Building Conceptual Understanding in Mathematics

NCTM, 04/10/2015, https://youtu.be/W1eLt0Dz8Fk

Conceptual understanding is essential for learning. It's an, it's important to understand why we're doing something and the why in the house. So oftentimes when students would memorize procedures or skills or facts and they don't understand the learning or the reasoning behind them, then they don't know when to

apply them, they don't know when or how to use them unless it's in the exact same situation. In Howard County we've focused on developing the concept of

rigorous as defined by common core is it sort of a three-legged stool: procedural

Fluency, conceptual understanding, and Application. We worked with administrators and teachers who understand that for generations we've been over emphasizing maybe procedures, you know. So if you were fast and fluent you were rewarded and you were progressed through the course sequence at an accelerated rate. It’s very important to me is that kids develop a conceptual understanding as well as a procedural understanding of these mathematical concepts. For example, in the example where perhaps you have two pieces of fruit for five dollars, you might want to know how much will twelve pieces of fruit cost so too often kids set up the proportion and then cross multiply and they have not a instinct about what they're really doing, why it works, or whether their answer is sensible. So for example the traditional algorithm in subtraction, just as a primary example, when you borrow - what do you borrow, do you know why you borrow? So literally you have students just you know crossing out numbers yet they don't know why or why do they carry? You know - why do you carry a 1? What does a 1 mean? Is it a 10? Is it a 1? If you think about the problem 199 plus 199, you can set it up as a standard algorithm. You can line them up very neatly. You can go 9 plus 9 because you're taught to add the ones first and do it, and then make another little mark for the one that you carry from 18. Which isn't actually a 1 it's a 10. Or you can think flexibly 199 plus 199.. well I know 200 plus 200 is 400 and I need to take away 2 because it's only 199 in each case. Oh the answer is 398 and that is the kind of flexibility the kind of thinking that you want to encourage. If you think about 12, and do you think about 12 divided by ,3 what you're really asking is how many threes are there in 12? So if you have a fraction division problem, for instance, 1/2 divided by ¼, really what you're asking is how many 1 fourths are there in ½? And the answer is simple: it's two. Because you know what these fractions look like and you can say there are two one fourths in ½, the answer is two. So it makes sense not only from a point of view of what division means but also it helps address the mystery of why when you divide a fraction less than 1 by a fraction less than one you can get a number that's bigger than either of those fractions more or less than 32. Simply applying a formula or an algorithm isn't going to get them to the place where they understand and grasp the whole problem. For example we used to teach keywords all the time and how keywords were so important when students were solving a problem. But you know when we use that word all together students were always taught that's addition. When we give them those shortcuts it seems like we're making it easier but if I said there are 28 students all together in this class, 14 are boys, how many are girls? Well I'm not adding any more - I need to subtract. But if we're simply just teaching all these little rules we're not encouraging students to really think about those problems. Let's be clear that there's more than one way

to learn things. You can learn calculus as a bunch of rules and a bunch of specific types of problems that you learn how to solve. Maybe you can even do well on an exam but that's not the same thing as knowing it so that you can use it. What common core is about from kindergarten up through the last year of high school is learning mathematics in a way that enables you to use it - that enables a student to encounter a new problem that they haven't encountered before and to apply the principles and ideas and computational skills that they've developed and work and solve that new Problem. That's really what a student needs to succeed in college.