Ep 52. The case for math practice and the power of Math Corps with Alex Kontorovich

This transcript was created with speech-to-text software.  It was reviewed before posting but may contain errors. Credit to Jazmin Boisclair.  

You can listen to the episode here: Chalk & Talk Podcast.

Ep 52. The case for math practice and the power of Math Corps with Alex Kontorovich

Timestamps:

[00:00:00] Introduction  

[00:02:23] The value of math practice 

[00:07:48] Engaging in math outreach 

[00:11:13] Rutgers Math Corps: Structure and curriculum 

[00:21:10] When research confirms what experience already knew 

[00:23:34] Why times tables fluency matters 

[00:29:36] Science of learning techniques in Math Corps 

[00:35:11] Using hand gestures and building classroom culture 

[00:39:30] Gamification 

[00:41:52] Why well-meaning schools still produce poor results 

[00:49:05] Discussion on A Mathematician's Lament 

[01:15:32] Conclusion and final thoughts 

[00:00:00] Anna Stokke: Welcome to Chalk & Talk, a podcast about education and math. I'm Anna Stokke, a math professor, and your host.  

Welcome back to another great episode. For this episode, I recorded both audio and video. So, if you're listening on a podcast platform, but you would like to check out the video, I put a link to my YouTube channel in the show notes. 

My guest in this episode is Dr. Alex Kontorovich, a math professor at Rutgers University with a strong passion for math outreach. We explored why practice in math is often undervalued compared to disciplines like music or sports. And Alex told me all about Math Corps, which is a summer math program for kids. 

Alex recently founded a chapter in New Brunswick, New Jersey. The program combines evidence-based learning strategies with high expectations, and the results have been remarkable. We also had an engaging conversation about A Mathematician's Lament by Paul Lockhart, sparked by a comment from a listener, whether you're a teacher, a parent, a mathematician, or simply someone interested in education, this episode has something for you. 

I hope you enjoy it. Now without further ado, let's get started. 

I am here today with Dr. Alex Kontorovich, and Alex is a distinguished professor of mathematics at Rutgers University. He has a BA from Princeton and a PhD from Columbia. He taught at Brown, Stony Brook and Yale before joining Rutgers. His research area is in pure math, specifically number theory, geometry, and dynamics, as well as formalized mathematics and applications to AI. 

He's very active in math outreach. He served as the Distinguished Visiting Professor for the public dissemination of math at the National Museum of Mathematics in New York City, and he also founded the Rutgers branch of the renowned Math Corps summer day camp. Welcome, Alex. 

[00:02:21] Alex Kontorovich: Thank you so much. It's such a pleasure to be here. 

[00:02:23] Anna Stokke: Yeah, I'm really excited to talk to you today, particularly about your outreach, and your thoughts on mathematics. So, I thought we'd start with something you posted on X, which kind of got an interesting response.  

So, you posted the following on X. So, you said “Piano teacher: ‘Your son is playing too much at home, please have him practice less; what's the rush?’ Soccer coach: ‘Your son is dribbling too much at home, please have him practice less; what's the rush?’ Math teacher: ‘Your son is learning too much math at home, please slow down; what's the rush?’ Two of these things would never happen. The third is commonly heard. Why does our culture allow it? (And what can we, politely, do to counter it? Besides ignoring the "advice"...).”  

So, Alex, why did you post that? Why do you think a math teacher would say something like that?  

[00:03:17] Alex Kontorovich: Well, because I heard it, and I was sort of flabbergasted that I, it's a very simple, I have a very simple modus operandi where, you, math is a skill. It's a skill that people learn. It's a skill that children are exposed to, and there are lots of other skills that children are exposed to and try to learn in all kinds of ways, like sports, like music, like art. 

And for some reason, when it comes to math, everything that we would do in any other field of human endeavour, any other skill environment, just goes out the window and you hear these things that you're like, apply that to any other context, and you would see just how silly that sounds. So, yeah, I was told, you know, that my kid, you know, “what's the rush?” “Why are you trying to, like, are you trying to get ahead with your kids that you're teaching them math at home?” 

And, and I just thought like, funny. I've never heard a piano teacher say that, and I've never heard a soccer coach say that. And why not? You know, these things, when you put them in the context of music or sports sound so insane. On what planet would a, would a piano teacher say, “What's the rush in attaining skill at piano?" 

And yet that is, I think, a pretty common. Like, yeah, of course a teacher would say, “what's the rush for math?” And I just don't understand why that's okay. 

[00:04:34] Anna Stokke: Yeah, I don't understand either. So, what sort of reaction did your post get?  

[00:04:40] Alex Kontorovich: Lately I've been, what's it called, hit and run, where I just post something and look away because I don't know that there's too much productive conversation to be had. Although, on the other hand, I meet people like you and, you know, I'm a big fan of your podcast, a long time— what's, what's the term—long-time listener, first-time caller kind of thing.  

So, the reaction was all over the place. People started arguing with one another about whether I was right or whether I was wrong. For me, it was less about the reaction, but more about just starting a dialogue to point out just how often people say things in the context of math education that they would never say in any other skilled activity. 

[00:05:19] Anna Stokke: So, did people defend that sort of thinking?  

[00:05:23] Alex Kontorovich: Yeah, I don't wanna put words in anybody else's mouth, but there was certainly one thread, I mean, again, I tried to look at this as little as possible, but there was a long thread, so my notifications kept going off.  

But one of the threads was about the fact that if a child starts learning math at home, it makes it harder for teachers in school to deal with kids at different ability levels all in one classroom, which is absolutely a valid criticism if your goal is for teachers to have an easy time with children in classrooms and not, if your goal is not for children to advance at whatever rate is right for them and, you know, reach their full potential.  

[00:06:05] Anna Stokke: So yeah, that's kind of what I would've guessed, people would be worried that it would just widen the gaps in the classroom, right? Because there are going to be some kids who can't practice at home, their parents can't help them, or they won't practice at home. And then if there are kids who are practicing at home and getting help, then they're going to advance more quickly, right? 

So that, that's kind of the argument I would've imagined. 

[00:06:28] Alex Kontorovich: And they're right. You know, the, what you said is really right about, life's not fair. Like nobody on the basketball team says, “Well, it's not fair that some kids are born taller than others and some are shorter and some practice at home and some have a basketball court in their backyard and some have to walk five blocks, so they're not gonna do it as frequently.”  

Like that is absolutely true. Those things are all true. They're the reality. And when kids come to the basketball court, everybody plays to the best of their ability and everybody hangs out and tries to get better from wherever they are. 

[00:06:59] Anna Stokke: Okay. So, are you still practicing math with your child at home? 

[00:07:03] Alex Kontorovich: Yeah, yeah. No, no one's going to stop me. I mean, I did say what's the most polite way to respond, which potentially has an impact in the sense of getting people to think about what they're saying when they say something like that. 

[00:07:17] Anna Stokke: So, what is the, what is the most polite way? 

[00:07:19] Alex Kontorovich: If there is one, I haven't learned it. Basically, with my kids, I tell them when you're in school, you, it's, “yes sir, yes ma'am.” You do whatever the teacher says. You do it the way the teacher says to do it. You don't complain. You don't say, “this is too easy.” You just learn to follow orders and listen to grownups, and then when you come home, we'll play with the stuff that's appropriate for you. 

[00:07:40] Anna Stokke: So, I thought we'd move into outreach because I wanted to hear a lot about your outreach today. And so first of all, why did you get involved in outreach?  

[00:07:48] Alex Kontorovich: That's a great question and kind of a funny story. I mean, I, grew up, just following that research math track and sort of thinking that, you know, yes you have to do teaching because that's part of the job, but that's a thing that you just get rid of and then go do your research and that's where the important stuff happens. 

And then I was at Yale at the time and applying for an NSF Career grant. And the career grant is, supposed to be like a larger NSF grant. It's a five-year grant for pre-tenure faculty and all NSF proposals need to have an intellectual merit component. So this is like, what's your research, why is it important? What's happening?  

And then Congress decided that we, they also want to know why taxpayer dollars are going to this kind of research, no matter how interesting it may be for humanity. Does it actually do anything for literal human beings now? And this is something called the broader impacts component. 

And the broader impacts typically is things like, you know, “I'm gonna develop a course that dis describes my research for PhD students and I'm gonna supervise PhD students and maybe I'll run some workshops on this cutting-edge research.” Like all of that stuff is really good and really important stuff to be doing and goes into broader impacts. 

Well, the career grant is, something where they want you to do something significantly more in the broader impacts direction. I heard of people starting schools in Nicaragua and you know, just doing all kinds of really amazing things around the world. And there was a lecture series in New Haven being run already by some colleagues of mine for, you know, New Haven area school kids, and I proposed that I would give, I think one, maybe two lectures a year in this series to the public.  

And, well, the NSF came back and basically said “Your broader impacts is extremely weak compared to the other broader impacts proposals that we received,” but they gave me the money anyway. But what happened a few years later is that I moved from Yale to Rutgers, and the grant moved with me, of course, and I needed to figure out something else I could do to satisfy the broader impact since I knew that, you know, I was not doing sort of what was expected for that component for a career grant.  

And at that time there was a new museum had just opened up in nearby New York City called MoMath, the National Museum of Mathematics. So I reached out to them and said, “Hey, listen, I have this grant. I'm supposed to give some public talks. Can I give a public talk by you guys?” 

To which, you know, then there was some back and forth. And eventually I was giving some other public lecture that they, I think in Massachusetts that they attended from New York City because they really want to vet who they're gonna bring before the public. They care very much about, making sure that when mathematicians are doing outreach, they're actually, they're not hurting the cause, rather than helping it, which is entirely possible.  

And eventually as a result of that, you know, that was sort of the beginning of our relationship with them. I gave maybe one talk a year for the first couple of years, and then I got more and more involved over time, and I realized I actually like it. I mean, I enjoy talking to people. I enjoy understanding math at sort of every level and every scale, and I find it very meaningful.  

And I still love, you know, proving a theorem that, you know, me and five of my colleagues around the world will understand, but I also like seeing a room full of people who have an aha moment, what at whatever level that aha moment is. So, I sort of got more and more involved in outreach. 

[00:11:13] Anna Stokke: I thought we'd now talk a bit about Math Corps, and so my understanding is that that's a tuition-free summer day camp, and it was started by some math professors in Detroit. 

And you recently founded a branch in New Brunswick, which is a city in New Jersey, that's where Rutgers is.  

[00:11:31] Alex Kontorovich: The other New Brunswick.  

[00:11:32] Anna Stokke: The other New Brunswick for our Canadian listeners, and yours is called the Rutgers Math Corps. I understand you do it at Rutgers. So, you ran it for the first time in 2024, and I understand you're running it again this summer. 

So can you give us an overview, first of all, what is Math Corps? 

[00:11:51] Alex Kontorovich: So Math Corps, as you said, is a summer program for kids, originally in Detroit, started by Steve Kahn and Leonard Boehm, mathematicians at Wayne State University. And basically they said, look, there's a research university next to, a stone's throw from a really struggling school district, and how can it be that blocks apart are people who are doing foundational research next to, a few blocks away are kids who can't add fractions. 

And, come on, we have to do something. Over 30 years, they iterated and iterated and found new ways to reach lots and lots of kids at scale. And came up with what I consider to be one of the, if not the best pedagogical approaches out there today. So a few years ago, they had an NSF grant devoted to understanding whether this can be scaled elsewhere, or was this something, you know, really special to them and their situation in Detroit. 

And a friend of mine, named Sarah Cook started a branch in Ann Arbor, Michigan. So, she's at U Michigan in Ann Arbor, that branch services kids in Ypsilanti nearby. And she was telling me about what an amazing program it was and how impactful it had been and how meaningful it was for her to participate in this. 

And I was just, you know, in tears after hearing her describe this program, and I had another NSF proposal coming up somehow. The NSF plays a very large role in driving me, even though I only come to them for research, but they drive me in this direction. It's kind of funny.  

And I couldn't help but write a broader impact section that said, you know what, if you gimme the money, and I asked for a little bit, you know, extra for, there's like a, you know, just my standard research component for my graduate students and postdocs and travel and stuff like that, and I said, if you give me this extra bit of money, if you give me the grant and this extra money, I'll open a branch of Math Corps at Rutgers.  

Like, what a fool. Who the hell am I to think I can do such a thing, that this is like remotely within my wheelhouse? But I asked, and maybe they're the fools because they actually gave me the money. So, the problem became that I then had to do it because the money was there. And so last summer was the first summer we did it, and this summer will be year two. 

[00:14:12] Anna Stokke: Oh, that's awesome. Okay, so how many students took part last summer? 

[00:14:18] Alex Kontorovich: Yeah, so for the inaugural summer we had 60 kids, and this summer we'll have 90. 

[00:14:23] Anna Stokke: Oh, that's a good number. And what ages? 

[00:14:27] Alex Kontorovich: So the program runs for incoming seventh, eighth, and ninth graders. And those are the, what we call campers. And then they're taught by incoming 10th, 11th, and 12th graders. So these are high schoolers, teaching assistants. We call them the TAs. 

[00:14:43] Anna Stokke: Okay. So how do you pick your TAs? 

[00:14:46] Alex Kontorovich: Okay. So, the TAs are chosen purely on merit. So first, when they apply, they have to write an essay. We look at the essays, but we don't really look at the essays too seriously until they come in and take a diagnostic exam. So that diagnostic exam ensures that every kid who's a TA knows their, you know, through algebra and maybe a little bit past algebra, cold. 

They know it inside and out. Step one: you have to have the skills. If you have the skills, then they come in for an interview. And during the interview we see is this someone who we think will work well with kids who sees the importance of the program, who understands what we're trying to accomplish here? 

You know, we wanna make sure that they understand the campers are not selected, we're not taking gifted and talented students. That's not what, what this program is. There are lots of programs that are like that, and they should exist and they should flourish. I want them to exist. this is not that. This is a program where we take all kids regardless of whether they're so-called weaker or struggling or if they're excelling. 

If they're excelling and they're not getting what they, what they need in order to reach their potential, we take them. If they're struggling and they want to improve and they, and they're not getting what they need, we take them too. So we take all the kids that we can, that we have, you know, resources for and staffing and so on, and we do the best we can with them.  

So we, so our, our kind of guiding principles are loving kids and believing in kids. And those, those are two very, they're not just like platitudes. They're, they're, technically defined things. And we spend a lot of time in our training with the TAs, defining those concepts, showing what they mean, what they don't mean, and explaining how everything derives, basically mathematically, like axiomatically, logically from those two principles. 

And one of the things that follows from that is you're gonna hire TAs that are the best people to teach your kids. And so, a) they have to have the math skills, and b) they have to be empathic, they have to be enthusiastic, they’ll be people that we want our campers to see as big brothers and big sisters. 

[00:16:47] Anna Stokke: Okay, so how many math profs are involved? 

[00:16:50] Alex Kontorovich: Very, very few compared to the number of kids that are being taught and at the scale that they're being taught. So, so this summer I think we will have four math professors involved. So there's two kinds of instruction. 

We call it the broccoli and the ice cream. There's  a, there's this battle in math education about whether kids need to learn foundational skills or whether they should have an opportunity to develop their critical thinking capacity. It's like, well, which one is Math Corps gonna do? And the answer is, yes. That's a false dichotomy. We're gonna have foundational skills and we're gonna have critical thinking. So the kids have 90 minutes in the morning of foundational skills, which by the way, just because you're learning foundational skills doesn't mean you're not doing critical thinking at the same time. 

And then in the afternoon, they get 60 minutes of the ice cream course, which just, for example, the eighth graders, our curriculum is that the seventh graders learn the real numbers. What the real number is, what it represents, positive numbers, negative numbers, fractions, decimals, everything lives on the real number line. 

Everything is sort of geometrized to the real number line. And once you have a really, really solid foundation of what every fraction means, what a mixed number means, what are equivalent fractions, how to convert between fractions and decimals, like they're all the same thing, they're just sitting right there somewhere on the number line, once you have that foundation in seventh grade, our eighth grade curriculum is operations with the real numbers.  

So our eighth graders, rising eighth graders will practice, you know, adding fractions, subtracting, decimals, multiplying, and dividing and so on, doing all these algebraic manipulations. And once their algebraic manipulations, once they really understand those deeply, then in ninth grade, of course, you do it with variables instead of numbers, then it's all the same. 

So that's, that's sort of the rough transition of the, of the curriculum. So our eighth graders in the morning are learning, are getting really good at adding fractions. And then the afternoon you say, “Okay, you know what, a half plus a quarter is, add another eighth to that, add a 16th, add a 32nd, add a 64th,” and so on to infinity. 

And boy, do kids love that word. Like you get kids thinking about infinity, they're just, you know, “What do you mean infinity? We're not allowed to talk about it. There's no such thing.” And here's a math professor saying, “Oh yeah, there is. Let's talk about it. Let's go there.” So, the afternoon has all of that, you know, steeped in creativity and productive struggle, and what can we really do while making sure that in the morning they're getting their raw skill and transitioning from places where they're working hard at something to facility and fluency. 

[00:19:21] Anna Stokke: Okay, so you chose the curriculum. It sounds like you were trying to get them to the point where they can like solve algebraic equations and, and work really with algebra, right? Is that, was that how you chose the topics? 

[00:19:35] Alex Kontorovich: Yeah. But our rising ninth graders should be prepared for high school by having a very solid foundation in algebra. 

[00:19:42] Anna Stokke: So, and how many weeks is the program? 

[00:19:44] Alex Kontorovich: So it depends in different places around the country. The Detroit program is six weeks. At Rutgers we, our first year we ran four weeks, which is like the minimum amount of time that you can really see very significant progress. And this summer, again, we'll run four weeks, but you know, if our financial, situation improves, then we'll hopefully be able to run six weeks like Detroit eventually. 

[00:20:08] Anna Stokke: And then how do the, how do you select, you call them your campers, right? How do you select your campers? 

[00:20:15] Alex Kontorovich: So the campers are selected, again, just based on the principle of where do we think we can do the most good. Ii's not about their test scores, it's not about a teacher's recommendation, they write an essay for us and they say, you know, “I want to come here. I really want to be here. I'm gonna work really hard. This is why,” you know, “I think you guys can help me.” 

And there's a story that Steve Kahn at, in Detroit, like likes to say, the founder, you know, he gets an application that says, “well, I'm no good at math and I'm not really good at anything and you're not gonna take me, but I'll write an essay anyway.” 

And that is exactly the kind of kid that he's gonna grab and bring in and give him a hug and give him, you know, breakfast in the morning and fill his mind with the ideals of the program, which are absolute excellence. Give me the best you possibly can and rise to your potential. 

[00:21:10] Anna Stokke: Okay, so you gave me a document that kind of describes some of the things you're doing and it sounds really great. And one of the things you mentioned in the document is that you actually incorporate science of learning techniques. So can you tell us a bit about that? So what specific learning techniques are most important in your mind? 

[00:21:30] Alex Kontorovich: Yeah, so this was really interesting to me because, you know, Steve and the people in Detroit have been running this program for 30 plus years, so they just iterated and iterated and iterated until they got to something that works as well as it does today. Then they hand that manuscript of how you do things over to me, and I see it with fresh eyes, and I ran it for the first time last summer. 

And afterwards, I was just sitting there kind of dumbfounded, like, “why does this work so well?” And I didn't understand why it worked so well. And I listened to podcasts like yours where I hear people talking about, you know, cognitive science research and neuroscience of learning, and I started just reading more of that literature and asking around. 

And every time I learned some interesting tidbit from that literature, I could tie it to a practice that we have in Math Corps that's kind of a unique practice. And by the end of this exploration, I had like a dozen things that we do that align apparently with this foundational research. So, just a few weeks ago, I was talking with Steve and I said, where did you learn all this amazing research? 

Like, do you read the cognitive psychology literature? Or how did you come up with this stuff? He's like, what literature? I just did what makes sense. If you follow the, if you actually love kids and you believe in kids and you try and try and try and things don't work and you try something else until you find things that do work. Just 'cause they work, that doesn't mean they can't work better. So you try something else and it doesn't work as well.  

And then you try something else and it works a little bit better. You just iterate. I mean, it's the way humanity has gotten anything we, we have is, is iteration. And if your goal, first and foremost is to make the best possible experience for kids, it will naturally lead to these principles. 

And so, the fact that I was able to connect them one by one to things that people discovered in the scientific literature just shows that it, it, none of this is, is rocket science. I mean, it's just you do the thing that you would do for your own children and amazing things happen. 

[00:23:34] Anna Stokke: Yeah, I agree with you that a lot of it's common sense. That's not the, the issue. In many ways, the issue is that there are a lot of people out there saying that things are true that are very counterintuitive, and you know, then, then you have to look to people that are showing that things like knowing your times tables is actually pretty important.  

Because like, to you and me, that's probably obvious. Is that obvious to you that you should know your times tables? 

[00:24:03] Alex Kontorovich: You know, it's a posteriori obvious, I don't think a priori it would be obvious. I sort of think of this, these two principles, again, loving kids and believing kids. I sort of think of it as like a two-by-two matrix. And, you could say, in this matrix is like believing in kids or not will be, let's say the columns and then loving kids are not, will be the rows.  

So in this, if you believe in kids but you don't love them, that's kind of like, like I understand things in, you know, archetypes or something. So there's like the archetypical, father figure like, like the overbearing father who says, “I don't care that your fingernails are bleeding. Keep practicing your violin until you have your scales perfect.”  

So that's believing in kids, expecting the best of them, but not loving them. And then there's in the other corner is the smothering mother archetype where, you know, you love kids and you want the best for them, but you don't actually believe that they, you don't fully ingrain in them a belief in themselves and the capacity to exceed whatever state they're currently in. 

And so, there's sort of a prisoner's dilemma here where either of those is actually worse than neither loving kids nor believing in them, which is kind of where schools default to.  

So that's, that's kind of in my mind, this prisoner's dilemma situation that we find ourselves in and Math Corps is purpose, is purposefully trying to break itself out into that first cell where you insist on both loving kids and believing them. 

So, I think I would classify not insisting the kids know their times tables as loving, but not believing. “Why do you have to know your times tables? It's annoying to have to learn them. There's this like process that kids go through where they have to learn it and if they don't learn it, what's the big deal?” 

Well, they'll take out a calculator. Like, yeah, you think a kid wants to—if you're trying to learn algebra, you're doing seven plus x in parentheses times eight plus x, and the first thing you're gonna do is take out a calculator because you don't know seven times eight, the thought, if you're trying to learn the distributive property, it's gone, it's out of your mind because now you're worried about seven times eight, you have to reach for a calculator.  

If things don't become absolutely fluent, just immediate, then you're setting a kid up for failure. 

[00:26:18] Anna Stokke: When you say not believing in kids, they think that they can't do it, that some kids can't do it. Is that what you think? 

[00:26:26] Alex Kontorovich: I mean, it's hard learning your times tables. Everybody who has ever crossed that path has, I don't know a single person who just looked at the times table and said, “Yeah, I got it.” It's hard. So if you, love kids but don't believe in them then you'll say, “You shouldn't have to do hard things”. Yeah, a calculator will do that for you. 

And if you believe in kids and you say, “no, actually I think you can learn this. And I think it's not gonna be that hard for you. You just have to literally sit down and do it. And it's gonna be a couple of weeks of pain and it'll be painful for me as the teacher to watch you go through this pain, but I know that there's, you know, no pain, no gain on the other side of this you'll be a much stronger math student and it will just help you in the rest of your life.”  

And of course you get stories of people who are professional mathematicians who say, “Oh, I don't know what seven times six is off the top of my head.” And you say, “Okay, but you're one in a million because the rest of us all do.” 

[00:27:16] Anna Stokke: Do you, do you know mathematicians like that? Do, do you like, I actually haven't met a mathematician who doesn't know their times tables. I like, I've heard that story. 

[00:27:26] Alex Kontorovich: I've heard that story, I've heard people say it. I never, you know, called people out on whether they really don't know seven times six. 

[00:27:34] Anna Stokke: Well, I mean, it would be hard to test them if they told you that and then you start drilling them, they could just not answer, right? So it would be, it would be hard to test that. But, about the difficulty by the way, so here's something interesting. So Amanda VanDerHeyden shared with me some research that she's done and, and this isn't published, this is just data she has like working with lots and lots and lots of children. 

It actually only takes two weeks to memorize your times tables. On average two weeks. 

[00:28:06] Alex Kontorovich: Yeah. But those are painful two weeks.  

[00:28:08] Anna Stokke: Oh, they’re a painful two weeks.  

[00:28:10] Alex Kontorovich: Imagine day six of that, day six of that, you've been struggling for five days straight and you've got four more days after that. I mean, you don't know how long it'll take you, but when you're in the middle of it, you just want to quit. And this is where believing in kids has to be balanced against loving kids. 

And you see someone struggling, you see someone doing something they don't wanna be doing, and you say, “no, I believe in you and you should believe in yourself. Nobody said this was gonna be easy. We know this is hard. We do it because it's hard.” The same reason we went to the moon. 

[00:28:40] Anna Stokke: Absolutely. Doing hard things is good. We need to, you know, I mean, if times tables are the hardest thing kids have to do, you know, life isn't that bad really, is it? 

[00:28:50] Alex Kontorovich: And kids love doing hard things. That's the thing that I'm so flabbergasted. It's the grownups who don't like doing the hard things they don't like, they don't have to do it themselves, they don't like having to watch kids struggle. This is exactly where the, that that smothering mother corner takes over. 

Where you see a kid who, you know is struggling to tie his shoe laces and you just bend down and tie it for him, like, no, no. He has to learn how to tie his shoe laces. And the only way he does that is by struggling. He's not, you know, if he's crying and he is upset or something, then you, then you of course you help him. 

So there's always this balancing act between these two things of when do you, when do you have to expect more of them, and when do you have to back off and say that “That's enough,” that's a very delicate thing. But the answer is not to default to, they don't need to know it. You're, you're setting them up for failure. 

[00:29:36] Anna Stokke: So, okay. So, let's go back to the science of learning techniques that you're using. So what are some of the ones you use? So is it retrieval practice, spaced practice? Are you doing daily reviews, things like that? 

[00:29:49] Alex Kontorovich: Pretty much all of it. I mean, The first thing that I start thinking about is fluency. I want these kids to have this stuff at their fingertips. So what does that mean? The economist Nobel Laureate, Daniel Kahneman has this great book, Thinking, Fast and Slow, where he talks about system one and system two processing. 

And system one is just, you know, like talking, talking is system one. It doesn't take any effort to talk. Everybody talks. You just do it and whatever comes outta your mouth, that's what it is. That's like an LLM, a large language model. Some operation in my brain is throwing out the next token out of my mouth based on the tokens that I've heard in the context before. 

Whereas if I'm doing some delicate mathematical computation, I'm moving very slowly, my, my brow is furrowed. My movement is hesitant and uncertain, and this is system two thinking. This is like really deeply thinking about something. Again, this false dichotomy is which should you do system one or system two? 

And of course the answer is both. We should be developing both. But, and here's the big but, when we're doing the so-called “school math,” that's when we should be transitioning things that start out as system two to system one. The first time you learn how to add fractions, it's really, really complicated, and you have to sit there and think about it. But then you practice and you practice and you practice. And in, I don't know, two weeks or two months or however long it takes, it becomes system one. It becomes fluent.  

You just see a pair of fractions, you know what you have to do, how to, to add them, you know what it means to get a common denominator, you know, yada yada. You just, you just do it. You don't even think about it. And then it becomes easy.  

And then if someone throws a fraction at you, you're like, “yeah, I got that. Nope, no big deal.” We insist that every kid gets to fluency on anything that they're working on a big question would be like, how do you do that? How do you do that in a classroom with, how do you do it with 60 kids? Well, the obvious answer, and, and it's kind of both the right and the wrong answer, is private tutoring.  

This is the, so-called Bloom’s 2 sigma effect. So Bloom was a education researcher and he observed, I guess he had an experiment where there was a control, which is just kids in a classroom and then some number of those kids were randomly chosen and given private tutoring. And private tutoring means you go to exactly the point where the kid is stuck on something, you scaffold that.  

There's like a step, and I imagine people walking up, walking up a bunch of steps, a flight of stairs, and there's a step that's like six feet tall. They can't get up, get over it, and you're like, “Why can't you get over it?” You're not, you're just not big enough. Your legs aren't big enough yet. So, what you do with that step is you just break it down into a bunch of smaller steps. You scaffold that process until each step is small enough that the kid can do it easily. 

And then you walk through that path enough times, time after another time again with space repetition, with retrieval practice, and with interleaving and it just becomes system one over time. So everything we do is geared towards that. So the Bloom 2 sigma effect is that students who receive this private tutoring at the end of the school year, whatever, I don't remember the details of this, of this study, had on average two standard deviations improvement in their outcomes. 

So if you just think of a normal distribution with like the 50th percentile in the middle, two standard deviations, is that 97.7 percentile. So it's, it's a massive, massive amount of improvement that you can do with private tutoring. And of course, like that's, that's again, a no brainer, that if you hire private tutors for kids, then they'll be able to progress faster than, make progress faster than they would if they were in a classroom with lots of different kids at different skill levels.  

The question is, how do you provide private tutoring to a whole bunch of kids with not a lot of money? And by the way, we spend about a third for this one month that we run the program at Rutgers. We be, we spend about a third of what the public schools spend in a month of school. So this is where the, the brilliance of the Math Corps comes in. 

It's a pyramid structure. So, we break down the camp into teams and every team has 10 campers, 10 middle schoolers rising, seventh, eighth, or ninth graders, five high school TAs, so it's two to one tutoring and one college instructor.  

So those 60 kids from our first year really means 40 middle schoolers and 20 high schoolers. Now, that would still be too expensive if you have to pay all those high schoolers for all of the tutoring that they're doing to these kids. That's, that's gonna be a high cost. So the, the next brilliant iteration in the Math Corps is that while the high schoolers, even though they're selected to be strong math students, they are strong math students, we still want them to get even stronger. 

So in the afternoon, they get their own broccoli course and ice cream course at whatever level they're at. And so, it's sort of like a half barter half, they get a stipend for their work. So, these are high schoolers who over the summer learn some really cool math from research mathematicians at a university in a university setting, get to get paid and spend the morning tutoring kids in this environment that breeds success. 

And of course it looks great on their college resume. So that's basically the structure by which, for a very low cost, we can get extremely high quality tutors in front of every single kid. 

[00:35:11] Anna Stokke: There were a couple of other things I just wanna mention about the program just because I think actually some of this might help even teachers, things they might wanna implement in the classroom. In particular, I wanted to talk about the hand gestures. and I did watch the video actually the one from that was made with the Detroit section. 

And so, I mean, I thought this was a really good idea. So for example, when kids were speaking or when they went up to the board and they were gonna present their solution, the other kids, they roll their hands like that, right?  

[00:35:45] Alex Kontorovich: That's the support signal. 

[00:35:46] Anna Stokke: Yeah, a support signal and I think that's a really good idea. 

And you know, they had kids talking about it and saying it really helped because they knew that their peers were all supporting them. It was just a gesture of support. So, I'm wondering if you can talk a little bit about the hand signals and if there are others that you use. 

[00:36:06] Alex Kontorovich: That's, I don't know where that innovation came from. I think that that came from Leonard Boehm if I'm not mistaken. But just imagine, we've all seen this scene where there's 30 kids sitting at their desks, there's a kid at the blackboard trying to solve some problem, and the kid is stuck and looks back and there's 30 kids who are bored and they have their head down and they have their eyes closed and they're doodling something on their page. 

And this kid is like, you know, this is the nightmare that people have of, “oh no, I forgot to put my pants on this morning,” or whatever, right? You have this huge feeling of all of these eyeballs on you and you're not performing, right? You, you missed a note in your, on your recital and now you can't get back to like, I just, “I'm freezing and I don't know what to play.” 

Now imagine the alternative. You look back and there's 30 kids who are all looking at you. And by the way, maybe it's 25 kids who are looking at you and doing the hand signal, but those other five kids see what the other kids are doing and they know that they should also be doing the hand signal. So it's a social thing that gets everybody involved and everyone's doing it. 

And you look back and there's a classroom of people who are rooting for you. They're saying, “Go, go Sally. Go, you can do this,” but not verbally.  Because verbally would interrupt what you're trying to say and what you're trying to do and how you're trying to think. So that's such a powerful idea. 

So brilliant. And I have no idea where it came from and I don't know why it hasn't, why it's not in every classroom. I don't think it would necessarily work in my college classrooms or with my PhD students. But for kids, they also love having inside jokes, inside secrets. And they love that you do weird things when you're in Math Corps that you don't do at home.  

We expect the highest behaviour from you, the highest, working as hard as you possibly can. And there's all this weird stuff like, like the hand gestures, the success, the silent clap, the silent applause where you wiggle your, your fingers up in the air. 

That's a way of not stopping the flow of the conversation, not interrupting the the sensation in the ears, not stopping someone from being able to talk over it. You can talk over silent applause all you want, but still the entire room is showing the person that they appreciate what was just said. 

So, that silent communication is just another one of these brilliant, you know, there's like a dozen brilliant insights that, that these guys have come up with that they handed over to the rest of us who have been trying to implement it elsewhere. 

[00:38:24] Anna Stokke: So did the students like the program? Like what kind of feedback did you get? 

[00:38:29] Alex Kontorovich: Yeah, that was the, the best part. You know, there, there was these two girls, especially last year, who were always giving me a hard time. They're like, “why do we have to do math? We don't like math.” And, you know, I, I knew they were, they were teasing me, and we would, we would just be joking around.  

And then we had our closing ceremony at the end of the camp we invite all the parents in and the kids put on, you know, a little demonstration of the kinds of things they've learned and talk a little bit. It's a celebration, we give out some awards. And, so I saw those two girls and I said, “so, you think you might wanna come back next year?” And both of them without skipping a beat, “of course.” 

Wait, wait a second, the entire, you were just telling me you don't, you don't like it, you don't wanna like, “No, no, no. We, we like it. We. You know, we wanna come back, we wanna do this.” So there's revealed preferences. People vote with their feet and they were there every single day, nice and early playing board games in the morning and hanging out and, you know, being too cool, great.  

You wanna be too cool for it? Be too cool for it. But they were doing the hand signals because everybody else was doing the hand signals and they were participating and they're coming back this summer. So, preference is revealed. 

[00:39:30] Anna Stokke: Can you tell us about gamification? 

[00:39:32] Alex Kontorovich: So, we have a morning assembly when kids, when they first arrive, there's an hour window for arrival. And the philosophy behind that is some families need their kids outta the house early in the morning to get to work, some families have a hard time, you know, getting things ready in the morning. 

And so, we don't say you need to be arrived by 8:30. We say you can arrive between 8:30 and 9:25, 9:30 our programming begins. You know, one of the pH, one of the principles, if you love kids and believe in kids, then you don't accept anything but their very best. And being on time is the very first indication that you're there to work, you're there to accomplish great things. 

We want every child to show us the greatness within them. So, you can't do that if you're not there. And so if you come in at 9:30, well our programming is started now. You get sent to the dean of students and there's a phone call home and there's a discussion about whether a single tardiness or a single unfinished homework is grounds for dismissal from the program, because otherwise we're not actually loving them.  

You know, we don't have the highest expectations of them. And most of the kids try to get there as early to, as close to 8:30 as they can because that hour window involves breakfast, we provide every kid with breakfast in the morning. No one should be trying to learn on an empty stomach, and we put out board games and chess and Monopoly, and kids just have a great time.  

They start the day playing with one another. We're also a tech-free school. There are no phones. They're asked to turn their phones off and put them in their backpacks the second they arrive. 

So everyone's actually interacting. No one's sitting there doom scrolling in the corner and then we start the assembly. You know, we, there's some words of encouragement. There's a joke from the Minister of Humour. So we take, humour and silliness extremely seriously. At the Math Corps, this is an official position, the Minister of Humour, and there are hand gestures for when a joke lands versus when a joke is corny.  

It's just great. You have to keep that air of we're here to, to play hard and, and work hard. And then the best part is we recognize every single child by name who got a perfect score on their homework the day before. So this is the gamification aspect. They get their name read aloud in front of the entire camp, every single time, every single day that they get a perfect homework score on the day before.  

So, of course, what does that incentivize. Like every kid starts sitting up wondering if their name's gonna be read. You know, they know what they did on their homework, but maybe they don't know if they got a perfect score or not. 

[00:41:52] Anna Stokke: And you mentioned the cost, that the cost is quite low. And actually in your document you wrote about the New Brunswick School District, and that's New Brunswick, New Jersey spending $30,000 per student per year, but getting poor results, like quite poor results. Something like only 10% of students were, were meeting expectations in math. 

Was that right? Am I remembering that correctly? 

[00:42:19] Alex Kontorovich: So, again, I wanna be careful about this because it's not that I want to blame the individuals involved in that organization and in the heroic efforts that I think they're trying to do in the face of insurmountable challenges. But these are just numbers that you can look up online. So you look up “What's the New Brunswick School District Annual Operating Budget?” 

It's something in the neighborhood of 300 million. “How many kids did they serve?” “9,000.” I can divide one by the other. They're spending on the order of 30,000 a kid, which if you say, you know, school year is like nine or 10 months, let's call it 10, 30,000 divided by 10 is 3000 a month, and we spend a thousand under a thousand a month. 

Again, I don't know that I would be making different decisions than the principal or the superintendent or a teacher in that school. I'm not going to go and blame these individuals. They're the, they're doing god's work, day in and day out. but you have to look at what the system's producing. 

And when you go on the New Jersey Department of Education, there's a link for every single school and they show their reading results and they show their math results. There's a meter that goes from zero to a hundred, a hundred being a hundred percent of the kids are proficient at grade level and zero being none. And the scale is all the way, I mean, it's far enough, it's close enough to zero that they don't actually tell you what the number is. 

They just tell you fewer than 10% at that point. So that means more than 90% of the children are being failed by this school district. This is both in the middle school and in the high school. Now, it's not that hard to run a thought experiment to see how such a situation could come about, right? Let's imagine that you're a principal. And again, it's not, I'm, there's not, I'm not trying to pin this on any individual. 

If I was a principal of a school and I'm being pressured to get kids to graduate, right? The four-year graduation rate is like one of the most important things for me. Well, how do kids not graduate? They don't graduate if they don't get credits. You don't get a credit if you fail a course. Okay, so let's recurse this backwards. 

How do you fail a course? Well, there's all kinds of components that go into what a course is, including homework and tests and quizzes and things like that. Well, giving some kids homework knowing that when they go home, there's not gonna be someone there to help them, whereas other kids will have parents around or older siblings or other kinds of family members or friends who can help them, that's not fair.  

So let's just, maybe we'll assign homework. We just won't collect it, we won't grade it, it won't mean anything. So those are points that can't contribute to a student failing. Of course, what happens if you don't practice at home is that you don't learn any skills. And then what do you do with exams? 

Well, you know, again, you have to grade them on a curve or something, you have to do something so that you just, you don't fail every kid. And then year after year after year, a kid goes from one grade to another, to another, to another, never actually mastering any skills. And then this doesn't show up until they have to take a standardized test for the state. 

And the state says 90 plus percent of the school is failed. And this is exactly what I mean by if you, if you just logically follow every incentive that's placed before you as a school teacher, as an administrator, this is the, this is what the system is meant, is designed to produce. The system produces exactly what it was designed to produce.  

And this is why the math course starts with the first thing that like, Marques symbols are loving kids and believing in kids, which means that if that's what it's producing, that you're not hitting either of those things. And the catastrophe is it takes really so little, it's so little in terms of resources, so little in terms of time in this one month that we have with them, we take kids who come in and they take a pre-test and they take a post-test. In the pretest they're at like a fifth grade level if they're a rising seventh graders.  

And in that one month we teach them all of fifth grade, all of sixth grade, and all of seventh grade. And they take a post-test and they get a perfect score on the entire seventh grade curriculum. Then they go into seventh grade into the schools, and over the next nine to 10 months they forget everything. Then they come back to us and they're like at a seventh-grade level and we teach them all of seventh grade and all of eighth grade in one month. 

Again, that just continues. Then hopefully they'll come back to us as high school TAs and we just keep them, you know, in that 97.7 percentile because of the private tutoring that they've been getting over years and years.  

I can't say we, because this is only year two of our program, but this has been the experience of Detroit and they, you know, I think Steve Kahn deserves a presidential medal of freedom for the work that he's done with tens of thousands of kids that have come out of Detroit and made fantastic—that doesn't mean, that doesn't mean that they became Nobel winning, Nobel Laureates or any, some nonsense like that doesn't mean that they went to college. If they finished high school and got a job or went to college and are doing something productive in their community, that is them realizing their potential and making something of themselves and having great lives.  

That's exactly, you know, I was born in Russia, like I came to this country because this country is, I have to specify since you're in Canada, but I mean, they're, you know, equivalent in this sense. These are lands of opportunity. These are places where if you invest in the right way early on and just keep compounding that investment over time, amazing things happen. 

[00:47:35] Anna Stokke: That was an intense discussion. Is there anything else you wanna say about Math Corps? 

[00:47:39] Alex Kontorovich: There's one little thing, which is the universality. You know, I was really concerned the night before we started our inaugural camp last year, I was sort of, you know, tossing and turning in bed trying to figure out like, “What if none of this works?” What if it work, it works in Detroit because their TAs were their campers and their college instructors were their TAs and their senior staff were their college instructors.  

Like they have this, you know, pyramid scheme where everybody's just getting better and better and over 30 years, of course it works. Everybody knows exactly what to do. It's only the new seventh graders, and they'll learn from everybody above them. We're starting from scratch with new campers, new TAs, new college instructors, a new executive director. I have no idea what the hell I'm doing. And so, I was like, “Will the system work?” 

And by day three, I was just walking through camp, peeking into the classrooms. Everybody, every kid was outta their seats. They were in groups of three at the blackboard all around the room. You know, three meaning the TA and the two campers just work computing away at the blackboard. And everybody knew exactly what to do and everyone was just, it just works. 

It takes so little from math, from a mathematician like me who knows nothing to just put this practice, put these ideas in practice and, and watch them flourish. 

[00:48:51] Anna Stokke: And you know, we, we might have people listening who might wanna do something similar where they live, right? 

[00:48:56] Alex Kontorovich: Yeah. This is a call to arms, mathematicians around the country, around the world. Please get in touch with Steve Kahn and the people in Detroit who can tell you the magic of Math Corps. 

[00:49:05] Anna Stokke: We'll put a YouTube video on the resource page that people can look at what's being done in Detroit anyway. And you're doing something similar where you are.  

So the next thing I thought we'd talk about, you know, I get lots of emails from listeners and I got an email from a listener and they thought that it would be good for me to critique A Mathematician's Lament, which was written by Paul Lockhart. 

So, I mean, I don't know if this is going to be a critique necessarily, but I thought we could just maybe have a respectful discussion about it. Before we do that, apparently you know Paul Lockhart, so maybe you could tell us who Paul Lockhart is before we start discussing that. I guess I'd kind of call it an essay, but you can get it on Audible, like you can get a little book. 

Am I right? 

[00:49:52] Alex Kontorovich: Yeah, I think it's a, it's a book and then it, if people just Google it, they can find The Lament, there. He has written several things, but The Mathematician's Lament is, I think we're referring to.  

[00:50:04] Anna Stokke: Yeah, that's the, that's the one. So, okay. So, can you tell us a bit about Paul Lockhart and his background? 

[00:50:09] Alex Kontorovich: Yeah, I don't know him personally. I think we've had some very short correspondence, but he was a PhD student in mathematics at Columbia with the same supervisor as myself. And then he did a postdoc at Brown, which is the same place I went for my postdoc. So, we're sort of, I'm following his trajectory and then he, I think he wound up teaching high school math in, at a private school in New York City if I'm not mistaken. 

So his Lament is, I think, a beautiful piece of writing. I can see why you would want to critique it. And let's, let's play the devil's advocate, let's play the opposite sides of that. Just to try to see where we have common ground and where not, but it basically starts with the premise, like, imagine the following scenario. You have music teachers who are told by the state that they need to learn, you know, the D scale in seventh grade. 

And so, they tell kids this is where you hold the violin. You put your finger down here and you pull, and then you put your finger here and you pull like that. And if a kid tries to do something else, or he's like, “no, no, stop.” You have to put your finger here. And you pull and you put your finger there and you pull. And that's music.  

And that's kind of the analog of the experience a lot of kids have with math, where a teacher, you know, that teacher has never heard of a symphony, doesn't even know, not only has not heard a symphony, has not heard of symphony. Doesn't know that there are things called symphonies that exist in the world. 

And of course, if they don't themselves know, how are they supposed to impart that on, on a kid? And so you might imagine a world where teachers, first of all are maybe professional musicians themselves, in the sense that, not in the sense that they're necessarily, obviously we're not gonna have armies of math professors teaching third grade as great or terrifying, as that would be. 

Who, who's to say that math professors are good teacher, right? many of, many of us are not. 

[00:51:56] Anna Stokke: Some are, some aren't. 

[00:51:58] Alex Kontorovich: Some are and some aren't. Exactly. but, certainly math professors understand that there is such a thing as research math, and certainly they understand that there's such a thing as higher math. Like one, one thing you could ask is any teacher at any level should have sort of, you know, a basic understanding of topics significantly higher than the level that they currently teach. 

So that means that if they're teaching high school calculus, they should know multivariable and linear algebra and maybe a little bit of real analysis or something. I mean, there's ideals and there's reality. So let, let's talk in ideals first and, and see if there's a path to that reality. But the thing that you would get from a teacher that knows those next steps is someone might ask a question, and that question might be well outside of that course. 

And this happens all the time in Math Corps when I'm talking to seventh graders and they ask something and they're like, ooh, oh, that's a good question. You know, I can't, you just noticed that the square of any number, whether it's positive or negative, is positive. And they say, “Well, do any numbers have, you know, squares that are negative?” And you just invented the complex numbers. And that's if I don't know what the complex numbers are, and if I have haven't been exposed to that, I say, “Oh no, that doesn't happen. Don't worry about it. Let's move on.”  

And if I do, then that's an opportunity to have like this really cool aside, it's like, this is not gonna be on the test. I'm not gonna ask you to learn these things. But just so you know, yes, there is such a thing. And it was a crazy story about how mathematicians, you know, were forced to find it because of solving the cubic equation.  

I mean, not that you would go into this much detail, but you know, when kids see that they, that they can push little buttons and get these cool ideas, then that's what they wanna do. They wanna push buttons, they wanna get cool ideas, they want to find things that does not have to get in the way of fluency and foundational skill. That should be just a part and parcel of the process of learning those things.  

So, I think when people read Lockhart's Lament and they say, you know, we shouldn't be teaching kids the times tables, that's, to me, a misreading of what Paul is saying. 

[00:54:04] Anna Stokke: Yeah, I don't know.  

[00:54:06] Alex Kontorovich: Please push back. 

[00:54:06] Anna Stokke: yeah. Yeah, so I read it recently and for sure the part about teachers. Yes, I get that too, that he definitely thinks that teachers that they need to have more of a background in math than they usually do. So he definitely is saying that piece. 

[00:54:25] Alex Kontorovich: One thing there, because if you look at what the Math Corps is doing. Our teachers, they actually don't have undergrad degrees. In fact, they don't have high school diplomas. They're themselves high schoolers.  

The point is that while saying that teachers need to have a lot of extra knowledge, I'm contradicting myself because our TAs do not have a ton of extra knowledge. They just have a lot of area expertise in the things that they're teaching. But here's the difference. Those TAs are themselves learning new math. 

And I think that's the big difference. For me, it's not that, you know lots of math, it's that you are currently yourself involved in acquiring new knowledge. I think that's because they're then more empathetic to the difficulty of acquiring new knowledge versus someone who acquired all the knowledge that they're gonna have 20 years ago, and is just year after year teaching the same thing and not actually learning anything themselves. 

I mean, this is what professional development tries to do, but in practice, professional development is like, you know, classroom management or things like this, that it rarely is a, how many high school or elementary, I mean third grade math teachers, how many of them are learning the next subject in their math trajectory? 

Or did they stop learning and that, and that's good enough, right? Maybe they took calculus in college. They don't need to know any more than that to teach third grade. Well, I would say if you know calculus, then on the side for fun, learn multi-variable, learn real analysis, learn linear algebra, learn whatever that next step is. 

And of course, research professors, I'm, confused about math every single day. That's my entire life is staring at a blank piece of paper and hoping some clarity comes that day, which very rarely actually happens.  

[00:56:00] Anna Stokke: Yeah, discovery math isn't that fun. It's only fun when, it's only fun when you actually discover something. But the lead up is, it can be quite painful.  

[00:56:11] Alex Kontorovich: Absolutely. Although I think that people who end up doing this kind of thing for a living are people who enjoy that struggle because they know that eventually somewhere, somehow there's gonna be that aha moment, and it's worth the wait. 

[00:56:23] Anna Stokke: Yes. 

[00:56:24] Alex Kontorovich: But, but anyway, so, so to the point of, of this question about how much expertise teachers need to have in practice, high schoolers who have not graduated high school, have all the expertise they need to teach up to rising ninth graders. 

On the other hand, what's important to me is that they themselves are on a journey and they're learning, and that I think comes across in how they teach. What do you think about that? You wanna criticise it? 

[00:56:47] Anna Stokke: Well, okay, so, so there's a few differences here. Okay. So first of all, in your program, you have people who have a lot of math knowledge overseeing it, and I bet you're giving them the lessons and they know what they're supposed to teach. Am I right? 

[00:57:04] Alex Kontorovich: So there is in every classroom where the high school TAs are working with their campers, there's also a college instructor who's typically a Rutgers math major.  

[00:57:13] Anna Stokke: Right. 

[00:57:14] Alex Kontorovich: Who's just circling around and if any, if they get stuck at anything, if they're trying to explain something but they're not really sure why it's true or, you know, they can jump in. 

[00:57:22] Anna Stokke: So they'll also correct the student. They'll correct them. You know, so this reminds me of, like when I teach a calculus course and I have a lab TA and I teach at an undergraduate university, so our TAs are all in the undergraduate program. So they'll have taken, you know, first year calculus, second year calculus, sometimes they'll have taken an analysis course, but I'll go around in the lab and I will hear them say things that are incorrect.  

And, you know, so I'm not so sure that you necessarily would want them teaching the calculus course, say. So I think there's a difference here in that, your program, you have people overseeing the program who actually know the math and they're able to oversee it. 

Your TAs also aren't, you know, they're not paid a high salary. They're not, you know, they're not hired as actual teachers that are teaching a class for the entire year, right. So I think there are a few differences. I agree with what you said at the beginning. I think you need to know a fair bit further than what you're going to teach because you need to be able to a) understand that topic deeply enough to explain it well, and b) be able to handle the curve ball questions that get thrown at you. 

[00:58:44] Alex Kontorovich: Yeah, and I would say those curveball questions to me, that that's the most fun about it because, when I'm saying standard stuff, you can find standard stuff, you can read it in books, you can find it on YouTube videos. The point of being in a classroom and in person is that you can throw a curveball at whatever is happening and watch how the person reacts. 

To me, it's the curve balls that make the classroom and not the, the actual material that I need to deliver. 

[00:59:08] Anna Stokke: But Alex, I think when the listener wrote to me, I think the listener was asking me to talk about different aspects of that essay. And so, so for example, some, some of the ways that I read it, so, okay, so he is sort of lamenting the fact that math has been turned into this industrial type thing. 

Like, you know, that it's, math in his view is really an art. So being a mathematician is like being a poet or being a musician, but students never see that. So students see math and people see math as being, I guess maybe a science or something that you use to like solve real world problems.  

[00:59:54] Alex Kontorovich: Or engineering. 

[001:00:26] Anna Stokke: Yeah, that's right. And he thinks we should just leave the real world problems out of it, you know? And just teach math, let students come up with their own conjectures and prove them if they can and, that type of thing. So first of all, do you, do you agree? 

Do you think math is like an art? 

[01:00:11] Alex Kontorovich: Math is absolutely an art, you know, music is also an art. And that doesn't mean you get to create art from the day you pick up a violin for the first time. Like you have to earn the right to make art. And, no one objects to a violin teacher starting with, you should know your c major scale and you should know where your fingers go and you should know how to pull the bow. 

Like there's all these technical components that have to be all in the system one, they're, they all have to be completely automatic before you can do that system two playful, like, let's improvise, let's play a jazz solo. Let's compose, a concerto, like music is absolutely an art form and it's extremely technical art form where the foundations have to be there in order for anything resembling art to come out. 

You just hand a bunch of kids a violin and say, “Go to town guys.” You know, what the hell are they gonna come up with? It's gonna be nonsense. So in the same way, yes, absolutely mathematics is an art form and maybe by, maybe by high school it's possible to have kids, if they, if they came through a system that gave them extremely strong foundational skills, then absolutely you can,can and should and must treat math as an art form. 

And by the way, even when you're learning foundational skills, math should still be an art form. You know, there's an art form to, every single thing we do. When we're doing our, our broccoli, our core foundational subjects, we talk about them as if they're novel discoveries. For example, in Math Corps, we have these great, impossibility statements. 

So, we'll ask a kid, you know, what's a half plus two thirds? And the correct Math Corps answer to what is a half plus two thirds is “not possible.” It's not possible to add a half and two thirds. And kids, kids know that it's possible. They know, they, they've seen this in school. Maybe they don't know what the answer is. 

They don't remember how to do it, but here's a math professor standing up in front of them saying it's not possible. Like, what the hell are you talking about? That immediately is exciting, like all kinds of feelings and emotions and, okay, where are you going with this old man? Like, “What, what are you trying to say?” 

And I said, you know, if I said, “What's two cats plus three dogs?” “It's just do cats plus three dogs.” You'd say, “Oh, maybe it's five animals.” Like, ah, wait, wait, wait. You did something there. What did you do? You converted dogs into animals and you converted cats into animals. And once they're the same thing, now you can add them. 

So there's like a, a foundational principle of addition. You can only add things that are the same. And once that principle has been, you know, you've been exposed to that principle again and again and again over all the years, then when you see two x squared plus x cubed, you're like x squared, x cubed, those are not the same thing, they're different.  

As opposed to, you know, five x minus seven x, you're like “x and x, those are the same things. So there's five of them and there's negative seven of them. I know how to add that.” So it just becomes second nature. That addition is something that can only be done to things that are the same, but that doesn't mean you can't somehow cleverly convert from halves into six and thirds into six and have some things where you can compare. 

There's a way of teaching that that feels discovery, that feels like it's discovery, it feels like it's an art form, but still gets across the foundational lessons and, and has those lessons land. 

[01:03:21] Anna Stokke: Yeah, so you're talking about motivating, why you're doing something, explaining why something's true, but you're not gonna leave out the piece where the student learns, you know, an efficient technique for doing it and giving them the practice to do that. You're also gonna give them good, interesting problems, but often you can't do those good, interesting problems if you don't have the skill. 

[01:03:46] Alex Kontorovich: That's absolutely right. And it, it's, for me, it's all about it exposing to system two, spending time working on system two, it's very important that we not just do system one, meaning the fast stuff. Let's, let's multiply, never mind, you know, why does multiplication work like this? Never mind, just do it. 

That's not okay either by itself.  

[01:04:04] Anna Stokke: Is it realistic, do you think, to expect that students or others would appreciate math as an art form? 

[01:04:12] Alex Kontorovich: Oh, absolutely. And, this goes to his point about the kinds of contrived problems that people make to make it sound like math problems apply in the real world. that's for grown-ups. Kids don't need that if kids, kids can, they love stories, they love, you know, infinity. They love all kinds of totally abstract things. 

And for some reason grownups have gotten it into their mind that unless you make something about concrete cookies and Sally came over and you have to split cookies with Sally, that kids won't understand what's going on. No, they love abstraction. It's us grownups who are scared to talk about that and are trying to force some application onto the kid. 

And the kids know it's not a real application. They can smell it, you know, it's like some kind of problem where the answer cannot possibly not be known to the people involved in the scenario. You know, it's like, why are we asking that question? What do you mean you don't know what date it is? Just ask her what date it is. 

Sally's gonna have a birthday in this number of weeks and it falls on this. Yeah, just tell me your, tell me when your birthday is. Why, why are you making me, you know, count something? So those kinds of contrived problems, I think are the kind of thing that, Lockhart is pointing to. 

[01:05:26] Anna Stokke: Oh, and I agree with Lockhart on that issue, by the way. You know, and I would go further. you know, like I've taught lots of students over my career. They usually don't like applications.  

And I think if you're always chasing the applications, you're just never gonna win that, right? If you're telling students,”Oh, it has to be applied and here's an application,” it's just gonna be hard to come up with applications all the time. And I think, you know, that math is, is is just fun, right? Like, can't we just do it because it's fun? 

And, and when I talk about it being fun, I don't only mean the type of like contest problems or you know, like it's fun for people just to like factor polynomials. Why can't that be fun? Why can't arithmetic be fun? Like, I think it's unreasonable to think that kids wouldn't have fun doing those foundational skills. 

[01:06:20] Alex Kontorovich: Yeah when you think of people that just, you know, dribble a soccer ball for no reason, it's just fun. You're like, Oh, you're practicing.” It's like, yeah, I'm practicing. But once it's something that you're fluent with, then it's like, you know, somebody rolling a quarter in their knuckles. 

Because they know how to do it. They just do it and they do it and they do it in the same way. Like multiply. You know, here's like when kids learn to multiply and they get good at it, they're like, “come on, gimme something, gimme a five digit number times a six digit number. Watch me do it.” Like they ask for those things. 

They want to do them because it's easy for them. And it's fun to show someone that they have this skill. Kids love attaining skills. I don't wanna know why grownups are scared of kids attaining skills. I would say one thing though, there there's like applications that are contrived applications and there's word problems. 

I do think there's a place for word problems as long as they're not contrived applications because word problems can get you thinking. Like, it doesn't just tell you, you know, this time's that. It says, here's a scenario, what's the right operation to do in this scenario? You know, like the, the act of there's a skill to converting a situation to what's, what's actually being asked in terms of a mathematical operation. That I would say there's a place for. 

[01:07:58] Anna Stokke: I'm really glad you pointed that out because I don't want people to walk away with, “Oh, don't do any applications.” No, it's absolutely, it's absolutely important to be able to do word problems, because that's a whole different skill, right? It is a really important skill to be able to translate the English language or whatever language you're working in, into mathematical symbols to make it easier to solve a problem. 

The other thing I wanna mention is his comments on the ladder myth though. So he seems to be suggesting that structure be removed from math class and he rejects what he calls the ladder myth. That math is arranged in schools by a series of topics that build on one another, and in his view, the teacher should throw out the textbook and throw out the curriculum and play around with mathematics. 

And students should be treated as budding artists posing and solving interesting problems. So, what do you think about that?  

[01:08:55] Alex Kontorovich: Given the, success that we currently have in the existing system, I guess almost nothing could be worse. So, you know, that's an experiment that one can run. I would like to think that something closer to what the Math Corps is doing is a better way of implementing what I hear Lockhart suggesting. 

Because then it's not, it isn't textbooks. It's a bunch of kids standing with slightly older kids who can show them something cool and they're motivated by it and they wanna do it. It is true, I mean, even our curriculum, you know, as I described the seventh to eighth and ninth grade progression, it is, a progression of skill levels because you're not gonna be able to manipulate algebraic expressions if you can't manipulate whole numbers.  

You know, maybe you can get him on the podcast. He's kind of a reclusive guy.  

[01:09:47] Anna Stokke: I can out what he actually means by all of it. 

[01:09:49] Alex Kontorovich: Yeah. It's hard for me to, to see what he would say about this. My version of what he's saying would be something like the following. You know, like when kids get to calculus in 12th grade, at the same time they learn the words derivative, integral, they, you know, they're looking, learning about slope, they're learning about infinity.  

Like, all of these things are hitting at once, and there's absolutely no reason that you can't do every single subject of mathematics at every scale. So, for example, kindergartners have no trouble solving systems of linear equations. They're doing linear algebra just with, you know, question marks and bananas and answers that involve only whole numbers, and, you know, small whole numbers and subtraction in addition and nothing more complicated than that. 

Because that's all they know, right? So you can't give them general linear equations because they don't know rationals, they don't know multiplication, whatever, but there's all kinds of problems that are very sophisticated mathematics whose answers are in the scope of what they can be exposed to, and they can understand.  

So I think we should be doing linear algebra at every single grade, you know, in all dimensions. And, getting evermore sophisticated with it as the student's understanding of the number system gets more sophisticated. We can be doing calculus at every single level, at every single grade. I just gave a lecture at, at mom math about how to teach calculus to kindergartners.  

You have some bar graph of how many cookies somebody ate on, on any given day of the week, and then you can have a cumulative bar graph. A bar graph that just stacks up all the previous days. Okay, that's called the integral. And then you can have another bar graph that that says, what's the difference from what you had tomorrow versus today? And do that for each day. And that we call the derivative. Like you can expose, it's not, the kids are afraid of vocabulary, grownups are afraid of vocabulary.  

Kids love, everything's new to them. Every word is new to them. They're almost infinitely adaptable to circumstances as long as you don't hold them down. I would agree if I interpret what Lockhart is saying as, we can be exposing kids to so much more sophisticated math at every single level, I agree wholeheartedly. You have to be very careful when you do that because the second you step one foot off of a system of linear equations, you might be asking them to manipulate fractions and they don't know what to do with that and they can't solve it. 

So it takes a lot of effort to get there. You can't just say, here, “you can solve, you can now solve every single system of three linear equations with three unknowns,” but you can make a lot of progress towards understanding what that is within the framework of what every kid at every grade level is capable of. 

 Do you buy that as an interpretation of what he is saying? 

[01:11:55] Anna Stokke: I reject his idea that, you know, this ladder business is a myth. Absolutely, it's gonna, you do have to sort of follow a progression when you're learning math, right? I mean, it, that's what makes it hard for people. 

[01:12:13] Alex Kontorovich: Yeah.  

[01:12:14] Anna Stokke: I actually think it's something that we need to recognize more.  

[01:12:17] Alex Kontorovich: The cumulative nature that everything stacks on top of everything. 

[01:12:21] Anna Stokke: Absolutely. 

[01:12:22] Alex Kontorovich: Yeah. If that's the interpretation of what he's saying, then I'm with you. I mean, there's, there's no other way around it. Math is built up from basic principles onto more and more complicated things, but that doesn't mean we have to wait a long, long time before we introduce kids to really complicated, interesting mathematics. 

[01:12:37] Anna Stokke: But then if you're gonna start intro introducing calculus in grade three, you're gonna have to have teachers who really understand that.  

[01:12:47] Alex Kontorovich: Yeah, I'm a student of, history in the sense that, you know, if you go back to new math, the new math revolution, that's what happened when you allowed research mathematicians to get involved in the educational system.  

Let's make sure whatever we do, we don't make things worse. Yeah. My take on it is that we are way too conservative with what education means, and we need to be running a ton more experiments like the Common Core, you know, I talked to the people who wrote the common, the mathematicians, on those committees. And they said their charge was to have an absolute baseline. 

This is like the bare minimum of what every child should have at every grade level. And the second it came out, it became the reach, it became the target. The Common Core is now like the gold, you know, schools, are, are proudly advertising the fact that they follow the Common Core, which when it was written, was meant to be absolute baseline, bare minimum of what every student should know by, by a certain age. 

But so the, the, the point of that is when we have something like the common core, which is meant to get the entire country on board and doing something in a unified way, that's a recipe for constriction of ideas and experimentation. And we should have just, you know, I'm all for the charter school movement and having lots of random experiments and people figuring out what works. I mean, that's what Math Corps is. Math Corps went completely around the system, just went off and did its own thing and look at what it's been able to come up with. 

Now, I don't know if that's, if that can be made to turn into a charter school, if it'll work like that, if it can be funded, if you'll find the personnel, like all of those are, are big, major questions. But we should just be running a ton of experiments and letting children and parents and families vote with their feed about where, where they want to put their kids and where they think their kids are getting a valuable education. 

[01:14:28] Anna Stokke: Okay. So that would be your solution. Just have a bunch of different types of schools that you know, if, if you want the discovery school, go for it. If you want the, if you want the Math Corps school, go for it. And people vote with their feet. 

[01:14:43] Alex Kontorovich: People vote with their feet, you know, it's, it's so hard to tell a priori. Anyone who tries to, thinks they know how to design a system that will work is either lying or a fool, as far as I'm concerned. Like, you just have to try it and iterate. That's how we get all the nice things that we have is iteration, and what's happening in school is absolutely not iteration. 

It's been, you know, Common Core and, and beyond and, and back. You can recurse a long way and see more and more unification. People are supposed to be like, “We know what's supposed to work. This is what everyone should do now.” And it keeps getting unified instead of getting sort of, there needs to be periods of unification and branching. 

The branching is discovering new things, and then when things work, they should naturally be allowed to flourish. That's kind of the unification is that everyone starts doing the thing that works. What we've been doing is things that don't work that get less, uh, that work even less and less and less. And here we are.  

[01:15:32] Anna Stokke: Well, Alex, I think we could talk for a lot longer about all these really big issues, but, I think we'll we'll end there. So this has been an absolutely fascinating conversation and I really enjoyed talking to you and hearing about your work and all your great ideas. So we'll wait for you to start your own charter school now. 

[01:15:54] Alex Kontorovich: Thank you so much. It's been such a pleasure to chat with you. 

[01:15:59] Anna Stokke: ​ If you enjoy this podcast, please consider showing your support by leaving a five star review on Spotify or Apple Podcasts. Chalk & Talk is produced by me, Anna Stokke, transcript and resource page by Jazmin Boisclair. Subscribe on your favourite podcast app to get new episodes delivered as they become available. 

You can follow me on X, Blue Sky or LinkedIn for notifications or check out my website, annastokke.com, for more information. This podcast received funding through a University of Winnipeg Knowledge Mobilization and Community Impact grant funded through the Anthony-Swaity Knowledge Impact Fund. 

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