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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.

Mike:

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.

Drill-Based Math Instruction Diminishes the Math Teacher as Well

Emma Gargroetzi posts an astounding rebuttal to Barbara Oakley’s New York Times op-ed encouraging drill-based math instruction. Gargroetzi highlights two valid points from Oakley and then takes a blowtorch to the rest of them.

I haven’t been able to stop thinking about her last sentence since I read it yesterday.

Anyone who teaches children that they need to silently comply through painful experiences before they will be allowed to let their brilliance shine has no intention of ever allowing that brilliance to shine, and will not be able to see it when it does.

I’m perhaps more hesitant than Gagroetzi to judge intent. Lots of teachers were, themselves, victimized by drill-based instruction as students and may lack an imagination for anything different. But I’m absolutely convinced that a) we act ourselves into belief rather than believing our way into acting, and b) actions and beliefs will accumulate over a career like rust and either inhibit or enhance our potential as teachers.

A math program that endorses drills and pain as the foundational element of math instruction (rather than a supporting element) and as a prerequisite for creative mathematical thought (rather than a co-requisite) inhibits the student and the teacher both, diminishing the student’s interest in producing that creativity and the teacher’s ability to notice it.

Teachers need to disrupt the harmful messages their students have internalized about mathematics. But we also need to disrupt the harmful messages that teachers have internalized as well.

What experiences can disrupt the harmful messages teachers have internalized about math instruction? Name some in the comments. I’ll add my own suggestions later tomorrow.

2018 Aug 25. I added my own suggestion here.

Featured Comments

Faye calls out the process of learning content and pedagogy simultaneously:

Many mathematics teachers do not have the mathematics content knowledge that they need themselves. The Greater Birmingham Mathematics Partnership has found that teaching teachers mathematics using inquiry based instruction results in increased content knowledge for the teachers and a change in their beliefs about how and what all children can learn, i.e., acting themselves into changed beliefs.

Chris:

Math teachers circles (www.mathteacherscircle.org/). They provide the space for math teachers to be mathematicians (in the same way a lot of the arts teachers I know are still practicing artists).

Another Chris echoes:

It wasn’t until I was asked to think about mathematical tasks and ideas for my own understanding that I could ask the same of my students. And then, it was unavoidable…there was no going back.

William Thill elaborates:

But when I can tap into the emotional and intellectual highs that emerge from playing with cherished colleagues, I am more likely to “set the buffet” for my students with more open-ended exploration times.

Martha Mulligan:

… watching yourself teach on video is a great experience to disrupt harmful messages about math instruction, like talking too much as the teacher. I know that many math teachers feel the need to provide the most perfect, refined, rehearsed explanation so that students can see what they are supposed to see in the way they are supposed to see it. I certainly felt (at time still feel?) that way. That practice diminishes the students’ roles of sense-making on their own. But watching a video of myself teaching was one of the most humbling things I’ve done and it changed my practice so much. I also watched them among other trusted teachers from whom I learned so much. Having time to stop a video, talk about, reflect on it, etc is very powerful. Even seemingly simple things like wait time and teacher movement/positioning can look very different than what we imagine we look like.

Alexandra Martinez calls out the limitation of reading narratives and watching videos of innovative teaching:

I think the most powerful way to disrupt teacher’s own experiences and expectations is new creative experiences with their own students. The evidence and reflection can support teachers in seeing what is possible. If we ask teachers to imagine what is possible through narrative, they won’t always believe it. But when they see their own students speaking and thinking as mathematicians, that evidence disrupts their established belief systems. So I’d say observations, modeling, Coteaching, pushing in, PLC planning with lesson study can all potentially do this.

Be sure, also, to check into Chris Heddles’ a/k/a Third Chris’s dissent:

I’m going to go against the grain and admit that I use drill as a prerequisite (or at least an opening activity) with many of my students.

Two New Interviews with Yours Truly

I have two interviews out right now that I want to bring to your attention – a PDF and a podcast, so pick your medium! Both sets of interviewers were fantastic – well-researched and probing – and I did my best to rise to the occasion.

First, Ilona Vashchyshyn interviewed me for the May / June issue of Saskatchewan Mathematics Teachers’ Society magazine [pdf]. Ilona read my dissertation and her questions drew together themes of online math edtech, mathematical modeling, and Math Teacher Twitter. Here’s me:

The consistent theme in my participation [in online math teaching communities] is the fact that my thoughts always seem perfect to me until they escape the vacuum seal of my brain. Once they’re out in the world, in a blog post or a tweet, that’s when I realize how much work they need. I can’t get that feeling any other way.

Second, I have an hour-long interview out today with Becky Peters and Ben Kalb of the Vrain Waves podcast. They dug deep into my blog’s back catalog and asked questions about decade-old posts I had forgotten about. Halfway through, they asked about the role of memorization in learning mathematics. I confessed to them that I never want to be stuck driving on the road at the same time as anybody who derides the value of memorization in learning mathematics. I know a bunch of you will disagree with me there, so listen to the piece, and then tell me what you think either here or on Twitter.