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- [Voiceover] I'm here with Bill McCallum, one of the authors of the math common core standards. And what I thought we would talk about today is decimals. So Bill, I think maybe a good starting point would be, what's the philosophy that the common core has taken towards decimals, and how might that be different from more traditional standards before the common core? - Well, the philosophy the common core takes is that decimals are really just a way of writing certain types of fractions. I think often in school mathematics decimals have been treated as some new type of number, and the emphasis of the common core is on the unity of the number system. You can already see that the way we treat fractions. We try to introduce fractions in terms of unit fractions, where a fraction is just sort of like a whole number except the unit, instead of being a one, is a third or a fifth or something like that, and then a fraction is just a certain number of those. So extending that to decimals, then you have certain sorts of fractions. Whatever unit that you use is a particular unit adapted to the base 10 system, like a 10th or a 100th. So decimals are just a way of writing fractions whose denominators are 10ths or 100ths, or 1,000ths, or whatever. - [Voiceover] Right, and that is a big departure. I mean, I remember when I first, I guess when we all first learned it, it is like this new just type of number, and later you learn to convert between decimals and fractions. But it really is just a way of representing, or writing, a fraction. So you're not really converting between the two, you're just rewriting it between the two. - That's right. The word "convert" makes you think you're actually transforming this beast into different sort of a beast, whereas 0.62 and 62 100ths are just two different ways of writing the same number. - [Voiceover] Makes sense. We have the standards here on the left side of the screen, and maybe we could just do a quick walk through just to understand how you all think about it scaffolding through the different grade levels. Right over here, these first three standards, these are fourth grade. How do you think about what should happen cognitively in fourth grade, and also how this might be different from what's traditionally happened. - OK, well one thing you should remember is that in fourth grade mostly you're adding fractions with the same denominator. The complete sort of algorithm for adding fractions with different denominators doesn't really come till grade five. But in grade four you do have this special case where you have 10ths and 100ths, so it's like getting into this idea of adding fractions with different denominators, particularly with regard to whether the denominators are just 10 or 100 or, well, just 10 or 100. So it's both foreshadowing the full-blown addition of fractions, it's also giving you the tools to start talking about two-digit decimals. - [Voiceover] I see, right, that does make sense. By doing this, yeah, you're getting a preview of unlike denominators and decimals, and then there's just a little bit of, this is, of course, the conversion between fractions and decimals-- - It's what we just talked about, yeah. - [Voiceover] And I shouldn't use the word "conversion," I guess. The rewriting of decimals. (laughs) - Whatever, you know, I'm not super fanatical about people and start worrying, "Oh my god, "the common core says I'm not allowed to use these words." That's not really the point. (laughs) The point is to try to use the language that suggests a certain view of things, so you can use that word for it. - [Voiceover] Alright, thank you for the pass. And then some comparisons when you're seeing something in decimal notation, up to at least the 100ths. - That's right. - [Voiceover] And then as you go into fifth grade, I guess what's happening in the rest of fifth grade is students are adding fractions with unlike denominators, and so what's happening in the decimal world then? - Well, in the decimal world you're starting to fit the decimal notation into the broader scheme of the base 10 system that you've been familiar with from the very beginning of elementary school. So we used to have 10's, 100's, 1,000's, ones, now we have 10ths, 100ths, and 1,000ths. And we're beginning to learn to see how these all fit into one scheme. So the emphasis in those five point MBT standards, the first cluster that you're showing there, are about simply understanding place value, which you've been doing since the beginning of elementary school, but extending it to this decimal context. - [Voiceover] Right, you're extending to the-- - And the same goes for that second standard there. You want kids to understand when you multiply. Kids learn all these rules, multiplying by 10 you move the decimals point, like do you move it right or left, you know, I don't remember. But if you understand why you're moving the decimal point, or why you're adding zeroes to a whole number when you multiply by 10, if you understand that in terms of the position of the digit, having a greater value when that digit is moved to the left or a lesser value when it's moved to the right, and that the change is always by a multiple of 10, then you can understand those rules and explain them. - [Voiceover] One question I have-- Oh, sorry-- - One thing here is we're beginning to use the notation 10 squared, 10 cubed. At some point that's just easier than talking about 10's, 100's. - [Voiceover] Yeah, we're talking about powers of 10. - It's supposed to point to some regularity in the notational system. - [Voiceover] Right. One question I have when I see common core standards like "explain patterns in the number of zeroes," I completely get what you're saying, like if you talk to a child or a learner of any age, and you said, "Hey, why are you doing that? "Why are you moving the decimal over one?" you know, or "Why are you adding a zero?" But what's your sense of kind of what a good assessment, and I know I don't want to ask you to write it in real time, but.... - Good question. I think some standards are hard to assess, because standards about understanding and explaining, they pose a challenge to assessment. So the best way to assess this would be to be able to talk to the student and get them to explain, and then be able to ask questions. We can't do that with large-scale assessments. But I think you can write items that, although they don't directly solicit and explanation, they get at whether the student has that underlying understanding or not. - [Voiceover] Yeah, yeah. And it's a tricky thing, because if you do it a little bit off it can become a very bizarre type of problem. - That's right. I mean, I wish we lived in a world where people didn't feel they had to have every single piece of mathematical knowledge that we want kids to have be assessable on a high stakes, machine-administered assessment. Because that's just an over-constrained world. I hope that as we start thinking about assessment and standards in the new way, there might be some room for breathing there. - [Voiceover] Right, right. No, I think that's a super good plan. I think sometimes some of the dings on the common core are badly written items that are trying to assess things like this, by doing it the weird way. - (laughs) For some people, you know, maybe it would have been better to focus on some other core aspect of it. - [Voiceover] Yeah, yeah. And then these other fifth grade standards, just to move one, these kind of extending what you're doing now to the 1,000th place, where you're rewriting and comparing to the 1,000ths. And then you're also kind of starting to use place value to start rounding. - That's right. And so these are just really extensions of those grade four standards we looked at, you should think of it just as a progression across those two grades. - [Voiceover] Right, and then there are all the arithmetic operations that they've been doing without decimals, they're now extending to decimals. - That's right, and this is the same grade where they're adding fractions with unlike denominators, so it fits for them to be applying that understanding to thinking about how to add decimals. Although, when you're adding decimals you also reference back to the algorithm for adding whole numbers written in the base 10 system. One reason people think decimals are different from fractions is they think fractions are different from whole numbers. Because the way you add fractions is so strangely different from the way you add whole numbers. So the whole set of standards we've been looking at is designed to try and bring all that together, so that kids see how all those pieces fit in. - [Voiceover] Makes sense. And when we look at this last fifth grade standard which we just talked about, it's kind of, you know, being able to do all your operations with decimals, essentially, at least to the 100ths. And then in the sixth grade, kind of the only standard that directly talks about decimals are fluently add, subtract, multiply, and divide. And so even at Khan Academy, we were trying to write items that were like, how does "fluently" differ from the standard that we saw at the end of fifth grade? - So there's another important difference there, or addition, which is using the standard algorithm for each operation. Until now, although there have been algorithms, there hasn't been a specification of a standard algorithm. This is something that we thought should be a culminating fluency standard, where kids should eventually, although they might use different algorithms or strategies when they first start adding, it's very helpful for them to understand how addition and subtraction work, how the number system works. Eventually you want them to sort of have this efficient algorithm and be fluent with it. - [Voiceover] Is it bad if they're using the standard algorithm in the fifth grade, to kind of-- - Nope, not at all. This is a common misinterpretation, so I'm glad you brought it up. You could be learning this the standard item from grade two for addition, you don't have to, the standards allow flexibility. This is an issue on which different people have different opinions, the standards allow those different approaches to flourish. So I think some people would want to start off with all sorts of expanded algorithms, other people might want to start off with the standard algorithm. There is a requirement that you sort of understand what you're doing, and that's why you give some time for that understanding to develop. - [Voiceover] Right. So the sixth grade standard is kind of, I guess it's not even a slight constrain, it's just kind of a minimal viable, (laughs) like just make sure that.... - And I should add, there is a using the standard algorithm there is a standard requiring fluency in adding and subtracting using the standard algorithm for whole numbers in grade four, multiplication and division of whole numbers in grade five, and so this is actually drawing on two previous fluency standards. - [Voiceover] Great, great. Awesome, well I think this is super helpful, thanks for clearing things up for us! (laughs)