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 = x
^{2} + 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.