Podcast: Measuring the Whole Spectrum of Mathematics Achievement, With Richard Rusczyk
Teaching students to combine basic ideas to solve novel, difficult problems is imperative to lay a foundation for STEM pursuits. Rob and Jackie sat down with Richard Rusczyk, founder of the Art of Problem Solving Initiative, coauthor of the original Art of Problem Solving books, and cofounder of the Mandelbrot Problem Solving Competition.
- AoPS and Beast Academy Math programs for Advanced Students. Art of Problem Solving. (n.d.). Retrieved December 9, 2022.
- Robert D. Atkinson and Merrilea Mayo, “Refueling the U.S. Innovation Economy: Fresh Approaches to STEM Education” (ITIF, December 2010)
Rob Atkinson: Welcome to The Innovation Files. I'm Rob Atkinson, founder and president of the Information Technology and Innovation Foundation.
Jackie Whisman: And I'm Jackie Whisman. I head development at ITIF, which I'm proud to say is the world's top ranked think tank for science and technology policy.
Rob Atkinson: And this podcast is about the kinds of issues we cover at ITIF from the broad economics of innovation to specific policy and regulatory questions. And today we're going to talk about STEM education. I think you all know STEM. The M in STEM is for math, and that's a pretty fundamental skill, especially for students who plan to go into STEM or STEM-related occupations when they graduate. Yet, a US average math scores on the NAEP test, which is the sort of standard test that the government has, they still show that the average scores of the US are below proficient. Perhaps even more importantly, the advanced scores have largely plateaued in the last 10 years. In other words, the top 10%. So it's not just that we need more people who know math, we need more people who are great at math.
Our eighth-grade top 10% scored lower than Korea, Taiwan, Singapore, Japan, Moscow. So these countries not only have overall better average scores, their top performers are doing better, at least on these test scores, than our top performers. And when people talk about this, and there's a lot of recognition that we can do better, it's largely around issues like, well, we should just pay teachers more, or if we just had smaller class sizes, or we'll just have more math requirements, like that somehow is going to solve the problem. And almost nobody talks about how we need to change how math is actually taught. And that's what we're going to talk about today.
Jackie Whisman: Our guest is Richard Rusczyk who founded the Art of Problem Solving Initiative, which is a nonprofit that runs the Bridge to Enter Advanced Mathematics program for high potential students from underserved communities in New York City and Los Angeles. He's also co-writer of the original Art of Problem Solving books, Rob owns them, and a co-founder of the Mandelbrot Problem Solving Competition. Prior to this, he was a participant in national math counts, a three-time participant in the Math Olympiad summer program, and a USA Mathematical Olympiad winner. So he's legit, and we're happy to have you here.
Richard Rusczyk: Well, thank you very much for inviting me.
Jackie Whisman: So in your books, you've proposed a new approach to learning math. Can you summarize that to get us started?
Richard Rusczyk: For our primary curriculum for middle and high school, we'd start every section with the problems that they'll be learning in that chapter, in that section. We want the students to have an opportunity to discover the mathematics for themselves. Now, we do some scaffolding in those questions. We do kind of gently lead them, hopefully, to be able to figure out a lot of the concepts on their own and then follow up by reinforcing what we intended for them to discover. Now, sometimes they find completely different ways to solve the problems, and that's fantastic. But this is what we need our students to learn how to do, how encounter new problems and discover ways to solve them. And that's what we mean by problem solving, is solving problems you've never seen before.
Jackie Whisman: So why this approach? And maybe we can also talk about what's wrong with the traditional method.
Richard Rusczyk: Well, this is how I learned math. So the way I learned math and how to solve hard problems, I participated in a lot of math competitions when I was in middle school and high school. Now, my math classes, I could get through my math classes with a very simple strategy. I could just memorize everything. It doesn't work for me so well anymore now that I'm 51, but when I was 14, I could memorize everything in sight, memorize all the tools, all the tricks. The test problems looked exactly like the homework, so that strategy worked just fine. Of course, nowadays, computers are much better at that strategy than we are. If there's a problem that the world has seen before, we can teach a computer how to solve it. The computer's going to get it right every time.
When I started taking math competitions and encountering these problems that involved ideas that I had never seen before, at least I thought I had never seen before, I had to develop the ability to take those basic skills that I had, combine them in interesting ways, and solve those problems. And it took me a long time to develop that skill. It's not something that just came naturally to me, but it wasn't until college that I really understood how valuable that skill was. I went to a very aggressively average high school in northern Alabama. Even our football team went five and five every year. Maybe three quarters of us graduated. Not a whole lot of us went straight to a four-year college. I was going to Princeton, so I was worried about competing against students who went to magnet schools or who went to fancy private schools. I was worried that they had special training that I didn't have.
And I saw very quickly, right in the first semester, that I had it backwards. I had special training they didn't have. I had spent these years struggling with these hard problems, many of which I couldn't solve. And when I got to college, well suddenly I'm in these classes where on the tests, there are five problems, you've got three hours. That's nothing like high school. And the problems don't look anything like the homework. That also is nothing like high school, even in the fancy schools, even in the magnet schools. But I had struggled with those types of problems for a few years before I got to college. And it wasn't just math. Physics, chemistry, engineering, computer science, economics, philosophy, anything that required combining basic ideas to solve novel, difficult problems, I had that skill walking in the door.
Most of my classmates didn't. And a lot of them left science as a result. And that's what we need to prevent. Because those students, they could do it if they just learned to fight through it the way I did. And many of my college classmates did. The first year they were in college, they were really struggling. By the end, they fought through. By the end of college, they could do what I could do, and some of them could do things that I couldn't do. But those students who left along the way thinking they weren't smart enough anymore, that... that's a huge loss. Those students were every bit as capable as I was. They just didn't have the opportunity to develop the problem-solving skills that I developed in middle school and high school.
Jackie Whisman: Yeah, because it really seems like it's not really always about the math. It's about the problem solving, and I think that's why you've named your organization and your books as you did. It's not really about finding the answer to the problem. It's the way that you're teaching the brain to work in finding the answer. Because I remember in high school, I was always resistant to math. And I'm like, well, I'll always have access to a calculator. And it wasn't until adulthood that I realized, well, it was about figuring out how to solve the problem, not about really the answer.
Richard Rusczyk: Yeah, that's right. Your calculator's not doing math. It's doing arithmetic.
Jackie Whisman: No.
Richard Rusczyk: Yeah. When I'm teaching math, I don't think of myself as teaching math. I really do think of myself as teaching these more general problem-solving skills. Math is a great tool to do that. It's not the only tool. We can do that with a lot of different tools. But math has that very quick feedback mechanism, and also it underlies physics, it underlies chemistry, increasingly biology, economics. It underlies all of these other things. But sometimes people ask me if I'll be disappointed if some of my students don't become mathematicians. I'm like, oh no, no. I'd really be disappointed if most of them do become mathematicians. We need these thinking skills. We need these skills everywhere.
Rob Atkinson:Having a son who's now older and in computer science did a lot of math, but a lot of what he did, he did on his own just because he was interested in it. Now, having a daughter who's a junior and she took AP history at our local high school, and I have to say, I don't know. I have to say I was just like, to me, the only thing different about AP history is you had more facts to memorize. And facts that I... And I know a lot about history. I wrote a book, a peer-reviewed scholarly book about American technology history. I didn't know half of these things, don't need to know half of these things.
What I needed to know and what I wanted my daughter to know is how do you analyze these issues? How do you think them through? How do you figure out what meant what and how did it all... So we kind of did that on our own with industrial history. So your point to me is, as you pointed out, Richard, is more than math. Almost a different way of doing pedagogy, at least when you're at a point like high school. [inaudible 00:08:45] get it earlier, okay, but at least in high school. It's a pretty core skill that they don't seem to really focus on.
Richard Rusczyk: Yeah, absolutely. History... For me in high school, history was ‘This is ridiculous’. We're just memorizing a bunch of things. History is just boring and stupid. And then after over the last 20 years, I have read a lot of history books. It's fascinating. A really good history book... Like a high school class, you should give students two of those books in a year and just really dig deep in them. That would be way better than memorizing all the dates and all the names and all the presidents. I agree with you. It is the same problem there.
Rob Atkinson: Yeah. It's frustrating. We were talking earlier, before we were recording, I ran across you because my daughter was talking about how frustrating normal math is and how she wanted to really understand the underlying reasons, what's the logic behind these things so that then she could take that knowledge and apply it to any problem she sees. And so I went on YouTube, and I found you. And you described in this particular interview your experience in high school where you... And I don't know if you could just share that, because I thought it was such a cool experience. You do well in high school, you're getting As, you go to this Olympiad...
Richard Rusczyk: Yeah.
Rob Atkinson: Eh. Do it again, eh.
Richard Rusczyk: Yeah, that was... Yeah. No, no, no. I mean it was very foundational for me. And it's something a lot of my students go through and believe that they're the first one that's ever had this happen to them. I'm like, no, it happens to everyone. So when I was in middle and high school, the regular classes, memorized through everything. The early levels of math competitions, I could memorize my way through a lot of those too. I had this huge stack of formulas. I'd just memorize them all and I could do a lot of the more basic problems on these competitions. And my sophomore year, I qualified for this training camp. This is a training camp for the top 24 students on a series of math tests in the country. And from that group, they select the top six that will represent the United States at the International Math Olympiad.
So now I'm coming out of Alabama. There's a math contest circuit there, but no one from this math contest circuit had ever been invited to this camp. So I go to this camp thinking I'm certainly the smartest person in the world. And then I get to the camp. I'm there for five weeks. You have a practice test every other day. It's three or four problems, three or four hours. And so I see 50, 60 questions while I'm there, and I get exactly none correct, zero correct over this training camp. So it was clear I was not the smartest person in the world. So I go back home, and my junior year I'm like, I know what the problem was. I didn't know enough math. I need to know more math those other students did. So I spent that next year memorizing more and more and more formulas.
My stack got larger and larger and larger. I was going to get back there. I was going to show them what I could do. So I got back to the camp, and I showed them what I could do, because they had the same thing. I saw 50 or 60 questions during those five weeks, and I showed him exactly what I could do. I still couldn't solve any of the problems. And I was still definitely not the smartest person in the world. And I went back after that. First I was a little demoralized, but then it was really nice that this happened to me. In a math competition, it's not that important. A lot of kids have this experience that I'm describing in college, but for me, I could step back and look at how I was learning and think I don't need to learn more formulas. Maybe I need to know fewer and need to have the skills and have the mindset to be able to develop new formulas, new approaches when I needed them.
So I spent my senior year going back through my stack of formula sheets and trying to prove some of those formulas, trying to prove some of these theorems. And I couldn't prove them all, that's for sure. I couldn't prove a whole lot of them, but I could prove some. And that transition worked. My senior year, I was one of the Olympia winners. I was an alternate for the US team. So I'm still not the smartest person in the world, not even remotely close. But I developed those skills that would take me through college and then into my career. And those are the skills that we need kids to develop much earlier. And the thing that's so disheartening when you look at what's happening in math education is this sort of perseverance, this sort of, I'm going to keep going until I get it, kids have this. It's a natural, innate thing.
Kids want to solve problems. Look at a little kid. The three-year-olds, the four-year-olds, they keep going at things. And the thing is, those three-year-olds and those four-year-olds, they're tougher than the 18-year-olds and 19-year-olds. Yes, they'll quit, they'll cry. They see adults doing these amazing things, riding a bike, chopping vegetables, whatever it is. They want to do it too. They can't do it because they're three or four, but they'll try. They'll fail, they'll cry, they'll quit. But they're four, and they'll do something amazing. They'll come back 20 minutes later and try again because, therefore, they haven't learned yet how to quit for good.
And somewhere between age four and age 18, we train that out of kids by getting them to expect, in their math class, they got to get it right every single time. And if they can't get it right every single time, that's the ceiling. They can't go any further. They're finished. And that's what we need to pull out of our math education system and keep that sense of challenge, that sense of exploration all the way through. So the first time that these kids hit it again at 18, well it's not the first time they're hitting it when they get to that challenge.
Rob Atkinson: I'm just curious, how much do you think that... In my daughter's case and I think in what you're talking about for a lot of kids, I'm going to learn it in a way where I memorize the formulas and I see these problems and I can fill in the right bubble chart. Oh, it's going to be an A or a C or a D. Then I'll get a good score. The teacher's super happy because the teacher is rewarded on, hey, my kid's done really well. The principal's super happy. The district is super happy. The state is super happy. It seems really hard to sort of say, wait a minute, we're going to do something different here. Because problem solving is a very different thing, and it's a different set of skills and outcomes. And how much of that sort lock in of the testing regime we have and all that do you think is part of the barrier to moving to where you want?
Richard Rusczyk: Yeah, the testing regime is definitely part of it. And there's also the idea that the students who are getting 100s on the tests are doing well. The ceiling is fairly low. The student who's going to be a future scientist or a future engineering engineer, their ceiling in the classroom is not high enough, so the teachers don't have a feedback mechanism. So I received an email a little bit after I started Art of Problem Solving. I received an email from a classmate of mine that I didn't actually know in school, and his email kind of explained why. He explained that he had gotten perfect scores on all of his middle school and high school math tests. And then he went into Princeton, and the first-year math class was, he described it as the most bewildering experience of his life, and it poisoned him on math for years.
And I saw this a lot at college, students who walked in thinking, I'm going to be a scientist, I'm going to be an engineer, I'm going to do amazing things. And then he hit that first math class, they hit that first physics class, and they're out because they didn't have the sort of training that I had. And this gets back to what you're talking about here. There's no feedback loop. Because that student didn't go back to the teacher and say, yeah, what's up? You didn't prepare me for that. Because the teacher doesn't see that later failure. That teacher still thinks of that student as a success. And then when you think about what's measured as success in the classroom, a lot of it is just a single state test. It's a single bar instead of evaluating schools, evaluating teachers on empowering students to achieve their potential wherever they start.
So when you have this single measure, this single bar, students that are far from the bar, either way above or way below, the system doesn't have a whole lot of incentive to focus on them. It's the students who are just below the bar. You want to get them over. And the students who are just above you want to keep them above that bar. All of the others, kind of, they don't matter as much. So as we construct incentive systems to evaluate schools or to report on schools, to inform parents or inform administrators, we want to see the whole spectrum. And that means you need measures that see the whole spectrum. And our basic tests do not do that.
Rob Atkinson: One... And then I want to let Jackie ask a question. But one thought, we did a big study with a colleague of ours, Merrilea Mayo, and I think it was the longest report ITIF has ever done, like 250 pages. We don't usually do that. And it was called Refueling the US Innovation Economy. It was looking at all the analysis of STEM and all this. One of the things that struck us the most on that is the narrative is we need a pipeline to be bigger, so get kids in kindergarten to do that.
The reality is the pipeline is pretty big. It's just that the dropout rate is pretty high, as you noted there. If we could just keep 20% more, 30% more kids who are really this stuff in high school all the way through graduation, we got a lot of people who could do this. And your story there about these kids going to Princeton or wherever and just going, oh my God, I don't want to do this. I'm going to go transfer over to political science or something. And the colleges don't seem to really fundamentally care. That's the other thing that I find bizarre. It's like, eh, what difference does it make?
Richard Rusczyk: Yeah, the colleges don't care at all. I mean, the colleges... Certainly when I was a student, which was a while ago, but I don't think this has changed much, the colleges don't do much support of students for thinking through the challenges of the first semester or thinking through the career trajectories that they might have. And I think there's a lot of work colleges could do there.
Rob Atkinson: Yeah.
Jackie Whisman: I wonder... I mean, maybe I never used advanced math because I went into a field that didn't require math because I wasn't very good at it. But I'm wondering if everybody really needs to take advanced math courses or why not a statistics focus instead of calculus?
Richard Rusczyk: Oh, definitely. Statistics is wildly underappreciated, undertaught in schools. I think, broadly speaking, education systems should have two goals there. The first goal is broad literacy. We want everybody to understand a baseline set of things in a set of core disciplines. And that is what a lot of our system is built around. We can talk about how well they even accomplish that. But there's a second goal, and that is deep mastery in something that really resonates with the student. So if I were designing an education system from scratch, let's say I'd take the day, I'd split it in half and I'd be like, all right, the set of things for a student that doesn't really resonate with them, well, we're going to get you up to a certain baseline over here in the first half of the day. So you're really into history, you're really into deep reading and really understanding how the world has transpired or hoping you can understand how history has transpired?
All right, first half of the day you're going to get your core skills in math. We're going to get you up to a baseline, we're going to get you through. We're to understand exponential growth so when there's a pandemic, you're going to understand what's going on. We're going to get you to understand some of these core things. But you're not going to learn calculus. You're not going to learn linear algebra. That's okay. Second half of the day, you're going to do some really deep reading in history that the students for whom history is going to be a baseline thing for them, they're not going to do that, and that's okay. But the second half of the day, you're going to get to go really, really deep in something you love.
And that means you're not going to take 16 AP classes. You might only take four, but you're going to be really good at those four things. Far more important even, you're going to learn how to get really good at something. Because that is a transferrable skill. We don't know what these kids are going to need to be able to master in 20 years. We know they're going to have to be able to master something new at that time. So that ability to excel, that's what we need students to develop during high school. But we don't need them to excel in everything. We don't need them to even be very, very good in everything. We need the baseline, and then we need them to be able to excel in the areas that most resonate with them.
Rob Atkinson: So I was asked to give a talk on STEM education to, I don't know, a group of national STEM Educator Association thing, a couple hundred people. And I said, essentially, I said, I don't think everybody... I said exactly what you said. Everybody needs a baseline, I get that. But after the certain point, to me, you ought to be able to identify what it is you love, what it is you are passionate about, and as long as it's academic-y. I'm not saying, hey, I'm passionate about pogo sticks or something. And then you ought to be able to go deep into it.
And I think we really... It's almost, to me, it's disrespectful to children, teenagers, as human beings to say, we actually don't care one iota what it is you are interested in. Because I know in my daughter's experience in high school, and I look at Montgomery County High School, the number of electives you can take now because the state government, oh, well, they have to take this and now they have to take that. These are all important. So they just cram it in, and the kids really have very little choice about saying... One of the things is when my son was homeschooled, he was really, really interested in Lisp programming, which is a language.
Richard Rusczyk: That's wonderful. I wish someone had showed that to me when I was young.
Rob Atkinson: Yeah. And so he's like, there's this national Lisp conference at MIT. And he says, I want to go. I'm like, fine, put you on the train. He goes up there, he spends a week on it. He wasn't going to get that level of depth in high school, but he loved it. And then he started reading, learning Python, all this other stuff. That was his thing. And now he's, I think, going to get a PhD in AI and machine learning. So couldn't agree with you more. I guess so the question I would have then is how do we connect this... I guess maybe the question is is there a path for change? Is it individual schools who are more experimental? How do we break this mold?
Richard Rusczyk: I mean, I think part of it is going to be a few brave school leaders who just do it and just say, we're going to do things this way. And I think part of it's how we communicate about things, like to say not all students should take calculus. People may code that in different ways. To add to that message, every student should have the opportunity to take calculus if they want to. Because when people will hear, oh, not every student needs to take calculus, then you get starting to talk about equity and all of these sorts of things, which are real problems. But that's about access. There are a lot of students out there who don't have access to the sort of resources that we provide at Art of Problem Solving, and we're trying to solve that as well.
So I think part of it is a communication thing. But I think for getting people to think differently, to think about this as baseline of literacy plus creating the space for students to excel in areas that they deeply care about, you're going to need a few school leaders who make that happen. And you're going to need them... Some of them will be in unconventional schools, but you're also going to want to see that in some conventional schools as well to serve as a model. So a superintendent or a principal that is willing to say, all right, we're going to structure our day a little bit differently, and we're not going to care quite as much about the tests. And you know what might happen? They might end up doing better on the tests anyway because the kids will be more dialed in.
Rob Atkinson: So one last point I would make is, if we think about this as a pretty interdependent system, it just didn't kind of happen, there are all these forces on it that led us to where we are. One of them, I believe, is college admission requirements. We're trying to do a little study, I haven't published it yet, but we are looking at math admission requirements for colleges like University of Wisconsin. They don't allow statistics to count, and they require four years of math and three years of this and... So in some ways, the high schools are prisoners of that because they know that they want their students to be able to go to good colleges regardless of their socioeconomic status. So in some ways, it's kind of an overall system thing. We got to look at it at all the levels and how do we break that down to allow exactly what you're talking about to be the norm.
Richard Rusczyk: I wonder if high schools can game that system, if they choose to. You can define that literacy path in ways that will hit those requirements. Now, the students need to know going in, the schools need to work with the kids to get them to really understand, if they're going to go into science, anything science-related, they're going to need more than the baseline.
Rob Atkinson: Sure.
Richard Rusczyk: But honestly, if they're going to go into science track, they're going to need more than what the schools are doing now anyway. So that should be an easy message. But as long as the students, again, have access to these sort of opportunities, I think the schools might be able to game that system.
Rob Atkinson: I love that. I love that way of thinking. We should do everything we can. Richard, this was great. I'd love to keep talking, but we are limited on our time. Really, really want to thank you. I hope everybody who's listening, if you have a kid who's going into high school math, really pick up the Art of Problem Solving series. You can pick it up on Amazon. It's where we got it. And it's a great just a textbook for just really sort of going through this. One of the things my daughter said to me the other day, she said, dad, I wish that high schools had textbooks for math. Because that's the other thing, they don't, at least in Montgomery County. They have workbooks and papers. And this way I can see the entire progression. I can kind of see where I am and I can go through it. Just even having a textbook like what you're talking about would be a change. So again, Richard, thank you so much. I really, really enjoyed talking with you.
Richard Rusczyk: Thank you so much.
Jackie Whisman: And that's it for this week. If you liked it, please be sure to rate us and subscribe. Feel free to email show ideas or questions to [email protected]. You can find the show notes and sign up for our weekly email newsletter on our website itif.org. And follow us on Twitter, Facebook, and LinkedIn @ITIFdc.
Rob Atkinson: We have more episodes and great guests lined up. New episodes drop every other Monday, so we hope you continue to tune in.