Critical ThinkingEducationPedagogy

You Cannot Not Have Conceptual Understanding

In education, we seem to take some delight in shoveling a confused mix of folksy connotations into sciencey-shelled words and phrases. Some of my colleagues would call the result edujargon, though I think that word allows us to feel too smug about our own obtuseness—as though the problem is that the field of education is so darned technical.

Anyway, I’ve been itching to pick on one such phrase lately, conceptual understanding, and I think I’ll start here with the on-picking. In doing so, I want to be clear that I’m not reaching for what ‘conceptual understanding’ actually means. Rather, I’d like to suggest that what emerges from the everyday way we talk about this concept is a meaning that, at best, doesn’t make good sense.


Always with the Algorithms
Take a look at the student work at the right and marvel at how his handwriting perfectly replicates a web font. Then decide what he might have done wrong.

Addition problem showing that a student solved 62 plus 39 with 41

If you showed this work to any gathering of K-8 math teachers, it wouldn’t be long before someone suggested (after emphasizing that there’s no way to tell from one example) that the student probably doesn’t have a strong ‘conceptual understanding’ of addition—or something along those lines. And a compelling reason to make this judgment, assuming that the student’s error is consistent, is that the sum is less than one of the addends. That is, the answer is impossible given the meaning of what the student is doing (combining two positive quantities), yet the student seems to be unaware that there even is a meaning to be had in adding (again, assuming the mistake is consistently made).

Conceptual understanding, then, has to do with meaning. That is certainly the takeaway I wanted you to get from the hypothetical above, but this is also a point to which others have returned when writing about this concept:

A procedure is a sequence of steps by which a frequently encountered problem may be solved. For example, many children learn a routine of “borrow and regroup” for multi-digit subtraction problems. Conceptual knowledge refers to an understanding of meaning; knowing that multiplying two negative numbers yields a positive result is not the same thing as understanding why it is true.

Daniel Willingham

Thus, it is important to not only teach students how to go about solving problems, but also to teach them the frameworks in which those problems obtain meaning. Just one meaning of addition, for example, is a combination of values or quantities, and just one advantage of being familiar with that meaning is that it allows us to discern the reasonableness of sums. This is what we generally talk about when we talk about conceptual understanding.


But Let’s Go Deeper
As constructivists, though, we must all concede that human beings are meaning-making machines. They simply can’t help themselves. We see meaning in places where no given meaning is present. Pareidolia, Rorschach tests, and apophenia are some of the more easily googleable examples of our ineluctable fondness for injecting our thoughts onto a random and uncaring universe.

Of course, being a meaning-making machine does not mean that the machine works correctly all the time. Nor does this observation say anything about the quality of the meanings thus generated. What comes out is typically far from a model of logical perfection or mathematical purity.

So, why do we talk about students as having no conceptual understanding or weak conceptual understanding? What in the heck are we talking about when we say things like this? The student who solved the addition problem incorrectly above does have a conceptual understanding of addition. He has constructed a meaning for addition. It’s just that he has, with the help of his schooling, his parents, and his community, constructed the wrong meaning–an attenuated meaning.

A stellar example of what misguided conceptual understandings look like is Robert Kaplinsky’s “How Old Is the Shepherd” task. Here we have a problem whose only requirement, really, is to grasp its meaning in order to solve it:



It is also a good example to use to point out the bone-headedness of talking about conceptual understanding as something that can be weak or non-existent. Of the 75% of students who completed the task incorrectly, what did virtually all of them do? If a lack of something caused all of these students to misfire, this lack also caused all of them to misfire in the same basic direction: to take the numbers in the problem and operate on them.

No. There is conceptual understanding here—a conceptual understanding that was given to them and reinforced over and over in the years leading up to this. And there is plenty of meaning that these students are working with here. It’s just that it’s the meaning that their parents are comfortable with, not the meaning that will be of much use in their own futures.


Why This Is Important
The distinction between conceptual understanding as something you can have a certain quantity of (possibly 0) and conceptual understanding as the unavoidable result of mental activity lighting up and organizing your interior and exterior worlds is important primarily because it represents a divergence in diagnoses that ultimately leads to a divergence in treatment, as it were.

The latter view is almost certainly better for business, for example. If I can convince you that you have a hole to fill, I can more easily sell you something to fill it. It is also a view that can help steer attention away from more systematic failures of schooling—including what teachers are taught in universities—onto the isolated U.S. classroom teacher.

Conceptual understanding, as we have seen, is never not there; it is a complex and emergent property of knowledge shaped by a student’s interactions with the whole of her schooling, her parents, and her community. We should at the very least be aware that our own conceptual understanding of conceptual understanding, including the way we talk about it amongst ourselves and with our students, has consequences.

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J.D. Fisher

J.D. Fisher

Josh Fisher is an author and designer for K-12 mathematics curricula, currently in Pittsburgh, Pennsylvania.


  1. March 8, 2015 at 7:14 pm —

    The students who went wild with the shepherd problem numbers did what they were taught to do: They got information, they got a problem, they tried to get the answer. To most of them it obviously didn’t occur that adults could just hand them a nonsense question, probably because many of them have been told over and over again that yes, this problem can be solved, everything you need is right in front of you, get started!

  2. March 9, 2015 at 4:15 pm —

    I was once offered (and accepted) a job promotion based on a project (actually a test) where I submitted that not only was the math not solvable but *why* it wasn’t. (I was told that I was the only unwitting applicant to explain the why.) The sneaky part was, the project was presented as *having* a practical solution to be applied to the company’s financial projections, so we all wasted some time trying to make it work before deciding either a) we weren’t up to solving it or b) there was no way to make it work. Not knowing it was a test added to that, but yeah, we do have an idea fostered in school that you can solve everything, just work with your data.

  3. March 9, 2015 at 4:38 pm —

    It’s interesting that both of you outlined this alternative (“bad”) conceptual understanding with a single characteristic–that all problems can be solved.

    I would add to that outline “all problems can be solved by operating on numbers.” What else can we add, specifically with regard to fleshing out what a maladaptive conceptual understanding looks like?

  4. March 9, 2015 at 9:05 pm —

    For some time now I’ve been tutoring SAT & ACT, and this issue comes up almost exclusively with word problems. In most school systems the questions are worded the same way all the time so usually, plug & chug algebra or arithmetic works. Word problems are the stumbling block because you actually have to understand what’s going on, and you’d be surprised how many students from very affluent school systems still stumble.
    My point is that some questions *do* have “not answerable with the information given” as an option, and most students have never seen this before!!! Data Sufficiency ought to be a topic in every High School math course including geometry & algebra, but certainly starting with arithmetic.

  5. March 10, 2015 at 6:01 am —

    I think that “teaching critical thinking” and “teaching that teachers aren’t infallible” would be good starting points. This means you need to foster a classroom climate in which students are comfortable with challenging you.
    And of course confronting them with questions that cannot be solved.
    I teach languages, where obviously often many answers are possible. Still I try not to give students hints as to when all acceptable answers (IMO) have been given, for example in a sorting exercise. That way I can find out if they still have some gaps or false concepts and I also foster a climate where they can say “no, that’s it” (which can also tell me something about their level, for example if they only know one very common meaning of a word and not another one).
    Word problems in maths have their issues. For one thing, they are adding an unfair burden on students with bad reading comprehension and language issues. OTOH, they’re also very helpful for students who need some concrete application (for me, maths in maths was often “meh”, but when applied to physics or economics it suddenly all made sense). Including problems with “not enough information” once in a while would surely help, but I admit that’s difficult when 30% of your students are actually struggling with 48+15=

    • March 16, 2015 at 6:05 pm —

      There is a similar phenomenon in language instruction, at least in the case of grammar worksheets, since students can often fill in the correct answers for things like verb conjugations without actually understanding the meaning of the sentences they are completing. You could just as easily provide students with a nonsensical sentence and a blank to fill in with the proper form of a verb/adjective/etc. and many would happily do so even if the result is gibberish.

      • March 17, 2015 at 3:45 am —

        Not exactly, since the students in the grammar class are demonstrating that they’ve mastered the very skill you’re expecting them to master.
        While it makes of course more sense for everybody involved if they understand what they’re writing, my teaching objective at that point is probably: “Can form the simple past of regular verbs”, not “Can write about the last weekend using the simple past”. The issue with the word problem is that I’m testing reading comprehension. A child might be perfectly able to do “94-37”, but the moment you phrase it “Susi has 94$. She buys a toy for 37$.” you’re adding a whole new level of skills needed.
        With the grammar worksheet the student has all the tools they need: They know the pattern, they get the verb, and even if I make those verbs up on the spot they can still do the exercise. Not that I’d recommend it.

        • March 17, 2015 at 7:47 am —

          I guess I mean more that, since sentences like that are kind of like word problems (in that students can approach them by guessing the conjugation without understanding the sentence), you could intentionally design a problem like the one in the video. I guess to be strictly equivalent that would have to be a paragraph where students would be expected to fill in blanks based on a nonsense context. It’s still not entirely the same, but it gets at a similar problem.

          • March 17, 2015 at 9:14 am

            I think we’re talking past each other.
            My point was that word problems, all of them, do not only test the mathematical ability, but also reading comprehension. In my example “Susi has 94$. She buys a toy for 37$.” the students have to be able to understand the words “has” and “buys” in order to get to 94 – 37. This adds a layer of difficulty to the math problem. Just imagine a student with a disability or a student who doesn’t speak the language well yet.
            Sure, in language instruction I could make students form the comperative and superlative forms of “brillig” “slithy” and “momsy”

          • March 17, 2015 at 11:27 am

            Ah but I don’t mean nonsense words, mire like grammatically correct but meaningless sentences (colourless green ideas sleep _____? *Furious )

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