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Lesson by lesson podcasts for teachers of Illustrative Mathematics®.
(Based on IM 9-12 Math™ by Illustrative Mathematics®, available at www.illustrativemathematics.org.)
Hey there. Ready to make standard deviation really stick for your algebra one students? We're gonna deep dive into that lesson plan you sent over. And, I think this is gonna be a lot more engaging than how we were taught standard deviation back in the day.
Speaker 2:Definitely more exciting than just formulas. We really want your students to go beyond just calculating and actually grasp what standard deviation really means, you know, like, to actually get it.
Speaker 1:And that's what I love about the approach in this lesson. It starts right off the bat with this activity called InfoGAP, African and Asian elephants.
Speaker 2:Oh, I love that activity.
Speaker 1:Way more interesting than I was expecting. Right? So for our listeners, can you give us a little rundown of how this activity works?
Speaker 2:Okay. So imagine this. You've got your students paired up. 1 gets a card with, like, a problem about elephant weights. Maybe something like, is there a big difference in how much captive elephants weigh versus wild elephants in Asia?
Speaker 2:And then their partner gets a data card with all sorts of info, you know, like the mean weight, the median, standard deviation, all that good stuff.
Speaker 1:So it's like they each have a different piece of the puzzle.
Speaker 2:Exactly. But here's where it gets good. Okay. The student with the problem. They don't know which piece of data on that card they're gonna need to solve it.
Speaker 2:So they've got to actually talk to their partner, request specific information, give me the standard deviation, give me the mean, and they have to explain why they need that number.
Speaker 1:I love that because it forces them to think critically about what each of those statistical measures actually tells them. Yeah. You know, instead of just plugging numbers into a formula, they've gotta figure out what's actually relevant to the problem.
Speaker 2:Right. And let's be real. Elephants are way more interesting than a dry textbook example.
Speaker 1:Oh, absolutely.
Speaker 2:Plus, it kinda subtly highlights how standard deviation comes into play in, like, actual research, you know, comparing groups and understanding how much things vary.
Speaker 1:Totally. Okay. So we've got elephants, which is already a win in my book. But beyond that, something I noticed is the lesson really makes an effort to connect standard deviation back to the mean.
Speaker 2:Oh, yeah. You can't really talk about one without the other. Right? And there's even this section, I think it's called, are you ready for more, where the lesson has students compare apples and oranges, but in a statistically sound way this time.
Speaker 1:Because they're comparing how many standard deviations each fruit is from its own mean Yeah. Which is basically like sneaking in the concept of standardized scores, like, c scores.
Speaker 2:Exactly.
Speaker 1:Without explicitly going there yet because it might be a little too early, but it's, like, planting the seed.
Speaker 2:It is. And it's planting the seed early so that when they do get to z scores later on, it's not a totally foreign concept.
Speaker 1:It's like laying the groundwork. Yeah. Okay. We've got engaging activities, real world connections, but let's be honest, even with the best laid lesson plans, there are always potential pitfalls. And the lesson plan actually acknowledges this too, especially when it comes to some common misconceptions that students have about standard deviation.
Speaker 2:Oh, yeah. And you know the big one, calculation without comprehension. They can calculate it perfectly, you know, get the right number and everything, but then they have absolutely no idea what that number means in the context of the problem.
Speaker 1:It's like they're missing the vocabulary. Right. Right? Like, they can do the math, but they can't speak the language.
Speaker 2:Yes. Exactly. And I think this lesson plan does a really good job of trying to bridge that gap by constantly grounding those examples in things that students can actually relate to. Like, there's this example about race car drivers.
Speaker 1:Oh, yeah.
Speaker 2:And it shows 2 drivers who have the same average lap time
Speaker 1:Mhmm.
Speaker 2:But then totally different standard deviations.
Speaker 1:And it really hits home the point that it's not just about some abstract number. Standard deviation really tells you something about how consistent the data is.
Speaker 2:Oh.
Speaker 1:You know, like, are we talking about a driver who's consistently fast or someone who's kinda all over the place?
Speaker 2:Exactly. And you can take that idea of consistency versus variability, and you can bring it into all sorts of other examples too. Like, think about students' grades. You could have a student who has a really high standard deviation in their test scores.
Speaker 1:Interesting. Okay.
Speaker 2:And it might be because they're brilliant in some areas, but then really struggle in others.
Speaker 1:Yeah. It gives you a much more complete picture than just, like, looking at their average grade.
Speaker 2:Right. Exactly.
Speaker 1:Okay. So that's a really good way to, like, bring it back to something that they can connect with personally. Now, the lesson plan also points out this other big misconception, which is that students really struggle to visualize what standard deviation is actually showing them. So they can calculate it, they get the number, but then connecting that number to the shape of the data is a whole other challenge.
Speaker 2:It's like they're missing the visual vocabulary.
Speaker 1:Yes.
Speaker 2:Like, you can say, okay. Well, a standard deviation of 5 means this, and a standard deviation of 1 means this. But until they can actually see what that looks like Right. It's not really gonna click.
Speaker 1:And that's where I think the lesson's emphasis on comparing dot plots is really helpful.
Speaker 2:Right. Oh, absolutely. Seeing those data points spread out visually is so important. I actually like to take it a step further with my students, and I have them create their own dot plots from a given standard deviation.
Speaker 1:Oh, interesting.
Speaker 2:Because it's one thing to look at it, but when they're actually the ones plotting the points
Speaker 1:Right. Right.
Speaker 2:It's like it makes that connection so much clearer.
Speaker 1:It's like they're really building that understanding from the ground up.
Speaker 2:I love that. You know, it's funny because I think even as teachers, we can sometimes fall into this trap of thinking, if we just explain this clearly enough, they'll get it. But the reality is there are all these little subtle nuances that can really trip students up.
Speaker 1:You're so right. And there's actually a few other gotchas, I like to call them, to watch out for that maybe aren't specifically mentioned in this lesson plan. But one of them is the importance of units.
Speaker 2:Units. You mean, like, inches or pounds or
Speaker 1:Yeah.
Speaker 2:In the case of our elephant friends, tons.
Speaker 1:Exactly. Sometimes students, they get so caught up in just calculating that number, they forget that it always relates back to those original units of measurement. So, like, a standard deviation of 2 means something totally different if you're talking about the height of trees and feet versus, like you said, the weight of elephants and tons. It's like they're speaking the language, but they're missing the grammar.
Speaker 2:Yes. Exactly.
Speaker 1:Okay. Yeah. I can definitely see how that would lead to some confusion.
Speaker 2:And then another one is, overgeneralizing from a single standard deviation value. So students might see a large standard deviation and automatically think, oh, this data is all over the place. It's super spread out, lots of outliers.
Speaker 1:But it could just be that it's a wider range, right, even if it's still relatively evenly distributed.
Speaker 2:Exactly. It's not just about the absolute value. We have to think about the context, the scale of measurement, what's considered typical for that type of data.
Speaker 1:So it's like, we don't want them to just calculate the number. We want them to actually look at the bigger picture, look at the context, the units. Think about the overall distribution.
Speaker 2:Become like statistical detectives.
Speaker 1:Yes.
Speaker 2:That's a great way to put it. Okay. We're giving them the tools, but they have to know how to, you know, interpret the clues too. And speaking of clues, I think this brings us to maybe the most subtle misconception, which is that standard deviation alone tells the whole story. Story.
Speaker 1:Ah, that temptation to just oversimplify things. But as we've been saying, context is key.
Speaker 2:It is. Like, a large standard deviation might seem like a big deal, but if you don't know the mean, it doesn't really tell you much. Are we talking about, like, a large spread but around a really high average, or is it a wide scattering but around a really low average? Those are 2 totally different things.
Speaker 1:Right. And the lesson plan had that great example about the starting salaries for different college majors.
Speaker 2:Oh, yeah. Yeah.
Speaker 1:And if I'm remembering right, it was the social science majors who had the smallest standard deviation
Speaker 2:Right.
Speaker 1:But also the lowest average salary.
Speaker 2:Exactly.
Speaker 1:And then you had, like, the engineering majors, higher standard deviation, but also a much higher average starting salary.
Speaker 2:It shows you can't just look at one number in isolation.
Speaker 1:Right. Exactly. It's like looking at one brushstroke and trying to understand the whole painting. You need the full picture. So big takeaway for our listeners, standard deviation and the mean, they're partners in crime.
Speaker 1:They work together to give you that complete understanding of the data.
Speaker 2:I love that. Encourage your students to always ask those questions. What's the typical value that this data is clustered around? Is it high? Is it low?
Speaker 2:How does that then change how I'm interpreting this standard deviation number?
Speaker 1:Yes. Those are the questions that are gonna turn them from just number crunchers into actual statistical thinkers.
Speaker 2:Exactly. And I hope, you know, that this deep dive has given our listeners not just the what of teaching standard deviation, but the why and the how.
Speaker 1:Absolutely. I think we covered it all from those really engaging activities and real world examples to some of those more subtle things that trip students up. But before we let everyone go, I wanna leave our listeners with a little challenge. We've talked a lot about teaching standard deviation, but how would you explain it
Speaker 2:Yeah.
Speaker 1:Without using any numbers?
Speaker 2:Oh, I like that. It's like the ultimate test. Can you really convey this complex idea in a way that anyone can grasp?
Speaker 1:Exactly. So there you have it. That's your challenge for this week. Until next time, keep exploring, keep asking questions, and keep diving deep into that fascinating world of data.