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.
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.
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.