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 (@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 Maths (@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 canâ€™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 canâ€™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

— Thatâ€™s 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

https://twitter.com/jetpack/status/1037804944550768640Student can round 84,899 to the nearest thousand (thanks to silly rhymes), but canâ€™t place 84,899 accurately on a number line to see that 85,00 really is the â€œnearest thousand.â€

— Meghan Oliva (@MsOlivaEIS) 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? â€“ 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 Ã· 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 canâ€™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 doesnâ€™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. Theyâ€™ve heard language spoken; theyâ€™ve watched basketball; theyâ€™ve eaten meals; but they probably havenâ€™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 youâ€™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 donâ€™t think this is consistent with the idea that conceptual and procedural fluency co-develop â€” 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 canâ€™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 â€“ calling it â€œcancelingâ€ would even be ok.) Example: -4 + 5 + -9 + -5 + 4 + 9

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

## 90 Comments

## Daniel Peter

September 5, 2018 - 4:08 pm -In a conversation with a prominent PhD in cognitive science from the Ontario Institute for Studies in Education (OISE) he said the emerging consensus in cog sci is that conceptual understanding is demonstrated by the multiplicity of representations. Take a grade 8 child who understands linear relationships can represent the growing patterns with concrete materials/drawings, using a t-table, through graphing the line, through describing the relationship in words and with formal algebraic expressions/equations. And they can describe how the representations are connected to each other such as describing the constant in the concrete materials, where the line crosses the y axis and in the expression. Those who can only manipulate the algebra and do not see it as a pattern lack conceptual understand.

## Harry O'Malley

September 7, 2018 - 6:52 am -I don’t believe more representations necessarily means more understanding. As I said in my comment below, understanding only comes from linking symbolic representations (graphs, tables, expressions, equations) to concrete, visceral representations. If a student can accurately convert an equation into a graph, this implies no understanding (unless the graphical representation helps them translate to the concrete, visceral one and they are using the graphical representation as a proxy for that.) Concrete, visceral representations of ideas are the only ones that make sense to humans. We’ve invented symbolic representations to make it easier and more efficient to think and communicate about real, tangible things, but unless you are imagining real objects when you are looking at symbolic ones, you don’t understand what you are looking at.

Also, “explaining” the connections between representations isn’t what we should look for to know whether or not someone has made a proper connection between representations. Translating one representation into the other, not explaining the connections between the two, should be the skill we look for. This is an easier thing to measure and a more fundamental behavior. Explaining the connection between two representations, let’s say a table and an equation, can be thought of as translating the table representation into a verbal representation, then translating the verbal representation into elements of the equation representation. This is messier for two reasons: 1) the imperfection of verbal representations muddy the translation; 2) it brings three representations into play instead of the two that were intended…

## Howard Phillips

September 7, 2018 - 5:47 pm -to Harry O’Malley:

This is the forgotten clue for all of maths education, from kindergarten to PhD

## Ben Orlin

September 5, 2018 - 4:21 pm -A student who…

1. Can find the perimeters of rectangles, but arrives at different answers when labeled lengths are added or removed (e.g., labeling just one of an opposite pair vs. labeling both).

2. Can find the zeroes when given a factorized quadratic (e.g., (x-3)(x-2) = 0), but doesn’t know whether the same method will work for a factorized cubic (e.g., (x-3)(x-2)(x-1) = 0).

3. Can multiply an integer by a fraction (e.g., 7 x 1/9) only by rewriting “7” as “7/1” and then “multiplying across.”

A related game is to imagine what “procedural fluency must come before creative application” would look like in other domains:

1. “A foreign language learner must not attempt dialogue until mastering all of the grammatical forms that may arise in the conversation.”

2. “A basketball player must not make a given pass until the play has been rehearsed repeatedly with the team.”

3. “A cook must not prepare a dish until each step (whipping the eggs; baking for the right amount of time; etc.) has first been practiced in isolation.”

These all sound really silly! That’s because practice and application always come interwoven, not one-first-then-the-other. You learn grammar *and* you converse; you do passing drills *and* you scrimmage; you practice culinary technique *and* you try out recipes for dishes you enjoy.

Featured Comment## Karen Campe

September 5, 2018 - 6:52 pm -Featured Comment## Ben Orlin

September 6, 2018 - 4:44 am -Yes, good call! Karen’s example comes up more often, and gives a clearer illustration of the “procedure w/o concept” phenomenon.

## Harry O'Malley

September 7, 2018 - 10:04 am -In basketball, the ultimate goal is simple: win when you play in a basketball game. I wonder what would be an example of “playing the basketball game” in mathematics that would be so engaging that lay-people would choose to watch it for fun, like they do basketball.

## thaslam

September 7, 2018 - 12:30 pm -I have a book called The Math Book. If we ever have extra time a the end of the day, I open it to a random page and do a little “story time” for lack of a better term. Today I showed students some cool stuff with the Fibonacci sequence. Last week I talked about Gauss and him summing an arithmetic sequence quickly–to his teacher’s surprise. A few weeks ago we cut a mobius strip in half and showed them that it remained one piece. Sometimes its more interactive than others–usually depending on the time we have–but there is always a lot of theatrics and suspense built into the discussions. The students love them. I don’t know that anyone would chose to watch if they weren’t a captive audience anyways, but that is probably the closest thing I can think of to a spectator friendly aspect of math, which is really the antithesis of a spectator sport–I’d rather be playing the game!

## Harry O'Malley

September 11, 2018 - 2:08 pm -To Michael Pershan:

I, too, engage in “story time” in the exact same vein as you, although it is often just from memory about something I’ve recently heard or was wowed by. I agree that it is entertaining.

That makes me think of something like Numberphile. Definitely popular and is usually nothing more than someone doing math. I wonder why it is engaging?

I invented a game called Function Wars. In short, two players play against each other according to certain rules to see who can get a function to output a value of 100 first. I did a tournament with my class of 32, March Madness style. Extremely engaging. Once it got down to the final two, the whole class watched them play it out on the big screen in the front of the room. There were cheers, moans, and cotton-mouth level adrenaline from the players. It was as thrilling as any sports game.

## thaslam

September 11, 2018 - 5:09 pm -I’m intrigued. You’ll have to explain the rules of this game in more detail.

## Karina Araiza

May 21, 2019 - 7:11 am -Woo!## Michael Pershan

September 5, 2018 - 4:21 pm -Featured CommentRittle-Johnson, Bethany, Robert S. Siegler, and Martha Wagner Alibali. “Developing conceptual understanding and procedural skill in mathematics: An iterative process.” Journal of educational psychology 93.2 (2001): 346.

In my teaching, I don’t always find it useful to distinguish between procedural/conceptual knowledge. I try to teach in a way that blurs the line between them, so that I’m continuously teaching kids how to handle a new type of problem with understanding. I don’t know if that makes sense or if that’s a good way to explain what I try to do, but it’s one way that I’ve thought about this.

## William Carey

September 5, 2018 - 4:34 pm -> I try to teach in a way that blurs the line between them, so that Iâ€™m continuously teaching kids how to handle a new type of problem with understanding.

Featured CommentCertainly an interactive process.

## Kyle Pearce

September 5, 2018 - 4:44 pm -I think kids who are good at memorizing unrelated ideas can be taught procedures pretty easily. I think it is a big reason why the “math gene” myth exists. It IS true that some can more easily commit unrelated information to memory than others and thus I believe those who struggle, also believe they “can’t” do math.

If mathematics is being taught as if it is just unrelated rules, steps and procedures, then it would make sense that so many believe they fall in this “bad at math” group.

## Chris Heddles

September 6, 2018 - 5:14 pm -I’ve found that this hinges on student preparation.

– For students who have most of the prerequisite learning and skills for the content to be taught, concepts are a useful, easy way into the new material.

– For students who are significantly behind the expected starting point, procedures are far easier to start with as they can be started immediately whereas concepts often requires that the students learn the prerequisite material first.

As many of us receive students in our classes who carry learning gaps that extend back many years, our experience is that rote skills are easier to teach than concepts.

## thaslam

September 6, 2018 - 7:43 pm -You might be interested in the research regarding the CMI (comprehensive mathematics instruction) framework from researchers at BYU and Utah’s K-12 public schools. Just google it.

## Harry O'Malley

September 7, 2018 - 9:52 am -Hey Michael. To give you some feedback, referring to your line “I don’t know if that makes sense or if thatâ€™s a good way to explain what I try to do” I would have to say that it does not make any sense to me. If you don’t distinguish between procedural and conceptual knowledge how can you blur the lines between them (there are no lines to blur)? It sounds more like you don’t have definitions for procedural fluency or conceptual understanding to begin with, so you’re not sure which of those words to use to describe the different elements of your instruction…

## Matthew McGovern

September 5, 2018 - 4:36 pm -I think teachers rely alot on anecdotal evidence of success in their classroom. They will all agree that they “teach” conceptual and procedural knowledge at the same time. However, actually seeing if the kids have learned both of these forms of knowledge can give different results than what the teacher thinks. The classic video I love is “How old is the shepherd” https://youtu.be/kibaFBgaPx4

Most teachers will watch this video and proudly exclaim “Not my kids!”

## Mike

September 5, 2018 - 4:42 pm -Featured CommentI think we as teachers assume a higher level of conceptual understanding automatically exists just because someone can do some procedural work. Finding and dealing with that difference is a growth area for me.

## Kyle Pearce

September 5, 2018 - 4:47 pm -Great post, Dan.

We focus on the 5 proficiencies outlined in Adding It Up in my district and despite being in our 3rd full year of heavily focusing in this area when we engage in professional learning, there are still many who struggle with conceptual understanding vs. procedural fluency. Even when the definitions are consistent, some *believe* they understand what it means, but their descriptions suggest otherwise.

I’d also agree that the rush to the algorithm kills any chance for equity, creativity and passion in mathematics.

## Clara

September 5, 2018 - 5:04 pm -One of my personal aggravations is teachers who â€˜giveâ€™ kids the (b/2)^2 to find the vertex, before theyâ€™ve done ANY other work with quadratics. (Alg 1, looking at the key characteristics of graphs. Iâ€™ve seen it used to teach kids to graph. Aaargh!)

## Martha Alibali

September 5, 2018 - 5:42 pm -Alert: Research!Crooks, N. M. & Alibali, M. W. (2014). Defining and measuring conceptual knowledge of mathematics. Developmental Review, 34, 344â€“377. doi: 10.1016/j.dr.2014.10.001

https://alibalilab.wiscweb.wisc.edu/wp-content/uploads/sites/371/2018/09/CrooksAlibali2014.pdf

## Lara Triona

October 2, 2018 - 8:44 am -Thanks to Alibali for sharing your 2014 article with Crooks about two potential definitions of conceptual understanding of math (https://alibalilab.wiscweb.wisc.edu/wp-content/uploads/sites/371/2018/09/CrooksAlibali2014.pdf).

I noticed that some of the “fluency without understanding” examples could be considered the lack of number sense (such as “Kids that know 7+7 by memory, but have to count 7+8 on their fingers, or know 8×5 is 40 but have no idea that 8×6 is just 8 more.” https://twitter.com/KristaPottier/status/1037890779606343681 and “Student can round 84,899 to the nearest thousand (thanks to silly rhymes), but canâ€™t place 84,899 accurately on a number line to see that 85,00 really is the ‘nearest thousand.'” https://twitter.com/MsOlivaEIS/status/1037843325712248832) .

How do you see number sense fit in these definition?

Is it a foundational to conceptual understanding or one of the related math concepts to understand (see Tosto, et al. 2017 https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5611774/ )?

## Daniel Knox

September 5, 2018 - 6:47 pm -I had a few students in calculus yesterday who could use the power rule to find derivatives of polynomials super fast. Then when I asked them to find the slope of the tangent line to another polynomial at a certain point, there were a some blank stares. You better believe we will take a couple steps back and make sure the conceptual understanding is there.

Thanks for this post and for the reminder.

## Tyler Haslam

September 5, 2018 - 8:19 pm -My example: a student who knows how to find the area of a rectangle and how to multiply binomials but is unable to find the area of a rectangle with binomial side lengths.

…maybe that’s more a lack of application skills, but it’s all conected.

Anyway, my gut feeling is that conceptual understanding has–at it’s very core–some deep tie to conjectures, both formulating them and (dis)proving them, even if its subconscious.

Looking at everyone’s examples I see a clear ‘theorem’ that once accepted resolves the lack of conceptual understanding.

## Marcia Weinhold

September 5, 2018 - 8:26 pm -A student with procedural fluency can tell me that (x-2)(x-1) = x^2 -3x +2, but does not consider that the factors x-2 and x-1 can be seen as two lines (graph them) and their product can be found by point-by-point multiplication of y-values for a given x-value. This concept leads to more likely understanding of polynomials as a ring in higher mathematics.

## Barbara Oakley

September 5, 2018 - 8:51 pm -Dan, the definition you are bringing forth for conceptual understanding may be that used by education researchers, but the reality is, it’s not how the vast majority of math teachers themselves understand the term.

I like the definition you are alluding to for conceptual understanding. But most students could not reach that level of conceptual understanding without plenty of practice, often coupled with the development of procedural fluency. Your examples of “A student who has procedural fluency but lacks conceptual understanding” could all be addressed by asking the student to practice problems related to those ideas–exactly the type of practice I advocated in the NY Times op-ed.

I find it difficult to understand why you continue to advocate policies that do not give children the practice and procedural fluency they need to attain the conceptual understanding we all are seeking for those children, even as you disparage those who do bring up much-needed reminders about the value of practice and procedural fluency. As is clear from many of the postings on this blog, the result of the many-decade-long de-emphasis by the NCTM on practice and procedural fluency is the increasing development of a two-tiered system, where mathematically-attuned parents from homes that can afford it seek out extra supplemental practice for their children. The disadvantaged, on the other hand, are left to their own devices, with an approach to math at school that does not give them the practice and opportunities for the development of procedural fluency that go hand-in-hand with the development of deep conceptual understanding. The result, sadly, is further inequality.

## Sharon Soule

September 6, 2018 - 7:19 am -I was surprised when I read the quote from Dan’s blog because it is *exactly* the definition of conceptual understanding that I use in my classroom. Yes, procedural fluency is important, but if we never make the explicit connections between say factoring a quadratic to find its roots (procedure) and the understanding that those roots are the x intercepts of the graph of that quadratic (conceptual), why are we even bothering to teach them the procedure?

I agree with Dan that the implication of most articles that argue for more drill is that those of us on “the other side” are just trying to make our classes fun so students will like math. I’m not trying to make my classes fun at all; rather, I am trying to engage students with the math because an understanding of how the concepts connect and are interwoven is what makes the subject of mathematics inherently fun. Math is cool, not because I can put on a dog and pony show, but because it is full of patterns that we can learn to recognize and connect to other patterns in other places where we might not expect to find them. The more we find those patterns, the more we connect them in interesting and unexpected ways, the cooler and more engaging and more fun math becomes for us.

And, yes, sometimes it is necessary to develop a certain level of procedural fluency to make connections, but most times, you can use the drill time strategically to say “do these problems and then let’s look for the pattern that the answers make.” Once you establish the pattern you can say, “let’s do a few more problems and see if the pattern continues.” Just making students perform problems over and over again without making the connections is what turns them off to mathematics.

## Dan Meyer

September 7, 2018 - 3:45 pm -Practice is necessary but insufficient in the development of conceptual understanding. In many cases, routine practice is the

causeof the weak conceptual knowledge that concern the scores of educators here and on Twitter.Your claim in the NYT that students could use more practice doesn’t bother me. (Though US performance on PISA, a exam that tests conceptual knowledge, isn’t a great support for that claim.) It’s your critique of conceptual understanding, a strand of mathematical proficiency you didn’t seem to understand in the NYT or in my comments, that brought about my criticism.

## Chris Heddles

September 5, 2018 - 9:48 pm -From what I found in my own teaching is that the key prerequisites for students to “play” creatively with a new concept are confidence and prior concepts – in that order. Generally speaking, the prior concepts should be accompanied by considerable procedural fluency relating to those concepts (but obviously not the new concept).

For example, a few contributors above have mentioned polynomials. While there is no need to gain fluency with algebraically factoring polynomials prior to exploring the pattern, there is a need to have a strong understanding of the following concepts (along with the associated procedural fluency):

– factoring in general

– use of the Cartesian plane to represent relationships between two variables

– straight line graphs of equations in the form y=mx+c

With these prerequisite concepts firmly in place, students can confidently explore the behaviour of linear factors being multiplied together to create polynomials. Of course, many of our students arrive at our class missing a lot of this prerequisite learning. As teachers, we then need to deal with this reality and help the students as best we can to learn the prescribed content. There are two main options for this:

1. Start the student from wherever they are and build them up ‘properly’ to the current content. This is the ideal but takes huge amounts of time and effort from both teacher and student. It’s often not possible to have a student even start the current content before they are required to be assessed on their learning of it (thanks to externally-imposed schedules).

2. Have the student practice/learn the procedures for the current content even though they understand nothing of it. If they are lucky, they will pick up enough to scrape through the upcoming assessment tasks. This approach is facilitated by assessment tasks that are often written so predictably that it is possible to pass by applying procedures that are not understood.

Despite our best intentions and frequent misgivings, many teachers often use some variation of option 2. In many cases, students are so used to it that they will actively fight any attempt at option 1 because that is scary, hard work and would lead to them failing badly in the short term.

The least bad thing I’ve been able to do for most students is to do a bit of both. Enough of option 2 to keep the system happy while encouraging students to work like Trojans on option 1 to catch up their gaps from previous schooling over the course of multiple topics. This also helps their confidence.

## Eric I

September 5, 2018 - 10:35 pm -I think this video is a good example of a child who was taught procedurally but is missing the concepts. Her knowledge is incredibly situationally specific and brittle.

https://youtu.be/_ofQ_WnQiZ4

## kmorrowleong

September 5, 2018 - 11:06 pm -Featured Comment“canceling” would even be ok.) Example: -4 + 5 + -9 + -5 + 4 + 9

## Barbara Oakley

September 6, 2018 - 1:22 am -The entire exercise here of finding examples of procedural fluency without conceptual understanding is a clear example of motivated reasoning to enhance the value of “conceptual understanding” devoid of procedural fluency–precisely the type of problem I alluded to in the New York Times op-ed. See Will Emeny’s work on the value of procedural fluency in many areas of K-12 math education: http://www.greatmathsteachingideas.com/2014/01/05/youve-never-seen-the-gcse-maths-curriculum-like-this-before/.

## Tim Stirrup

September 6, 2018 - 2:51 am -Does Will mention procedural fluency there? It seems to be an analysis of why mastering number skills is important, not procedural fluency. He does mention it elsewhere on his blog. e.g. here http://bit.ly/2wPzW21

“Iâ€™ve never understood the â€˜drill-and-kill versus teach-for-conceptual understandingâ€™ argument. In that, I mean itâ€™s not a dichotomy. Both reinforce each other. Students who try to problem solve with fluency but no conceptual understanding canâ€™t apply their knowledge to new contexts. Students who try to problem solve with conceptual understanding but no fluency fall into working-memory overload in doing the basics and lose sight of, or canâ€™t form, the strategy for solving the problem.”

## Barbara Oakley

September 6, 2018 - 3:46 am -Will Emeny described the chart to me in my hour-long discussion with him as being indicative of the value of procedural fluency. See also Paul Morgan’s rebuttals of Dan’s continued mischaracterization of Paul’s research–https://blog.mrmeyer.com/2018/drill-based-math-instruction-diminishes-the-math-teacher-as-well/#comment-2446885. Paul’s comments provide insight into research that reveals the interwoven importance of procedural fluency and conceptual understanding.

## thaslam

September 6, 2018 - 11:40 am -How so? The blog is about pinning down what is meant by conceptual understanding and the exercise drives that home (as is the intent). Nowhere has anyone stated that conceptual understanding devoid of procedural fluency was preferable …its just far less common since procedural fluency is the easy part of the conceptual-understanding/procedural-fluency/application trifecta that is rigorous math.

Also, sometimes to enact change you have to push the pendulum. While i don’t agree with any argument that conceptual understanding should be a greater focus in math classrooms, I do agree that, generally, it still needs more focus on classrooms than it is getting (as does application). That said, I do value your piece and its effort to keep that pendulum from swinging too far in the other direction.

Thank you :)

## Holly Pope

September 7, 2018 - 10:20 am -If you read Adding It Up, that Dan referenced in his post, you will see that procedural fluency and conceptual understanding are only two of the five components that make up a comprehensive understanding of mathematics. Most people outside of the field of education don’t even know about the other three strands: adaptive reasoning, productive disposition, and strategic competence. All five components need to be attended to during instruction or student mathematics understanding will be weak. Furthermore, procedural fluency is more than being able to quickly recall facts and remember procedures. It includes knowing procedures (note the plural), knowing WHEN to use them, being able to estimate, and doing so FLEXIBLY, accurately, and efficiently. There is no dichotomy between procedural fluency and conceptual understanding, except that which is created by those who are outside of the field of teaching and learning mathematics.

## Paul Morgan

September 7, 2018 - 10:37 am -Dan writes that our study “attempts to discredit conceptual understanding by linking it to â€˜movement and musicâ€™ (p. 186) in math class.â€

This is incorrect. Instead, as we stated in the studyâ€™s introduction, movement and music have been suggested as ways to teach students math, including concepts. It is easy to find suggestions of exactly this. For example, see this NEA link (http://www.nea.org/tools/lessons/music-and-math.html). NCTM is currently offering teachers grants to use music to teach math (https://www.nctm.org/Grants-and-Awards/Grants/Using-Music-to-Teach-Mathematics-Grants).

Dan incorrectly attributes to us a claim made by others that movement and music might help students learn math. Instead, we empirically evaluated this claim.

We empirically evaluated whether instructional practicesâ€”that teachers themselves reported usingâ€”were associated with math achievement gains by 1st grade students. One type of instructional practice was using movement and music. Others were teacher-directed and student-centered instructional practices.

We examined for achievement gains by 1st grade students who had struggled in math during kindergarten as well as by those 1st grade students who had not. We adjusted the estimates for many other explanatory factors.

We found no evidence to indicate that use of movement and music were associated with achievement gains (Table 5). In contrast, teacher-directed instruction was consistently associated with achievement gains including both by students who had previously struggled and those who had not. Student-centered instruction was associated with achievement gains, but only by those who had not previously struggled. Our finding that student-centered instruction may not benefit students who struggle is consistent with other work (https://www.tandfonline.com/doi/full/10.1080/01425692.2015.1093409).

Yet we found that 1st grade teachers were more likely to be using movement and music as well as manipulatives and calculators when leading classrooms with greater shares of students who had struggled (Table 3).

We should be using instructional practices to teach math that are empirically supported. This is especially true for students who are already struggling. Mischaracterizing a peer-reviewed study examining which instructional practices look to have more empirical support only detracts from our shared efforts to help all students learn math.

## SteveH

September 6, 2018 - 3:41 am -This “zombie” argument is fallacious. Where are those students who are what you propose as the model? What exact curriculum did they take? Are they prepared for STEM degree programs? What are their SAT I and II scores and their AP/IB grades? Where are they? Integrated math lost the battle in high school. The only winners are AP and IB math. K-8 has no opt-out of the low expectation CCSS slope to no remediation in College Algebra, so the only kids who get to STEM programs now are those who get mastery help at home or with tutors. My son was one of them.

This is not some sort of argument for what NON-STEM prepared students should be offered. The position here is that this is best for even the top math students. I could argue that there needs to be more stepping back to see the forest for the trees, but simplistic, in-class, time-wasting concepts-first methods that assume engagement or curiosity driven mastery do not create STEM students, and they do not create students who have any particular level of understanding that can drive the solution of problems in other areas.

All traditional math classes start each unit with concepts (def: an abstract idea; a general notion), but there are many levels and types of understanding. Full understanding is a subtle and long term process that is never completed. There is no such thing as a student who can DO math who has a fundamental failure of understanding. Any limited understanding can be improved. However, a student who can’t DO the problems, but can spout off words is nowhere. That one student can count up is just a silly argument. Where is proof of transferrence?

Go ahead and expect students to derive the formula for the area of the triangle. Are you talking about Green’s Theorem and line integrals? My traditional math classes taught me how to calculate areas for simple polygons. What level of understanding do you want? To characterize that traditional math requires little or no understanding is completely wrong, but you already know that. Your key fallacy is that you don’t want to make traditional math (the only thing that works) better. You want to fundamentally change the process to slow it down to spend more time on in-class concepts and engagement. Anyone should be able to slow a process and do a better job, even for traditional math, but that’s not what you want. You want ownership of pedagogical turf.

And what happened to my comment in the last thread that appeared and then magically disappeared?

## thaslam

September 6, 2018 - 11:28 am -SteveH,

I couldn’t disagree more.

First of all, The traditional vs. integrated is about content sequencing, not delivery methods. Both are CCSM and both prescribe a core focused on rigor: the pursuit with *equal* intensity of conceptual understanding, procedural skills and fluency, and application. To say any of these three aspects is less important than the others is to demonstrate a misconception of the nature of mathematics. Too many people in this country think that math is nothing more than a set of procedural tools. They donâ€™t realize that those tools are used for something and that an appropriate application of those tools (or better ones) requires some understanding of where those tools come from and why they do what they do. The â€œzombieâ€ argument is certainly not fallacious. There are many students who perform memorized procedures without any thought as to weather their use is appropriate. A good mathematician is one who decides which tools at their disposal are best suited for the task. Part of this “conceptual understanding” is knowing when, where, and why we are using a particular tool/method. â€œMasteryâ€ is not synonymous with procedural fluency. Mastery goes beyond that. A student who has truly mastered a mathematical concept understands the concept and the procedures it encompasses (which, by the way, are only a subset of the concept–there are many other aspects of mathematical concepts than just the procedures) and are able to apply the concept in a useful way.

As for AP/IB, ACT, ect. One issue we have right now is that it is hard if not impossible to assess any aspect of rigor other than procedural fluency without expending a lot of time and resources. To use Danâ€™s example, any mathematician would agree that a student who uses a discriminant to determine if x^2+2 has real or complex roots has not yet obtained mastery. But without seeing how a student goes about answering the question â€œDoes x^2+2=0 have real or complex solutions?â€ we do not know if they are at that mastery level or not.

…And Iâ€™d hardly say that a curriculum devoid of conceptual understanding has been very successful at pumping out STEM students that are at a mastery level anyway. On the other hand, our University offered a summer high school program were high school students were invited to participate in a course on number theory (basically Math 4100). The course was derived from the Arnold E. Ross Mathematics Program (which originated in 1957), and is very conceptual. Here is an example question from day 1:

Consider the following number systems: Z, Q, R, 2Z, Z/3, Z/6, Z/8, Z/11. Each of

these systems has two operations, addition and multiplication. Consider various algebraic

properties and determine for which of these systems those properties hold. Some properties

to consider are: if a^2 = 1, then a = 1 or a â‰ 1; if b*c = 0 then either b = 0 or c = 0; if

d â‰ 0, then the system also contains d^-1; etc.

Make a list of several such basic properties, noting which systems have which properties.

Which of these systems would you consider to be the â€œmost similarâ€ algebraically? Why?

You can see that from day one, the students started building their own conceptual understanding. Granted, we didnâ€™t leave them out to hang–eventually as a class we pooled our lists and sorted out what defining characteristics each system has and from that built the axioms from the ground up. We did not give them a list, they made the list and we, with students as mathematicians, sorted it out until we settled on what an â€œIntro to Number Theoryâ€ course would introduce for axioms of the integers. THIS IS WHAT MATHEMATICIANS DO!! To say these concept-first methods were â€œtime-wastingâ€ is truly â€œfailing to see the forest through the treesâ€.

The only true rote procedures in the entire course were division with remainder and Euclidâ€™s Algorithm. Everything else in the course is either conceptual, or application of E.A or the division algorithm. To say that mastery meant being able to do division with remainder and E.A. is to sell the entire 4100 course–and number theory itself–short. A mathematician uses tools at their disposal to discover conceptsâ€¦ P-E-R-I-O-D! To say conceptual understanding is over emphasized in todayâ€™s math classes is to say that mathematics is over emphasized in todayâ€™s math classes. You canâ€™t make â€œtraditionalâ€ math better without bringing about a balance of conceptual understanding, procedural fluency, and application. both research and experience demomnstrates this.

In the sense of a graduate level math class, the problems students DO *are* conceptual problems. They are not practicing rote procedures throughout on homework sets. The last class that I can remember where there were more than a mere handful of algorithms to learn was maybe PDEâ€™s. Sure thereâ€™s an occasional algorithm that shows up almost all courses (like Page-Rank) but the *procedure* of the algorithm is hardly the meat of the course. It is the concepts that surround it that are of interest. Anyone who thinks that algorithms are the end-all-be-all in mathematics has not delved deep enough into what mathematics is. They are simply a means to an end, the end being understanding mathematical concepts or applying the procedure *correctly* to some application.

Take Greenâ€™s Theorem. If a future engineer/mathematician knows how to use it. Great! Do they understand the necessary conditions on the boundary curve? Do they realize that it is a generalization of the FTC and a special case of Stokesâ€™? Can they apply it to answers questions in electromagnetism? Can they apply the theorem to build a platometer or antenna? If the answer to all of these questions is â€œnoâ€, then I would ask, why the h – – – do they know Greenâ€™s Theorem anyway?! In fact, I would say their mastery of the theorem is minimal â€¦hardly more than a computer programâ€™s mastery–perhaps less since the program is far more infallible.

As for your argument that â€˜traditionalâ€™ math [education] is the only thing that works, research–even plain observation–would demonstrate otherwise. 90% of the people I meet admit to me right off, and quit emphatically â€œI canâ€™t do math!â€ These arenâ€™t millennials who are participating in this progressive trend (which by the way isnâ€™t that progressive, nor is it a trend since conceptual understanding and application have long had equal standing with procedural fluency in nearly every other countries education system), these people are products of what you would call a â€œtraditionalâ€ curriculum. And the 10% that claim to be good at math, turns out they know just a bunch of random facts/procedures. They think because they know some algorithms/formulas/theorems that they are good at math. After a few minutes of conversation, it becomes very apparent that they have never done anything beyond regurgitating what a teacher spoon fed them or what they read in a book or a blog. I have yet to meet someone outside of my circle of mathematician friends who is able to carry on a conversation about *real* mathematics. Iâ€™m getting rather lonely. Itâ€™s not that we want the process â€œslowed down to spend more time on in-class concepts and engagementâ€ We want more *real* mathematicians in this country than we currently have.

Finally, I can see why your comment would have been deleted. If it was anything like this one it was full of subjective opinions about what you think math ed should look like, not research based evidence. Opinions do not determine fact and they do not determine best practice, no matter how strongly we believe them. If you truly believe in math, then you should believe the math behind the studies. If I want to hear anecdotal dribble about how an engineers son struggled with real mathematics, I’ll go talk to the 10%!

## thaslam

September 6, 2018 - 11:30 am -…and I apologize for the harshness of some of my comments.

## Lola

September 6, 2018 - 5:19 am -Heh – I just realized from Ben Orlin’s comment that “running passing drills without ever watching a game of basketball” is almost exactly what happened to me in elementary school gym class. Now I can draw on that memory of bewilderment to help understand my students! I love analogies that help me like this :D

## Scott Adamson

September 6, 2018 - 6:51 am -Great examples.1. Dividing fractions such as 3/2 divided by 1/2. A student can only “keep, change, flip” rather than to reason that there are 3 “copies of” 1/2 in 3/2.

2. I saw this yesterday in an 8th grade classroom…the problem was something like this: The average weight of 36 7th graders is 75 pounds. The average weight of 44 8th graders is 85 pounds. What is the average weight of all of these 7th and 8th graders? The student says, “I added 75 pounds and 85 pounds and then divided by 2.” My point is that there is fluency in the sense that the student recognized that an average was needed and “fluently” added the two weights and divided by 2. But, the student failed to recognize that the number of 7th and 8th graders was not the same…thus lacking conceptual understanding of average as an equal distribution of weights (in this case) among some number of students (in this case).

3. Calculus students faced with the integral of (x – 3)^2 dx can sometimes only respond by going through the procedure of u-substitution. On the other hand, they could just think about their more conceptual understanding of derivatives (or maybe this is procedural, too?!?) and just determine that the anti-derivative must be 1/3 (x-3)^3 + C. My hypothesis is that students were first taught the procedure called u-substitution and then are not able to access any “number sense” or in this case, “derivative sense” and must go through the routine/procedure of u-substitution.

## SteveH

September 6, 2018 - 4:53 pm -“The traditional vs. integrated is about content sequencing, not delivery methods. …the pursuit with *equal* intensity of conceptual understanding, procedural skills and fluency, and application.”

Equal? That’s not what it is in practice.

“Too many people in this country think that math is nothing more than a set of procedural tools.”

“People” can think what they want, but that’s not how AP/IB math is taught.

“There are many students who perform memorized procedures without any thought as to weather their use is appropriate.”

Appropriate? Were they able to solve the problem? What additional litmus test are you applying? All mastery includes understanding and there are always more levels and nuances to learn from P-sets.

“A student who has truly mastered a mathematical concept understands the concept and the procedures it encompasses (which, by the way, are only a subset of the conceptâ€“there are many other aspects of mathematical concepts than just the procedures) and are able to apply the concept in a useful way.”

“Useful?” Truly mastering a concept is the end result of mastery of the skills from individual homework, becase introductory concepts are only words and any in-class group oriented project wastes too much time stretching thought in only one direction. There is no proof of transference.

“As for AP/IB, ACT, ect. One issue we have right now is that it is hard if not impossible to assess any aspect of rigor other than procedural fluency without expending a lot of time and resources.”

That’s because you don’t appreciate all of the layers of understanding that are required to do homework and pass tests. Perhaps they are layers you now take for granted. If you have another method of testing rigor in mind, then you best share it. Then you can compare your students with those in AP/IB curicula. Right now I don’t see any alternative curriculum.

“â€¦And Iâ€™d hardly say that a curriculum devoid of conceptual understanding has been very successful at pumping out STEM students that are at a mastery level anyway.”

Really? What do you propose that will work better? Where are your curriculum examples/results? I might be your biggest fan – honestly!

“The course was derived from the Arnold E. Ross Mathematics Program (which originated in 1957), and is very conceptual.”

Go ahead and create a full curriculum based on this. Just make sure it’s opt-in.

“Anyone who thinks that algorithms are the end-all-be-all in mathematics has not delved deep enough into what mathematics is.”

Strawman.

“As for your argument that â€˜traditionalâ€™ math [education] is the only thing that works, researchâ€“even plain observationâ€“would demonstrate otherwise. 90% of the people I meet admit to me right off, and quit emphatically â€œI canâ€™t do math!â€ ”

Where is your research on a full curriculum? Does “works” mean getting “people” to say that they can DO math? That’s wasn’t my definition of works. I was clearly talking about STEM preparation. AP/IB sequences are the benchmarks for producing these students. Where is your specific alternative?

## thaslam

September 6, 2018 - 7:33 pm -“Equal? Thatâ€™s not what it is in practice.”

You are correct, procedural fluency is still overwhelmingly emphasized.

“Appropriate? Were they able to solve the problem? What additional litmus test are you applying?”

Getting an answer by brute force methods when another method is clearly more efficiante shows a lack of mastery. I cite the great mathematician Paul Erdos’ comments about ‘good proofs’. It’s like using a screwdriver to hammer in nails. It gets the job done, but I wouldn’t consider it a sign of mastery.

“‘Useful?’ Truly mastering a concept is the end result of mastery of the skills from individual homework, becase introductory concepts are only words and any in-class group oriented project wastes too much time stretching thought in only one direction. There is no proof of transference.”

I think your problem is you are confusing “conceptual understanding” with “foundational knowledge”. Regardless, there is research that shows students who had a curriculum that equally valued conceptual understanding outperformed students who had a curriculum focused solely on procedural fluency on standardized tests–and these tests where heavily procedural!

“Thatâ€™s because you donâ€™t appreciate all of the layers of understanding that are required to do homework and pass tests.”

I don’t think you appreciate how easy it is to answer those types of questions. Obviously a CAS system has no conceptual understanding, yet it can answer any question on just about any test up through a traditional 1050 and beyond.

“If you have another method of testing rigor in mind, then you best share it.”

As a wise professor once told me when I asked him the same question, “the mind is analog” and therefore can only be measured approximately by digital means (i.e. multiple choice test such as ACT, half of AP, etc.) Even FRQs if too narrow fail to reveal how much a student understands.

“Really? What do you propose that will work better? Where are your curriculum examples/results? I might be your biggest fan â€“ honestly!”

Not only do I have personal observation (as do many colleagues of mine) there is research, perhaps most famously by Jo Boaler,but you seem the type that would renounce that research because it proves your argument wrong.

“Go ahead and create a full curriculum based on this.”

I don’t have to it already exists, since 1957. And has obviously been quite successful since it has survived 60 years and been adopted by many colleges.

“Strawman.”

…not if the entire mathematical community agrees with that statement. …Engineers don’t count :P

(Just kidding, I have lots of friends who are engineers. But to be honest, they themselves are procedurally heavy–which is probably why they are always asking me conceptual questions)

“Where is your research on a full curriculum?”

Again, look at Jo Boalers research. There is also a large body of work cited in NCTM’s book, Principles to Actions. My personal favorite is and probably the first person I ever heard attest to the equal importance of conceptual understanding and procedural fluency: Henry Picciotto. You can find his work at MathEdPage dot org

“Does â€œworksâ€ mean getting â€œpeopleâ€ to say that they can DO math?”

No “works” means getting people to be able to actually do math (beyond regurgitating algorithms that computers can do more efficiently)

“Thatâ€™s wasnâ€™t my definition of works. I was clearly talking about STEM preparation.”

Again, I point you towards all the data that shows students from traditional curriculum aren’t prepared for college math. (This phenomenon pre-dates the current math reform movement and therefore cannot be a product of it. It is a product of the ‘traditional’ way.)

“AP/IB sequences are the benchmarks for producing these students.”

Every successful AP/IB program I’ve is balancing procedure with understanding and application. You are just proving my point. My comments about them were regarding assessments and how measuring conceptual understanding is inherently problematic.

The bottom line is this: no one here is saying conceptual understanding trumps procedural fluency. I’m sure if you asked any teacher or math educator of any worth they would agree that they are both equally improtnat. To say otherwise (in either direction) is to ignore research. …Which, by the way, you have yet to offer any that concludes a balance of both is detrimental, or even that emphasis on procedure is beneficial. I’m sure you’ve been jaded by some unfortunate teacher who overshot the mark, but I’ve been involved in the mathematical, math educational, and math teaching communities long enough that your argument holds no weight for me. It’s not that I don’t value your opinion, it’s that I don’t value anyone’s opinion over research and my own personal observation. Sorry.

## SteveH

September 7, 2018 - 3:33 am -“procedural fluency is still overwhelmingly emphasized.”

In procedural learning lies all of the subtle understandings. Are P-sets only for speed?

“Getting an answer by brute force methods when another method is clearly more efficiante shows a lack of mastery.”

Strawman. That’s not what AP/IB do.

“.. there is research that shows students who had a curriculum that equally valued conceptual understanding outperformed students who had a curriculum focused solely on procedural fluency on standardized testsâ€“and these tests where heavily procedural!”

Which specific multi-year curriculum is this? What specific curriculum was it compared to? Was it compared to AP/IB math or was it for K-8 at a non-STEM level? Did it slow down coverage? Anyone can slow down and do better. Anyone can have properly-trained teachers and do better with any curriculum.

“I donâ€™t think you appreciate how easy it is to answer those types of questions.”

I don’t think you appreciate all of the subtle variations in problems that cannot be solved by concepts – general notions. When I taught college math and CS, nobody could pass my courses with rote knowledge or brute force techniques. Change a problem slightly and that requires a lot more than rote brute force.

“…research, perhaps most famously by Jo Boaler”

You forgot the quotes around research. I have to stop giggling. Prove me wrong?!?

“The bottom line is this: no one here is saying conceptual understanding trumps procedural fluency.”

Really!?! Where is the proper grade-by-grade enforcement of mastery in K-6 that gives students a chance of getting ready for a STEM degree without help at home? Math education has to be more than a “trust the spiral” process that puts the entire onus of mastery (above low slope CCSS) on students.

“…you have yet to offer any that concludes a balance of both is detrimental, or even that emphasis on procedure is beneficial.”

Strawman. Educators never do that research. They wouldn’t get the funding even if they were so inclined.

“I donâ€™t value anyoneâ€™s opinion over research and my own personal observation.”

How about the reality of the low slope of CCSS to no remediation in College Algebra? How about the reality of the dominance of AP/IB? How do you think students get ready for those tracks in high school? Ask us parents of the best students. We focus on mastery of unit skills. I did get my son to use GeoGebra in middle school where he studied trig functions to model saw-tooth functions, but that was neither necessary or sufficient.

I don’t see anyone here talking about improving AP/IB – and I could make many comments on that. This is all about replacing them with something that is never explained in longitudinal curriculum form. It’s easy to cherry-pick reasearch, but quite another thing to prove it in reality. Where is your reality? Again, I might be your biggest fan, but don’t give me K-6 examples because traditional math has been gone from there for decades. Our lower schools improved results when they moved from MathLand to Everyday Math. I tutor some of those students to help them make the non-linear transition to the high school AP math track.

## Robert Kaplinsky

September 6, 2018 - 7:02 am -Here’s a blog post I made with videos that show students who can procedurally solve a problem but have very limited conceptual understanding: https://robertkaplinsky.com/what-does-it-mean-to-understand-mathematics/

## Erik Von Burg

September 6, 2018 - 7:14 am -Example: A student who can successfully calculate 2/3 divided by 1/4, but they cannot provide a logical estimate.

## Fran

September 6, 2018 - 7:15 am -A student who knew all their multiplication facts up to 12×12 but had no thoughts about how to go about finding an answer to 13×13

## Christopher Danielson

September 6, 2018 - 7:24 am -Excellent thread here.We have so many concrete examples of kids who have mechanical procedures down pat, but lack understanding (as in your Twitter list from the other evening), and we have detailed research descriptions of them (poor Benny, and IMAP videos are rife with kids who nail a rule without any underlying basis of understanding). But I can’t cite any examples of the reverse. I know the Mathematically Correct crowd likes to mock the idea of a kid who understands but can’t do anything, but I’m lacking concrete examples to examine.

Perhaps that’s a bias of mine, right? I haven’t gone looking b/c the other type confirms my own preconceptions. Got anything for me? Where are those research examples or videos of kids who “understand” in a way that I would recognize, yet are unable to perform calculations?

## Michael Pershan

September 6, 2018 - 8:13 am -I taught a student who could completely explain to you the meaning of a multiplication problem (e.g. 7 x 4) but could only attempt to solve it laboriously and inaccurately. Would that count for you, CD?

## thaslam

September 6, 2018 - 11:57 am -It certainly exists. Its just much less common a problem. the reason being that procedural fluency is usually easier for students to obtain and for teachers to impart. That’s not to say it doesn’t exist. when it does happen, its usually from lack of continuing down the path uncovered by the conceptual understanding.

My first example: I observed a class of 10th graders writing recursive equations for linear, exponential, and quadratic discrete functions. Their teacher had taught them (and I had the impression that most understood) that quadratics have a linear rate of change and therefore the recursive function of a quadratic would be something of the form f(n)=(n-1)+mn+b

However, though most students understood this concept, very few were actually able to write down the formulas.

My second example: This one is personal. Steve H.’s comment about Greens Theorem made me realize that, although I conceptually understand what Green’s Theorem is all about and were it comes from and what its applications are, my Calc III days are so far behind me and my integration skills so deteriorated, I doubt I would actually be able to apply Green’s Theorem in any meaningful way in my present state of abilities.

…which shows the equal footing that procedural fluency has with conceptual understanding and application. Any of the three without the others is pretty pointless. One is not more important than the other, but traditionally conceptual understanding and application have had to take a back seat when in a rigorous math course all three should be up front driving.

## Christopher Danielson

September 6, 2018 - 12:05 pm -Michael, yes. Your example works for me. I’m wanting more rich descriptions of these kids. I want to watch them work, and read deep dives on their thinking.

I would also like to offer up the example of YOU (I assume, and probably most readers here) and finding square roots by hand to arbitrary numbers of decimal places.

## thaslam

September 6, 2018 - 12:30 pm -Great point Christopher Danielson,

the art of finding a [irrational] square root by hand is hardly mentioned today (possibly for good reason) though I think conceptual understanding isn’t diminished because of it.

Trig ratios are another realm. I once had a student ask how the calculator knew that sin(20 DEG) was 0.342020143326…

I was impressed that their conceptual understanding was deep enough to realize that this was even an issue. They understood how one could geometrically find things like sin(45 DEG), sin(30 DEG) etc, but they were able to pick up on the fact that the tools at hand were not enough to obtain what the calculator had shown them. They even understood that simply drawing a 20 degree right triangle and measuring couldn’t possibly lead to that kind of accuracy. After class I pulled this sophomore aside and we had a brief conversation of the angle sum and angle difference identities that revealed conceptually how one could arrive at the result.

## Chris Heddles

September 6, 2018 - 5:38 pm -I think that the reason we struggle to find examples of understanding without fluency is that it's far less durable than procedures without concepts. Because conceptual development builds so strongly on prior learning, students who understand something without fluently being able to do anything with their understanding will struggle to learn the next concept. This leads to a very short window during which they understand but can't process. After this, they become just another student who neither understands nor has any fluency with the next material. Students can (and do) go on for years memorising the algorithms for each topic as they need to do in order to pass assessments without ever gaining conceptual understanding.

That said, statistics and probability offer the best examples I've seen of students who understand but can't calculate. Plenty of students can explain and interpret statistical concepts of mean, median and standard deviation without having any ability to do calculations around these concepts. Some can even understand the idea of null hypothesis testing without actually being able to do such testing.

I'm not sure why statistics and probability have this pattern but I suspect that it's the relatively high penetration of technology to visualise and calculate the concepts for students. This allows them to play with the concepts without procedural fluency in ways that are not available to them in other topics.

## katenerdypoo

September 6, 2018 - 7:35 am -Here’s a very personal example:

When I took the PSAT as a 10th grader, we got back the test, our answers, and the correct answer. My mother looked through everything and pronounced to me, “you got every question involving percents wrong,” and asked me a simple percentages question, which I couldn’t answer. Despite being in the top of my honors algebra II class, I had somehow failed to learn anything about this middle school topic.

So she taught me that percentage/100 = part/whole, and to cross multiply and solve for x. Since I was very good at algebra, that method worked for me and I never got percents questions wrong again and my PSAT score the next year rose a ton.

But in reality, I still didn’t understand percents and had literally no other method than to set up a proportion to do even simple things like calculating a 20% tip (…or even a 10% one!). I didn’t come to understand how they worked until I became a math teacher myself, more than 10 years after my mother’s crash course intervention.

In other words, I had complete fluency (I could answer every question correctly) but no real understanding.

## Sean

September 6, 2018 - 9:26 am -Thanks for this Dan

1. Benjamin Piper, Wendi Ralaingita, Linda Akach & Simon King have done interesting work on this topic (figuring out how to generate gains in procedural and conceptual) for early-elementary students in Kenya over the past several years.

See here:

https://www.tandfonline.com/doi/pdf/10.1080/19439342.2016.1149502

2. One neat thing is Table 2, which shows a “procedural index” and a “conceptual index.”

They map actual skills to both indices (e.g. timed number identification is in “procedural” and “ability to solve word problems” is in conceptual.) You may disagree w/ how they group things, but the simple act of defining it with actual skills that flow from a valid test has been really helpful.

3. Their study revealed that procedural moved faster than conceptual.

## thaslam

September 6, 2018 - 12:13 pm -Thanks Dan for the great/interesting post.

I’m sure we all realize that conceptual understanding and procedural fluency–along with application {why dose that one always get left out? :( } are equally important in a rigorous math classroom. One interesting nugget I found while reading NCEE’s “New Standards Performance Standards: English Language Arts, Mathematics, Science, Applied Learning”, is they make the comment that some procedures require fluidity, while others require mere familiarity:

“Some skills are so basic and important that students should be able to demonstrate fluency, accurately and automatically; it is reasonable to assess them in an on-demand setting.

There are other skills for which students need only demonstrate familiarity rather than fluency. In using and applying such skills they might refer to notes, books, or other students, or they might need to take time out to reconstruct a method they have seen before. It is reasonable to find evidence of these skills in situations where students have ample time, such as in a New Standards portfolio. As the margin note by the examples that follow the performance descriptions indicates, many of the examples are performances that would be expected when students have ample time and access to tools, feedback from peers and the teacher, and an opportunity for revision.”

I you google:

“Some skills are so basic and important that students should be able to demonstrate fluency, accurately and automatically; it is reasonable to assess them in an on-demand setting.”

you can get the whole article. I found it to be a very interesting document from 1997 that was precursor to CCSSM

## Heidi Dixon

September 6, 2018 - 1:56 pm -I’ve seen this go the other direction as well. I’ve had students with wonderful conceptual understanding. They understand the model and what it means, but they are struggling to translate the model into symbolic notation. They battle a complex algorithm, multiply 2 by -2 and get four somewhere along the way. They understand but they can’t get anything right on a test and they struggle to write a proof. But they KNOW why the theorem they’re trying to prove is correct. These have wonderful potential to be creative with mathematics but can get discouraged in a class that values only procedural fluency and getting the correct answer to a calculation. Conceptual understanding, symbolic/notational translation and algorithmic fluency require different cognitive skills. Strength in one doesn’t imply strength in the others.

## Barry R Smith

September 6, 2018 - 2:47 pm -I see a lot of procedural understanding without conceptual understanding when working with inequalities. For example:

1) Can switch |x-c| < r to -r + c < x < r + c but doesn't understand they both describe the interval of points whose distance from c is less than r.

2) Knows to reverse an inequality sign when multiplying or dividing by a negative number and not when adding or subtracting, but doesn't know the reason — that applying a linear function ax+b to both sides will reverse the inequality when the slope is negative and not when it is positive because in the first case it is decreasing and the second it is increasing. They may also think you reverse inequalities when taking reciprocals, without knowing why, when really this only happens if both sides are known to have the same sign. And a lot of these students think it will be legitimate generally to square both sides of an inequality.

3) A good student may solve an inequality |x-4| < |x-2| by considering cases, like when both x-4 and x-2 are nonnegative. But in this case, they deduce x-4 < x-2, hence -4 < -2 and then ignore this case, as if the lack of an x in the latter inequality meant that this case led to no solutions.

## Nicola

September 6, 2018 - 4:43 pm -Thanks for the discussion Dan. I think so much of this conversation is tied up in vocabulary and peopleâ€™s different use/understanding of terms (e.g. What is conceptual understanding? What is procedural fluency?) Incidentally, the first example in this post of a student who has procedural fluency but lacks conceptual fluency could be confusing to some. If you are referring to the term â€˜counting upâ€™ as starting at 1999 and counting up by ones to 2018, as the term is commonly used in elementary schools, then obviously counting up is not an efficient strategy for this problem. I presume you are actually referring to a student either â€˜adding up in chunksâ€™ from 1999 to 2018, or using the â€˜make a tenâ€™ strategy and mentally adding one to 1999 to make 2000 and then adding 18 more to find the difference of 19. Either of these mental strategies would be more efficient than a standard algorithm for this particular problem, however because of your use of the term â€˜counting upâ€™ some readers may mistakenly think that you are advocating for students to count up from 1999 to 2018 by ones.

In a similar vein, I think some of the terms Oakley used in her article were not the best choice (e.g. â€œAll American children would benefit from more drilling.â€ Would substituting â€œmore opportunities to practiceâ€ for â€œdrillingâ€ have made a difference? Oakleyâ€™s choice of words may reflect the fact that she doesnâ€™t work in education (and who knows what role an editor played in her article), however her overall message that students need regular opportunities to practice math concepts and strategies has merit. Obviously, conceptual understanding and procedural fluency are both vitally important. We need more discussions on how to interweave the two in ways that are engaging for students, rather than pitting one against the other as commonly happens in these discussions.

## Jane Taylor

September 6, 2018 - 5:13 pm -A student who can calculate slope but has no idea what it means as the rate of change in a real context.

## Halil Tasova

September 7, 2018 - 6:00 pm -Similarly, students can know that the slope is negative when the line is directed downward left-to-right (just focusing on the perceptual feature of the line) but does not know that the change in y values decreases/increases while the change in x values increases/decreases.

## Harry O'Malley

September 6, 2018 - 5:40 pm -I have developed a definition of conceptual understanding that I believe is simple, useful and resonates with what everyone knows it is in their gut but can’t put it into words. Conceptual understanding of a discipline’s content is the ability to translate that discipline’s abstract symbols into concrete, visceral reality as well as translate concrete, visceral reality into that discipline’s abstract symbols. For example, the ability to see the number 12 and mentally translate that into a picture of some apples or the temperature of an ice cube is evidence that you conceptually understand 12. Going the other way, seeing a bunch of apples or feeling the temperature of an ice cube and mentally translating that to the number 12 is evidence that you conceptually understand 12. If you see the expression (x-2)(x+3)(x+7) and feel temperature changing over time, first warming up, then cooling, then warming again, that is evidence that you conceptually understand the expression (x-2)(x+3)(x+7). Going the other way, if you feel the temperature of something changing, first warming up, then cooling, then warming again, and you translate that experience into the expression (x-2)(x+3)(x+7), that is evidence that you conceptually understand (x-2)(x+3)(x+7). And there is no such thing as having conceptual understanding of something. There are only degrees. There are an infinite number of visceral experiences that can be represented by the number 12 and until you’ve translated 12 into all of them, you’ve got more to learn. 12 bee stings. 12 lumens. 12 kelvin. 12 tattoos. 12 screws. 12 donkeys in a car. 12 planets. 12 galaxies. 12 million bee stings. 12 milliseconds of excruciating pain.

And one beautiful thing about this definition is that it can be taught rather explicitly and can be practiced, especially with modern technology making the technical parts of the translation process much easier….

## Rachel Kirchner

September 6, 2018 - 9:37 pm -Thanks for your helpful examples of fluency without conceptual understanding. Here’s another one that I saw in my class earlier this week:

12 – 8 3/4 became 11 4/4 – 8 3/4 = 3 1/4, which works, of course. But a complete lack of understanding about how we could also do 12 – 8 = 4 and then subtract 3/4 to get 3 1/4 or that an another way would be to add 1/4 to 8 3/4 to get 9, and then add 3 more to get 12.

## Eric Nelson

September 7, 2018 - 2:12 am -The question of procedural fluency versus conceptual understanding in math has been studied extensively by scientists who study how the brain solves problems and how it learns. Hereâ€™s part of what they say and where they say it.

Scientifically, there are two distinct types of conceptual understanding: Explicit and Implicit. Explicit means you can explain why you do what you do. Implicit means you intuitively understand the right thing to do without being able to explain why.

We want majors in any field to be able to explain why the field does what it does, but for those who use a field in their work as a tool, to solve problems all you need is implicit understanding. For example, we all use English as a tool, and we are all able speak by fluently applying the precise rules for pronunciation, vocabulary, morphology, syntax, and semantics in our dialect with virtually no ability to consciously say what the rules are. Our brain knows the rules because human brains evolved to learn and apply rules fluently â€“ which means unconsciously –without needing to be able to consciously articulate them. See Steven Pinkerâ€™s â€œThe Language Instinct.â€

We teach math for the first 15 years of schooling so that scientists, engineers, carpenters, nurses, architects, etc. can solve problems in their field. They need implicit understanding of math as a tool: To be able to use math to do socially useful work. They donâ€™t need much explicit understanding.

We need a small percentage of math majors, compared to the nurses and engineers society needs, because explicit understanding is not needed for most productive work.

Thatâ€™s the science behind what Dr. Oakley was saying, and she got the science right. To become procedurally fluent, you must gain implicit understanding of a topic by first moving into memory (memorizing) facts and procedures, and then you must apply the acts and procedures in a variety of distinctive contexts â€“ and do so repeatedly.

Humans cannot use explicit understanding without implicit understanding because of limits in the working memory where the brain solves problems. See http://www.aft.org/pdfs/americaneducator/spring2012/Clark.pdf

What cognitive scientists call the â€œcentralâ€ strategy to improve student problem solving is â€œmemorization to automaticity.â€ In the 2008 Report of the National Mathematics Advisory Panel (NMAP), cognitive experts Geary, Renya, Siegler, Embertson, and Boykin write,

â€œ[T]here are several ways to improve the functional capacity of working memory. The most central of these is the achievement of automaticity, that is, the fast, implicit, and automatic retrieval of a fact or a procedure from long-term memory .

Note â€œimplicit?â€ Thatâ€™s from page 4 at : https://www2.ed.gov/about/bdscomm/list/mathpanel/report/learning-processes.pdf , a document that for math educators should be â€œMath Education 101.â€

What may be surprising is how well students must memorize fundamentals to be adequately prepared for scientific problem solving. The NMAP authors add,

â€œ[During calculations,] to obtain the maximal benefits of automaticity in support of complex problem solving, arithmetic facts and fundamental algorithms should be thoroughly mastered, and indeed, over-learned, rather than merely learned to a moderate degree of proficiency.”

â€œOver-learnedâ€ means that facts and fundamental algorithms are memorized until they can be recalled fast, perfectly, and repeatedly, consistent with what Dr. Oakley wrote.

For a discussion of implicit versus explicit understanding (with minimal scientific jargon), see http://www.cogtech.usc.edu/publications/clark_automated_knowledge_2006.pdf

As educators, I think it is helpful if we learn what science says about how young people learn and how the brain solves problems. Explicit vs. implicit understanding is an important distinction.

To be experts, and not just tool users, in our specialized field (education), we need explicit and implicit understanding of how the human brain works.

— Eric (rick) Nelson

## Ryan M.

September 7, 2018 - 2:20 am -A geometry student who doesnâ€™t understand that the distance formula is just a different version of the Pythagorean theorem and argues with the teacher about using one or the other.

## Jonas Oskarsson

September 7, 2018 - 2:37 am -Maybe worth thinking of:

Is your conceptual understanding of “conceptual understanding” more instrumental than conceptual?

## Dan Meyer

September 7, 2018 - 3:47 pm -Hey folks â€“Â this is a comments thread that got away from me and which I wouldn’t dream of shutting down. I’ve gone through your comments and added excerpts to the main post. I added similar comments from Twitter to the main post. I’ve also added styling to threads here that I think are well worth your time, including some dissents. Give the whole thing a skim again. I’ve found the whole exercise extremely fascinating.

## Halil Tasova

September 7, 2018 - 5:52 pm -Can determine a “concave up/down” graph represents an increasing/decreasing rate of change but has no idea that amount of change in one quantity increases/decreases as other quantity changes in uniform increments.

## @sscycle67

September 8, 2018 - 6:43 am -Students who count decimal places (rather than estimating, which usually makes the placement of the decimal a ‘no-brainer’ in the end, given some number sense) when multiplying decimal numbers with zero understanding about the quantity of their answer and whether or not it even makes sense.

## Eric Nelson

September 8, 2018 - 12:05 pm -“Adding It Up” was up to date in 2001, but in the decades since, thanks in part to new technologies, science has made substantial progress in understanding how the brain learns and how it solves math problems. May I offer a brief summary?

For millennia, of necessity, students learned math by memorizing facts and procedures, but after calculators arrived in 1970, instruction changed. By 2002, most U.S. state K-12 standards required calculator use in 3rd grade. Most state standards after 1990 also required â€œdecreased attentionâ€ to â€œmemorizing rules and algorithms,â€ â€œmanipulating symbols,â€ and â€œrote practice,â€ as recommended by the 1989 NCTM standards. These changes optimistically assumed that during problem solving, the brain could hold and process non-memorized information with the same facility as well-memorized information. Unfortunately, that assumption has proven scientifically to be mistaken.

Between 2001 and 2010, cognitive science reached a consensus that when solving mathematical problems, the brain must rely almost entirely on the application of facts and algorithms that have previously been â€œautomatedâ€ (over-learned, thoroughly memorized). Thatâ€™s not what anyone wanted to hear, but the brain is a product of natural selection — which is not required to heed our preferences. And consensus science is by definition our best source for deciding what knowledge we can rely upon to be true.

Unfortunately, between 1990 and 2010, K-12 math standards in most U.S. states assumed that with access to calculators and computers, memorization in math could be de-emphasized. Change that recognized the importance of â€œautomaticityâ€ has taken time, especially for students caught in the transition from old to new standards.

As a result, many current students have measured deficits in their abilities to solve calculations. In the international OECD PIAAC testing in 2012, US citizens aged 16-24 ranked 22nd among 22 tested nations in â€œnumeracy.â€

Even the more recent Common Core and derived stated standards ask students to calculate rather than recall subtraction and division fundamentals. Cognitive studies are in agreement that this is a serious mistake.

If an answer must be looked up or calculated, it occupies space in working memory (where the brain reasons) that is limited. In contrast, for information quickly recallable from long-term memory, space in working memory is essentially unlimited. During the steps of problem solving, if more than a few elements of information are needed that cannot be recalled from long-term memory, some non-memorized information drops out of working memory, and confusion tends to result.

Dr. Oakley is accurately stating the findings of science: Explicit conceptual understanding is not necessary for mathematical problem solving (though implicit understanding is). Focusing on explicit understanding takes time away from what science says students must do to solve math problems: Move facts and procedures into long term memory via overlearning, then use that information to solve problems in lots of different contexts.

â€œMemorize then applyâ€ constructs the cognitive wiring that is the physiological substance of implicit conceptual understanding. As neuroscience tells us, when neurons that hold facts and procedures fire together during thought, they wire together to form conceptual frameworks.

In 2018, none of the above should be news to math educators. If it is, then professional organizations are not doing their job of keeping members informed of the discoveries of science that have fundamental impact on their work.

For detail and citations on the science above, see https://confchem.ccce.divched.org/content/2017fallconfchemp8 .

But science teacher professional development should not be discussing quantitative problem solving with more content and accuracy than math professional development.

— Eric (rick) Nelson

## thaslam

September 9, 2018 - 4:29 am -I don’t think anything I saw at your site is news to math-ed. However, choosing to use the term “well-memorized” is begging the question. “Well-learned” is more appropriate. There are more ways to learn something than memorization. Math ed research has found that conceptual understanding underplaying procedural practice is more effective than procedural practice alone. A good example is the log identities you mention on your site. Students practicing those identities seem to always forget them eventually–by themselves they are just some abstract rules. With conceptual understanding of how those identities are derived and their relation to the power identities, procedural practice is more successful because students have something more concrete (power rules) to build upon and students have a way of rebuilding the procedural knowledge themselves if it gets lost. This is something I can attest to myself. In my youth, it wasn’t until the conceptual understanding was there that I could keep the identities straight in my head, regardless of all the practice.

## Eric Nelson

September 9, 2018 - 6:27 pm -Danâ€™s original question was the right one. How should conceptual understanding be defined?

Cognitive science (the science that defines and measures conceptual understanding) is in consensus on its definition. Conceptual understanding is made up of of observable, measurable cellular material. It consists of neurons in the brain, their stored information, and the wiring between them.

When you solve a math problem, science can SEE conceptual understanding occurring â€“ in real time â€“ using new technologies such as fNMI, PET, and MEG.

In 2018, this science is fundamentally well understood.

When you read a problem, your vision moves the problem data into â€œworking memory.â€ Working memory then searches the neurons of your long term memory (LTM) for matching data elements. If it finds matches, the matching neurons â€œactivateâ€ and â€œfireâ€ — send out a measurable electric current â€“ to up to hundreds of other neurons they are â€œwiredâ€ to (connected to via synapses).

If the current is strong enough, those connected neurons â€œactivateâ€ and â€œfireâ€ as well. The relationships stored in the neurons of memory that are activated can then be recalled into working memory, where they are applied duing the steps of solving the problem. Thatâ€™s how mathematical problem solving takes place.

Those activated â€œconceptual frameworksâ€ — the neurons and wires among them â€“ are the source and substance of your conceptual understanding. The key question is: How do we help students construct those conceptual frameworks?

That process is also well understood by 2018 science. First, fundamental facts must be moved into long-term memory and made recallable â€“ they must be â€œmemorized.â€

AFTER fundamental facts are recallable, they must be used in procedures that are memorized and used to solve problems. As recallable facts are used in procedures, the facts and procedures in memory are wired together (â€œneurons that fire together, wire togetherâ€) , and implicit (tacit, intuitive) conceptual frameworks grow in number and strength.

But neurons cannot grow conceptual connections until AFTER facts and procedures are stored in them AND those facts and procedures are made recallable for processing to at least some extent.

And according to science, making math facts and procedures recallable takes considerable time and effort (over-learning/ retrieval practice).

So, in answer to Thaslamâ€™s comment: For the initial (non-expert) learning in a topic, â€œmemorizationâ€ (storing fundamentals in neurons and making them recallable) must physiologically occur BEFORE â€œlearningâ€ â€“ which is gaining an intuitive sense of WHEN to recall a relationship by the brain relying on the strength of neural electrical impulses.

For more on conceptual understanding, I recommend the short book â€œLearning How to Learnâ€ by leading US neuroscientist Dr. Terry Sejnowski (and co-authors). It is jargon-free — yet clearly explains the neuroscience of how students construct conceptual understanding.

Thanks, Dan, for asking for the scientific definition of “conceptual understanding.”

— Eric (rick) Nelson

## thaslam

September 10, 2018 - 5:52 am -Again, Rick, you are missing the point. We know about neurons, working memory, and long term memory, transfer and all of that. This is not 2018 research, this is old news and in fact the very premise of the conceptual understanding argument; however, â€œmemorizationâ€ âŠ‚ storing fundamentals in neurons and making them recallable. As pointed out before (but you donâ€™t seem to be listening–you just keep pushing your â€œLearning How to Learnâ€ course), research shows that storing fundamentals in neurons and making them recallable occurs much more effectively when what is being learned is relatable-to/built-upon/coupled-with something already understood. First, so we are clear on our definitions, here is the definition of what we are talking about in the field of math education with regards to â€œconceptual understandingâ€ and â€œprocedural fluencyâ€ (from NCTMâ€™s Principles to Actions):

Conceptual Understanding-the comprehension and connection of [mathematical] concepts, operations, and relations. (example: understanding subtraction is the inverse of addition)

Procedural Fluency-the meaningful and flexible use of procedures to solve problems. (example: being able to effectively and efficiently subtract quantities)

Both of these are â€œFACTSâ€ and both fall under the cognitive science definition of â€œconceptual understandingâ€ or a â€œconceptual frameworkâ€ of neurons. But math ed research has found that distinguishing between these two types of FACTS is beneficial (as mentioned in Adding it Up), so you need to stop hijacking our definition and respect what the definition means to our field–itâ€™s not math edâ€™s fault that cognitive science coined a term by using a phrase already in use to mean something more specific.

Here is the research from math ed supporting the conceptual understanding coupled with procedural fluency practice. (I read yours, now please, kindly, read ours!):

(The following excerpt is also from NCTMâ€™s Principles to Actions–I hope they donâ€™t mind my pasting it here)

â€œMajor reports have identified the importance of an integrated and balanced development of concepts and procedures in learning mathematics (National Mathematics Advisory Panel 2008; National Research Council 2001). Furthermore, NCTM (1989, 2000) and CCSSM (NGA Center and CCSSO 2010) emphasize that procedural fluency follows and builds on a foundation of conceptual understanding, strategic reasoning, and problem solving.

â€¦

â€œWhen procedures are connected with the underlying concepts, students have better retention

of the procedures and are more able to apply them in new situations (Fuson, Kalchman, and

Bransford 2005). Martin (2009, p. 165) describes some of the reasons that fluency depends on

and extends from conceptual understanding:

â€œTo use mathematics effectively, students must be able to do much more than carry out mathematical procedures. They must know which procedure is appropriate and most productive in a given situation, what a procedure accomplishes, and what kind of results to expect. Mechanical execution of procedures without understanding their mathematical basis often leads to bizarre results. Fluency is not a simple idea. Being fluent means that students are able to choose flexibly among methods and strategies to solve contextual and mathematical problems, they understand and are able to explain their approaches, and they are able to produce accurate answers efficiently. Fluency builds from initial exploration and discussion of number concepts to using informal reasoning strategies based on meanings and properties of the operations to the eventual use of general methods as tools in solving problems. This sequence is beneficial whether students are building toward fluency with single- and multi-digit computation with whole numbers or fluency with, for example, fraction operations, proportional relationships, measurement formulas, or algebraic procedures. Computational fluency is strongly related to number sense and involves so much more than the conventional view of it encompasses. Developing studentsâ€™ computational fluency extends far beyond having students memorize facts or a series of steps unconnected to understanding (Baroody 2006; Griffin 2005). A rush to fluency, however, undermines studentsâ€™ confidence and interest in mathematics and is considered a cause of mathematics anxiety (Ashcraft 2002; Ramirez et al. 2013). Further, early work with reasoning strategies is related to algebraic reasoning. As students learn how quantities can be taken apart and put back together in different ways (i.e., decomposition and composition of numbers), they establish a basis for understanding properties of the operations. Students need this early foundation for meaningful learning of more formal algebraic concepts and procedures throughout elementary school and into middle and high school (Carpenter, Franke, and Levi 2003; Griffin 2003; Common Core State Standards Writing Team 2011).

â€œIn meaningful learning of basic number combinations (i.e., addition and subtraction within

20 and multiplication and division within 100), students progress through well-documented

phases toward fluency (Baroody 2006; Baroody, Bajwa, and Eiland 2009; Carpenter et al.

1999). Students begin by using objects, visual representations, and verbal counting, and then

they progress to reasoning strategies using number relationships and properties. For example,

to solve 8 + 4, a first grader might count on from 8 early in the school year, whereas later in

the year the same student might reason that since 8 + 2 is 10, then 8 + 4 must be 2 more than

10, or 12. A third grader might initially use repeated addition to solve 4 Ã— 6 and then progress to reason that 2 sixes are 12, so 4 sixes must be double that amount, which is 24. This approach supports students, over time, in knowing, understanding, and being able to use their knowledge of number combinations meaningfully in new situations.â€

â€œLearning procedures for multi-digit computation needs to build from an understanding of

their mathematical basis (Fuson and Beckmann 2012/2013; Russell 2000)…

â€œSimilarly, a high school student who does not understand the distance formula may have trouble accurately recalling it and applying it appropriately to problem situations.

By contrast, a student who understands that the formula is an application of the Pythagorean

theorem (i.e., the distance between two points can be thought of as the hypotenuse of a right triangle) can use an understanding of this underlying relationship to solve a problem involving the distance between two points correctly (Martin 2009).

â€œClearly, students need procedures that they can use with understanding on a broad class of

problems. This raises questions regarding how students can move most effectively toward

fluency with general methods or algorithms, as well as what defines an algorithm. Fuson

and Beckmann (2012/2013) argue that a standard algorithm is defined by its mathematical

approach and not by the way in which the steps in the approach are recorded. They suggest

that variations in written notation are not only acceptable but indeed valuable in supporting

studentsâ€™ understanding of the base-ten system and properties of the operations. They also

emphasize the importance of understanding, explaining, and visualizing: â€œStandard algorithms are to be understood and explained and related to visual models before there is any focus on fluencyâ€ (p. 28).

â€œFor example, the conventional algorithm for multi-digit multiplication is difficult to understand, whereas the three alternative methods shown are more transparent with respect to the central mathematical features of place-value meanings and properties of the operations (Fuson 2003). The diagrams show the multiplication of tens and ones and the relative size (in area) of the partial products. The accessible algorithm shows a clear record of the four pairs of numbers that are multiplied. This progression also supports students in establishing a basis from which to apply and extend these understandings to operations with rational numbers and algebraic expressions.

â€œIn moving to fluency, students also need opportunities to rehearse or practice strategies and

procedures to solidify their knowledge. However, giving students too many practice problems too soon is an ineffective approach to fluency. Students need opportunities to practice

on a moderate number of carefully selected problems after they have established a strong

conceptual foundation and the ability to explain the mathematical basis for a strategy or

procedure. At that point, providing students with practice on a small number of problems,

â€˜spacingâ€™ or distributing these over time, and including feedback on student performance

support learning outcomes (Pashler et al. 2007; Rohrer 2009; Rohrer and Taylor 2007).

â€œSimilarly, practice with basic number combinations should occur after students can explain

and justify their use of efficient reasoning strategies. A word of caution is important in regard

to timed tests. The premature and overuse of such tests may hinder studentsâ€™ mathematical

proficiency and lower their confidence in themselves as learners of mathematics (Boaler 2012;

Seeley 2009). Practice with basic number combinations should focus on solidifying studentsâ€™ use of an efficient strategy for specific number combinations (Rathmell 2005; Thornton 1978).

Isaacs and Carroll (1999) suggest that practice be brief, engaging, purposeful, and distributed. For example, practice can target specific strategies, such as making a ten for addition or doubling a known fact for multiplication, and can be embedded in problem-solving tasks and

games (Crespo, Kyriakides, and McGee 2005).â€

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## Eric (rick) Nelson

September 26, 2018 - 6:44 am -Thaslam —

Much of what you have quoted above (not all, but most) involves people with no expertise in how the brain works claiming that the brain can do what cognitive experts say the brain of students cannot do. That does represent how U.S. math standards have been set since 1989.

And what has been the outcome? By the international standards, at or near the end of their schooling, how are U.S. students doing?

In 2012, the OECD tested skills in 22 developed-world nations including the United States. In “numeracy” (solving problems with mathematical content), U.S. 16-24 year olds ranked 22nd out of 22: Dead last. See http://bit.ly/2tMzJ08 .

In a world where a nation’s technical and engineering skills are essential for self-defense, that outcome devastating for our children and grandchildren. Your math standards that deny science have been tried for 30 years. It is time to ask cognitive scientists to join in making decisions on what our math standards should be.

— rick nelson

## thaslam

September 26, 2018 - 11:02 am -Then show this research by these “cognitive experts”, because as of right now no one who makes your argument on this page has followed up with anything.

(you provided research regarding direct instruction vs discovery learning, that is a completely different conversation. Your report by Geary et. al. actually supports Math ed’s stance: “Instruction focusing on conceptual knowledge of fractions is likely to have the broadest and largest impact on problem-solving performance. Procedural knowledge is also essential, however, and although it must be learned separately, it is likely to enhance conceptual knowledge and vice versa.” as does the Clarebout/Elen article: “The development of competence requires foundation knowledge, a conceptual framework, and the ability to organize and retrieve knowledge.”).

As for Nelson’s article, That is not a scholarly article–lots of spelling errors, poorly written, agenda driven. You make the claim then try to prove it, scholarly articles present the findings and then comment on them–you have no findings, just a slew of claims with citations jammed in for ‘support’. …Two times you make your claim: “The cognitive science of 2017 suggests that with ‘decreased atention’ (sic) to ‘memorizing rules and algorithms,’ skill in calculations would decline” and “The 2010 CCMS assumes the brain can work efectively (sic) with facts before they have been well-memorized. Science in 2017 says the student brain cannot.” still have no reference! Also, claiming that U.S. is last in 2012 doesn’t help support your claim at all, since many of these practices weren’t widely adopted by that time–I’d say based on empirical observation, they still aren’t widely adopted. Not to mention there are a lot of other reasons why the US struggles in educating its masses.

I’d be a fool to follow you and your (still unproduced) science without a better argument or SOME ACTUAL SOURCES!!!

## Eric (rick) Nelson

September 27, 2018 - 7:18 am -Thaslam â€”

Could you (or any readers) possibly provide the filename of the post with the typos? I donâ€™t doubt you are seeing them, but the file I am downloading at http://bit.ly/2tMzJ08 does not have those errors. That said, with tech, I do expect disasters.

I believe readers looking at the sentences you cite as â€œwithout referencesâ€ will see plenty of them.

On the need for memorizing facts as a necessary foundation for building conceptual frameworks, permit me one additional reference.

In the theory of learning described by Jean Piaget, moving facts and procedures into long-term memory (LTM) is termed â€œassimilation.â€ Resulting changes to the brainâ€™s conceptual frameworks in LTM are â€œaccommodation.â€ Herbert Simon and his colleagues write that a

â€œcareful understanding of Piaget (shows) that assimilation of knowledge also plays a critical role in setting the stage for accommodation–that the accommodation cannot proceed without assimilation.â€

— from Anderson, J.; Reder, L.; Simon, H. (2000). Applications and misapplications of cognitive psychology to mathematics education. Texas Educational Review, Summer (emphasis added)

The article, downloadable from https://files.eric.ed.gov/fulltext/ED439007.pdf , takes some hard shots at the 1989 NCTM standards which I suspect you will take issue with.

But how math can be applied to a variety of problems is research for which Dr. Simon was awarded a Nobel Prize. Readers will find the article to be especially interesting.

For the record, I think everyone was highly disappointed to hear from science, 10-20 years after the 1989 NCTM standards were written and adopted in most states, that the standards asked students to do what their brains could not. Absent clear science at the time, the NCTM made an optimistic guess about how the brain worked. It worked out the way optimistic guesses about science usually do.

Everyone hoped the CCMS would fix what was wrong. It was disappointing to find the CCMS, though better aligned with science than the NCTMâ€™s standards, are not well aligned.

If we must have “state” standards, they must work for students both mathematically and cognitively. But for kids and the nation, it would be better if standards in education were set by front-line practitioners who learn from and respect scientific research, in the manner doctors set the standards of medicine.

— Eric (rick) Nelson

## Andrea Harrison

September 9, 2018 - 11:47 am -Not a math teacher, but I do teach science. This resonates with me deeply. I continually find that students can tell me that a hypothesis is â€˜an educated guessâ€™ but when asked to explain what that means, give me a blank stare. We have to get away from asking students to memorize facts and move into organic, authentic experiences that give them a conceptual understanding of the world around them which they can then attach our content and equations.

## Eric Nelson

September 9, 2018 - 7:03 pm -In response to Ms. Harrison and many of the comments so far:

If K-14 students need to explicitly (verbally or in writing) explain WHY they take some of the steps they take to solve math and science problems, shouldnâ€™t math and science teachers be required to correctly explain the rules of grammar they apply for at least some of the sentences they write? If you canâ€™t â€œdiagram a sentence,â€ are you incompetent in communication?

Actually, science says you donâ€™t need to be able to recite the rules of English linguistics to speak, write, or teach. Your brain knows the rules implicitly or you could not talk or write, but like most of our knowledge in memory, it is unconscious. If we can correctly solve the problems of speech, or math, or riding a bicycle, explaining verbally why we do what we do is a waste of our limited cognitive resources.

Science says the goal of most learning is to be able to do the right thing intuitively, without having to stop and think about why. Members of our species who had to stop and think during confrontations at the watering hole were weeded out by natural selection.

NSF figures tell us that of the 1.8 million US BA graduates in 2015, only 1% (about 18,000) were math majors. Less than 1% (about 15,000) were chemistry and physics majors.

Majors need to explain the why of their major. But every K-12 student takes math, and all engineers and health care workers take chemistry and physics — because they will likely need what is taught in those courses as a tool to solve problems.

Experts on the brain agree that solving problems requires intuitive, often subconscious understanding. They say your brain simply does not have time to gain explicit (explaining why) understanding for all of the fields in which you need to solve problems in life and “on the job.”

Isn’t that consistent with our real life experience?

— Eric (rick) Nelson

## Peter Gerdes

September 16, 2018 - 2:28 am -Truth is for 99% of students (non-future mathematicians) we don’t give a crap about understanding of mathematical concepts or shouldn’t. What we should care about is putting them to work in math class in a way that develops their general ability to reason and think clearly since that’s what they will actually need in real life. This generally corresponds to what we call by the shorthand ‘understanding the concept,’ i.e., if we demand they reason about the notions involved in a way that demonstrates the ability to solve unexpected questions in novel contexts.

## Kelli Wasserman

September 29, 2018 - 9:02 pm -I would argue that procedural fluency cannot be achieved without conceptual understanding. Achieving procedural fluency requires number sense, and I would argue that you cannot have number sense without understanding.

Additionally, what is the point of being able to use procedures without understanding where they came from or why we are using them? Would it be so we can mindlessly calculate? I am with Conrad Wolfram on this one, the world does not need human calculators…

## Kelly

December 6, 2018 - 7:51 pm -I had students that can solve for the domain of a rational function but do not understand why they have to set the denominator to zero first. They know exactly how to solve for it but they did t not know why setting the denominator to zero was important. Dan what would be some lessons you would more likely teach students the procedure first in a math class?

## thaslam

December 7, 2018 - 7:49 am -Here’s something scary, I was helping a Pre-calc student with some homework on solving rational adding and multiplying rational expressions and solving rational equations. Their teacher told them to they have to limit the domain when they multiply, but not when they add or solve. In fact, one of the examples the teacher did for them showed the solution to be 3, but the denominator of one of the terms was x-3. Talk about fluency without understanding!

## Dan Meyer

December 9, 2018 - 9:21 pm -Thanks for the question, Kelly. In this situation, it feels like students are (a) focusing on a narrow feature of a given equation, (b) manipulating it in a way they don’t really understand, and (c) getting a result that they

alsodon’t really understand, all without really understanding how the algebraic representation of the function connects to the domain.So one possible move here is to give them a rational function and several x-values they need to evaluate, some of which should be outside of the domain of the function. They need to know what it looks like to

successfullyevaluate a function for an x-value and then what it looks like tounsuccessfullyevaluate the function for an x-value. In some case they’ll have wasted effort, only realizing deep into their evaluation that the x-value doesn’t work.That leads to a useful question: “How do we know

in advancewhich x-values work and which ones don’t?”Thanks again for getting my brain moving a bit tonight. Really interesting question.

## Gaby SÃ¡nchez

June 21, 2019 - 12:48 pm -An example of procedural fluency without conceptual understanding is when students use the butterfly method to determine which fraction is greater 7/3 or 7/4 (True story. It happened to me today)