Part 3 of our conversation with Deborah Ball, professor at University of Michigan School of Education.
Here is a transcript.
– Welcome to PLtogether Lounge Talks. I’m Adam Geller, founder and CEO of Edthena. Today we’re talking with Deborah Ball. She is a researcher, a professor at the University of Michigan School of Education, the founder of Teaching Works and another role, which many of you know her for, but really relevant to what we’re talking about today. A practicing math teacher. Deborah thanks so much for joining us for Lounge Talks.
– Good to see you Adam.
– Good to see you as well. Well, today I wanna talk a little bit about math. Folks know you as a math scholar and someone who has a lot of opinions about how to teach math. So let’s you know, I guess you know, the math curriculum is very big. There’s a lot that we can definitely not get to the whole curriculum and in 10 minutes, but you know, if I’m a math teacher and I’m thinking about, okay, all my kids are at home all my students are at home. How should I be thinking about adapting what I’m teaching? Like, how do I pick and choose? You know, if I’m teaching third grade, how should I be prioritizing?
– That’s a great question, Adam. And I’ve been trying to think about this and I wouldn’t pretend to have answers for this, for sure. But I think there’s some opportunities right now that we might wanna take. I think that most of us who have been around for a while and been teaching for a while have seen the curriculum in the elementary school and certainly the other levels just get layered with more and more and more, very little ever leaves the curriculum and everyone talks about how more things just get added. This is probably a moment to take a step back and think about what are the things that really could get more kids hooked in to enjoying math to thinking mathematically what are the opportunities here and I can say one thing that trying to stuff, the regular curriculum, whatever that is, in our schools, stuffing into zoom is probably a recipe for disaster on a number of fronts, and I don’t wanna dwell on that right now because the reasons why in the immediate we probably tried to do that. But if, as we think forward, I think there’s other kinds of opportunities. For example, we often know that for children to engage in some pretty significant mathematical problem solving, that takes time and in school, we’re often so pressed for time, you have a math period in a math class, and it’s followed by some other class. And there’s just isn’t the time that it takes to do what we call persevering in mathematics. But, you know, here we have an opportunity to consider some problems that kids could work on over a longer period of time and come back together on zoom or asynchronously to think about how the problem is working out. And there another is a collection of such problems that are pretty easy to get into, but not simple to finish, that I’ve seen from the work that I do with children, children get really engaged in including children who have found themselves as not being interested in math. I’ll give you one example, that is pretty straightforward and kind of simple on the surface, but is of kind of example, it might not sound all that exciting, but I’ve seen kids really take off on it. And it’s as simple as this write equations for time. So, you know, kids will start off, if they’re a little, they’ll write things like, you know, eight plus two equals 10. And then, you know, they start writing things, and sometimes a little bit of a nudge, they realize, oh, like, it doesn’t have to be adding, I could subtract, but within moments, and depending on the age, and what kinds of tools they have to bring to bear, they’re trying to see like, what kind of fancy things can they write with, like multiple numbers in it or different operations, and the more sort of arithmetic and other calculational tools they have at their disposal, like factorials, or, you know, any number of things, they can become more and more intricate. Pretty quickly, kids of many ages begin to realize like, hey, this is gonna go on for a while, which raises a really interesting question like, you know, is there a finite number of solutions to this problem? Well, you know, actually there isn’t. And we don’t give kids very many experience with solving problems that have infinitely many solutions and grappling with the infinite is actually something children get into. So this is the kind of problem that a teacher could set up for a group of children in her class or even a broader group. They could go off and work on it, they could come back together, there could be a little bit of provoking like, hey, you know, has anybody tried new problems that involve division? Or has anybody tried any problems where they see a pattern where they think they can generate a lot more? And we’ve worked on these kinds of problems with children in the summers and the summers, often we work with children who have begun to think of themselves as not good at math. But they get hooked into these problems and begin to realize there’s just a whole different way of thinking. So what I’m arguing in part is there’re problems that have a lot of space in them that depending on the tools kids have, they have many different things they can do. They take time to do in school time is not our friend often but you know, at home time looks pretty different. And these are also interesting problems because many of them contain opportunities to practice very ordinary skills of mathematics. But the bigger picture of the problem is something a lot more fundamental, like the idea of solution space that’s infinite. You can also think of problems that have no solutions. And I can give you many other examples. But this is a moment to consider problems that have a lot of math packed into them, but aren’t actually quite so easy to do in school and would work quite well for a range of credit levels of current academic skill, and would allow kids to stretch and not hit the ceiling right away, not trip, but be able to expand both their sense of like playfulness, but also development of ideas.
– So I think you kind of mentioned the idea of students during the summer and that kind of space of in some ways less, at least the students feel like it’s less structured learning. And now it’s like endless Summer for them, right, like of lots of opportunity for less structured learning. If I’m a teacher, and I teach math, and I’ve tuned in, and now you’ve of whetted my appetite for this type of problem it’s okay if not, but is there something I should google or phrase that helps me find problems that are like this one that you just described? If it’s not something that I have as part of, my resources already?
– Yeah, that’s a really good question. I don’t know that there’s a single site. But over the last, you know, a few decades, you know, math reform kind of comes in comes and comes and comes. So depending on whether you’ve taught, you know, five years, 10 years, 20 years, you’ve seen these cycles come around, they have different names. But going back as far as even the 1970s there were some really interesting problem books that were called something like basic skills with a problem solving orientation. I don’t think that’s quite what it was called. But it was, you know, in the era of basic skills, a guy named Robert Works and edited a whole bunch of problems that I ended up kind of learning when I was a very beginning teacher and realizing oh, like these can be adapted out the train problem, which I’m not gonna go into right now a problem that we’ve done every summer with kids very complicated problem that has no solutions. Originally born out of a very simple problem that Robert Works suggested, which involved trying to figure out whether it was possible to create all the sums out of a very small set of numbers. There’re problems that they’d been around, I think one has to kind of Google problems with them many solutions or problems with no solutions. Another source of the comprehensive school mathematics program, which was a curriculum in the 1980s, developed in the 70s and into the 80s and 90s. It’s still around, not very many districts used it, but it has very non-standard sorts of problems, that if you do the crosswalk to the Common Core, or whatever the standards are in your locality, easy to make that mapping. But what’s interesting about these problems is that they’re what I would consider the layered they have basic skills, and then they have basic concepts, but they involve a question that or a stuck kind of point or something where you can be like see something from a different angle so that the same time you’re doing some basic drill, you’re actually doing things that are far more interesting and not everything’s numerical. It’s also useful to be thinking about problems that are outside the space of letter number and calculation like problems involving space and pattern, problems involving probability. Like most people never get opportunities to really think like, how do you know how likely something is and their dice problems where you get really caught off guard about what the likely outcome is. So I think building some kind of like, collective space to collect these problems. We don’t need thousands of these problems. We need like 50 to 100 for now, I mean, it’s better to curate a few really good problems that kids can get hooked into. And that at least is one opportunity, things that take kids some time that they actually get interested in. And I think that is something for us to think really hard about rather than worrying about whether every single piece of the current curriculum we work with gets somehow managed to be transmitted to home. I think that’s not a recipe for helping kids. You know, feel like they enjoy mathematics, learn things not sort of get stuck. I think we really have to flip and use this opportunity definitely.
– It’s kind of like in science, you know, there’s this whole category of problems, discrepant events. And I, you know, as a science teacher, you learn, that’s the name for that problem set. So I think I was not a math teacher but I think if I was listening to that I heard there’s a whole category of problems called problems without solutions that I should go be looking for some very helpful before we wrap up this conversation, you know, kind of sticking in this realm of thinking about the learning at home, and adapting the curriculum and picking certain problems. Is there an example of how teachers could create learning opportunities about math that they couldn’t, for lack of a better word assigned, but have to do with going out and exploring the world? Are there any examples of that? I mean, you know, at a really young age could be count the trees in the park or something simple, right? If you’re really just at the counting phase, but I’m not curious more at that middle or later elementary where there might be something where you, you go out and you explore and you collect data, which seems like a valuable skill to be developing.
– Yeah, you’re actually kind of bringing up two different trains of thought here. One would be creating collections of things. And that leads to interesting questions about categorization and structure mathematical structure is one of the key practices in the Common Core. And I think for a lot of us, when that got introduced, we were uncertain, like, what does that mean for little kids to identify mathematical structure? If you have a collection of things or a set of problems and kind of think what structures are that you could create like categories or things that seems similar, there are things you can do with collections that you can try to figure out like what’s similar within this collection? What’s different a little kids could think about that. You can try to think about what does it mean to compare collections? And that took me to another kind of thinking that you’ve raised for me and that question, which is, we don’t spend enough time thinking about what this sort of basic question of how much or how many what that question is about when we ask it a little kids, we’re always assuming accounting type of answer. But we asked questions about how much or how big or how many were all kinds of different measurements, for example, is it more or less cold today? Is it you know, is something taking up more space than something else? Is something longer than something else? Is something taller is something more numerous? And we know from Piaget that kids mixed so sometimes they think something’s more because it takes up more space. There’s a lot of playing around with questions of comparison of quantity and measure, you know, what does it mean for something to be later than something else? That’s also a measurement, measurement of time, volume, weight, these are all different kinds of measurement. And with a little bit of just sitting back and thinking many, many people could be stimulating children to ask many more questions about find something that’s more than something else. What makes you say it’s more? And open your mind to think that they might not be comparing number but they might be comparing how much space something picks up or how much it weighs or how long it takes to carry it somewhere and invite children to think of lots of different ways to compare, in some sense, the question how much or how big among and think of reasons for that, which that gets you mathematical reasoning, it opens the space of what measurement actually means. But I do think that for this to happen outside of school, teachers might have to have the space to be able to be released from thinking they’re just taking their curriculum and putting it into zoom into meetings for kids, and thinking, are there some things that are worth spending a little time on that they could explain to families and that kids could kind of have some time and space to think about, and I wouldn’t worry so much that the kids are going to, fall behind this, in fact would give kids an opportunity to be enriched mathematically in ways we often don’t give them time for.
– Listening to you, I feel, my understanding of math education constantly stretched and I realized how much my experience of it as a science educator does not prepare me to be in front of students to teach math because yeah, I mean, well, but, you know, I think here you’re bringing up some really valuable examples of ways that teachers can adapt and create these kind of rich learning opportunities that are centered in you know, for talking about elementary level like math concepts that can involve the parents can involve going out in the real world, but still tie back to those key skills and in some ways translate from the kind of, you know, the academic way of describing it as you did into the, in some ways to still version that’s gonna get sent home to parents as like, hey, it’d be great if you did this, and this and this with your students and the next two weeks, and then, you know, let’s have them come back and talk to each other about that experience. So I think you’ve, you know, you kind of broaden what ways to think of what’s possible. A fast talk but we will continue together in just a moment, Deborah Ball thank you so much for joining us for another PLtogether Lounge Talk. If you’re joining us for the first time, you can listen to more conversations at pltogether.org. Deborah, thanks again for joining us.
– Thank you Adam.