her math knowledge helps her connect student thinking together and illustrate for the students the enormous value in their ideas.

For the teachers like me who are motivated by a love of mathematics, teachers who want students to love mathematics as well, those five practices give them a rationale for understanding their students as people. Students are not a blank screen onto which teachers can project and trace out their own knowledge. Meaning is made by the student. It isn’t transferred by the teacher. The more teachers love and want to protect interesting mathematical ideas, the more they should want to know the meaning students are making of those ideas. Those five practices have helped me connect student ideas to canonical mathematical ideas, helping students see the value of both.

Neither a love of students nor a love of mathematics can sustain the work of math education on its own. We work with “math students,” a composite of their mathematical ideas and their identities as people. The five practices for orchestrating productive mathematical discussions, and these ideas for putting those practices into practice, offer the actions that can develop and sustain the belief that both math and students matter.

You might think your path into teaching emanated from a love of mathematics, or from a love of students. But it’s the same path. It’s a wider path than you might have thought, one that offers passage to more people and more ideas than you originally thought possible. This book will help you and your students learn to walk it.

That Isn’t a Mistake

I’ve seen this particular incorrect answer from dozens of students over the last several weeks.

The work for 10 and 15 marbles is incorrect, but it isn’t a mistake. If I label it a mistake, even if I attach a growth mindset message to that label, I damage the student, myself, mathematics, and the relationships between us.

Mistakes are the difference between what I did and what I meant to Do.

For example, I know that words in the middle of a sentence generally aren’t capitalized. I meant to type “do” but I typed “Do.” That was a mistake.

What we’re seeing in the table above, by contrast, is students doing the thing they meant to do!

When I call that table a mistake, what I’m actually saying is that there’s a difference between what the student did and what I meant for the student to do. Instead of seeing the student’s work as a window into her developing ideas about tables and linear patterns, I see it as a mirror of my own thinking.

And it’s a bad mirror of my own thinking. It doesn’t reflect my thinking well at all!

It’s a bad mirror, so I call it a mistake. “Mistakes grow your brain,” I say. “We expect them, respect them, inspect them, and correct them here,” I say. And if we have to label student ideas “mistakes,” maybe those are good messages to attach to that label.

But the vast majority of the work we label “mistakes” is students doing exactly what they meant to do.

We just don’t understand what they meant to do.

Teaching effectively means I need to know what a student knows and what to ask or say to help her develop that knowledge. Calling her ideas a mistake transforms them from a window into her knowledge into a mirror of my own, and I am instantly less effective.

Our students offer us windows and we exchange them for mirrors.

The next time you see an answer that is incorrect, don’t remind yourself about the right way to talk about a mistake. It probably isn’t a mistake.

Ask yourself instead, “What question did this student answer correctly? What aspects of her thinking can I see through this window? Why would I want a mirror when this window is so much more interesting?”

What Does Fluency Without Understanding Look Like?

In the wake of Barbara Oakley’s op-ed in the New York Times arguing that we overemphasize conceptual understanding in math class, it’s become clear to me that our national conversation about math instruction is missing at least one crucial element: nobody knows what anybody means by “conceptual understanding.”

For example, in a blog comment here, Oakley compares conceptual understanding to knowing the definition of a word in a foreign language. Also, Oakley frequently cites a study by Paul Morgan that attempts to discredit conceptual understanding by linking it to “movement and music” (p. 186) in math class.

These are people publishing their thoughts about math education in national publications and tier-one research journals. Yet you’d struggle to find a single math education researcher who’d agree with either of their characterizations of one of the most important strands of mathematical proficiency.

Here are two useful steps forward.

First, Adding It Up is old enough to vote. It was published by the National Research Council. It’s free. You have no excuse not to read its brief chapter on procedural fluency. Then critique that definition.

Conceptual understanding refers to an integrated and functional grasp of mathematical ideas. Students with conceptual understanding know more than isolated facts and methods. They understand why a mathematical idea is important and the kinds of contexts in which is it useful. They have organized their knowledge into a coherent whole, which enables them to learn new ideas by connecting those ideas to what they already know. Conceptual understanding also supports retention. Because facts and methods learned with understanding are connected, they are easier to remember and use, and they can be reconstructed when forgotten. (pp. 118-119.)

If you’re going to engage with the ideas of a complex field, engage with its best. That’s good practice for all of us and it’s especially good practice for people who are commenting from outside the field like Oakley (trained in engineering) and Morgan (trained in education policy).

Second, math education professionals need to continually articulate a precise and practical definition of “conceptual understanding.” In conversations with people in my field, I find the term tossed around so casually so often that everyone in the conversation assumes a convergent understanding when I get the sense we’re all picturing it rather differently.

To that end, I think it would be especially helpful to compile examples of fluency without understanding. Here are three and I’d love to add more from your contributions on Twitter and in the comments.

A student who has procedural fluency but lacks conceptual understanding …

  • Can accurately subtract 2018-1999 using a standard algorithm, but doesn’t recognize that counting up would be more efficient.
  • Can accurately compute the area of a triangle, but doesn’t recognize how its formula was derived or how it can be extended to other shapes. (eg. trapezoids, parallelograms, etc.)
  • Can accurately calculate the discriminant of y = x2 + 2 to determine that it doesn’t have any real roots, but couldn’t draw a quick sketch of the parabola to figure that out more efficiently.

This is what worries the people in one part of this discussion. Not that students wouldn’t experience delirious fun in every minute of math class but that they’d become mathematical zombies, plodding functionally through procedures with no sense of what’s even one degree outside their immediate field of vision.

Please offer other examples in the comments from your area of content expertise and I’ll add them to the post.

BTW. I’m also enormously worried by people who assume that students can’t or shouldn’t engage creatively in the concepts without first developing procedural fluency. Ask students how they’d calculate that expression before helping them with an algorithm. Ask students to slice up a parallelogram and rearrange it into a more familiar shape before offering them guidance. Ask students to sketch a parabola with zero, one, or two roots before helping them with the discriminant. This is a view I thought Emma Gargroetzi effectively critiqued in her recent post.

BTW. I’m happy to read a similar post on “conceptual understanding without procedural fluency” on your blog. I’m not writing it because a) I find myself and others much less confused about the definition of procedural fluency than conceptual understanding (oh hi, Adding It Up!) and b) I find it easier to help students develop procedural fluency than conceptual understanding by, like, several orders of magnitude.

2018 Sep 05: The Khan Academy Long-Term Research team saw lots of students who could calculate the area of a kite but wrote variations on “idk” when asked to defend their answer.

2018 Sep 09: Here’s an interesting post on practice from Mark Chubb.

2018 Sep 27: Useful post from Henri Picciotto.

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Featured Comments

Karen Campe:

Can find zeros of factored quadratic that equals zero, but uses same approach when doesn’t equal zero. E.g. can solve (x-3)(x-2) = 0 but also answers 3 and 2 for (x-3)(x-2) = 6.

Ben Orlin:

The big, weird thing about math education is that most pupils have no experience of what mastery looks like. They’ve heard language spoken; they’ve watched basketball; they’ve eaten meals; but they probably haven’t seen creative mathematical problem-solving. This makes it extra important that they have *some* experience of this, as early as possible. Otherwise math education feels like running passing drills when you’ve never seen a game of basketball.


Today a student correctly solved -5=7-4x but then argued that -4x +7=-5 was a different equation that had to have a different answer.

Michael Pershan:

This has definitely not been my experience, and I don’t think this is consistent with the idea that conceptual and procedural fluency co-develop — an idea rooted in research.

William Carey:

I really like that way of talking about it. The way I think of it is a bit like exploration of an unknown continent. One the one hand, you have to spend time venturing boldly out into the unknown jungle, full of danger and mistakes and discovery. But if you venture too far, you can’t get food, water, and supplies up to the party. Tigers eat you in the night. So you spend time consolidating, building fortified places, roads, wells, &c. Eventually, the territory feels safe, and that prepares you to head into the unknown again.

Jane Taylor:

A student who can calculate slope but has no idea what it means as the rate of change in a real context.

Kim Morrow-Leong:

An example of procedural fluency without conceptual understanding is adding up a series of integers one by one instead of finding additive inverses (no need to even call it an additive inverse – calling it “canceling” would even be ok.) Example: -4 + 5 + -9 + -5 + 4 + 9

2018 Oct 13 NCTM offers their own definition of procedural fluency in mathematics.