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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.

Featured Tweets

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.

Watch the Four ShadowCon Talks from #NCTMAnnual and Sign up for the Follow-Up Conversation

Image of the ShadowCon Auditorium

On Thursday at #NCTMAnnual, four speakers urged teachers to reflect on their power not just to help students encounter mathematical knowledge but to change how students define themselves in relationship to math and to each other.

Video of each of their talks is online right now and each presenter invites you to join them in a follow-up conversation about their ideas over the next month. (More below.)

  • Lauren Lamb told us about her experience learning mathematics as a young woman of color, how she often felt invisible in her classrooms and unrepresented in her textbooks. She described the ways her teachers did and didn’t involve her in her own mathematics education.
  • Javier Garcia contrasted the ways we talk about students (as though they’re incomplete, fallible) and mathematics (as though it’s complete, infallible) and made a case that teachers should reverse those two descriptions.
  • Nanette Johnson impressed upon the audience the fact that each of them will leave behind a legacy for their students, an indelible imprinting of their efforts, either positive or negative.
  • Andrew Gael revealed the potency of our presumptions about student competence, and how students often live up and down to those presumptions. What we believe about student competence affects how we work with those students, which affects their opportunities to develop competence.

Great talks each one. Each one well worth your time.

But what happens to talks like these after they’re over? The ShadowCon Hypothesis is that the ideas from even great talks rarely survive contact with the reality of classroom instruction; that absent any kind of conversation or community organized around their implementation, those ideas are too easily put in a box labeled “Nice to Think About” or “Maybe Later.”

Each of our speakers agree with that hypothesis and each one wants to participate in a conversation with you over the next month. Sign up for a course you’d like to think and talk more about. We’ll place you on an email thread with a couple of random, interesting colleagues. Then you’ll receive a new discussion prompt once per week for the next four weeks, starting May 7. On a weekly basis, the speakers will summarize the most interesting ideas and answer the most perplexing questions from across all the groups.

It’s going to be a very interesting month.

[image via Cassie Sisemore]

Where You’ll Find Me at #NCTMAnnual

The icon on my airplane’s wifi signal indicates I’m somewhere over Wyoming right now, en route to Washington, D.C., for collaboration and conviviality with thousands of math teachers from all around the United States. I’m looking forward to reconnecting with old colleagues and meeting new ones so let me tell you where we’ll find each other. If we’ve met, let’s catch up. If we’re just meeting, let me know what you’re working on or wondering about.

Wednesday

Desmos Preconference Workshop

A picture of the Desmos preconference at TMC in 2017

My team will be running a morning workshop and an afternoon workshop on our newest, hottest technology and activities.

Also: Emdin’s opening session; NCTM Game Night.

Thursday

ShadowCon

ShadowCon will be at 6PM on Thursday in Ballroom B.

Zak Champagne, Mike Flynn, and I have recruited four interesting speakers – Lauren Lamb, Javier Garcia, Nanette Johnson, Andrew Gael – each offering their own variation on a similar theme. The presenters and the organizers collaborated on these ten-minute talks for the last several months. The process was a joy and the resulting talks are really exceptional. We’ll also introduce a new way to continue the call to action of those talks long after they have ended.

Desmos Happy Hour & Trivia

The Desmos Math Trivia Happy Hour is Thursday April 26 from 6:30-9:30PM at Clyde's of Gallery Place in the Piedmont Room 707 7th St NW, Washington DC.

Beep beep! Right after ShadowCon, I’m speed-walking straight to NCTM’s Top Rated Happy Hour Event. Then we’ll commence NCTM’s Highest Grade Trivia Competition. I can’t divulge any of the categories but if you were to brush up on your naughty words that rhyme with math vocabulary, I don’t think you’ll regret the effort.

Also: Sessions with Stiff, Rosen, Briars, Usiskin, Cirillo, Pelesko, and time at the Desmos booth, trying to convince people to buy our free calculators and free activities. (Pretend you don’t see me, friend. Pretend you’re on the phone with your mother. Pretend I didn’t invent those exhibitor-dodging moves. We are having this conversation, friend.)

Friday

Full Stack Lessons

I know my talk is at 8 AM. I know that. But I’ll be bringing coffee for at least me and one other person in the room. Maybe three more people if I can find one of those coffee carrier trays.

Here’s the description:

Two teachers can take the same idea for a lesson and experience vastly different results in class. This is often because one teacher taught from the full “stack” of questions and the other taught from just part of it. We’ll look at the contents of that stack and learn how to put the full stack of questions to work in your classes.

Also: Zager, Martin, United Airlines.