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It All Adds Up! John SanGiovanni Explains Numeracy
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As ISBE’s team tours the state collecting feedback on the second draft of our Illinois Comprehensive Numeracy Plan, we wanted to share this fun conversation with math education expert, John SanGiovanni, who has authored more than 20 best-selling books on the topic. He sees math — and numeracy — in daily activities that everybody does, and says we all do more math than we give ourselves credit for. Learn more about our plan at www.isbe.net/NumeracyPlan.
This transcript is computer generated.
Speaker:Hello, we are the Illinois State Board of Education and we have Illinois Schools. I'm Dusty Rhodes in the Public Relations Department. Inspired by the success of our literacy plan, ISBE is now developing a numeracy plan. But if you're like me, you may be asking, what does numeracy even mean? So we sat down with John San Giovanni, the best-selling author of more than 20 books about math, to talk about my favorite kinds of numeracy: basketball, shopping, and pizza. So the first thing I want to ask is are you a person who has always found math to be easy? Are you one of those mathy people?
Speaker 2:Oh, that's a great question. And it depends on how you define math. School math? No. I wouldn't say I always found math to be easy. Um, math in my world, I found that to be interesting. Maybe not always easy, but something I was interested in playing with and and kind of exploring, if you will. So math at home, you know, playing with Legos, seeing patterns, exploring, you know, fellow shot percentages I get older or thinking about batting averages and, you know, finding scores of sports and things. Um, that was something that I've naturally gravitated toward at home. School math, um, no, I wouldn't say it was easy for me. I I could consolidate and follow rules. The problem is that sometimes the rules didn't um they they seemed long and slow, and there seemed like there were other efficiencies that I could could have, even though those weren't the words I was using as a 13-year-old, uh, as a 13-year-old boy. But today I recognize that um those ways we're thinking about math were viable. They were just withheld from many students. And what I find to be really interesting is when I talk to my friends while we're playing golf or you know, in the neighborhood having a conversation about, you know, doing work around the house or yard work. Most adults think intuitively about mathematics that doesn't always align with the way that they were taught uh such ideas.
Speaker:So, as you know, Illinois has launched a math and numeracy plan inspired by the success of our literacy plan. Is that, I'm gonna try to sound mathy. Is that a balanced equation?
Speaker 2:Oh, that's that's a that's a good mathy attempt. Um finish your equation, sound like you were more of an expression. If you don't get that, don't worry about it. Um, so and we think about positioning students to uh to to have opportunities um and to access opportunities in college and career and just in life in general. And and we know that literacy is much more than just mechanically reading. Um the idea of numeracy is much more than the notion of just mechanically uh computing, right? It's about reasoning, thinking, sense making, decision making, um, creating and innovating. So to answer your question, I think that numeracy is a natural parallel to literacy. So long as we put it into the context of it's more than just calculation.
Speaker:Okay. So the title of your talk at our numeracy plan summit was numeracy from fluency, what does that mean? So what does that mean?
Speaker 2:Computational fluency is the ability to manipulate numbers for efficiency and doing it in a flexible way so that you're accurate. It's about not necessarily manipulating rules, but instead choosing strategies due to your understanding of the math, due to the way you see the math or think about the numbers involved, um, and navigating whether or not you use certain procedures. I can give you a really good example. If you think about a basic problem like 49 plus 27, many of us were taught to write that down, to line it up and use an algorithm. And there's nothing wrong with an algorithm, but a fluent student may not do that. A fluent student may think of it as 50 plus 26. They may add the tens and then the ones and put everything back together again. Um, they may manipulate the numbers in some other way by adding 50 plus 27 and taking one off. The moral of the story is that fluent students understand the operations, understand how to decompose and compose numbers, and then manipulate them if in efficient ways.
Speaker:Is that something that we currently encourage or tolerate? Or do teachers today have like, this is the way I want you to solve this problem?
Speaker 2:Yeah, so um it's a good question. Much of the work is still uh maturing, if you will. The notion of many of the strategies I just described, um, they're they're not new. They've been around for thousands of years, right? Uh, or at least hundreds of years. We just necessarily haven't taught them to American students and instead try to combine every human to use, or at least every American to use, the exact same procedure for for everything. And again, our standard American algorithm um is fine. Um, but there's alternative strategies that are used throughout the world and and really within our own country. The moral of the story is, or to your to your question, um, do our teachers know about these strategies? Many do, and many are still learning them. Are they incorporated into our curriculum resources? Yes. Um, in textbooks, yes, but we still have work to understand them and um, you know, continue to do a better job teaching them.
Speaker:You said that fluency is an equity issue. What does that mean?
Speaker 2:Fluency becomes an equity and access issue in a lot of different ways, and not maybe the traditional ways we think about in terms of equity and access, but giving students choice to make decisions about how they compute numbers, how they compare fractions, how they solve systems of equations. So teaching them different approaches and then letting them maneuver through those approaches or choose the ones that are better for them for those situations. That's an equity issue. And it may not sound like one, but let me reverse it. When we confine every student to think the exact same way when we fully recognize that humans don't think the same way is an equity issue. Another equity issue is when one classroom teaches a variety of strategies and then arm students with making choices about how they're going to do math, and the classroom across the hall withholds those different strategies and requires students to go about doing mathematics in a very specific way, um, that becomes an equity issue. To not recognize, again, that humans think in different ways and not only respect that, but to promote that idea becomes an equity issue. Um, so those are just some of the ways that we could characterize computational fluency as an equity issue. Um but I'll think of another one, and that is having access to a teacher who has had lots of mathematics preparation, um, who has and pursues professional learning on their own. And then having access to a teacher who who hasn't had those opportunities or or what have you, that that becomes an equity issue. So I think there's so many different wrinkles to this that it becomes quite a grand um conversation or a grand topic for for equity and access.
Speaker:Okay. So that's not really dependent upon how much financial resources your school has.
Speaker 2:Well, it can be. Okay. It can be because if your school doesn't have access to financial resources for current professional learning, right? Or for current instructional materials and has to make do with, you know, antiquated, outdated, or incomplete resources of professional learning, among other things. Um, yeah, that it could be a financial consideration.
Speaker:I like the slide about the 40 problems. And I mean, obviously I wasn't hearing what you were saying about it, but it seemed to me that you were saying instead of having students solve all the problems and grade it for the answer, have them circle the ones that could be solved in this way and star the ones that would be solved in a different way. And and it was a more a worksheet about choosing the right strategy.
Speaker 2:Yeah, um, well, the the point of that that slide, and for any of them listening, that was looking at a traditional worksheet of you know, 40 problems of computation. And for many adults, the way they experienced math was to sit down and to do that in five minutes, 10 minutes, not that it had to be time, but it was just naked computation over and over and over again. And it was can you execute a specific procedure that has been um you know directed to you? And that real mathematicians don't always do that, they'll they'll encounter a calculation and decide what which procedure is better for them in that situation. And so part of that slide was about taking traditional resources and not forcing students to complete them in ways that don't make sense mathematically, but more importantly, um thinking about does the practice you put in front of students go after a very narrow aspect of math? For example, just carrying out one procedure repeatedly versus uh something that's stimulating and and and uh causes the doer to really think about what they're doing and to make choices about what they're doing. And and historically, much math practice has been, you know, that naked computation, 55 problems on a paper or whatever. And that's insinuated the students that that's all the math is. And some have been lucky enough to break out and realize that math is about creating and innovating and solving problems and making sense of the world. But many have been denied that because of the experience they had in third grade math, for example.
Speaker:What are some common mistakes that very well-meaning math teachers make? And give me examples from various grade levels.
Speaker 2:Okay. Um, first and foremost, I'm glad you said well-meaning math teachers, because I know that every teacher in every school across our country, across Illinois, um, wants the best for their kids, just like their parents do. Right. So I want to put that out there first. Um, I think that often teachers um expect or suspect that the best thing to do for the students are to deliver the math the way it was delivered to them, or more importantly, um just to make it as uh what's the word I want to say, is just as straightforward as possible. Right. Um, and enter math through one specific computation and and don't necessarily see broader rules and generalizations. So for example, going back to that 49 plus 27 problem in a third grade classroom, a teacher might say, if you just do this standard algorithm, combine it up and teach it and repeat that over and over and over again, you will always get the right answer. And the impetus was on getting the right answer and not just the efficiency and and and other aspects of math that do matter. And when I say efficiency, I don't mean time, right? In that same third grade classroom, if students are taught different strategies for computing that, they now have tools that they can use in different situations. For example, you know, we talked about 49 plus 27 just becoming 50 plus 26 or manipulating in other ways. That idea holds true no matter what the numbers are like. But let's not talk about numbers, let's or no computation, let's talk about fractions. Uh, fractions for many adults are uncomfortable and they're a great example of very proceduralized rote instruction. But fractions are numbers that we can think about and reason about, and they don't have to be um, well, they don't have to be the experience that many of us had. Let me give you an example. Um, if one fraction is one fourth, and the other fraction is, I don't know, seven ninths. While that can be hard to think about common denominators, another strategy could be to think about those fractions and how they relate to benchmarks such as one half, like one's greater than a half, the other one's less. And so that's a reasoning strategy. So in fourth grade, teaching students ways to reason and make sense of not just the math but the world is much more powerful than having a very narrow process that can be convoluted and really complicated. So that's an example in fourth grade. I spoke to third grade, um, thinking about all the way down in first grade that students see how to make 10 as six and four. But you could think of 10 as six, three, and one, or potentially six, two, and two. And so helping students see that um there's a lot of different ways to decompose ten is different than some of the traditional approaches we've taken in the past. How's that for some examples?
Speaker:That's good. Um I always think about fractions as pizza, and um that's how far I go.
Speaker 2:Actually, can I can I speak to that? Not you, the pizza, not you the pizza, but this is where this becomes an equity because of some of the very narrow ways that many of us were taught mathematics. Many adults have damaged dispositions towards mathematics and don't see themselves as a math person. Everybody is, right? Don't see them as somebody themselves as somebody who can do math well, but yet everybody can. Uh, I guess the point is that the experience that many of our teachers and just adults in general have had with math over the course of a lifetime is something we're trying to undo because having power over math positions kids and adults to do all sorts of great things and have all sorts of opportunities. And I bet you probably reason about math all day long and don't even realize the high-level math you're doing, if that makes sense.
Speaker:I I do get kind of insulted when I was shopping, and the signs on a rack will say, you know, 30% off, and then it'll give a price breakdown of every price and what it is. It's a good one.
Speaker 1:I think you're speaking, yeah, yeah, yeah.
Speaker:And what no, I actually kind of need that one, but when it says half off and it still gives a breakdown, like I can't divide by two. Are you kidding me?
Speaker 2:That that's a great that is a beautiful example, but even when it's 30% off, you were taught to do something very explicit for finding a percentage per se. Um, but even think about 30% off, thinking about well, if I could find 10% and triple it, then I can take something off. Yeah, that's what you do. By the way, that's a brilliant way to go about it. It always works, but for many adults, they were never trained to do that.
Speaker:And I do that by moving the decimal point.
Speaker 2:Absolutely, absolutely. And so why I'm saying this to you is you you're speaking to an example of what adults have learned to do on their own through experience. But what would have been different for them if somebody actually exposed them to that when they were in sixth grade?
Speaker:I will tell you, I have two sons, and my older son is one of these people who math came very naturally to him. He um he sailed through accelerated AP calculus. My younger son was like struggling to get through algebra two and just kept saying, I'll never need this. What is this for? And he's adopted. But I feel more kinship to him. I feel like my older son is like a space alien, and I don't know how he got him. This is true, but I do know that all my life I have used like math cheats, like um, you know, the clock. The analog clock.
Speaker 2:It shows you like a pizza, it shows you fractions kind of yeah, it shows you multiples of five when you skip count by fives all the way around it, right?
Speaker:And it's you know, quarter past an hour, it's half past an hour, and money, yeah, coins have been a math cheat.
Speaker 2:So I want to clarify for you, you did not cheat. You did something else that has not been historically done, but we are doing a better job, and we have to do an even better job. You actually saw connections, right? And you use connections and context to make sense of the computations and what have you. For example, when someone says a quarter past the hour, you see a quarter of a circle, and and you can understand what's happening there. And then if I said, well, what's 16, or excuse me, not 16, but 6D divided by one fourth, you can make that connection right back to a clock and oh, it's 15. And the moral of the story is you you didn't cheat. There isn't any cheating. You you made connections and and and and made sense of relationships. In fact, if we did more of that, we might have students in a much better place mathematically as a whole.
Speaker:Well, I'm wondering because you keep coming back to 49 plus 27. And I mean so in my head, I'm like, out of all the numbers you could choose, you keep choosing these two. And and I'm wondering why. And oh, okay. And I I mean, my theory is it's it's very easily, you know, money, a quarter or two quarter.
Speaker 2:I mean I just use that problem as a talking point. So let's just do the same. So the idea of turning into 50 plus 26, 49 plus 27, I mean, that's not really unique to that problem. I could make it 38 plus 28, and I could easily think of it as instead of 38 plus 28, I could turn it into 40 plus 30 and then take four off. I could make it 40 plus 28 and then just take two off. So the idea of making the 10 and having more left over, or um adjusting numbers to make them friendlier to think about and then compensating for what you did. Um, it's easy to articulate with 49 plus 27, but it works in many situations and not just whole numbers. For example, if I made it four and nine tenths or four point nine plus two and seven tenths or two point seven, I can manipulate those decimals in the same way and manipulate the whole numbers. Um, so to answer, I I'm using that problem as is one that's easy for me to just talk about off the cuff. But these patterns apply uh all number types. It's just when do I, right? If it's 30 plus 20, there's no reason to do manipulations. That's just 50. And so it's making choices about when do I uh do these things.
Speaker:And that's a thing that you want to encourage more educators to do, to teach students to have multiple strategies, right?
Speaker 2:Right. I want us as a profession to do a better job of two things. One, teaching uh different strategies or different ways for doing the math. And two, uh helping students make decisions about when they use a strategy, as opposed to saying, here's a problem, solve it all these different ways. Well, that's ridiculous. That's not how anybody would go about it. Instead, we want them to interrogate a problem saying, well, this one is better to do this way because I see the numbers in these ways. Um, and I think that that's something we really um we want to do better with relative to computation, but not just computation. I mean, it's, you know, comparing fractions or proportions for that matter. Finding percents, to use your example. And listen, there are rules of mathematics, but more so around notions and patterns, relationships, and generalizations rather than, you know, do this first, do this second, do this thing third. I say all this to you because yeah, I think one of the things we aspire to do in in mathematics is do a better job embracing creativity and individualism. Um, and again, not saying that four plus four does not equal eight. Well, I want to be very clear, it does, it always will, right? But the ways to think about how math can be executed and represented and such, um, we need to expand our work there as math teachers. And to be very clear, math education profession and preparing our teachers and such, we have some work to do there as well. Because I know that many teachers I work with, when they experience new strategies, like, that's the way I always thought about it, but I thought I was doing it wrong. And so, you know, some work to do there.
Speaker:So I was wondering how like revolutionary this idea is. Like, how long have you been pushing this? And do you get pushback? And how's it being received by the field?
Speaker 2:Yeah, well, I want to be clear. The conversations that my co-author, uh Dr. B. Williams and I talk about, they aren't new. In fact, people have been writing about them for a really long time. Um, and they're not ours. Um, they're manipulations of math properties and relationships that have been around forever. Um, I think it gets into like their different beliefs about what math is and how it should be done. And those ideas continue to be at odds. I think that um pushback is natural and not a bad thing. I think we should have academic discourse, and just in the world in general, we should have discourse. Course and debate. Like that's a good idea, right? We also have to recognize that there are alternatives to doing things, and that there are many different ways to go about things that we believe to be straightforward, even like mathematics. And the proof that I have is that we have countless adults who are taught explicit ways of doing math, who don't use those same strategies in their everyday life. You're one of those examples and thinking about percentages. And that we have countless adults who don't see themselves as mathematicians, yet they estimate reason solve problems every day. They've just been misled about what it means to do math.
Speaker:Yeah, and you you said you grew up like looking at sports scores, and that's all math.
Speaker 2:You know, I would compare a box score course of the week. Um, you know, in Illinois, you're a Cubs fan, you're a White Sox fan, maybe a Cardinals fan. I don't know. But you think about two batters and a batter that's gone 10 for 22 in a series, and another batter's gone 10 for 19 in a series, like series of three baseball games, right? And I knew that one guy had 10 hits and 22 at bats, and the other one had 10 hits and 19 at bats. The latter was a better batter because they had the same number and fewer number of hits, fewer number of at bats, and you just you you you can recognize that really quickly, right?
Speaker 1:Yeah.
Speaker 2:Well, that's comparing fractions through common numerators, which we were told you can't do, but yet I just did because I rationalized and reasoned about two players having the same number of hits and different numbers of at bats. The point is, I started to recognize these things as a 13-year-old and started taking shortcuts and cheats, like you called it. And it just turned out to be 20 years later I realized they weren't cheats, they were just other ways of doing math that I just didn't get to experience when I was 13.
Speaker:Okay, I'm an NBA fan. And but that's a great example.
Speaker 2:Same with foul shots, right?
Speaker:Yeah. And field goals and threes.
Speaker 2:You can go ahead, keep going. You I'm sorry.
Speaker:No, they just made a new rule that you know the the heaves, like when the clock is coming down at the end of a quarter, and so you don't have time to advance the ball, you just heave it from however far away you are. They've changed the rule, so now it's going to count as a team attempt instead of on the player, because players don't want it to affect their personal field goal average.
Speaker 2:Yeah. So that I didn't know about that rule change, but you think about it, right? You're just throwing something. Yeah, but but let's just say you do that five times over the course of a season, and I don't know what the numbers are, but that could affect your percentage from the field by you know two percent, right? Well, that two percent over the course of a couple of years, I mean, it could cost you money, right? Um, it affects your statistics. I I I never really thought about that. And people are like, yeah, but it's only one time, but one time over the course of you know, a season could make a difference. I didn't know they changed that rule.
Speaker:They've just changed it, and the commentators are already breaking down which players refuse to take those. I think they said Kevin Durant is one, and then what players do take those, like Luca Doncic, uh I know Anthony Edwards does, as players who are more about, you know, they're gonna try for the team, even if it affects them.
Speaker 2:So yeah, but they may not understand the mathematics of how much it affects them, and that could change their tune. But at the same time, think about this. In the past few years, baseball has changed their extra inning rules, where when you start the 10th inning, there's a runner on second base automatically, right? But if you're a pitcher coming into that situation, you're automatically have somebody on base that you didn't put on base necessarily, which could inflate or change your earn-run average. And so baseball changed the rules about how that runner that's that run is counted and who it counts against. And for very similar reasons, I suspect. Yeah, what a great, great, great uh connection.
Speaker:But um you get so excited about math. I just the tone of your voice, you get so much more excited about math when it's in sports.
Speaker 2:I do, I do. That is that is that is true. I mean, there's other places where I get excited about the math as well, but that that's one that that I certainly gravitate towards.
Speaker:And I bet a bunch of students do too.
Speaker 2:That's actually why this has become an equity conversation as well, right? Because for every one of us who gets excited about math relative to sports, there are students who don't get excited about it relative to sports, but they have something that they do get excited about. And if we can't find those contexts, they start to disengage in the math, not because they aren't capable, right? But because they don't see the purpose. But if I'm somebody who loves to bad example, shop, and now I'm thinking about percentages relative to discounts, now I see a purpose and I'm excited about it again. And again, those are two very, very generic examples, but there's so much more to math than just those two. How do we find those? How do we recognize those different um activation or engagement contexts for our kids?
Speaker:Uh you have written a lot of books about math. If a person listening can, you know, go out and just get one, which one should they get?
Speaker 2:Oh, that's impossible. That's a which child do you love the most? I mean, we all know there's a real answer there. Yeah. Um wow. So I think it depends on what your interest is and what your focus is. Um, you know, if you're really pursuing um or you're really interested in learning more about problem solving, then I think you know, my writing about problem solving would be one. I mean, fluency is, you know, something we're talking about right now, something that's not well understood. And I think that the uh figuring out fluency series would be something to explore. Um I guess those would be two of my short answers, but I think if you're someone who works with new teachers and or you are a new teacher and you have questions about teaching math that aren't about fractions, about instead, how do I make an anchor chart in math? Um, I think answering your biggest questions um would be a book. So yeah, I don't have an answer. Sorry. I have a lot of answers.
Speaker:Fine. That's fine. That's totally fair. Okay. Thank you very much. I've learned a lot and I've enjoyed learning it.
Speaker 2:Thanks. I appreciate it, and I enjoyed talking with you. And keep in mind that you do a lot of great math yourself. It just may not look like the school math that you thought you had to do.
Speaker:That was math educator and author John Sangiovanni. You can learn more about our numeracy plan at isbe.net slash numeracy plan. That's isbe dot net slash numeracy plan. Thanks for listening.