← The Report Card with Nat Malkus
The Report Card with Nat Malkus
Mathematical Flexibility and Teaching Middle School Math (with Jon Star)
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- [00:12]
A
Welcome to the Report Card with Nat Malkus, the Education Policy Podcast from the American Enterprise Institute. Math is one of the subjects that gets the most attention in American education. But how well do we actually understand what good math instruction should look like? Should math classes mainly consist of students solving problem after problem? Or should math classes also include opportunities for discussion and group work? Should students learn a topic and then move on to the next topic after they have achieved competency? Or should teachers strive to teach each topic deeply, giving students many different strategies for solving problems? And if math education in America were dramatically improved, just how good could it be? To discuss these questions and more, I invited John Starr onto the podcast. John Starr is the Carl H. Forzheimer Jr. Professor of Teaching and Learning at the Harvard Graduate School of Education and a middle school math teacher. John Starr, welcome to the Report Card.
- [01:21]
B
Thanks, Nat. It's my pleasure to be here. Looking forward to the conversation.
- [01:25]
A
John in conversations about math education, people often will contrast conceptual understanding with procedural fluency. Are those two things separate in your mind?
- [01:41]
B
So yes and no. They are separate because in mathematics, the there really is a distinction between the things that we know, like principles and ideas versus procedures or algorithms or strategies that we execute. So there is a difference between those ideas and things we do. They just feel like fundamentally different things, but at the same time, they're closely tied to each other in the way we think about math working or what we want students to know. So, for example, there might be a procedure that I know and I know how to execute it, but I know why it works, or some underlying ideas that are related to it. So those are tightly connected. We do argue a lot about which of those is more important, and I'm not sure those arguments are particularly productive. They're both important. We argue about which one should come first in learning, and that's also not a very interesting argument, because we know it doesn't quite matter which one comes first as long as they're connected to each other. So these are big battles.
- [02:46]
A
So, John, just for the uninitiated, how would you just briefly define the two if somebody's not familiar with. I mean, conceptual understanding is pretty clear, but what's procedural fluency?
- [02:59]
B
Well, there are certain strategies that we know about how to do certain, let's say, computations like how to multiply two numbers or how to subtract or divide numbers, and there's a set of steps for doing that, and someone who's fluent with those steps can execute them and can do so accurately and maybe even quickly. And Efficiently or without thinking too much about it. And that's what we mean by procedural fluency. It's sort of the steps that you execute to do a computation or something that's kind of automatic, that you don't have to think too much about it.
- [03:32]
A
So can one of these be developed without the other? I mean, for instance, can students develop procedural fluency without conceptual understanding?
- [03:43]
B
Definitely. They certainly can do so. So imagine that you have a set of steps that you're taught about how to divide two numbers, and you practice those steps. You can do them in the right order, you can do them accurately, and you may not have any idea of what they mean or why they work or any of the broader context around them, but you can do them. You can sort of compute using those steps. That would be procedural fluency without any underlying conceptual understanding. And our math education in the U.S. we think that historically, that's what we did too much of, is that we taught students a set of procedures that they needed to memorize and practice and learn, and they didn't know what they meant or why they worked. And we felt that that was a serious limitation to the way we taught math. It left students with very fragile understanding of the math that they couldn't apply in real life or in their workplace. And. And so we've been trying to better connect together the conceptual and the procedural.
- [04:42]
A
So let me repeat back what I've heard, because I want to just make sure this comes through. You know, if you have a division problem for whatever, a fifth grader or something like that, and you see the symbols there, and you can manipulate the symbols, and you know how to do the basic arithmetic, you can get to the quotient. But the conceptual understanding is. Is where you see those operations in the context of how they might be applied in a lot of other instances or uses. So you sort of understand how the procedure that you're doing might be applied to a range of concepts. Am I getting this right?
- [05:21]
B
So I think almost. I'd say that if you understand a little more about why the procedure works and what's underneath it, then that understanding will let you apply that procedure in new settings. So what you described is sort of the benefit of those connections, but it isn't the knowledge that you're connecting. So let's imagine that you have 10 objects, and you want to compute 10 divided by 2. We know the answer is 5, but there's two different ways we can think about the division operation that you're doing. We can either say those 10 objects, you're dividing it into five groups and how. Or in this case it would be two groups. How many objects are in each group? And you would say, oh, that's five. Or I could say, I want to divide the 10 objects into groups that have size two. How many groups are there? So in one case I'm answering the question how many groups are they? And the other I'm answering, how many objects are there? And in either case I get five. But there are two different ways that I could think about it. And if I can picture that and explain that, then I know something about division and I know something about the algorithm that I did and why it worked.
- [06:38]
A
John, I think you might have already answered this, but does conceptual understanding need to be taught before procedural fluency?
- [06:46]
B
No. That's a common misconception that has persisted throughout, especially the past 30, 40 years of the ways we've been thinking about math. It's captured in the way we talk about conceptual understanding. We often say foundational conceptual understanding, which conjures up this picture of a foundation of a building. Like you have to get the foundation in there right or the building's going to crash down. And so there's this perception that that's the case. But there's been a lot of research on the relationship between conceptual knowledge and procedural knowledge and how they develop and what we believe. And there's good evidence for this, is that as long as they're connected, as long as when you learn a concept, you're connecting it to a procedure, and when you learn a procedure, you're connecting it to a concept, and you're sort of going back and forth essentially with your focus, sometimes on the concept, sometimes on the procedure, then that will develop into the rich understanding that we want students to have. There's nothing required or mandatory about doing the concepts first. And sometimes that's a good idea to do, but in other times you can do it with the procedure first.
- [07:51]
A
Now, John, you are a professor of math education. You're also a middle school teacher, is that right?
- [07:59]
B
That's right. So before I became a professor, back when I was young, I was a math teacher in middle school and high school, but left that behind to become an academic and a professor. And that was my life for a long time. But I guess about eight or so years ago, I decided I wanted to get back in the classroom, both because I love it, but also it really informs the work that I do in my research. And I wanted to make that happen. Also. I work a lot with beginning teachers, with pre service teachers, and I felt like I wanted to be someone who was directly experiencing math teaching as I was teaching people how to teach math. So, yeah, I started up teaching in the middle school again, and I'm there every day teaching math this year to eighth graders. But I've worked with grades five to eight generally over the past several years, and often it's the highlight of my day that I get to start off working with students in math.
- [08:50]
A
So, John, first of all, I just think that's cool. I was also a middle school teacher, and so have sort of an affinity for this age group. But I asked that to sort of frame the next question, which is, how do you develop conceptual understanding in students? And I kind of mean that, like, with your students. How do you do it?
- [09:09]
B
Yeah. Well, I'll say two things that I do a lot of, and there's many ways that other teachers do this, and these two ways. I like both because I found them to be effective. But also, I've personally done research about these two things, and I'm confident they work. So the one is, I get students to talk a lot. I do a lot of discussions in my class, and that can take many different forms. It could be quite Socratic, or it could be really a meaty discussion where students are in groups or they're talking to each other. Um, I'm involved in these discussions. I'm asking lots of questions, but I really want to. For them to be talking and making meaning and verbalizing their thoughts and listening to each other, and I really think that has a key impact on the development of their understanding. So that's the first thing.
- [09:56]
A
Yeah, well, I just think a lot of people will hear that and say, what are they discussing in math class? So I just want to chase that with what? What are these discussions about? I mean, what sorts of things do you discuss in math class?
- [10:10]
B
Yeah, well, I ask them questions that are not questions you can answer with a yes, no, or with a number. So a simple question is just, why or why do you think that's true? Or is there another way to do that? Which is what I'll talk about in a minute, or which of the ways to do this do you think is the best? And why? Or how is your way different from the way that the person sitting next to you did it? So. So this question, why do you think that's true? Or what underlies what you just said justify what you just put forth as your idea? I think that's what we want students to do. If I'm giving them just a problem like, what's three times two? Then maybe there's not that much that they can do other than give me that numerical answer. But if I'm giving them problems that are a little more rich, a little more involved, then there's always questions I could ask about, how did you get that? Or why is that true? Or is there another way I could do that? And I think students like to be involved in these discussions. Many of us, when we were growing up had a very boring experience in math. And maybe it was because there weren't opportunities to engage socially in this way and talk to each other and argue with each other and debate. That's fun.
- [11:29]
A
How much of discussion do you think is about building understanding, and how much do you think is just about getting students interested in the material?
- [11:41]
B
Well, I'm not sure how important it is to separate those two, because if they are understanding things, however I'm doing it, then that's enjoyable. That feeling of building knowledge, that's great. So I think it serves both of those purposes, but for me, in terms of someone who thinks about student learning, I'm asking them questions and engaging them in discussions because I want them to essentially reach into their heads and pull out the knowledge that they're wrestling with and they're making sense of and put it out there for us to look at and talk about. And then they have to reexamine their own ways of thinking and then put it back in there. And I think that's what it means to understand is to just be sense making about what you're hearing and what others are hearing, and then eventually come to some personal conclusions about what you think it means to add or subtract or multiply or divide or solve equations. Whatever we're doing in math, teaching conceptual
- [12:44]
A
understanding is often tied to constructivism. So I have sort of a double question here. First, can you briefly describe constructivism for the audience, for the uninitiated? And then why is it that teaching conceptual understanding might be linked to constructivism?
- [13:03]
B
Constructivism means lots of different things to different people. And as a research enterprise, it's sometimes hard to pin down exactly what we're talking about. But one thing that people talk about is the role of the teacher in constructivist learning environments, where if the perception is that in the past we used to have the teacher just lecturing or providing all the information or being the authority, maybe constructivism would postulate that we want to put more of that authority in the hands of students, where they're not just receiving the knowledge, but they're Participating in a more student centered kind of way. So that's a key principle about constructivism. There might be some people that would argue that certain kinds of activities are very central to constructivist teaching, like group work or partner work or projects or other sort of things like that. I think from my perspective, conceptual understanding is a goal all the time for all of our students and something that everyone can engage in. And I don't think that constructivist teaching has an inside lane, if you will, toward conceptual understanding. I feel like a good teacher teaching in just about any way that that teacher teaches well can develop conceptual understanding. And some excellent teachers teaching in a constructivist way do so. But teachers who teach in a more Socratic way, or even a more teacher directed kind of way, they can also develop students with deep understanding. And I think that is a point that not everyone agrees with. I know in my field, but I feel like a good teacher teaching well can develop conceptual understanding, regardless of which of these camps they're instruction is characterized as being similar to.
- [14:54]
A
I don't want to invite too much controversy, but what do you make of constructivism as an approach?
- [15:01]
B
Constructivism taught well is wonderful. And there's days in my class where someone walking in would say, wow, that's a really constructivist lesson that you just did. But there's other days where they would walk in and they would say, that's not really a very constructivist lesson. But I would argue that if I do well in both of those lessons, then I've achieved my learning goals for students. In terms of understanding. I think that I'm centrally thinking a lot about learning. But as you pointed out earlier, motivation is really important too, and students really being engaged and interested in the math they're doing. And it is true that there are some things that we can do in the class that might serve as much of a learning purpose, as much of a motivational purpose as they do a learning purpose. So maybe when I'm talking about engaging students in discussions, I see the learning benefit to that, but I also recognize that's very motivational and interesting for students and that serves my learning goals as well. So there's a lot about constructivist teaching that not only is addressing learning, but also motivation. And again, not that that's the only way to achieve your learning goals and motivational goals, but we found that a teacher teaching well in a constructivist environment that could have very positive motivational consequences for students. And that's wonderful. But I think that the point that I always try to make when I'm Talking about this issue is that there are people that make a distinction between constructivist teaching and whatever you call the opposite of that. There's different names of that. I would say that there's another distinction that we should talk about, which is good teaching and not so good teaching. And those are not the same thing. In some ways, I think of it as more of a two by two, that I could have good constructivist teaching and not so good constructivist teaching and good the opposite of constructivist teaching and not so good the opposite constructivist teaching. And I just want good teaching. And I'll take it in either of those boxes. And those different boxes might have different affordances in different situations, in different lessons with different students. A good teacher has both of those in their repertoire, and they can teach in both of those ways.
- [17:06]
A
Well, more on the procedural side, one thing you'll sometimes hear is that students should see a worked example and then do a problem pretty similar to that worked example and then, you know, rinse and repeat a number of times. What do you think of that sort of approach?
- [17:26]
B
So I am trained as a psychologist and there's no denying that worked examples work. That when students see worked examples, they help them learn. And textbooks used to have lots of worked examples in them. And I think, though we've learned a lot about better ways to teach, I think, than we taught maybe 50 or 100 years ago. The textbooks had a lot of worked examples and the student could learn a lot by studying those worked examples. I think our textbooks have reduced significantly the amount of worked examples in them over the past generation. And I know that a lot of psychologists have concerns about that because we know that worked examples work. So in the argument around constructivist teaching, sometimes worked examples are put out as a not constructivist facet of teaching or curricula because you're sort of giving the students the information rather than having them discover them themselves. But I think that's a bit of a false dichotomy that worked examples. I'm not sure how we argue against worked examples as a sort of productive way to learn. There might be better and worse ways to use them in a classroom setting. Like if I give a worked example, then maybe I want to engage students in a conversation about how you're making sense of this worked example. Talk to your partner. What does this mean? What are you noticing in this worked example? Let's discuss it. What is that? How does that fit into these camps? I'm having a discussion, but it's around worked examples. And I do that a lot. And so I think that's an example of where these boundaries blur a bit in a way that is probably productive.
- [19:07]
A
John, there's been a shift. I can't quantify to what degree it has been, but for curricula to move from what 15 or 20 years ago would have been totally based on textbooks, or overwhelmingly based on, to curricula delivered digitally. I'm just curious about how those two sort of avenues for curricula, either a textbook versus digital curriculum, might have impacts on students.
- [19:38]
B
Yeah, great question. And this is something I think about a lot. And I'm going to make a generalization knowing that there's lots of different kinds of digital curricula out there, some of which might be subject to what I'm going to offer as a critique, and some not so much. But what I like about physical textbooks in classrooms is that they present a coherent picture of the math for the students. We have been guilty in the past of telling students topic after topic after topic to learn and not emphasizing how those topics are connected. So it's sort of like topic of the day, and then tomorrow it's a different topic and they have nothing to do with each other. And textbooks are designed to illustrate that coherence. They have chapter structures that you can trace an idea through multiple sections in the book and how it develops. And teachers should know about that coherence. They should know what we're doing now in September. The reason that you need to know this is because in March we're going to be doing this other thing, and that's how it's related. And so that helps things make meaning for the students. And that's key. If an online curriculum doesn't have that and it just appears to be Finish that problem. Here's another problem. Finish this problem. Here's another problem. It's just a series of problems, or even a series of problems and worked examples. Then it's not clear what are the overarching ideas or topics that this is all about. It's just a series of disconnected skills, and I think that's making it harder for students to make sense of what they're doing. And that's why I'm often in schools where I ask them, what book are you using to teach math? And they say, oh, we're just using whatever, or we're using the stuff in our file cabinet or we're using this program. And I feel like that often doesn't work so well.
- [21:27]
A
And I think textbooks often sort of get a bad rap because, you know, they're Long and big tomes and so forth. But. But there's also a question here on what it means for teachers. So if you're a teacher and you can bring that coherence to sort of a decentralized curricula, great. I would imagine that's a pretty tall task for a lot of teachers. Whereas a textbook seems like it has some navigable coherence in it and it's kind of in one place. Am I getting that right? Does that sound reasonable?
- [22:01]
B
I agree with that. And I think it's an important skill still an important skill, I'll say, for students to learn how to use a textbook, how to learn from a textbook. And so in my teaching, we work explicitly on this. So what's the structure of the textbook? Where do you find the homework section and where do you find the worked examples and what's in the glossary, and which answers are in the back of the book and how do you use it to study for a test? These are all learning skills that help you be more successful in math, and they need to be taught explicitly. They're not just something that someone's going to pick up just by picking up a big heavy book. And there may be online programs that replicate that experience wonderfully. But I think there's others where it just feels like it's a series of skills that I'm working my way through. And I don't really have that big picture at all. It's hard to get at. And that's something the student misses. That's something the teacher misses too.
- [23:00]
A
John, one of your main interests is flexibility. When you say flexibility, what do you mean?
- [23:07]
B
So in the context of math, what I mean by flexibility is that a student knows multiple ways to solve problems or a problem, and that they can think about which of those ways to solve a problem might be the best. Where best can mean lots of different things. It can be the most efficient one, or the one that gets the right answer, or the one that meets whatever kind of problem solving goal I have. Like, I want to be very careful. There can be multiple ones that are best, but they're perhaps better than ones that are not the best. So it's not that there's one best. It's that we want to be thinking together about what are the different ways to solve this problem and what are the pros and cons of those different ways. Math problems can always be solved with multiple strategies and. And in building understanding of math, I'm not just interested in students being able to get the right answer. That's important, but I'm also interested in them understanding what they're doing because that will help them be better problem solvers with future problems as well. And for me, flexibility is really key to that, that I want them to know multiple ways, and I want them to know which of those ways might be particularly good or bad given a certain problem.
- [24:20]
A
So there's multiple ways or procedures that you, you want kids to learn. Are procedures things that you either know or don't know, or are there gradations to how deeply a student can. Can know a procedure?
- [24:36]
B
I think there are gradations to how a student can know procedures where they may know some procedures deeply and others very superficially. Knowing something superficially might be just memorizing the steps. And memorizing steps is a pretty risky strategy for learning because if you forget one of those steps, then you have no way to reconstruct it. You don't really understand what was going on. You're stuck. If you understand a little more deeply about what's going on with those steps, why they work, or what are some of the underlying ideas that make those steps true, then if you forget the procedure that helps you maybe reconstruct it. If you encounter a problem where that procedure, as you learned it doesn't work exactly, and you have to modify it, maybe that helps you modify it. You see a brand new problem, you're not sure which procedure to use, maybe that helps you decide how to go about it. So that's what I've argued in my work, is that procedures are something that are integral, important, critical to learning mathematics. And as are concepts, and they're both important. And we need to think about what it means to understand procedures, what it means to deeply know and be able to know the whys behind the steps that you execute. And that's part of mathematical understanding. And that's what I mean by flexibility.
- [26:01]
A
John, on the flexibility, a clear idea. Can we measure it clearly?
- [26:09]
B
I think we can, and I don't want to get into the weeds about how we measure it in our specific research studies. But if flexibility for me is that a student knows multiple strategies and they know which strategy is the best for particular problems or which strategies are better than others, I could just ask them that. I could say, hey, here's a problem, Solve it, then I could say, can you solve it again, but using a different way? And if they can, then they know two ways to solve it. I can ask them, which of these two ways do you think is best and why? I can interview them and ask them that question I could give them other problems, some of which there is a better way and some of which there isn't a better way, and I can see how they solve it. So in a nutshell, what we do is we give students problems, we look at how they solve them, and we have ways of analyzing those results to make determinations about which kids are flexible and which kids less so.
- [26:59]
A
In math, teachers can try to get students through the content so they can cover every topic they're supposed to cover, and then they also can try to make students into better problem solvers. Is flexibility more about the latter?
- [27:15]
B
Problem solving is this phrase that we toss around and I'm not sure what it means anymore. And so I tend not to use that phrase because are we talking about their ability to solve problems or certain kinds of problems? Is that what problem solving is? Or is it about certain real life applications? Or are we talking about transfer or critical thinking or 21st century skills? What is problem solving? So whatever, perhaps I might argue that if I am taking that phrase literally and I'm talking about their ability to solve the problems that they encounter in a math class class, then I think that ability is enhanced by not only understanding the big ideas underneath the math, but also knowing multiple problem solving strategies for problems and being able to talk intelligently about why those strategies work and which ones are particularly good and which ones not so helpful on what kinds of problems. That's what it means to understand problem solving strategies. And that I think is what makes you a good problem solver.
- [28:24]
A
I want to push a little bit on the flexibility. So let's say you have a mathematically flexible student and she encounters a tough problem that's a little more difficult than anything she's seen before, what's she going to do that's different from a more non flexible student?
- [28:42]
B
This is what I would want her to do. And this is sort of the thought process I hope happens. Maybe implicitly she sees the unfamiliar problem and she's not sure how to approach it. So she looks at this problem structure, the problem's mathematical structure, and she might say, oh, I see there's certain terms there, I see there's certain constructs here that we've talked about before, certain words in the problem, and she thinks, what prior problems have I seen that have that same or similar structure? And then for those similar problems that I've seen before with that structure, what strategies did I use on those problems? And if you can get to that point where you say, oh, these are the strategies that I use On a problem with this structure, then you have a good guess, perhaps that those same strategies might be helpful on this problem as well. And so you can try them out and see if they work and then repeat. And that's what it means to be flexible. That I've sort of have a series of strategies that I really understand, that I know. And those strategies are linked in my mind to features, mathematical features of problems. And that's going to help me when I see other kinds of problems that aren't familiar. But I learned to look at that structure of the problems, and that shows me that I'm understanding what I see mathematically and connecting it to the strategies. So I think that's flexibility in action for me.
- [30:17]
A
And so as a teacher, how does one teach flexibility?
- [30:23]
B
A key way that we found promotes flexibility is just exposing students to multiple strategies for solving problems. So they solve a problem either with a way they came up with or a way that you've shown them, and then we consider other ways to solve that problem. Again, it might be I show them a second way, or it might be they develop a second way. But I want them to know that there are different ways to solve this, that it's not. The message I'm sending is not when you see this problem, here is how you should or must or are required to do it. Follow these steps. It's more that we're saying there's lots of ways to do this. Which one do you think we should explore? I couple that with discussion. I want students to be talking about these strategies, comparing, contrasting, evaluating these strategies and discussing them. And that's what we've shown does produce improved flexibility. There's others as well. It does all boil down to setting a culture in your class where you say to your students explicitly and implicitly, I'm interested in the right answer. But that's not all I'm interested in. I'm interested in how you found the right answer. And we're going to have a conversation about that how? Because I'm interested in that. That's important to me. Mathematics cares not just if you get the right answer, but how you get the right answer. Mathematicians like to talk about this thing that they use the word elegance a lot to talk about mathematics. And elegance is really hard to define. And it has this sort of aesthetic dimension. It's sort of about beauty, and that's what elegance is all about. But at its core, that's something that mathematicians talk about. That's something we think about mathematically. And I'd like students to be doing that as well in K12 to be thinking about the strategies and the methods they're using and which ones are clever, which ones are efficient, which ones are beautiful. It's not just about getting the right answer.
- [32:20]
A
So John, in, in teacher training and preparing teachers and the teachers that we have. This is an impossible question to answer, but I'm going to ask it anyway. How good is the average math teacher in America? Like just the. The sort of modal kind of instruct math instruction that's generally given in middle school along this dimension of teaching flexibility. I mean, are most teachers attuned to this? And it would seem to me that the strawman example would be, well, we have this formula and you plug it in and now I'm going to teach you the next formula. And it's disconnected. Just how would you handicap that landscape?
- [33:02]
B
So I think we're getting better. If you went back a long time ago, maybe the argument would be that teachers are teaching in that way that you just characterize as purely procedural or memorizing this. Do these steps. Don't think about what you're doing, just practice, practice, practice. And in reaction to that, we focused our energies on teachers teaching more conceptually and perhaps that pendulum swung far or too far and flexibility is kind of eyeing a more middle ground. I think think there it's sort of saying procedures are important. We do need to focus on them. We also need to do concepts. It's not one or the other, it's both. And we do need to have students be working on understanding the skills that they're doing in this flexible way. So I wish I could say that we have made a lot of progress in that but it's something we're still working on and teachers face a lot of challenges in their day to day work. And it may seem like this is a really hard ask of them that teachers may say, and teachers do say this to me. I'm having a hard enough time teaching my students one way to solve this problem. And here you want me to teach them multiple ways to solve this problem. And I hear that, but I think that is exactly what we're trying to nudge teachers toward doing. And students that we've worked with really welcome this shift. If a student doesn't really understand the method the teacher taught them, but they perceive that that's the only thing they can do, that you have to do it this way, that's the way the teacher did it, that's the way I must do it, then they're kind of stuck. They've got nothing to fall back on. But if they know that there's actually lots of different ways to solve this problem and some ways are going to make more sense to them and maybe that's okay that they do it the way that they thought of, that's very empowering and that that changes the dynamic of the math class completely.
- [35:11]
A
John, we had the folks from Math Academy on not too long ago. I'm not sure if you're familiar with Math Academy. One of the big things that they talk about that is a problem for math students is if they have holes in their prior knowledge, their, their prior capacities. And it seems to me that if you don't have sort of a well developed flexibility around some of these skills, that a gap that you might have earlier a week, that weakness you might have in sort of a something lower on the, the tree of skills might be all the more brittle. Does, does that sound right to you?
- [35:50]
B
Definitely. There's too many students have these gaps, these places where they didn't develop the knowledge that they need or they didn't develop it sufficiently fully. And that has repercussions later on for sure. Math is a hierarchical subject that it builds on each other, it builds on itself. And that's a huge challenge for teachers. And it seems like we've always faced that challenge. But certainly Covid and the disruptions around Covid made that much, much worse.
- [36:22]
A
John, does math or progressing in math well, make students into better problem solvers or critical thinkers? And I'm using air quotes around that because they're often used, but does it make people better problem solvers or critical thinkers more generally?
- [36:41]
B
Not automatically. Sometimes. We have had this idea that math is a special domain or one of a series of special domains where just by doing math you will get better at very general skills. Critical thinking, logical thinking, problem solving, just by learning the day to day math. And I don't think it's as automatic as that. This is a perspective that goes way back in classical times. This was the perception of math that, that your mind was viewed as like a muscle and that math was this excellent exercise that you could do and it made you better at everything else just by doing math. There's pretty good evidence that that is not the way the mind works, that there isn't this universal type of exercise like math that helps you do better at anything else. Psychologists talk about this as a transfer problem where it's not the case that if I learn math, then all of a sudden I'm going to be good at anything else. Whether it's music or science or philosophy, just by learning math, that's not the way it works. Rather, if I want students to make connections between the math they're learning and more general problem solving skills or logical thinking skills or critical thinking skills, I have to build that bridge for them. I have to walk them across that territory to teach for transfer, essentially. And so I think it's definitely a goal that our students should become critical thinkers and good problem solvers. But what it means to teach math with that goal in mind is not as clear cut as it might seem. What do you do when you're teaching math so that your students will become more critical thinkers? Well, in my classroom, that's asking questions, that's getting students to talk and debate and share multiple strategies. That's how I accomplished that goal. If I didn't do that, I don't think they would develop those critical thinking skills.
- [38:42]
A
What about the transferability within math? So you have taught a Algebra 1 student very well. They're very capable. Does that transfer to sort of higher capacity when they reach geometry?
- [39:00]
B
I'd sort of give a similar answer that it doesn't happen automatically. It's not the case that a student who succeeds in algebra is automatically going to succeed in a later math class. I have to, as I said, build that bridge. And the way that I build that bridge is by emphasizing the coherence around the math curriculum. So I'll give you an example, and you can tell me if this is too mathy, but when students learn Algebra one, they're working on a particular type of mathematical relationship that we call linear. Linear relationships is all what algebra 1 is. And we do a lot with linear relationships. We graph them, we look at symbolic representations of them. We use them to make predictions about future values. And there are some linear relationships in our world, even though most of the things we study are more complicated. But that's what you do in algebra 1. The sets of things that you do with linear relationships that I just mentioned, like solving equations and graphing and looking at predicting future values, it turns out that you do exactly those same set of things in algebra 2 and in pre calculus, just with more complicated types of mathematical functions. So instead of linear, you're doing things with things that we would call quadratic relationships or cubic relationships, or logarithmic relationships, or trigonometric relationships. And it gets more complicated as you're looking at those harder types of mathematical functions. And so you have to do more and more, but at its core, you're doing the same thing that you did in algebra 1. Just building on it. And I don't think what I just said is something that many students ever see when they take these classes. They walk into algebra 2 and it's this completely new thing that they say. This has no connection to anything we've ever done before. Rather than it feeling like, oh, this all fits together. I see exactly why we're doing all of this. It just mirrors what we did in grade eight. So that coherence is so key to helping them transfer to that more advanced math, I would argue.
- [41:13]
A
All right, John, it's time for grade it. I'm going to give you some topics. You give me a grade. You're a middle school teacher. This should be easy. A to F. And a brief explanation of your grade. Are you ready? Ready. Math competitions.
- [41:30]
B
Solid B. Math competitions are great for students who are excited and motivated about the math, and it pushes them to develop some thinking and advanced skills around hard problems. Not for everyone, and that's totally fine. But for the kids who are into that, it really does help them, and that's great.
- [41:50]
A
Creativity as a concept.
- [41:55]
B
C. I don't know what creativity really means. It's something that in the research literature, people talk about a lot, but I'm not sure what definition makes sense to me. I don't know how you measure creation. Creativity. So I'm a little confused about creativity, but I'm working in that one, actually.
- [42:13]
A
The potential of AI to improve math education.
- [42:19]
B
B minus. So I have a lot to say about AI. But I'll try to give you the brief answer here, which is that I think AI is a very different kind of technology than any other technology that we've asked the same question about in math. And the answers that we've had from previous rounds of technology don't work so well with AI. So we have to think completely differently about the impact of AI on math teaching and learning in schools. And I don't think we've started to have that conversation yet.
- [42:50]
A
Middle school students. And I say this in the context that some people say middle school students are very difficult. But what's your grade?
- [42:58]
B
A plus. I love middle school students. It's true that among teachers, people say you either love middle schoolers or you hate middle schoolers. Essentially, you're either cut out to teach middle schoolers or you aren't. And I think I am. I really like my middle schoolers. They are so much fun, strong.
- [43:18]
A
Second, the Mathematician's Lament by Paul Lockhart,
- [43:23]
B
A very interesting publication. I'd recommend people look at it talks a lot about the ways that mathematicians think about the math that we teach in schools. In an interesting piece,
- [43:38]
A
NCTM
- [43:40]
B
B nctm, for those who don't know, is the National Organization of Math Teachers in the US and they put out a lot of excellent curricula and are really active on the policy scene. I think that I would love to see them have a stronger research base for some of their recommendations that they do, and I'm not sure that's always there.
- [44:03]
A
Relatedly, the state of knowledge in math education research, I mean, have we figured this out yet?
- [44:11]
B
I'd give us a B plus on that. We have actually learned a lot in the past, let's say 30, 40 years. But historically, math education as a field has been around for about 100 years in the US and I think we've learned a lot. I think we have a tough task ahead of us to try to improve the teaching and learning of math. But we are working hard on this and there's some really great people who are in this game trying to figure it out. So I'm optimistic.
- [44:39]
A
The average US Mathematics curriculum, what gets taught,
- [44:45]
B
I'm going to go B on that. I think that we have made some real advances recently. There's some excellent curricula out there, especially at the elementary school level, that are doing a wonderful job that we didn't have a generation ago. I think our high school curriculum is still largely traditional and it looks the same as what people saw it 30, 40 years ago. But I think we're moving in the right direction on curricula. And I think curricula is very important. It's very important to have good curricula. And we're, we're making strides in that area.
- [45:15]
A
I'd say Chinese math education,
- [45:21]
B
I feel like I'm giving a lot of Bs here, but I'm going to give a B there as well. I do a lot of work in China and think a lot about the way Chinese are. The Chinese math education system is structured and the Chinese do amazingly well in international comparative assessments. So they are doing something right. It's an open question exactly what they're doing right. And I'm not sure we always know, but that's what I find promising about the Chinese math education scene is there's something there for us to learn from. And it's not something that's obvious, perhaps, but that's what I'm interested in trying to figure out.
- [46:00]
A
Calculators in the classroom A. I like
- [46:06]
B
having calculators in the classroom. I recognize that there might be a certain age whereby you don't want to have calculators be regularly used, very young kids. But they're a tool and I want students to learn how to use that tool. And maybe there's times in my class that I say I don't want to use calculators, but I think in general, if student needs a calculator, then they should use a calculator. And I'm going to make sure the questions I ask them are still good, Good, hard, meaty, interesting questions. Even if they have a cop debater.
- [46:36]
A
All right, John, thanks for playing great. I want to ask you a couple of questions about teaching math. You're a middle school teacher and a professor. How much of a difference do you think your research background makes to your teaching?
- [46:51]
B
It makes a lot of difference. I feel like when I was just learning to be a teacher in my 20s, I was doing the best I could. It's a very challenging thing to learn how to do well. And my first years of teaching, I was struggling and trying to figure it out myself the way that all new teachers are. But now I actually know a lot about Math teaching through 20, 25 years of studying it. I like to joke that after 25 years of studying math teaching, I feel like I finally know enough to be a good middle school math teacher. And so I bring my research with me in class, class every day. Whether it's this focus on multiple strategies or this focus on discussion, students are surprising me every day and raising new questions for me as well. So it's actually mutually informative. It's not that my research leads me to think about how to teach, but the things, the problems of practice I experience in my teaching, they raise questions for me that I want to subsequently research. So it's really a back and forth forth that's so interesting to me.
- [47:54]
A
What do you think matters more for how you teach your mindset as a researcher, or particular findings or ideas that come from the study of math education?
- [48:06]
B
That's an interesting question. I'm not sure I've thought about that. I'm going to think about both of those alternatives one by one and see where I land. So on the mindset point of view, what drew me to becoming a professor of math education is that in my initial teaching years, there were students who I worked with who were struggling to learn math. And I was so fascinated in about what was going on in their heads, like what is clicking and why is it not clicking in their head. And that was what I was so curious about. In my teaching, and that drew me to go into research to be able to explore that essentially for time. So I still had that mindset. I asked lots of questions. I like to hear what students have to say, and I like to think about what they say and how they're making sense of the math. So that is a mindset that I bring with me, and that plays a huge role in my teaching. So that would be a vote for that side. On the other side, I do know a lot about the research out there in math teaching. And when I encounter situations in my class that are challenging or that don't go the way that I hoped, then I do draw upon that knowledge to move in a different direction, informed by the research. So I'm not sure which of those is most important. They actually both play a big role in my teaching.
- [49:26]
A
So, John, if I went to observe your classroom, what's something that you think that would stand out maybe from the average classroom? Or what's something that I would observe that I might not expect?
- [49:39]
B
I think you might notice that in a typical class, I talk less than you might have expected a teacher to talk, and that the talking that I do is a lot of questions rather than a lot of presentations or statements to students. And I do a lot of prompting students to listen to each other. So if I have asked you to explain how you're thinking, then I might turn to someone else in the class and I might say, susie, did you hear what Nat said? Can you tell me what you heard him say in your own words? And then I might go back to you, and I might say, nat, did you hear what Susie said? Did she hear you right? If not, can you tell us again? So those types of things I'm doing all the time in my classroom, and it's my way of getting students to talk and. And to listen, to think to each other. And I don't know, but I would guess that maybe that's different than what you would have expected to see in my class or in a math class.
- [50:44]
A
As a math teacher, I'm sure that there's a certain content you're expected to cover over the course of a year. I know when I taught, we often did not get through the full scope and sequence of the curriculum. Do you try to cover the material more quickly or more in depth, even if you don't get all the way through it? How do you approach that?
- [51:08]
B
There's a certain degree of depth, that of depth that I feel is really important, that I do prioritize, and it might Vary topic by topic. So maybe I'm going to triage a little bit and think what is the. Of these ideas that I have to cover, which of them do I feel like is most central? Either because it's just a foundational mathematical idea, or because I know it's going to become useful in a month or a year or in two years. And I prioritize depth on those ideas. There might be other topics where I realize this is a bit more of a one off or this is something that mastery isn't expected at the grade I'm teaching, but it will be covered again later. And maybe I sacrifice depth to go for speed in that type of topic. So I'm making those determinations on a day by day basis. And it draws upon deep knowledge that I have about the curriculum and about what's covered at what grade and what are the big ideas of math that are important and that's important. That does drive a lot of my daily decision making.
- [52:14]
A
Do you think that the standard progression of one subject per year, typically with about 50 minutes spent in a class per day, makes sense? Or do you think that we should try to cover more and spend more time per day on math?
- [52:36]
B
Well, I love math. I think math is really, really important in schools. But I know there's many other subjects that are important as well. So I'm not the one that's going to say we should take time away from another subject and increase math or even argue that math is the a number one subject that matters the most for kids in school. But that said, I do feel like kids should have math every day and that the pacing decisions about what to cover should be something that we are willing to revisit and reconsider with particular classes. Maybe there's an opportunity to go faster and to cover more as long as we're not sacrificing depth. But again, I think for many of our students there's this really important depth that they do need that's going to serve them well later. And I'm not sure we should be sacrificing that to race ahead. There are a lot of people that might feel that if a kid is able to go faster, we should let them go faster. We should let kids go as fast as they're capable of to progress through the curriculum as fast as possible. And I'm not a big fan of this. It tends to result in it feeling like a race, like a race through the curriculum or a race to calculus. Whoever gets there first wins. And there's this perception that the faster you get to calculus, the better your chances of getting into a particular kind of university are. And that's a real push among many. And I just think that that's not the way we want to be thinking about math and math outcomes for many different reasons. One is that race often sacrifices the depth and the rigor and the reasoning that we want in math. That's actually the whole point of why we're learning math. It's not just to get through the curriculum. And I don't want to sacrifice that. And furthermore, I'm actually not convinced that taking calculus earlier is directly related to increased success at college or increased chances to get into college. I think it's a lot more complicated than that. And so I would love us to be working a little more to get kids to deeply engage in the math and cover what we need to cover, but not be as concerned with moving too fast.
- [55:01]
A
You do math research. Are there any particular findings that you think more math teachers should be aware of?
- [55:10]
B
I would talk about my own research in that regard to be completely self interested here, but the work that I do about multiple strategies, I would love it if more teachers were aware of that. This idea that if you can get kids to produce multiple ways to solve problems and compare and contrast and discuss and debate and evaluate different strategies for solving problems, that just has good consequences on many different levels. And I would love for teachers to do more of that and to think about what that looks like. And similarly, the other one that I do a lot of is about discussion. And I think that it isn't easy always to have a discussion in math classes, but the benefits are huge for kids. Engagement for their math learning, for their confidence, for their motivation. And so I would love teachers to be more aware of that series of studies and also some specific recommendations about exactly how to do that. Because even if a teacher knows that research says this is a good thing to do, actually doing it is really hard. And oftentimes we're not giving teachers specific guidance. And I would like for teachers to be given more specific guidance about how to do these particular things.
- [56:21]
A
So you would say that even for a skilled math teacher teacher, most math education research is pretty accessible. Not that accessible. I mean, is the bridge built to the profession?
- [56:35]
B
It's not very well, no. There's not a lot of easy ways for math teachers to engage with the research. And there's also not a lot of incentive for math teachers who engage with the research. And I think we can do better on both of those fronts. Certainly there are Journals and conferences and organizations that are trying to get research into the hands of teachers. But for some researchers, they're not making it in a format that could be easily understandable by teachers. I think as researchers, it's partly our obligation to seek out outlets that we can use to get things in the hands of teachers, whether it's journals for teachers or conferences for teachers, or just giving talks for teachers. And it's also incumbent upon us to think about whether our research is valid in a real instructional context. Did we do this research in a lab in a highly controlled setting? And thus we should be very cautious about making claims of whether it actually works in the classroom, or did we do it in more authentic or ecologically valid settings in classrooms such that we can go to teachers and say, I actually tried this in a real classroom and. And I have evidence it works. And that's why I feel like I'm recommending this for you.
- [57:48]
A
John, let's say that every American math classroom had a very well informed, your ideal math teacher. You know, what you hope math teachers would be. How much better do you think American math education would be?
- [58:05]
B
A lot. I place a lot of stock in teachers and in good teaching. And in that idealized vision, if everyone were teaching excellently, then we would improve things a ton. It wouldn't be perfect. Of course. There's many other factors that play a role in student success in schools besides how good a math teacher they have. But I am someone who thinks that that math teacher is extremely important to students success in math.
- [58:34]
A
Obviously, there's a lot of assessments like NAEP or. Or PISA that we use to determine how good American students are at math, both nationally but also internationally. But in terms of what classroom instruction looks like, how good do you think American math education is?
- [58:56]
B
Average is what I would say that we. We certainly are working hard on this and we've made strides. But if you look at. At how other countries are doing on some of these assessments, then their students, and many of them are doing better than our students. And there may be many reasons for that, but one is that in some of these countries, I think they're doing a better job of teaching math than we are. Now, this is a really tough question to answer because we're a huge country. Other countries are huge countries. There's so much variation in what goes on. And in certain pockets of the country, we know that there's excellent math teaching that's being consistently given to students. In other pockets of the country, we feel less confident about that. And that's True for other countries as well. But it is the case that many countries are doing better than us. And it's a fascinating question about why. Why are some countries doing better than us on these assessments? And what role does the teaching play in that? I'll give an interesting example for me, which is that we have certain ideas about what we think is good to do in terms of teaching math or in schools generally. So we may say class sizes that are small, that's better than class sizes that are big, or the teacher not lecturing too much, that's better than a teacher lecturing. So we have these ideas, some of which are based on research, some of which are just our beliefs and intuitions. And when you look to other countries, sometimes you find that their students are doing better than us on assessments, despite having these features that we feel like are not the right thing to do. So they have huge classes, they have teachers lecturing, yet their students are doing so much better than our students. And it raises this question about why. What's going on in these classrooms that is resulting in these excellent performance if it's going against the grain of what we think is best here? And it's a very complicated and nuanced answer to that question. And in some cases we actually don't know know. But I find that very interesting. That brings me back to China, that this is a, a case of what happens in China. Class sizes are huge. Teachers are very often lecturing or teacher centered classrooms. They're doing very well. We don't think that's the way to go. But what can we learn from what they're doing that helps us understand what
- [61:12]
A
we can do better in America? If you had to pick sort of your highest leverage to improve American math education, would it be teacher quality? And if you had to choose one, would you lean on teacher quality instructional materials and curriculum or something else? What's the highest leverage?
- [61:40]
B
Well, I'm going to pick one of the options you mentioned and not pick something like eliminate poverty, which might actually be the highest leverage thing we could do. But I put my money on teacher quality and I know that in my field there's people that are lately thinking that high quality curriculum materials are the thing that they answer. And I recognize that that's a key part of the equation and that's something that we want to improve. Sometimes I worry that our, our current focus on curriculum is accepting a reality that some people might have arrived at, which is that it's really, really hard to change teachers and teaching. And we've been trying that and we haven't been successful. And so let's shift gears and think if we can get the gains we want by making the curriculum so excellent. I guess I'm still feeling that I want us to work on improving teaching and that being the lever, I feel like that a good teacher, an amazing teacher, can do wonders with just about any curriculum that they're given. And I'd like to figure out how we can get more teachers to be like that.
- [62:58]
A
Thanks for listening to the Report Card with Nat Malkus. And special thanks to today's guest, John Star. Also, I want to thank our producer, Ellie Lucas. He makes this podcast possible. That's all for this episode. I'm Nat Nopus,
- [63:30]
B
Sat.