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- [Man] I'm here with Bill McCallum, and we're going to look, today, at the Common Core Standards for statistics, especially in the sixth through eighth grade, the middle school years. And, I guess as a starting point, Bill, what I remember statistics in late elementary, and early middle school, it was initially, find the median, and the mean, and the mode. It was kind of formulaic, and it wasn't until much later, probably late high school, even early college, that you start to really think about statistical questions. It seems like the Common Core's bringing a lot of the statistical thinking into the middle school. - That's right. At the same time, we took a lot of those exercises about mean, median, and mode, which, quite honestly, were a little bit "(unclear audio)" like. You know, you'd have these exercises where you had to find the median of a set of six numbers in it. Those were taken out of elementary school mostly. And when we do start statistics in middle school, it's more about looking at data, and understanding how to get a grasp on sets of data. So, the mean and the median, at least, then come in the context of understanding large sets of data that you might use technology to examine. - [Man] Right. So, really getting to the meat of large sets of data, you want one number, or set of numbers to represent the central tendency, or the dispersion of a data set. So really, not just teaching students, hey, add up a bunch of numbers, and divide by the number/numbers or things like that. - That's right. The power of statistics is when you look at large data sets, you can't just see what all the numbers are, and obviously, these days when you have access to technology that enables you to do that spreadsheets, or graphical programs, then that's really the way you want to go to get an understanding of statistics as the study of the these data sets, the study of real world data, and questions about that data. - [Man] Yea. And as we kind of jump in a little bit into the standards, it looks like the first standard right here is really about immediately getting to what questions can you answer with statistics, or kind of put you in a statistical mindset versus ones that are, I guess, in some ways, more deterministic, or... you're not looking at data. - The key, of course, being this idea of variability. If you're asking a question about any quantity that is variable in the population, and you want to try and get a handle on that, then it's a statistical question. - [Man] Right. And, then students also get a sense of looking at data, visualizing the shapes, and central tendency, plotting data in different ways, and summarizing them, which is, in some ways, the point of descriptive statistics. - That's right. You don't need to summarize small data sets. That's why those exercises with small data sets were sort of dumb, but large data sets, you need to get some idea of a measure of center of a set, and also how spread out, how variable the data is around that center, and then as you get more sophisticated, you're looking at distributions that might not be symmetrically distributed around the center, and so on. - [Man] Right. What's interesting is even in college, people learn the measure of center, and then they say, okay, now let's look at standard deviation and variance, but it looks like the Common Core is approaching as, hey, how would you measure variability, really trying to get the students to almost discover these tools. - Well, the standard deviation is a little bit hard to... it's always a problem when you're teaching it 'cause students want to know, well, why am I squaring all these things, and then '(unclear audio)" square root. There's a good mathematical reason for that, but it's not one that's accessible to middle school students. On the other hand, it's fairly intuitive if you have a distribution, just look at how much all the data deviates from your measure of center. Take the absolute value, 'cause you want to measure that deviation, and then take the average of all of those deviations, so that's what we call the mean absolute deviation. It's something that arises naturally if you just first ask the question, how am I going to get a sense of how spread out this data is. - [Man] Right. And then later on, when they do see standard deviation, it'll make a lot more sense. And this is just another way of measuring it, and has better mathematical properties, and whatever else. - Yep. yep. - [Man] And then as we go into seventh grade, This is... You know, I keep saying this, but I don't remember doing this until I was in early college, but really starting to think about what does it mean to sample. What's a random sample and a non-random sample? What inferences you can make and cannot make based on your sample? - RIght. And, I do want to emphasize that the statistics in the Common Core, particularly in middle school is nothing like what you would be taking in college. It's meant to be a fairly hands on, engaging set of activities with data sets enabled by technologies, so that you're not memorizing a lot of formulas, or you're not doing long computations by hand, but you could ask the spreadsheet to do it for you. You do want students to understand those computations and what's going on, but you're not getting into the minutiae of statistics, particularly in sixth grade. It's really, I think you have to understand, it's sort of a light touch here. - [Man] Yea. Yea. - Getting into the ideas. - [Man] Right. Right. Right. Completely fair point, but these are fairly deep ideas, I mean, even as Khan Academy were developing a lot of the items to give practice and assessment on some of these, we, every now and then, found ourselves doing a debate, 'No you actually can't infer that, or you can't infer that.' - Right. Right. Well, of course in some states, the statisticians will say, statistics is not mathematics, there's more to it than mathematics. Statistics is really more like science. There are questions you answer in sort of this reasoning, inductive reasoning in statistics, which is different from the deductive reasoning in mathematics. However, it belongs in the curriculum, and mathematics curriculum is a natural place to put it. - [Man] Right. Right. Right. And so in seventh grade, we're starting to draw inferences, we're starting to make comparative inferences based on the statistics of various populations. And then we're also starting to... and obviously probability and statistics are kind of two sides of the same coin, not to make a pun out of it, but also starting to go into probability models. What does it mean to have a probability? It kind of struck me when we were looking at some of the items that we developed at Khan Academy is there's a lot for students to really get a sense of what is a probability of 1/4. What is a probability of 90 percent? So, it has meaning to them. - Yea. Yea. So the work with probability, again, is work where you look at frequencies of events and you think about what that tells you about chance. There's also this idea of a probability model where you just decide, for example, that the model for a number cube would be that there's a probability of 1/6 that each side showing up, and that's a sort of model you put forward to understand, roll a dice. Then, you might compare that with actual observed frequencies. - [Man] Right. And there's a lot of, as you said, hands on activities for students to understand this. I think some of the standards even call out using physical objects to gather data about frequencies of events. - Right. And then those frequency of compound of events you use devices like organized lists, or tables, or tree diagrams to calculate out those probabilities. Again, with the emphasis on understanding what's the actual space of possible outcomes, rather than getting too bogged down in lots of formulas. - [Man] Right. - That you don't "(unclear audio)" calculating. - [Man] Right. And we see all this here. Then in eighth grade, it kind of starts to come together. You're starting to build the tools to really do a real data analysis by variate data. To me, that means starting to see if you can draw correlations between data. So, once again, I know this isn't mathematically as sophisticated as what we might remember in our freshman college statistics classes, but in a lot of ways, conceptually, it is trying to build those muscles, even maybe more so, than some traditional statistics classes might have. - Yea, I think the idea here is that if students going into those college level statistics classes had had more experience working with data in a more intuitive way, then they would be better positioned to understand the mathematics that they're exposed to in college. So, the idea is not to have a super mathematical treatment here, but to be thinking about... starting out in sixth grade with single variable data. In grade eight, you move to this idea of looking at scatter plots, thinking about of lines of best fit, and learning to have some judgement about would it make sense to use a linear fit. How to deal without lines, and so on. - [Man] Yea. Yea. Awesome. No, this is really helpful. Thanks a bunch, Bill. - Sure! Nice talking to you.