Deborah Loewenberg Ball, University of Michigan
Deborah Loewenberg Ball spoke to Emily Freitag about effective mathematics questions to accelerate student learning. Deborah Ball is the William H. Payne Collegiate Professor of education at the University of Michigan, a research professor at the Institute for Social Research, and the director of TeachingWorks. She uses elementary mathematics as a critical context for investigating the challenges of helping children develop agency and understanding, and of leveraging the power of teaching to disrupt racism, marginalization, and inequity.
Watch the full conversation or read the abridged Q&A below.
EF: I’d love to hear your story as a learner, and how that frames your perspective on what learning entails and how we can think about accelerating learning.
DB: When I learned that you wanted to ask me about my own experience as a learner, I had to step back for a moment and realize something I hadn’t really noticed before. I have a pretty deep skepticism about how wedded we are to the notion of grade-level. There are some reasons why we use grade-level, but there’s a lot to say about what’s extremely problematic about the concept. An example would be that to say someone is at grade-level or not requires us to decide to prioritize and privilege particular things over others. There isn’t really any way to say that a child is at grade-level or not even within a subject area, it doesn’t make sense, let alone overall. But then I realize that it does actually go back to my own experience as a child, where multiple times I was aware that grade-level didn’t capture anything about my own situation. I don’t think I ever could have said that, but I had multiple experiences in which I was a misfit as it pertained to grade level.
I hadn’t realized before that maybe my skepticism dates back to before I was skeptical about grade-level as a teacher. One example would be as a child who came to school, having learned to read very young, I was a big problem for my first-grade teacher and ended up with some extremely unpleasant experiences that positioned me very poorly with other children as a first grader. And I know that that’s a story about the privilege of already being able to read as opposed to the idea of being “behind”, which is what grade-level does to kids too. But the being so-called “ahead” also was painful. There were other instances like that, where my own experience about what I understood and didn’t, wasn’t well-calibrated to what grade I was supposed to be. And I multiple times found myself at odds with the grade level structure. So that is one thing.
It’s also the case that I probably knew from a very early time that a great deal of children’s learning occurs not in school at all. I don’t think that schools have ever been well prepared to pay attention to what children learn in their own culture, in their own environment, from their families. Clearly, schools privilege particular kinds of knowledge that middle class and white children do often get at home. But if we zoom into that a little bit more, I think we could understand that schools are set up with not only grade-level as a rigid idea, but a very flat idea about what it is that children can do that doesn’t attend well at all to the differences among families and students.
I’d always thought that my view came about because I taught in a school where children were so diverse, that the typical structures of schooling like grade-level or being “behind” or “ahead” didn’t apply in my teaching situation. But now I’m realizing that dates back to my own experience as a child.
EF: I think so much of our impression of school really is formed in our own student experience. Take us now then into your experience as an educator and as a researcher and let’s start in a place of optimism—what do we know about what works?
DB: I was an elementary teacher, and I will choose to talk about math since I do currently do a lot of my work in and around elementary math, and more generally in elementary children and their learning.
It is a real opportunity right now to kick off the kind of stranglehold that grade-level has in math and start thinking more deliberately about, What does it mean to be mathematically competent? What is mathematical thinking? What are the mathematical ideas? And to loosen that grip a bit more when we’re thinking about what children may or may not be learning right now when they’re not in the normal organization of schooling. Right now we’re being pushed to do that thinking and it might carry over to allow us to break free of the very linear and hierarchical and to some extent extremely narrow, way that we’ve constructed mathematics in school.
But we also want to say that that is a very tall order in some ways because in order to think more broadly about what mathematical competence is, and what children might just be able to think about it and do, you actually have to understand something about math to recognize and to notice those kinds of things. It would require us to break out of thinking, let me see if I can find a worksheet online, that will make sure that they’re continuing their work on addition facts, or that they’re thinking about linear equations, or whatever the content is. If we just go to the internet and look for worksheets, we’re going to be buying back into that very narrow structure of what it means to be smart at math, what it means to be good at it, or what math even is.
What we know but have rarely been able to act on is the idea that mathematics actually is a whole set of ways of thinking and kinds of problems, kinds of questions that little children, beginning at a very young age, and certainly going on through elementary school and further, have an affinity for, that don’t correspond well to thinking about whether kids are “behind” or “ahead.”
For example, every time children organize things categorically, and children do this all the time, they’re thinking about attributes, they’re thinking about structure. The Common Core standard has as one the eight mathematical practices: finding, identifying, and working with mathematical structure. In general, that practice hasn’t been foregrounded much in some of the work on the Common Core, but when it has, my observation is it’s often quite trivialized. When in fact, the mathematical structure is one of the key ideas to be looking for structure. Categorization is one of those structures and there are many others as well. Children are able to categorize, group, think about arrangements, and think about order in novel ways. That takes me back to my point about understanding some math yourself. Because once I say that it’s possible to reduce that to a very narrow set of things like shapes or color, children are quite capable of thinking of different ways to organize and different ways to order things. And the very notion of doing that is mathematical. There are a lot of possibilities when thinking about orders of magnitude for slightly older children. We don’t really think very well with children about not just counting or adding or computing, but, How do you understand how much something is or how big something is? By getting so wedded to thinking about how we use that greater than and less than symbol and keeping it so narrow, we really neglect the possibility of children beginning to wonder how big something is or how small something is. Or what does it even mean to know how big something is or how much something is in their everyday lives, in which their learning is now situated? Can we figure out what some of these really key mathematical questions and ideas are, and help children to be supported in thinking about those things in their everyday life right now? Could we think about things that could be available that families could access, but don’t rely on parents becoming the teachers of fraction concepts?
By saying that, I don’t mean that families shouldn’t help. But it’s actually not that simple to listen to children’s mathematical ideas and begin to develop those. So, the more we could be understanding what children think about and are able to think about and create spaces for them to do that and reinforce that, we’d be doing a lot of good when it comes to accelerating. That’s what we can learn and take forward. So, let’s say we don’t know exactly what schooling might look like not only next fall, but in the future. Well, we’ve never really done the kind of hard thinking about what we mean by “homework” in math either. So it’s thinking about, what kinds of things can kids do outside of the context of being in groups with other children, and with the teacher? What are other mathematical things they already think about that they could be doing? Because the difference between being in school and being at home is at home there’s a wider range of territory that you can experience, from a spatial point of view, from a probabilistic point of view, from a numerical or combinatorial point of view. Once you have an awareness as an adult that math is all of those things—it’s about space, it’s about patterns, it’s about probability, it’s about counting and enumeration. And think of what are those spaces where little children are constantly operating and create experiences to talk, draw, represent some of their ideas. That might inform us about what’s useful to do for homework also.
What we really want to talk about here is acceleration. So we worry a lot that if regular schooling as we’ve known doesn’t occur in the same ways all of the time, what’s going to happen particularly for children who often don’t get the opportunities for complex thinking and high-quality content? Is this only going to magnify inequities? And the answer to that is clearly yes, because those inequities already exist. And the more we subscribe to these ideas that we think of as advancing equity but actually are going to probably further narrow and constrain children’s development, the worse it’s going to be.
So I think the answer is something we have known, which is that children are capable at a very young age of working on very complex problems. This came early for me when I taught first grade. I taught an extremely multicultural, very multilingual school. I had the opportunity to play around with some curriculum materials that are pretty different from a lot of the things that we typically see. And as I ventured into those and asked my children to think about questions that these materials supported me to ask, I was blown away by how six-year-olds, some of them who weren’t very fluent English speakers or hadn’t had those kinds of opportunities before, completely jumped into problems. There were a number of problems where on Monday, I would say, “I’m thinking of a mystery number, and here’s one clue,” and they would instantly start trying to make a list of numbers. And then on Tuesday, I would say, “Well, I’m going to give you one more clue,” and they would hang with these problems for a whole week. And by the end they’d isolated the mystery number. So across the week, it was like a process of elimination, trying to think about what I know, what I don’t know, using number concepts—a clue could be something like, “my number is even.” Use concepts you want them to know, but in the space of figuring something out. Kids like puzzles, they like to think about things that are uncertain or unclear. They like problems that don’t have single answers, and may even have infinitely many answers. There are logical things like that to begin making available that wouldn’t require a huge amount of support. Children could actually be doing them, and they’d probably like them a lot better than being funneled into working on worksheets. They would answer your question about acceleration because they would be having a much broader diet of mathematical experience. That’s something which I’d really like to figure out collectively about how we can support opportunities for that. And that not only would that help us in the current situation we find ourselves in, but would open the constraints that we’ve forced math into.
EF: Your comment about just how innate this love of problem-solving is really resonates. We’ve been talking to people about the metaphors we carry in our heads about how learning works that actually might not be that productive. What are your thoughts on that? And then, more broadly, tell us what you think about math as a construct and how people might recover a fuller view of it?
DB: The question of what math is has a lot of different answers and there are quite a few people who’ve done very substantial work to uncover.
If I return some basic concepts of what I would think of as mathematics, I think of particular kinds of domains of question-asking. So questions that have to do with how likely something is—we call that probability, but the whole realm of trying to quantify the likelihood of something is a really natural kind of question. We know from research that, in general, adults aren’t any better at reasoning about probability than children are. And part of that is because we don’t teach it much and we also don’t teach it particularly well. It’s actually quite difficult to learn to reason about probability. And we know a lot from psychology about how recency influences whether you think something’s probable. There’s so much more for children to learn to think about, What does it mean to ask if something is likely to happen or not? And to learn that there are actually tools for trying to get on top of that, and systematize that to be able to answer that question. That’s one realm, that question of, Is something more likely? Is an example of a core mathematical question.
Another is space—how things fit together, how things relate to one another, how things are organized. Like that activity of packing the trunk of your car when you have a ton of stuff. How many things to fit into a trunk is a mathematical question. It doesn’t mean you have to do a formula. But the reasoning about how you pack a bunch of objects into a confined space is itself a mathematical question. For space, the subject matter we often refer to is measurement or geometry. And that’s fine. But space also goes to a basic human question about how we arrange things, about the size of things, or how we know that one thing is bigger than another, The different kinds of questions about bigness are really rich and important. And I think we end up thinking mostly about weight and length with children, but there are tons of other questions that have to do with bigness. We blithely assume that they’re going to understand all our kinds of measurement, but measurement of time is a really different kind of thing than measurement of length, measurement of volume is different than measurement of weight. As you get into it, you learn more technical things you can do.
There are fundamental human questions like the probability question. Then obviously, there are things about how we enumerate, how we count, how we record, how we combine, how we split up. Then later, things that are more algebraic— that’s a whole realm of thinking about how you how you pattern, how you see relationships, how you look at proportions.
Core domains of mathematics each come with a whole bunch of basic human questions that are asked in a lot of different ways and answered in different ways. What I don’t know is whether there aren’t also some other domains that in western mathematics we haven’t really paid attention to. I think it’s likely that that’s true. That involves a lot of structuring, a lot of kind of naming of things. In teacher education, in our own experience becoming teachers, we don’t ever see that bigger picture. And that makes it really hard to appreciate what children are thinking about and to be able to connect with them. Nobody gets that bigger picture. So you’re like blindly trying to help children with little minutia. It’s nobody’s fault—the curriculum almost encourages it more because we’re so fearful that it’s all linear.
Math is often presented in hierarchical, linear ways. And yet, there’s nothing about it that makes it linear. There’s no reason that you have to learn to add before you can learn to divide—those things are not in a linear order, and yet they’re locked into the school curriculum.
EF: I’m taking away so much from this conversation about the relationship between what we do in school and just a bigger world that is fundamentally a mathematical world, and how we can connect those more effectively. Any final thoughts?
DB: What we know is that when you layer mathematical learning opportunities in the form of problems, there are ways to bring into children’s awareness things they may not have gotten instruction on, but they don’t have to go back. A favorite problem of mine, which is actually good across many grade levels, is to ask children to write equations for ten. First graders can do that really easily. But it’s actually a very challenging problem for older children because there are so many different ways you can make equations for ten. The more operations you know, the more ways you have of working, and pretty quickly, even very young children realize there are infinitely many such equations. Just super exciting. Think about what happens when you’re working on it. First of all, you can work on it, depending on what tools you have available to you in terms of arithmetic skills, or other kinds of reasoning patterns skills, you’ll be able to generate different kinds of things. But you’ll never finish, which is really good for learning perseverance. But also mixed in is the idea that you could write 100 equations and then you’re like, “Wait, I can think of another one!” is amazing. And while you’re doing it, you are practicing what we often think of his basic computational skills. So you are learning of things that you might not have had instruction in. But meanwhile, you’re asking a bigger question, which is, how many such equations are there? So that’s an example of doing some really basic, putting recognizable math arithmetic together with a question that’s much bigger and more stimulating to the imagination of a child. A problem that really has no end. There’s a lot of examples of layered problems where there are opportunities to be filling in things that children might not have had instruction in or the instruction didn’t connect, at the same time that they’re pushing up into things that are more abstract, more exciting, and more open to many ways of being solved. Those two things don’t have to be separated.