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.
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.
A little help here. It is not that conceptual understanding is vaguely defined, it is that an emphasis on conceptual understanding is hard to realize in teaching contexts that require learning to be shown through traditional tests. If we take conceptual understanding to be an understanding of something that hones to the fundamental principles of that idea making it transferable to other contexts then we have such challenges in our teaching. An example- a conceptual understanding of numbers is that we can take apart numbers and put them back together again in different ways to get the same result. This shows up in simple arithmetic- tens bonds - in multiplication- partial products- in division partial quotients- what makes this worthwhile is that it moves us beyond allegiance to staid algorithms in our work with numbers. This conceptual understanding works across many many problem types- which is why having such a conceptual understanding of number is important. The problem is not in definitions of concepts or conceptual understanding, the problem is making this real in a classroom. If conceptual understanding allows us to transfer and use this understanding across contexts then we have to make space for kids to work out such understanding across varieties of problem types- an open approach to teaching that does not work so well in schools where understanding is most often measured in discrete recitation of procedural understandings. Focusing on conceptual understanding forces the curriculum to be more open- which is why IB programs may be more able to hold conceptual understanding at the heart of its activity.
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.
A little help here. It is not that conceptual understanding is vaguely defined, it is that an emphasis on conceptual understanding is hard to realize in teaching contexts that require learning to be shown through traditional tests. If we take conceptual understanding to be an understanding of something that hones to the fundamental principles of that idea making it transferable to other contexts then we have such challenges in our teaching. An example- a conceptual understanding of numbers is that we can take apart numbers and put them back together again in different ways to get the same result. This shows up in simple arithmetic- tens bonds - in multiplication- partial products- in division partial quotients- what makes this worthwhile is that it moves us beyond allegiance to staid algorithms in our work with numbers. This conceptual understanding works across many many problem types- which is why having such a conceptual understanding of number is important. The problem is not in definitions of concepts or conceptual understanding, the problem is making this real in a classroom. If conceptual understanding allows us to transfer and use this understanding across contexts then we have to make space for kids to work out such understanding across varieties of problem types- an open approach to teaching that does not work so well in schools where understanding is most often measured in discrete recitation of procedural understandings. Focusing on conceptual understanding forces the curriculum to be more open- which is why IB programs may be more able to hold conceptual understanding at the heart of its activity.