About ten years ago math education made a big shift toward conceptual understanding. I want to try and describe the shift from my perspective and what I saw. This isn't an authoritative or deeply researched take, just my perspective. I didn't see everything happening in every school, but something interesting about my career is that I've worked in charter, private, and public schools over the last twelve years. I’ve also had a lot of chances to observe and talk to different teachers from a wide range of schools. What I’m describing isn’t the vibe I get from Twitter or teaching conferences, it’s from talking with regular teachers in lots of different contexts.
The Shift
The average teacher today cares a lot more about conceptual understanding than the average teacher ten years ago. This wasn't a new focus; there were plenty of teachers focused on conceptual understanding before. Some teachers didn't change much about what they did despite a lot of rhetoric around them. The shift was that the language of conceptual understanding became the accepted way to talk about teaching math. It became common not only at conferences but in district professional development, curriculum adoptions, department meetings, and all the places schools talk about math teaching.
It's hard to get teachers to change their practice. Why was this shift widespread, when NCTM and others had been shouting about conceptual understanding for years? I think there were two major causes. First was the Common Core, a once-in-a-generation nearly-nationwide shift in standards that prompted a wave of curriculum adoptions. Second, and I think more important, was a new testing regime as states overhauled accountability systems in response to the Common Core. If NCTM went shouting from the rooftops that conceptual understanding was important most teachers wouldn't even notice. If the standards called for a larger focus on conceptual understanding some teachers might pay attention but most still wouldn't change their practice. But the vast majority of teachers get some sort of pressure to raise test scores or keep test scores high. When a new wave of assessments came out with questions that looked different, that attempted to assess students in conceptual and non-routine ways, teachers and administrators paid attention. From what I saw, that was how the language of conceptual understanding made it to regular teachers and administrators who weren't attending conferences or on Twitter or anything like that.
I'll give away my stance here. I'm in favor of conceptual understanding. I think it's critical to math learning. Learning math without a conceptual foundation means the learning is unlikely to stick or transfer to new contexts. That said, I think the shift over the last ten years has been a net negative for math education. I want to lay out where I saw that shift go wrong.
Vague Messaging
What does conceptual understanding mean? If you walk into a typical district curriculum adoption committee and ask the teachers and administrators in the room if conceptual understanding is important, you’ll hear a resounding yes. But if you take those people into different rooms and ask them what conceptual understanding means, you’ll probably hear a bunch of different answers. You’ll hear about multiple strategies, multiple representations, understanding why procedures work, explaining reasoning, rich tasks, discussions in math class, growth mindset, productive struggle, and more. That’s not one thing, that’s a whole bunch of different things.
NCTM is often referenced in these conversations, but when I google “NCTM definition conceptual understanding” I can’t find a clear statement from the organization of what conceptual understanding means. Search around, you’ll find a few different definitions, all of which are vague and most of which are different from what you’ll hear regular teachers say. Many people will cite the Common Core Standards for Mathematical Practice but there are too many of them and they are too generic to be useful. What teachers need to know is, what does conceptual understanding look like for subtracting two-digit numbers? What does conceptual understanding look like for proportional reasoning? What does conceptual understanding look like for solving systems of equations? Conceptual understanding often seems to be this vague and ephemeral thing, you know it when you see it. That doesn’t help teachers figure out how to teach.
There was this sense, in particular around 2014 as the new tests were adopted, that conceptual understanding was important, and we had to do something differently. But it wasn't clear what that something was, and it led to this faulty logic of we have to do something → this consultant or whoever is telling us something → we have to do this.
Slogans
The second issue with the push for conceptual understanding is that it got communicated as much through slogans as it did through substantive descriptions of what effective teaching looks like. Here are the types of things I heard describing the shift:
Math isn’t about memorizing facts, it’s about solving problems.
We have to move beyond drill and kill worksheets.
Real mathematicians grapple with problems even if they aren’t sure what to do right away.
These are nice slogans and these ideas were very catchy for lots of math teachers. There were plenty more along those lines. The issue was that they oversimplified math teaching. Sure, there’s more to teaching math than memorizing facts, but knowing your times tables is still important. Sure, endless repetitive practice isn’t very effective, but that doesn’t mean practice isn’t important at all. Sure, real mathematicians are willing to struggle with problems, but is the best way to teach students to struggle to just give them lots of problems they haven’t been taught how to solve?
I don’t mean to imply that the answers here are simple. They’re not. That’s the idea — teaching math is complex. Effective teaching can’t be simplified to a few slogans. When this “new way” of teaching math got communicated through these slogans and others like them, a lot of the critical nuance got lost.
Why?
Why is this worth writing about and understanding? To be a bit cliché, if we don't learn from history we are doomed to repeat it. There was this shift that I think could have been positive, but got lost in vague messaging and sloganeering. Here is a very tentative prediction, but one that worries me.
Vague messaging and sloganeering could also describe the rhetoric I see in education around AI. I constantly hear people talk about how education will have to change and adapt. No one seems to have very clear suggestions for what exactly that will look like. There are lots of catchy slogans floating around, like "we shouldn't ask students to do anything a computer can do" or whatever. I think it’s a lot of nonsense, but some of that language is starting to stick.
I don't know if this will cause a major shift in math teaching. Standards reform is probably off the table, but this makes me want to keep a close eye on any changes states make to their accountability testing. Maybe it will fizzle out. Still, I feel nervous. The forces advocating for change or calling schools antiquated are well-funded and well-connected. I hope they don’t win this one.
It is even more confusing in the international school environments.
I like conceptual understanding and how IB framework uses it. It makes sense, and is useful. The trouble is, as you mention, it is not solid. Everyone understands and applies it differently.
Great article!
Oh my, what an interesting can of worms you choose to open here!
Dylan, you say you value conceptual understanding, but you also do not define it. You are right, that when memorization is divorced from understanding, then students don't remember it for long and cannot transfer their skills. But what is this "conceptual understanding" that you speak of?
I don't claim to have a good understanding of how things play out in schools across America. But speaking for the teachers who I work with, here is what I've noticed. First, when it comes to using IM or other constructivist curricula, the biggest problem is that some teachers don't understand the math itself. They don't recognize the math that is involved in these programs because all they know about math is a set of procedures, and those procedures are not always being taught. Those teachers complain a lot about what they are being asked to teach.
Slightly different, there are teachers who just don't believe that concepts exist in math, or who don't believe they matter at all. Sometimes, these teachers can't imagine what it would be like to NOT understand the concept. Why bother teaching something so obvious, when we still have these non-obvious procedures to teach! This is very similar to what you write about in your next post about memorization: people remember having to memorize the quadratic formula because that was hard. They don't remember memorizing "radius vs diameter," even though that is more important. Many people remember learning the procedures in math, but they don't remember learning the concepts, even if they did indeed learn them. So they assume math has no concepts worth learning.
So this is the background against which the common core and the new testing regimes developed. Yeah, if your teaching staff doesn't know the math or how to teach a concept, then it's going to be really hard to move towards conceptual learning in the classroom. If the principal doesn't understand it either, then no wonder you end up with slogans, etc. I don't know if a strong definition from NCTM would have made any difference, though you are right that they should have one. The problem with the slogans is that they don't explain what to do. They tell you what NOT to do -- don't drill, don't let your students give up. But they don't tell you WHAT to do. But a slogan never really does -- that's where extended, meaningful PD comes in. The trend that I see here in MA is that a lot of schools now have math coaches who support teachers, and that is where you will see the real change over time. But just maybe, the slogans were necessary to start the ball rolling. Maybe some teachers started to say, "Well, if you don't want me to drill, then what DO you want me to do?!?" Then they are open to working with a coach.
I think we are in a different place than we were 20 years ago, when I started teaching. Twenty years ago, the only resources I could find was MARS and Marilyn Burns. Now, there are huge numbers of quality resources available that understand that conceptual understanding is central to learning. And I encounter more and more teachers who agree with me, and who have practices that push in this direction. I think that the movement is slow, but the ship is finally turning.
One more thing. Conceptual understanding is NOT problem solving skills. Those are different things. Conceptual understanding means that the procedure happens for a reason. Often it means that you can explain the math going on in a problem using a model, a drawing, or some sort of analogy. Problem solve skills means that you are willing to look carefully at a problem, and attack it from multiple dimensions until you find a way. Here in Boston suburbs, we have the Russian School of Math, an after school program that many kids sign up for. I used to teach at a private school where many of the kids went to RSM. When I would meet a new student and try to assess them for math placement, I noticed something. If a student knew a formula or algorithm, and could apply it in 20 different contexts, but could not tell you one bit of where it came from, that student certainly learned at RSM. Multiply fractions! We got it! We know the algorithm. Okay, but I'd ask, "What does "1/2 * 1/3 mean? Why is the answer less than 1/3?" The kid would say, "Because that's the rule." No conceptual understanding, but great problem solving skills.