Interview with Dr. Jon Star - Episode 68

Welcome back to Pedagogy Non Grata. I'm really excited to be interviewing Dr. John Starr today. John Starr is a professor at Harvard University and he is there. faculty director for philosophy of education I believe correct me if I'm wrong are you the faculty director for a faculty of education Yeah, so among the things I do at the Harvard Graduate School of Education, I am in charge of our doctoral program in education.

But in my teaching life, I teach courses on learning and teaching, particularly math teaching and learning. That's what my research is about. And my background is that I I used to be a middle school and high school math teacher, and actually currently am a middle school teacher in addition to my Harvard teaching as well. Wow, that's really intense. I didn't expect anyone teaching at Harvard.

would be also teaching at a middle school. That's very shocking. I love it, actually. It really informs all that I do and think about. I realized a few years ago that I hadn't been in a classroom for a while in terms of being a full-time teacher and that I really needed to do so to stay fresh. And so that's what I'm doing these days.

That's incredible. I don't think I've I've interviewed a lot of researchers and professors on this podcast, and I don't think I've ever interviewed one who's both a professor and a teacher at the same time. I don't know how you find the time to do both. That's really incredible. I have to admit I'm nerding out a little bit to to interview you today because I've recently been doing a lot of research on the subject of conceptual math And a lot of the papers I read and cited

were written by you. So it's a bit strange to read someone's work over and over again and cite their work over and over again, only to later on then interview them on the topic. I'm almost nervous that I took the wrong conclusions from your papers, so we'll get to find out about that today. So is there anything you want to say about yourself before we start the interview? No, I sort of gave a little information about my background. I think that it is important that... that um

That perspective that I mentioned in terms of what I do in terms of my research and teaching life, it does inform my opinions on everything we're going to talk about today. So these are not just theoretical pie-in-the-sky notions that I come to from doing abstract research. research in my lab with the door closed. I come from a background of being a teacher.

work with pre-service teachers regularly as well as in-service teachers and I have been teaching middle school math now for the past several years and so these are issues that I live daily as do teachers and so it's something that I really think a lot about.

So I really am trying to bridge the research literature and the practical wisdom of teachers to think about what we should be doing in classrooms when we're teaching math. We just described the whole goal of my podcast right there. So it should be a good interview.

The first thing, I'm going to kind of dive right into this topic just because it's something we've been talking a lot about on the podcast recently, and I expect the listeners probably are already aware of the terminology.

If the listener is not aware of the terminology, it might help to go back and listen to the previous podcast. But to begin, I want to know what you think of the proceduralist, conceptualist, and activationist or iterative view of math learning. Do you subscribe to one of these theories?

Or do you have a different approach? So I would say I subscribe to the iterative view. But before launching into that, I guess I wanted to make kind of a... a top-level comment which is that until I thought about This conversation we were going to have I had never run across the labels that you're using for these camps so I'm not sure what that signals. One takeaway is that these labels aren't really used in the research literature that I read.

related to this topic, other than iterative. It might be that like proceduralist might be what some people in the research literature called procedures first and conceptualist might be concepts first. And then i'm not sure what i would even say interaction is i'm not sure i know what that is but but basically um it is sort of a indicator of the nature of the dialogue

in the research literature that these are not terms that are used. But for me, I think I would fall most closely into their iterative view. But I think if I had to state what the iterative view is, to me, it's really that learning that procedures can inform concepts and learning of concepts can inform one of the procedures and that there's some back and forth about that and built into the interview is that it really depends that it's really hard to have a blanket statement that says

for example it is always better to teach procedures first or is always better to teach conceptual knowledge first that the iterative view in my mind is the most flexible corresponding to the reality of math teaching learning, which is that it really depends, that in different topics it might be that

Teaching the procedure first is optimal. In other topics, it might be teaching the concept first is better. There might be variation by a child. There might be variation by prior knowledge. There might be variation by age of the kids. So I'm happy to talk about all of those. sources of variation for this question but in my mind as much as we want to come up with some general principles about what

impacts to learning math and how we should teach procedures and concepts. The reality is that it's quite complicated and I'm not sure the general principles that span the preschool through college math learning spectrum really exist along this dimension.

Oh, that's really interesting. So just to clarify, the terms I just shared or used, they're coming from Bethany Riddle's research review on the topic. And to clarify, the inactivationist view as she referred to it was the idea that conceptual understandings of math and procedural understandings of math were built wholly separate from each other.

Okay, so I probably should know those terms since I work so closely with Bethany, but it's not something that I thought, think about in those labels. So, yeah, so I think, I mean, in my mind...

concepts and procedures are definitely related in mathematics. But what's what's tricky is that the procedures that we have that do certain things mathematically depending on the sort of historical emergence of those procedures they bear a different relationship to the concepts that underlies them and i refer to that relationship

roughly speaking as a distance. So you can really ask yourself a question, what's the distance, if you will, between the procedure and the concept for a particular math topic. And there's some where they're quite close. They're really intricately connected. And there's some where they seem quite distant from each other. And as examples, and as a rough generalization, I think in secondary school math,

a lot of the procedures that we teach are quite close to the concepts that are related to them. So when we think about procedures for solving equations, for example, then when you're solving a simple linear equation and you're adding to the adding the same thing to both sides let's say then that procedural action of adding the same thing to both sides is related to a principle a concept

Some people call additive property of equality. So there's a concept, a principle that's underlying that procedural action. So they're really quite close to each other. But on the other hand, if you go to the elementary school curriculum, you find many more instances. where the procedure and the concept are quite far apart. So whether you take various procedures for working with fractions or for doing division.

The procedures in elementary school often emerged historically as these mercantile shortcuts. The merchants at the market needed to do a quick calculation, and this was a trick that they could do to get it. um having it be connected to concepts was the furthest thing from their mind this wasn't a schooling kind of thing it was just for practical commerce reasons and it was only later

that we decided that there might be an interest in teaching concepts related to that procedure. But the procedure developed quite independently from thinking about the concepts, so not surprisingly they're quite far apart. And as a result in elementary school, that's led to a lot of new alternative procedures being introduced in elementary school that are explicitly designed to be closer to the concepts. So another way to think about that distance for me is that some procedures...

I would say, are conceptually opaque, and some procedures are conceptually transparent. And in elementary school, A lot of procedures that we commonly teach are conceptually opaque. And if your interest is in connecting procedures and concepts, then maybe it makes more sense to teach things that are more conceptually transparent because they're kind of closer to each other.

That's really fascinating. Can you give us some quick examples of types of math where that link is more transparent? Yeah, so I've been thinking a lot about fraction division.

lately and this is in part because i'm teaching fifth grade and we're actually working on fraction division right now with my students but in generally this is this is something that comes up a lot in this debate so at least in many countries a common procedure for teaching fraction division and teachers have all sorts of shortcut names for this but it has to do with sort of flipping if you will the fraction that you're dividing by and then multiplying it by the fraction.

the other fraction essentially so flip and multiplier there's all sorts of tricky names for this the teachers introduced to help kids remember it and what's interesting to me is that that procedure that algorithm is something that works reliably works quite well no question about that but for me it's not conceptually related much to fractions

um really what that procedure is related to is the relationship between multiplication and division that it's sort of thinking of division the definition of division as multiplying by the reciprocal and so that procedure Happens to be used a lot of fractions, but it's really not at its core related to fractions It's really to me related to the relationship to multiplication and division and so

if i were teaching the relationship between multiplication and division then that procedure is actually quite conceptually transparent because it's related to the thing i'm teaching but if i'm teaching fraction division then it's a little more conceptually opaque What does that definition of division procedure have to do at its core with fraction division?

as an alternative to that let's say that when you're dividing fractions you do something a little different where you try to write equivalent fractions for each of the fractions you're dividing so they have the same denominator and if you do so then the denominators are the same and you can essentially just worry about dividing the numerators which is quite easy to do often and so that procedure which is less commonly taught and doesn't have any

um easy to remember tricks about how to how to use it um for me it was much more conceptually transparent in terms of what fractions are and what we're doing when we divide fractions is thinking about partitive and conceptions of division and what it looks like when you divide a number into equal sized groups.

No, I was just going to say that's interesting. I've never actually seen that way of doing dividing fractions. I'm not a math teacher by trade. However, I do teach math. So I've always done it the first way you've described.

I've never seen actually anyone do it, the second one you described. But in some ways, I think that's also more in line with the other fraction formulas we see. It's more similar to, say, how we add and subtract fractions. So in some ways, it might be easier, I think, for the students to get that. I agree with that. It fits within the space of operations with fractions generally. It doesn't feel like this extra outside thing that is quite removed.

From the underlying big ideas the fractions that's that's where I'm talking about this idea of distance suggestions just far away And I'm not sure it's necessary to set algorithm in order to do fraction computation problems, especially if my goal is to teach students a rather integrated way to think about fraction operations and the underlying concepts related to them.

So I think there's many examples of this. If we go to secondary school, I mentioned about solving equations. So there's... procedural steps that you can use or algorithms even that you can use to solve many forms of equations especially linear equations or quadratic equations and there's some ways that those procedures are well integrated into conceptual space of equation solving. I guess it's easier for me to imagine that distance being closely connected in that space.

When I'm solving equations using any one of these procedures or algorithms, I'm drawing upon conceptions of variable and conceptions of equality. various principles of equality, as I mentioned before. So there's lots of different things I'm drawing upon that are really inescapable, if you will, if you're thinking about the procedure and what makes the procedure work.

So I find that in some ways this debate feels most intense at the elementary school because you have this possibility of... huge distance between the procedure and the concept that raises all these questions about which we should do when they seem unrelated to each other which do you focus on i'm not sure we have these debates as much in the secondary community

That's really fascinating. I have to admit, as I was listening to you talk about that, I almost wish you had a book written on the subject now on the different types of formulas you would recommend for different types of math. Perhaps you have a book where you referenced on this, or is this all just ideas that you've come to that's not written down? Well, maybe I should write them. I don't have a book that I've written or that I can point to about this. I think...

In terms of alternative algorithms, there's many out there, and people have different ones that they prefer. But I'm not sure that I could definitively say that a particular algorithm is... better um but my point again is to raise this this condition this factor this uh parameter if you will that's about the distance between the conceptual and the procedural for a given topic and

ask math educators to consider whether it's possible to teach an algorithm that reduces that distance, if I have a choice. that it's really about minimizing that difference and whatever your your thoughts are about procedures and concepts then i would think that um people value both of them

They're both important. We're sort of disagreeing about the relationship between them and the instructional order of them. But if you think they're both valuable, then it seems like there would be some benefit to having them be connected at some level, and that connection is enabled if they're... Distance if you will between them is sure you know you're it's funny You're almost answering the the next questions that are popping into my head before I have a chance to ask them I think

You know, within the academic literature on this, I see a lot of debate on the order of teaching them. However, I see within amongst teachers, I do see a lot of debate, especially where I am, on this idea of what is more important. conceptual versus procedural and i i wonder if it's more of a value test than it is uh actually a theoretical test do you think one holds more value to students

Holds more value to students. I think students left to their own devices might value procedures more Students because procedures are the thing that lets you get things done and if you perceive that that's what math class is about is getting things done and solving problems and getting the right answer then you really can't do that without procedures procedures are your sort of

mechanism for doing problems essentially. I think in my teaching I have to work hard to make my students realize that I'm as interested in how they approach problems as whether they get the right answer or not. And if they believe that I'm very interested in how they approach problems, that strategy matters. then that's where I'm able to bring in the importance of concertos, that it's always about doing the problems for me.

Always about their ability to successfully complete problems, but I place great importance on their ability to make good choices about strategies for solving problems to solve problems in clever and efficient and innovative ways but also to be able to explain to me what they did and why it works

And so there really isn't a separation in my vision of this that I'm always interested in both and holding them accountable for both. But I certainly can understand that students will come in thinking that procedures are what allow you to actually get the answer. And so that must be the more important thing. Yeah. I have to admit that was a bias I held before starting to do my research on this topic. I tended to be more focused on procedure just because I saw it as the practical.

application of it in the sense that, you know, as an adult who isn't someone who studied math as a university, the math that I had needed to use the most in everyday life was more procedural in my opinion. I think once I started to read some of the literature on it, specifically some of the content by yours, I realized that I was too biased in this specific way.

Now, I will say that conceptual math has increased in popularity a lot over the last 10 years. Do you think there's a reason for this? Is it a reaction? Well, I speak most knowledgeably about...

the sort of development of these ideas in the U.S. context. So I can't claim to know a lot about other parts of the world, how this movement has been going. But I think that... in the u.s what has happened is that there has been there's a perception that sometime in the past our teaching focused too much on procedures

So there's this perception that sometime a long time ago or not that long ago even that we were really Drilling kids with procedures and not doing any work with concepts and as a reaction to that there has been a movement to focus more on concepts. Now, there's several interesting things about the observation I just made, again, which is that there's this perception that in the past...

The focus was much more on procedures and as a way to move the other way, to swing the pendulum the other way, to counteract that, we need to focus more on concepts. So one interesting thing about that is that that same argument...

has been repeatedly made throughout the past 200 years in sort of U.S. and North American perhaps math circles. So you go back a couple hundred years, and there's someone who's going to claim that... the people just before them were focusing too much on procedures and we need to focus more on concepts maybe they use different language but that's the spirit and then you fast forward another 20 or 40 years and someone is saying exactly the same thing

there's always this perception that people immediately before you were doing things too focused on procedures and that you need to be doing something different or better. A related and second observation about that point is that when you actually go back and look at what the people before did, it's really hard to find anyone who is explicitly... sort of proudly proclaiming their focus exclusively on procedures so in some ways in US lingo we would call this a straw man argument that

that there's this perception that someone in the past was doing something that was skewed or bad, and you're trying to correct it. But it's sort of a way to make your argument if you can sort of position them.

as an extreme in the past in the past they were doing this in such an extreme manner that we're trying to correct it but in reality no one was really doing it in that extreme way that's a straw man so i think what this illustrates is that We're always going to be wrestling in math education with this tension between the things we want students to be able to do or the skills and the things we want them to know.

which would be like the ideas of the principles that that's inescapable in the discipline of mathematics uh it's never going away and we're always trying to refine our our conceptions of those relationships and our delivery mechanisms if you will for how to impart that knowledge to students it's never going away and so i would say that in the u.s um this movement toward greater

conceptual math probably the latest incarnation of it began in the 1980s so it's not a new movement here that's been gradually moving in this direction for a while particularly at the elementary school level If anything, I would say the last decade has represented a slightly more moderate view in the U.S. about this issue where you do see

much more of a balanced view I would say in current elementary school curriculum than you did about 10 or 15 years ago so it may be that in some other parts of the world They're at a different place in the way this pendulum swings where they may be moving toward a more focused on conceptual. But in the U.S. at least, I'd say we went from a period in the 90s, let's say, or the 2000s where it was.

Quite focused on conceptual and the procedural aspects were were minimized such that teachers might sometimes feel bad or guilty or Worried about teaching procedures because they perceive that they were not supposed to But I think nowadays that's a little less common and teachers recognize that we need both and that the curriculum are trying to do best

In secondary school, I'm not sure I would say there's any such pattern. In much of North America, I think that what we teach in secondary school is pretty much the same as we taught 50 years ago in secondary school. I'm not sure it's really changed a lot. That's really interesting. I think the pattern has been similar in Canada, although I would argue that we are probably more focused on conceptual today than we were 10 years ago.

I kind of wonder if some of this actually stems from people's sort of perception of their childhood. You know, I think everyone can remember being in lower elementary school and doing worksheets, practicing specific skills. I think a lot of people have this sort of like...

sort of the opposite of rose-tinted glasses when they look back at that. They look back at it as this oppressive thing and then they think, oh, we need to take away this oppressive aspect of math, of skill and drill, for example. I wonder if that's where...

This idea that we're always too focused on procedure in the past comes from. It might be that teaching is an inherently conservative profession and that... The people that become teachers were students in the past, and there's a strong drive for them to replicate their experience receiving instruction when they were students when they get on the other side.

as teachers and teacher preparation programs are trying to get our new teachers to critically think about the instruction they received and really Wonder whether that is optimal for their students, but there's a strong draw toward doing what you experienced and so that that applies for parents as well parents may remember some of the procedures that they were taught and they it's not unusual for parents to be a little

unclear why their children are not being taught those same procedures in the same way the reality is that we do teach different procedures than we did before interestingly in my sort of

experiences looking at math teaching around the world. I think this sheds some interesting light on these debates. One thing I note is that these algorithms that we are often fighting about in North America Aren't the same algorithms that that people use in other parts of the world and in in we have these arguments about which which algorithms to use because which ones are more conceptually transparent using the language i talked about earlier in other countries they're using

Other algorithms that are equally conceptually opaque they're just another algorithm that emerged through some other historical commerce type process and so i'll be in a country and i often like to ask people to show me how they do long division in those countries and you would just be amazed how many different ways that's done

none of which are really an improvement, if you will, over our traditional long division algorithm. They're all just a little different. So there's nothing magical about these algorithms that we teach. And furthermore, each teacher... Often has a way of teaching these algorithms that the teacher has liked From some reason whether it's a certain name for it or a certain acronym or mnemonic or a certain like diagram like There's things that teachers call the butterfly method or whatever.

Each teacher has their own kind of way to do this that may exist locally, that maybe teachers in a particular region refer to it that way, but if you go outside that region, no one has ever heard of it. So there's nothing magical about these algorithms that we teach as being the best. in terms of doing the work. It's something that we should think critically about. Which algorithm should we teach and why?

as i mentioned earlier my money is on teaching the algorithms that are more conceptually transparent and that are in that sense closer to the underlying concepts that you really are most that's really interesting you know just just the word long division makes me think back to how hard that was to learn when i was a kid to be honest i i think actually because i that was the type of procedural math i struggled struggled with most as a kid i tend to really focus on it as a teacher

and then my students tend to actually find it they'll often tell me oh this is the easy thing and i'm like no it's so hard that's why you have to focus of course that should be i shouldn't be doing that i'm going to create a self-fulfilling prophecy but um

So one topic that has come up quite a bit here is this idea of teaching conceptual math implicitly or explicitly. Right now it's become really popular to teach it. Implicitly do you think it should be taught implicitly or explicitly? I'm not sure i have a really great vision of what that looks like in practice to teach it implicitly versus explicitly um so i'll give you my answer to your question but but with the caveat that i'm

I'm a little unclear on exactly what that looks like. But my answer is that I think everything should be taught explicitly. not just concepts but procedures everything is that I think our job as teachers is to make the things we want students to learn explicit and clear and that doesn't mean we're gonna we're gonna sort of dumb them down if you will

But we're going to make clear what we want students to learn and how we want them to be thinking about certain phenomena. Some of that explicitness... is going to draw upon students own conceptions or their own meanings that they bring it's not like i'm if i explicitly i'm not saying that the teacher is going to recite words and the students are going to memorize those words

That's not it at all. But in the same way that teachers write their lesson objectives on the board before they begin a lesson often, or they are using a textbook, which is very clearly delineated. kind of being explicit in what we're after for students to learn. Yeah, actually, I 100% agree with that. I think there's been some push for the idea that students should have activities and games.

where they're supposed to naturally come to understand the concepts without the teacher actually stating what the concepts are to the students. So, yeah. So here we would say... that this what you're calling explicit or implicit is more about uh inquiry or discovery i guess would be words that we would use based on what you just said that the idea being that some people think that um

To put this in the vernacular, some people think that kids need to have the opportunity to muck around and explore a bit. and that afterwards maybe you can talk about what you hope they got out of it.

but that mucking around first some people feel is essential whether other people feel like that's kind of a rather inefficient way to teach that if you have things you want them to know you should just tell it to them and not have them play this guessing game if you will uh and um try to figure out what they're supposed to get uh so that is that sort of at the core of what

you're calling implicit and explicit. Yeah, 100%. I've seen a lot of different names for these concepts. I think the most simple way of defining it is either direct instruction versus inquiry-based instruction or implicit instruction versus explicit. Okay. So, yeah, in that sense, then, you know, I would revise my answer that what I said before is I think everything should be explicit, but I don't think that that's...

endorsing the direct instruction all the time mode at all. I think that there are, for me, my answer to the question is always that it depends on my instructional goals in a given lesson. that I do both. There are days where I do direct instruction, there are days when I do more inquiry-oriented or discovery-based. It really depends on where I am in the unit, what are the topic I'm working on, what I think would be best in meeting my instruction.

objectives for the particular group of students that I'm working with. So for example, I can think of a situation where I'm working with students who are Have shown some pretty interesting insights about a phenomena that we studied last week And I think they actually could do a good job of mucking around in this new space try to come up with some interesting perspectives and that that would serve my goals both in terms of content but also might serve my goals in terms of motivation and um and

particular students needs in terms of exploration so I could see myself doing that for that for a given group of students but at the same time I can imagine other situations where the thing that I'm after is kind of complicated there's really no reasonable expectation they're gonna get it by themselves and that I will just teach it to them as a direct instructional kind of technique some people in the

Inquiry-oriented camp might call that just-in-time instruction or benchmark instruction, but it's providing students what they need at the time they're going to need it in a direct way and then allowing them to use it.

I totally understand our need to have a one-size-fits-all to these questions, whether it's how to teach procedures and concepts or this direct instruction versus inquiry-oriented instruction debate. I just don't think that that is how it plays out in teaching math for the most part. We need to have the ability as teachers to teach across the spectrum.

these debates and to make decisions as to which one is the best for given circumstances that we're in in our classrooms with the particular students we're teaching and the content we're teaching and I think that's what good teachers do.

They vary their instruction based on the hand that they're dealt and they're fluent if you will in all of these different things That's interesting. I feel like we could almost summarize what you just said by the word nuance Would it be fair to say that in your mind you're trying to balance the holistic needs of a child with your academic goals for the students?

I think that would be fair. I think the job of a teacher is really, really hard. And that's at the core of what we're asking teachers to do, is really know and understand. the students in their classroom and to know and understand the context that those students live in both in and out of school and also to bring to that their their deep understanding of the content.

instructional methods for teaching that content and figure out an optimal path forward that works for all kids and I think it's it's um maybe where this debate came from at some level is that when we try to one size fits all, let's say we try to kind of pure direct instruction or a pure focus on procedures, then Some kids were doing okay with that, but other kids were not. And it's really about us thinking about those kids that it didn't work for and what might work better.

And then is there a way that I could teach that broadens my reach a little bit, that doesn't just reach the top third or the top half of kids? doesn't just reach the bottom third of the bottom half of the kids, but really teaches all the kids and helps as many of them as possible meet my instructional objectives. That's a hard, hard thing to do, but that's exactly what we're asking teachers to do. Yeah.

I really like the way you summarize that. So the first, most of the questions I've asked up until this point have really been addressing sort of the academic controversies and debates that exist within this topic. I'd like to really just sort of transition the focus into some practical application questions for teachers who are listening to this podcast. So one of the things I'm curious about is strategies we can use that help to teach conceptual and procedural math at the same time.

So if I had to pick one strategy that I think is useful for teaching conceptual and procedural math, and perhaps this is a strategy that I would say... accomplishes other goals as well in the classroom. I think discussion and getting kids talking is the strategy that I would focus on. So in my classrooms,

I am always asking students a lot of questions, and I'm always expecting students to try to answer those questions. And that means that I... give students space to respond to my questions, that my questions are intentionally hard. uh they're not just yes or no or what's two plus three kind of questions but they're really difficult questions or i'll say why or how do you know or can you explain that i'll ask other students to explain what they heard

a colleague. So if one student gives an explanation to a question that I posed, then I might ask another student whether that... second student agrees or disagrees and why or if they could summarize what the first student said so i'm a huge fan of discussion For many reasons, but it certainly serves this goal of integrating conceptual procedural knowledge that if a student is solving a problem in a particular way and using a procedure and they know that at some point

I'm going to say to them, why did you do that? Why does that work? Or even just explain to me what you did and how it's different from Johnny across the room, what he did. that's going to force a certain depth of thinking in their minds that is naturally going to deepen their knowledge of both concepts and procedures and also potentially serve to integrate them. So for me, that is one core strategy is engaging kids in discussion and in teacher questioning. That sort of reminds me of...

the strategy called number talks. I believe there's two books written on the concept, although that might be more of just the trademark version of what you just said. So there's a subtle difference I want to make about... about the number talks world and I think the number talks literature is really excellent and it's also evolving so it might be that what I'm going to say is

Not completely up to date in the way that people who use NeverTalks are thinking about NeverTalks. But it's a point of emphasis that I like to make in thinking about this is that... When I'm asking kids to share strategies and discuss strategies in the number talks kind of way, I think what's important in my classrooms is that that generation of strategies is just the first step.

in the discussion that i actually want to have it's not the goal the last step so in essence what i'm saying is that i ask students to share their strategies and their thinking because that then enables me to ask certain kinds of other questions that are really rich so it's not so much that my goal is just to get a bunch of strategies

out in the air and hear from lots of students and get a lot of students talking. That's just the first step, the raw material, if you will, that then leads me to ask a subsequent question. So, for example, when I have taught algebra, then in addition to getting kids to share their strategies, then I might want to ask them questions such as like, which of these do you think is better and why?

What does better mean? Would you choose if the problem were this other problem? Or can you make up a problem where strategy one is better than strategy two? So there's a number of questions that I can ask that are really intended to foster kind of deep thinking about both the strategy, the procedure, and the concept. And it doesn't end just by asking students to share what's on their mind.

So, again, I think that there are some implementers of number talks where that's exactly what we do. And there's some implementers of number talks where it really just is about kids participating and sharing their strategies and not about...

Happens next hmm That's it that's an interesting distinction Okay, I'm going to just move on from that um so how would you structure a math lesson then in your what would you ideally use as your typical math lesson structure i know a lot of people in the past suggested the three-part math lesson would you would you separate yourself from this model or would you propose a different model so maybe not surprisingly given what

what i've said before i'm i value a certain kind of flexibility in thinking about the typical lesson structure that recognizes that i have daily objectives and goals and that the structure of my class in a given day should be designed to support those goals. So I feel like that any aspect of a lesson that People might believe is the sacrosanct is this is what you should always do every lesson I tend to go in the other extreme and Almost reluctant to identify anything

that I would say you always need to do in every lesson. Because for me, it always depends on what your goals are for the lesson and what students are bringing to the lesson. So for example, In many classrooms in the US, many teachers begin with an activity...

It's gone by different names. Sometimes it's called a warm-up or a do now. I'm not sure in the Canadian context if there's a name for this kind of thing. But it's, you know, students walk in and you want to get them focused and get them started and do a do now.

And then after that many teachers will spend some time going over their homework and then they might have a time where they're introducing new material and that's followed by some opportunity for independent practice. So that's a very common model. There might be some advantages to that model, and it's one that I use sometimes for sure. But I'm very happy to deviate from that model if I feel like doing so is in the interest of my students on a particular day.

I encourage flexibility. Throughout this flexibility there is a need to always assess what students know and so that sort of formative assessment always needs to happen.

my point is just that there's many ways that it can be done it doesn't have to be done in a given way and similarly I am a big fan of homework I do value giving homework and there is a need for students to have some feedback from their homework on a given day and so I think about how to incorporate that but there's many ways you could do it it doesn't have to look the same every time students in my classes come to know a small set of

structures. It's not like I have 100 different structures, maybe I have five or six different structures, and I use those structures throughout the year depending on what my goals are for the day. So it's not kind of... It's not a sort of array of unfamiliarity every day. There is some familiarity. It's just not unstructured across a given year, maybe five or six. Okay.

Wow. So again, what I hear you saying is there is no standard model. I feel like a lot of what you've said today could be summarized with the idea that everything should be looked at in the perspective of what you're trying to teach that day. and to not try to form one standardized model for your math instruction. I agree with that, yes. I recognize that that vision of teaching...

poses some real challenges for teaching people how to teach. But novice learners, novice teachers do need some sort of structure. and some familiarity to develop comfort in teaching in the classroom and i agree with that completely so what i'm proposing and what i'm describing is more of the mature model of expert teaching it's not the way i imagine people starting out so for my pre-service teachers certainly i have a model that we work on initially um and

I might pick a model that I feel like has some affordances that can be generally used. And I don't think there's one magic model that does that. There are many that are used. But novice teachers do need something they can hang on to. in the learning process, and I'm very supportive of that. But I think as they become more comfortable in the classroom, I want them to expand from that model to other models and develop a repertoire of teaching strategies that they use flexibly.

across different days and to not get into a rut where they're doing the same thing every day. Because I don't think that's good for anybody. Would you almost then recommend that as teachers progress in their teaching, they experiment with different models of instruction? to see how they can be used and implemented in different ways? I would love that. I would love departments and schools to give teachers opportunities to watch other teachers teach in support of that.

It's striking to me in the US that many teachers... who have been teaching a long time, let's say I'm in a conversation with a teacher who's been teaching for 20 or 30 years, and I might ask that teacher, when was the last time you saw someone other than yourself teach math?

over the past 20 years and i believe that most teachers in the u.s would say never that not since on my student teaching days have i seen anyone else teach mac that we don't have um a norm in many of our schools or an expectation that teachers are going to have an opportunity to see other people teach math and i'm not sure how we expect our teachers to learn and grow without that opportunity in other countries

It's a built-in part of the teacher profession that you watch other teachers teach. And I think that's something that in many places in the U.S. we lack. That's really interesting. I have seen that idea looked at in research in the past, but I've never tried it myself. I have to admit, I haven't watched another teacher teach in eight years of teaching myself.

Maybe that'll have to be something I'll try in the near future. I'll have to ask one of my colleagues. Hey, do you mind if I watch one of your lessons? i think it would be stressful it really leads to some interesting conversations if if that happens so um let's imagine that um you know uh i've been in many conversations with high school math teachers about this where um

Let's say that you watched another teacher teach a lesson that you had taught several times before in the past, and you were then debriefing afterwards. Then it's very likely that you would say something like, noticed that you did this thing a certain way and you described you know when you teach factoring out the greatest common factor this is what you do but when I teach it I do it this way and so

That could be the end of the conversation which would be sad But that could be the end where you sort of each agree to go your separate ways But if you could then go a step further and say why do you do it that way? and you could sort of interrogate each other as to what your rationales are for each way and which way you think is better, then I think that could lead to some really interesting insights. A lot of what teachers do...

especially as they get experienced and comfortable in the classroom it just becomes what's routine it's the way you've always done it i'm not sure i can explain why i do it this way but that's just the way i've always done it but if someone really pushes you on it and suggests an alternative then maybe you'll think well I'll try it that way and see if that has an impact.

um we all we all can become little experimenters or little scientists in our classrooms by seeing other ways that teachers teach and then experimenting trying things out ourselves making hypotheses about where we think will work better or not and then evaluating our pedagogical strategies based on how things grow. Yeah, I think that's a really interesting concept. I think John Hattie basically calls this idea a collective teacher efficacy.

My podcast partner and I, we talk about this idea as reflective practice as we bounce all of our ideas off of each other. But I will say that... It can be a daunting thing for a lot of people because I think people are always worried that they're going to embarrass themselves as not doing best practice and they don't want to be under the microscope. Similarly, they don't feel safe in having that sort of...

interrogation with each other. You have to find some way to either find people you really trust or build trust within a school for that to work. Agreed. Agreed, definitely. Okay. I'm going to just ask you...

What are some general evidence-based teaching strategies that you recommend teachers use for math? Out of pure self-interest, I'm going to talk about one that I personally have done a lot of research on. That's okay. Just because I'm especially knowledgeable about it, but also I'm a big fan of it, to be honest. So we do a lot of work on comparing and contrasting multiple strategies for teaching math.

I think it is something that really works well for teachers. So the idea is that we are interested in teaching students multiple ways to solve problems.

that that's encouraging the kind of flexibility that we want students to have the integration of conceptual procedural knowledge that they have multiple strategies is a good thing the ways in which we teach multiple strategies effectively breaks down to how we present those to students how we offer students the opportunity to engage with multiple strategies and we have a particular way that we do this that's first about presenting the strategies side by side

on paper on the whiteboard or whatever and then engaging students in a particular way of trying to understand the two strategies and making sense of them and so this sort of comparison. We compare in learning all the time. comparison is one evidence-based strategy that I think works quite well at all levels. And we see this incorporated more and more in curricula and in teaching methods, but it's something that

I would love to see even more teachers do focus on multiple strategies and allow students the opportunity to compare and contrast them. That's awesome. It's actually something we've looked at on this podcast before when we've looked at meta studies on.

Math interventions, it's one that's come up as a high-yield strategy a couple times. I've been trying to do it more myself in my classroom. I have to admit, sometimes it feels like a challenging or daunting thing to implement in the class, just because...

Sometimes it feels hard enough to get students familiar with one way of doing a math strategy, let alone trying to teach multiple math strategies in the same class. I don't think it has to be daunting, but it's probably something I just need more practice with personally.

Yeah, well, in our work, we've tried to come up with some easier to implement strategies for doing it, and that it can be challenging. For example... um some teachers think that uh if if comparing contrast in two strategies is a good thing then why not contrast three or four or five or six uh and we found that that's that's a little too much it's a little overload for students but

a reason that i think it's important to do and this is a comment particularly geared toward the secondary teachers is that a big part of math mathematics as a discipline is is the aesthetics of mathematics that we're all faced with those difficult questions that students ask about why are we learning this or why do you teach math or when do i have to know this and

Those are hard questions to answer, and it's another conversation to talk about how I would answer those questions. But a part of that... is really pointing to what's special about mathematics as a discipline and one special part of mathematics as a discipline is its focus on aesthetics what that means is that in mathematics we often are having debates about which

which method of solving a problem is optimal or better. In higher mathematics, People use the word elegant a lot as a way to capture that So mathematicians will have debates as to which proof is more elegant for a given topic and if you stop that debate and pin someone down and say what does it mean to be elegant? Can you define elegant? They really can't. It's a hard thing to define. It really is kind of a...

Again, an aesthetic, it's really nuanced, but yet it's important. And this debate that they're having about which is more elegant is really core. So we're not doing complicated proofs most of the time in high school and middle school and elementary school math. And so maybe it might feel like we're not able to have this conversation around elegance, but we can get pretty close to it.

by having a conversation around choices of strategies. So if student A does it one way and student B does it a different way, then I can raise that question. Which do you think is better? Which is more efficient? Which do you like more? Why? And that's going to uncover... some real important thinking about mathematics that we want students to come out of math class knowing.

It dovetails with my emphasis on discussion that I'm both going to be comparing and contrasting multiple strategies, but I'm also going to be engaging students in a discussion around those strategies and getting them to think about comparing and evaluating and generating new strategies. And for me, that's what it's really all about. That's really interesting.

the error of my ways already with using this strategy and i have to admit when i've used the strategy in my classroom i haven't found a lot of success with it but i have been typically looking at Yeah.

An example that I use, again, here to the secondary audience for this, that sometimes resonates with folks is, when you think about teaching linear systems, solving linear systems in the first-year algebra course, then... We teach multiple methods for solving linear systems in the US. We typically teach three methods for solving linear systems, which we typically call graphing, elimination, and substitution.

They might go into different names and other contexts. But let's take the elimination and substitution examples. I find that even teachers who really haven't thought about teaching multiple strategies. do teach these multiple strategies so this is a domain where they say yeah yeah i buy that i do need to teach multiple strategies i do want kids to know both elimination and substitution and furthermore um even teachers haven't thought too much about this

they will agree that we'd sort of like students to be able to say something about what are the features of a system that might make it most amenable to being solved with elimination. And what are the features of a system? that might make it more amenable to solve with substitution. So for example, in substitution, teachers and students will often note that if um one of the equations is given in the form y equals then maybe that suggests substitution might be handy or if one of the equations has

um one of the variables that has a coefficient of one maybe that makes it easier to do substitution although not always and so this is knowledge that we want students to know it aids their problem solving it sort of shows that they kind of get the essence of the method they can say well this is this one looks like it'd be easier to solve with elimination or even if they say it's about the same easiness between the two so that's kind of like a

meta-level conversation around the strategies that we do want students to know. It's not just about, can you do elimination? Can you do substitution? We want them to know a little more. We could debate about what you call that knowledge. for me i would still call that procedural knowledge it's about knowledge of the procedures what the problem looks like such that a given procedure is a good choice

But other people would say, well, no, that's kind of a deeper level knowledge. It's more about something that's conceptual, I guess. But either way, it's not just about executing the strategy. It's about thinking about what kind of problem is most amenable to using that strategy. That's a really interesting perspective and like always a very nuanced perspective on the topic.

i really want to thank you for for coming on to the podcast today you've been um incredibly articulate insightful and charismatic as you talk through these topics You've surprised me a little in just how little you've left with me to say. Normally I'm a very talkative person, but after multiple of your answers, I felt like, wow, I have really nothing left to contribute to that thought process, which is not normally true for me.

So I'm really happy to have you on the podcast today. If our listeners would like to find out more about you, where could they find you? So you can find me on the Harvard Graduate School Education website.

you can find links to publications um and also if you're interested in these these ideas that i was talking about with respect to discussion and comparison and explanation of multiple strategies then i have a project that um has explored that in some depth including some curriculum materials that we put together and usually you can find that either through my webpage or you can just google contrasting cases harvard and that usually ends up

at the right place. It's a collaboration that I have with Vanderbilt and we have materials posted on both Harvard University grad school education web materials as well as Vanderbilt University websites as well. I'm going to have to check that out myself.