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- [Voiceover] So I'm here again with Bill McCallum one of the authors of the Math Common Core Standards. Bill, I thought we would talk today about place value. - [Bill] Sure. - I guess a good place to start is what's your sense of how the common cores view of place value is either consistent or different with either the state standards or maybe international standards. - Well, it's not different. I mean place value is what it is. It's the underlying concept behind the way we write numbers and so the mathematics hasn't changed, there's nothing there. I think there are plenty of state standards and international standards that pay significant attention to place value. It's always been a part of the curriculum. I do think that we tried to make sure when we wrote the standards, to be really clear about this idea of understanding place value because if you look at some of the things that kids do in school, they can be done without understanding. You'll see exercises where there might be a worksheet which has a whole bunch of problems that are like -- You're supposed to write the number in the spaces on the right and it'll say something on the left like three hundreds, plus two tens, plus nine ones, and you're supposed to write three two nine, 329, in the answer space of the worksheet. Now if you think about those sort of exercises, you can do them without really knowing what you're doing just by, like you know, kids get it. They'll say, "Oh, I just take "the three, the two and the nine "and I'll put them in the spaces." Right? Or there might be questions where people will say what's the point to the two and they'll say, "What place is the two?" And kids will say, "It's in the tens place." Again, that's just sort of knowledge that you can have, but do you know what that means? Do you know that that represents two tens and that the next nine ones and that you're adding all of those together with the three hundreds? So really digging into understanding what the meaning of the place value is, how the place value system works, we wrote the standards to be pretty explicit about that understanding. - Yeah, but what is -- I guess it still puts a lot on the person evaluating it to make sure they don't fall into that trap. That they don't pattern match it, so that a kid can just -- In all of these things, kids are awfully good at figuring (laughs) out the pattern without necessarily understanding the real meaning. - This is where you do things out of order. Like what's two tens plus three hundreds, you know? And things like that. - And even, I know we've made some exercises where it's, what's 12 tens plus two hundreds? - Exactly. And that gets into the whole idea of re-bundling those tens into hundreds, right? 12 tens is more than 10 tens and so you can bundle 10 into 100 and create that higher level base 10 unit. - Right and that's much harder. You can't just pattern match as maybe there's some other ways, but yeah, we'll try to get around that. Just seeing how the progression works out. I mean, kindergarten, there's not really an expectation for much. It's really just to see that the teens can be broken up between a group of 10 although they're not calling it a 10 at that point just the number 10 and some number that's less than 10. It's really in first grade that some of the vocabulary and some of the deeper understanding seems to hit and it does seem a little bit more -- a little bit deeper than I remember when I was in first grade of actually thinking about a 10 as a bundle. Thinking about a one as a bundle, which is kind of a hard concept to grasp because that's just a one, that's just a number. What do you mean it's seven ones as opposed to the number seven? - Yeah, and 10 is not -- It is just a number, but it's a number that can be iterated, right? And you can have a whole bunch of them. You can have nine of them, that's called 90 and that's nine tens. So that then you're thinking of that number, not just as a number, but as a unit. The reason a unifying concept throughout the standards about thinking of numbers in terms of basic units on the number line. The first one being a one, like when you just start counting out along the number line and you count by ones, but then bundling those units into higher level units, that really is the foundation of the base 10 system. - It is a pretty deep concept, I think, for -- I have a five-year-old and I've been trying to kind of get, but it is a fairly -- I think there's some adults who don't fully appreciate that you can kind of do this bundle as a unit and even this idea of 10, 20, 30, 40 and 70 being seven tens, you know, that's starting to kind of touch on the underlying idea of multiplication. - That's right and that's a good example to pick because lots of kids will learn to skip count, 10, 20, 30, 40, 50 and so on. And that's good. You know, they should learn how to do that, but there's a step beyond that which is understanding that those numbers that you're naming are numbers of tens and that's really what this standard is asking the student to do. - Yep, and also in first grade you kind of do greater than, equal, or less than, which does, if my memory suits me, was earlier than what's typical but it's also something that I think is very reasonable for that age. I've seen very young students not have -- You know, they have sometimes trouble memorizing less than or greater than and there's mnemonics and the larger side and things like that, but the idea isn't difficult for them. - Yeah, and certainly there are standards where those symbols were introduced and the standards come to us to put an emphasis on introducing symbols early so that kids can become familiar with them. It's like part of learning the language so that they'll be prepared for algebra when they get there. - And as you go into second grade, that's when we're starting to add. We're going to three digits. We're adding the hundreds place. We are starting to round to the nearest 10 or 100. What's your sense, you know? At least for me, when I first got exposed to it when I was a child I remember when it was just ones and tens I'm like, "What's the point?" But then when I saw hundreds that I started to say, "Hey, okay, now I'm starting to get "why we care about place value." Why not just start with hundreds? - Well, so you don't want to start with hundreds probably because in kindergarten it's a pretty big number, but -- (laughing) I see what you're saying. Is you want to build up. Okay, here's the point I think you're making, which is it's only when you get to hundreds that you understand that it's always the same bundling factor that you use. You bundle 10 ones into a 10, then you bundle the same number of tens, 10 into 100 and you bundle 10 hundreds into 1000. And it's the repetition of that same factor which makes the base 10 system work that means you only have to use the digits zero through nine to name a number. Also it means that any algorithms you have are regular, they work the same way as you move across the places because the bundling is the same. So unlike when I was a kid and you had to add pounds, shillings and pence. Where there are 12 pennies in a shilling and 20 shillings in a pound and this sort of crazy system. - Yep. And actually I made a slight mis-statement before. In second grade you do the three digits and you're learning to read and write the numbers and you're comparing them with three digits, but it's in third grade that you actually start doing the rounding and it sounds like -- In third grade, that's kind of the incremental thing that happens in third grade, the rounding. - Yeah, well, also we're getting up to thousands and there's also a lot going on in third grade with multiplication, addition and subtraction is beginning to round out. The idea really is that by the end of second grade you want kids to have the base 10 system understood and from then on you might start dealing with more and more digits. - Right, and then as you go into fourth grade, this is where you're starting to really see kind of the relation between the values of one places to the left is 10 times more than a place to the right. - There's that regularity of the way the system works and then of course, you extend it to decimals, where you start, instead of bundling things up into larger units, you sub-divide them into smaller units, 10ths, 100ths, 1000ths. And you're always sub-dividing by 10. - Yep and that's in fifth grade when that really kind of hits home is when you're sub-dividing as opposed to bundling I guess. - [Bill] That's right. - [Voiceover] And once again, rounding with decimals. One thing, you know, just to finish up and this might be a crazy thought. You're the expert here, but I don't think I really understood the benefit of the base 10 system until I -- Or I understood even what a base was until I got exposed to base two or base 16 and that only happened ... I used to be in these kind of academic games I used to do in late elementary school. That was the first time I saw them, but then I was like, "Oh, now I get it. "We could have had a base 16 system "and we would have needed 16 different digits "or a base two system and we only need "a zero and a one." - Right. - Why not put that in because that did help me a lot. - It's great stuff and I think, like for kids like you, it's a great extension topic. I think we felt there wasn't room in the curriculum for it. There's something about a, I guess? It's one of those things where you first want kids to understand that 10 is really, really special. It's a special number. Then you want them to understand, well it's not so special. Actually it could have been an 11. It could have been a two. (laughs) But if you make that point before you've made the first point about how the system depends on 10 then I think you might end up not making any point at all, so it's just a question of the level of development of the student and our judgement was anyway that there was plenty for kids to be doing. If we're really adding this demand for understanding with the base 10 system without expecting them to go on to other bases. - Yep, that makes sense. - Again, for most kids that is, there's always going to be students who need depth and I think that would be a great way of deepening the curriculum for them rather than accelerating them on to algebra prep (mumbling). - That makes sense. Well, thank you. That was very helpful. - Sure.