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.

An AP Phyz S that can calculate the area of a circle but doesn't understand that pi is the ratio of C to d *for every circle*.

— Marcie Shea for Henrico School Board (@MarcieShea) September 5, 2018

How about students who can solve an equation in the form ax + b = c but not b + ax = c or c = ax + b or even understand they can substitute values for x to find an answer.

— Dr. Jennifer Jensen (@DrJenJensen) September 6, 2018

A student who sets up the percent proportion to find 100% of a number.

— Lara Metcalf (@LaraMathcalf) September 6, 2018

Students who can graph the equation y=2x+6 but days later, when all Ss are asked to select a random #, double it and add 6, the same student is blown away when the class plots the ordered pairs and it is a line not a scatter plot.

— Wendy Clark (@wendymairclark) September 6, 2018

A student who sees sqrt(7) * sqrt(7), multiplies them together to get sqrt(49), then takes the square root to get 7.

— Malcolm Eckel (@mceckel85) September 6, 2018

A student who can solve for b or m in the equation y=mx+b but has no idea that x and y are a point on the line.

— Angela Cooper (@angecooper) September 6, 2018

Students who know the square root of 25 is 5, but can’t tell me how big the square root of 75 is.

— Mr. Hoffman (@HoffmanMath) September 6, 2018

Student who borrows to solve 13 - 8 only to end up with 13 - 8.

— Mike Flynn (@MikeFlynn55) September 6, 2018

A student who finds two solutions of a quadratic equation but can’t see why only one would work in context. #fluencyisnotenough

— Jonny Balsman (@Algebro92) September 6, 2018

A kid who can factor x^2 - 6x + 9 but not x^2 - 9

— Nils Headley (@NilsHeadley) September 6, 2018

A student who can compute the derivative of a function but not be able to look at the corresponding graph and indicate where its derivative is positive.

— Daniel Scher (@dpscher) September 6, 2018

Thought of another one.

— Lauren Baucom (@LBmathemagician) September 7, 2018

Solve 7x+22 = 109. Then solve 7w + 22 = 109.

Ss with procedural fluency will solve both equations twice. Ss with conceptual and procedural will solve once, and recognize that variables represent the value no matter the letter.

@ddmeyer using the vertical line test to determine of something is a function but having NO IDEA WHAT A FUNCTION IS.

— Megan Parise Schmidt (@Veganmathbeagle) September 7, 2018

Kids that know 7+7 by memory, but have to count 7+8 on their fingers, or know 8x5 is 40 but have no idea that 8x6 is just 8 more.

— Krista Pottier (@KristaPottier) September 7, 2018

Student can round 84,899 to the nearest thousand (thanks to silly rhymes), but can’t place 84,899 accurately on a number line to see that 85,00 really is the “nearest thousand.”

— Meghan Oliva (@MsOlivaEIS) September 6, 2018

S who subtracts 16 -9 by borrowing from the tens column to make 16 -9. S who says 6 x 8 = 48 but when asked 7 x 8 says "I don't know that one".

— Kit Luce (@kitluce1) September 6, 2018

Students who can graph a line, given its equation in slope/intercept form but who do not understand that the graph is the collection of points that satisfy the equation.

— Scott Farrand (@scott_farrand) September 6, 2018

Students who can accurately subtract a positive # from the same positive # to get 0 but panicks and resorts to "adding opposites" when given negative #'s ex. -5 - -5

— YinYangScorp (@YinYangScorp) September 6, 2018

Student work from today as we solved absolute value equations. Conceptual understanding vs procedural fluency? – Day 17 #teach180 #iteachmath pic.twitter.com/PGLb2inG0F

— Paul Jorgens (@pejorgens) September 6, 2018

And a corollary to that:

— David Cox (@dcox21) September 6, 2018

Students who can do 1/2 ÷ 1/4 by "flip and multiply" but can't see that there are 2 1/4s in 1/2.

Student who can calculate 4.89 x 8.127 using the standard algorithm but doesn't understand that the product is between 32 and 45.

— Daniel Luevanos (@danluevanos) September 6, 2018

Here's one from my work on interviewing students. Watch Mallika Scott asking Marisa to solve the school bus word problem. It's about 1 minute: https://t.co/eV4zCX3Y9D It's also in the Video Library at https://t.co/uOH9FMT1dX

— Marilyn Burns (@mburnsmath) September 6, 2018

A student who "just knows" 6x6=36 but doesn't know how that might help them solve 6x5 or 6x7 or 6x60.

— Investigations3 (@Inv3_Math) September 6, 2018

Students who can graph the equation y=2x+6 but days later, when all Ss are asked to select a random #, double it and add 6, the same student is blown away when the class plots the ordered pairs and it is a line not a scatter plot.

— Wendy Clark (@wendymairclark) September 6, 2018

Students who can use the power rule to find the derivative, but can’t find the derivative of a constant

— Lori Bodner (@Bodnermath) September 6, 2018

**Featured Comments**

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.

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.

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.

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.

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

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.