All Things Being Equal: Why Math Is the Key to a Better World

[00:01] Announcer:

Welcome to Principal Center Radio, helping you build capacity for instructional leadership. Here's your host, Director of the Principal Center, Dr. Justin Bader. Welcome, everyone, to Principal Center Radio.

[00:13] SPEAKER_00:

I'm your host, Justin Bader, and I'm honored to welcome to the program Dr. John Mighton. John is a playwright, mathematician, author, and founder of Jump Math, a charity dedicated to helping people fulfill their potential in math. He's the author of three best-selling books and has been recognized as an Ashoka Fellow, awarded five honorary doctorates for his lifetime achievements, and appointed as an Officer of the Order of Canada. He holds a PhD in mathematics from the University of Toronto, and one of his books is All Things Being Equal, Why Math is the Key to a Better World, which we're here to talk about today.

[00:50] Announcer:

And now, our feature presentation.

[00:53] SPEAKER_00:

John, welcome to Principal Center Radio. Thank you so much. Let's talk about that thesis that is there in the subtitle, that math is the key to a better world. What led you to believe that, and what is the case for math being the key to a better world?

[01:08] SPEAKER_01:

Well, because I struggled in math when I was younger, I mean, I almost failed calculus the first time through university. I had a very fixed mindset. I thought you had to be born with a gift to do math. I have pretty radical beliefs about ability in math coming from my own experiences. In my 30s, I got the confidence to go back into mathematics and eventually did a PhD. And there's so much evidence now, both from cognitive science, from studies Jump has done, that math is far more accessible to people than mathematics.

[01:38]

you know, general population beliefs. In fact, I believe it's actually can be the easiest subject for kids. It's a subject you can close the gap between kids most quickly. And that's really the thesis of the book is that we've neglected math as a tool for social justice because we underestimate potential of children in math.

[01:53] SPEAKER_00:

There is this perception that there are math people and not math people. And if you're a math person, you figure that early, you're a prodigy. There's this idea in elite mathematics or very high level mathematics that you have to do your best work by the time you're 22 or something like that. So I'm very interested in your trajectory, coming back to math a little bit later, not super late in life, but certainly later than the undergraduate years when a lot of people go into mathematics. What was it like coming back to mathematics as, you know, I don't want to say the old guy, but as an adult learner who was no longer a teenager and getting into advanced math? You know, I think it's interesting to think of ourselves as educators, but also as students.

[02:36]

Talk to us a little bit about your experience as a student of advanced math as an adult.

[02:41] SPEAKER_01:

It was a great experience because I actually knew how to learn. I knew that you could become better at anything through deliberate practice. And things that were totally mysterious to me in high school, like why does a negative times a negative equal positive, or what's the root of a negative number, became clear when I just worked through those concepts at my own pace as an adult learner. And there were no great mysteries. There was nothing too hard for me. So it was a wonderful experience.

[03:06]

But I remember in first year, I think I bombed a test in group theory and I was almost bedridden. I thought I have to drop out now. All my dreams are shattered, even though I was doing quite well in everything else. You know, three years later, I realized I could teach that material on the tests that I failed. So it was clear to me it wasn't the level of difficulty of the subject. It was just having the time to digest it and practice enough and learn it deeply.

[03:35]

That was the only barrier. And I saw also universities are set up to make sure kids don't have that experience. You know, they're set up to weed kids out. They go too quickly. I tell students now, learn the course before you take it to the time Second time through is way, way easier.

[03:52] SPEAKER_00:

Yeah, let's talk about that issue of time, because often we conflate the speed or the ease with which someone learns new material with their ability, right? We say, if you don't learn it quickly, you know, we kind of assume, even if we don't say it explicitly, that if you can't get it the first time, or if you don't learn it quickly, then you just don't have the aptitude, you're not going to get it. And, you know, we kind of steer people in a different direction and certainly not into majoring in math or doing graduate work in math. In cognitive terms, why is time so important and why do we not really get that as a profession? You know, why do we not notice the importance of time in succeeding in math?

[04:32] SPEAKER_01:

Well, I do a lot of demo lessons ranging from gifted classes to behavioral classes. But universally, I tell the kids, I'm not a very fast mathematician. I don't care how fast you are. I say I make mistakes all the time. And I also say, if you don't get it, you can put your hand up and ask me to teach it again. It's my fault.

[04:49]

So if you set that up with kids and you remove that pressure of time and the hierarchies, if you structure your lessons so the kids can't compare themselves easily, and you always have some bonus questions on the board because it'll always be kids who are faster. But you don't make them an issue. You say, as long as you get a few of these questions, you're doing perfectly well. Then kids' brains can work because they stop comparing themselves. They stop being anxious. And also you keep them in a zone where it looks hard to them, but not too hard.

[05:19]

And that's the magical zone for learning. So if you take away the pressures of time, of competition and things like that, Within a lesson or two, you can see classes change. I've seen behavioral classes cheer for math after a couple lessons like that. So it's just a tragedy that we neglect the science of learning and the way the average brain learns when we design our lessons.

[05:41] SPEAKER_00:

Yeah, let's get into some of that science of learning, because I noticed that one of the endorsers of the book is Daniel Willingham, the University of Virginia professor who is a cognitive psychologist and one of our best translators of psychological research into educational practice. So I was very glad to see his name on the book and have that seal of approval, because certainly, honestly, we hear a lot these days That is basically opinion about math and people have kind of ideology about math. But there is a robust research base. There is robust literature about the science of learning, the science of math. And I wonder if you could take us into some of that and especially what we tend to not really take good advantage of. What's out there in terms of the evidence base that we miss as educators or that we do the exact opposite of?

[06:33] SPEAKER_01:

Yeah, I was very honored that Daniel Willingham said he found the argument of the book persuasive because he's one of my heroes. And he's one of the best translators of the research into practice that I've met. I think, you know, in reading, people have finally realized there's a science of reading and that, for instance, phonics and decoding are important. But when I ask people, what's the science of learning math, they don't really know. And I've talked to the hundreds and hundreds of educators, and I think that most people aren't really aware of some of the most important research in the science of learning math. So I'm happy.

[07:08]

It's a bit long winded, but I'll just tell you a couple of things. One is most every program that I'm aware of in North America is popular and successful is based on the idea that kids will learn to be real-world problem solvers by grappling with a lot of real-world problems. In fact, you can't even argue against that position because they'll say kids should have a lot of practice doing rich, authentic problems. Well, I guess the opposite would be poor and inauthentic. So it's hard for people even to escape the language that they're caught up in. But test scores have been flat or declining for the past two decades when that approach has just reigned across North America.

[07:46]

Give kids lots of complex real world problems, low floor high schooling problems. Let them find their level. Make them love math because you're finding highly contextualized problems for them. But that's been an abject, a complete failure, a complete failure. It only works with some kids. And so there's an article that came out in Science Magazine 15 years ago, and the title is The Advantage of Abstract Examples in Learning Math.

[08:13]

And I've never met an educator who's aware of that article. This came out in Science Magazine. There's a vast body of evidence that right from preschool to university level that we become better real world problem solvers by starting with more abstract examples and adding complexity and context just across the board. And so in my book, there's a whole chapter on that. What would problem solving look like if you actually approached it from the different direction, starting with more abstract representations and then adding context? So it's a serious problem that no one's aware of that because that approach wins hands down.

[08:51]

And it's connected to another problem. And this is the last thing I'll say on this. There's an article that came out in Nature magazine about four years ago called the 85% rule for optimal learning. And the research suggests that we are happiest and learn most efficiently when we're about 85% proficient at something. We've almost got it. We have to stretch ourselves a bit.

[09:14]

It can't look too hard, but it can't look too easy. It's also called the Goldilocks zone, deliberate practice. There's a whole bunch of different names for this research. But where does the average kid spend their time in an average math class? It's not in the 85% zone, more like the 40% zone. And so we constantly push kids out of that zone where they can learn optimally and are most engaged, partly because we ignore the research on concrete versus abstract.

[09:41]

So this is what the book is about. A lot of the middle of the book is really looking at myths around how kids learn math and what the science says.

[09:51] SPEAKER_00:

So to go back to the, what did you say, low floor, high ceiling problems, you said the research does not support that approach. Is that right?

[09:58] SPEAKER_01:

Well, the research doesn't approach trying to create great problem solvers by giving them lots of complex problems that are highly contextualized and where many different levels work. That's really the end of where you want to get to, I think. But, you know, I've surveyed hundreds of teachers and said, if you had to teach chess, would you teach kids the full game and give them a bit of coaching, let stronger players help weaker players? But let them play chess because that's the object, what you want to get to. Or would you start with mini games with just a few pieces that have been designed by experts to help kids develop mental representations to see how a piece moves, to see what's a strong position, weak position and add complexity. Almost all teachers pick the second.

[10:41]

Then I say, well, do some action research. Look at any textbook program or program that's available. Are the problems the same level of difficulty and complexity at the beginning of the year and the end of year? Is the density of language the same? Are the demands the same? If that's the case, then...

[10:58]

it's probably not in line with the science of learning because you just chose a program, teachers at university chose a program that introduces difficulty incrementally. So that's what I mean. All those things come together. If you don't gradually increase the level of difficulty, if you don't keep kids in that zone where they're working optimally, If you create hierarchies in the classroom, if we have one talent, it's for knowing where we are in a hierarchy. Kids as early as grade one can recognize if they're in the inferior group. So what happens if all the kids are doing these problems that have multiple levels and kids notice that they're not doing the same kind of work?

[11:35]

Then you have an added problem that they start to give up and stop paying attention. So we've done studies in Jump where you show that if you start more incrementally, you go back a few years You give everyone the foundations they need. You quickly move to grade level. You use bonus questions to entertain faster kids, but everyone is working on roughly the same stuff. And you increase the level of difficulty gradually and you keep everyone in that zone. Then you just get jaw-dropping results by the end of the year.

[12:02]

We had an article published in the New York Times. It was the most emailed article in the Times about one study we did where a teacher took a very ordinary grade five class, kids as low as the ninth percentiles on them, about three years behind. By the end of the year, the lowest mark was in the 95th percentile. A year later, they all wrote the Pythagoras math competition and all but three got awards of distinction. So that's the power of a more incremental approach based on the science of learning. Kids were doing well beyond the kind of problems they were expected to do, but the entire class was doing those problems.

[12:35] SPEAKER_00:

John, that sounds, I mean, I don't want to be reductive here, but that sounds like a fairly traditional way to teach math, right? To take the concepts, to break them down, to teach them systematically, to build knowledge, to build students' fluency with the different pieces and then put them together. That sounds pretty traditional, right?

[12:53] SPEAKER_01:

It's a bit of a complex issue. It is traditional. In one sense, there is a lot of evidence that explicit teaching or direct instruction works. I don't know why we've neglected that evidence. And also, people used to believe that basic knowledge was important, and there's a vast body of research showing you can't be a great problem solver if you don't have domain-specific knowledge. So in that sense, traditional teaching wasn't so bad.

[13:19]

Unfortunately, it was often wrote or the teacher may not have understood it well enough to do great traditional teaching. But there's two twists you can make that make it much more progressive, I think, and also effective. One is we call jump structured inquiry. So in traditional teaching, the teacher might show how an algorithm works and then give kids practice, which is fine. They actually end up being able to do it. But what if you just structure it as a series of Socratic questions where you're just asking questions all the time and helping kids make critical discernments, helping them see the connections?

[13:53]

They can actually discover the algorithms themselves and even beyond that, create their own. So that's one element you can combine Socratic questioning and structured inquiry with traditional, with more directed instruction or guided instruction. Also as kids become more proficient and confident, you can let them struggle more. You can let them figure out three things at once. So that's one thing you can do to really amplify the power of traditional teaching. Second is no one is really looking at the impact of hierarchies in the design of math curriculum.

[14:27]

And when we ask teachers, what would you do if you're teaching a kid who's like three years behind grade five? And you're tutoring them. Universally, they say we'd go back to that level. We'd scaffold. We'd give lots of practice. We'd teach to mastery.

[14:39]

We'd expect the child to learn it because we're teaching it really well. And then we'd raise the bar incrementally with bonus questions. We'd give them things that are pushing that zone. And the really encouraging thing is we've shown in jump studies that you can do that with entire classrooms. In math, it is relatively easy to go back to the right foundational level. Even in algebra, we've got confidence building lessons for algebra for grade eight kids that go back about five years.

[15:04]

But within one lesson, they're at grade level. If you know how to scaffold the math, you can get kids very, very quickly to grade level. And the only differentiation you need is extra questions for the faster kids, kids who are initially faster if you're scaffolding really well. And when the kids are initially weaker, see, I'm always getting perfect on my assessments. They don't care so much if someone's doing some extra questions. And immediately the damaging hierarchies stop having that impact.

[15:32]

So that's how that teacher, within a couple of years, got all the kids writing the Pythagoras competition. First by making them feel equal, and then they became, to a shocking degree, they weren't exactly the same, but in terms of curriculum standards, they were pretty equal. So it's just this crazy illusion we've created that you have to have hierarchical classrooms in math. And it's self-defeating because it becomes a vicious cycle when kids are convinced they're not good at math. They stop engaging, thinking, and eventually they develop anxieties and they can't even hear the teacher.

[16:04] SPEAKER_00:

Yeah. So I wanted to ask about a couple of things that have come up now. One of them is the idea of anxiety or math anxiety, because we hear a lot of people saying that various things that teachers do create math anxiety or trigger math anxiety. And certainly anxiety seems bad for learning. We don't want to be causing anxiety. And then the second issue that I wanted to ask about was productive struggle or the, you know, the idea that we need students to grapple and kind of muck around a bit and, you know, struggle without explicit support.

[16:36]

Sometimes talk to us about those two issues, if you would, because I see a lot of debate on social media about anxiety or math anxiety and productive struggle.

[16:46] SPEAKER_01:

Yeah, so I think anxiety affects teachers as well as students. If a child decides, so I've seen some studies, early as grade one, kids know if a teacher has different expectations of them, even if it's a great teacher. And when we train teachers in JumpMath, we often say, so what happens to a child who decides they're in the inferior group? And teachers will say, well, they'll stop working, they'll stop engaging. And as they become anxious because they realize they're not succeeding, I've seen they even develop a kind of guessing instinct. They just want to not look dumb.

[17:20]

They're barely hearing what you're saying at some point. So the anxiety makes it harder and harder for them to learn. And they sometimes pick up on the anxieties of their teachers, especially there's research suggesting female elementary students pick up on the anxiety of female math teachers. If they're math phobic, they're picking up those anxieties too. And I think all those issues are really compounded by the fact that we don't really understand what productive struggle means. I agree 100% that kids need to feel that what they're working on isn't too easy.

[17:52]

And that's that 85% zone. They need to feel that they, it's a challenge, but they also need to feel safe enough to embrace that challenge. And otherwise, it's just not going to work. If you push them too far outside that zone, and we've seen the evidence of that across an entire continent, kids are really struggling in math. They're not enjoying it. And pushing them out of that zone is not working.

[18:18]

So if productive struggle means keeping them in that zone where you vary one concept, you vary one skill, let them meet that challenge. There's a lot of research on mastery. We love mastering things. It's one of our deepest motivators. So if you allow them to do that, productive struggle isn't a fixed thing. Once they've been in that zone long enough, you can then broaden that zone.

[18:40]

You can let them struggle more. So I agree 100%. The end is to let them struggle, to get them to the point where they can engage in problems without support. But we keep mistaking the end where you want to get kids to for the means to get them there. And the research is just It's so clear that that zone of productive struggle is so narrow at first for most kids, especially if they've struggled most of their academic career. But if you do create a series of problems that look hard to them, even if they're just the tiniest variations, you suddenly wake up their brains.

[19:14]

Literally, I've taught a lot of behavioral classes. I've taught a lot of special ed classes. And you can't believe how quickly the kids start to learn when you keep them in that optimal zone.

[19:25] SPEAKER_00:

It almost strikes me as kind of a swimming analogy. A lot of people were taught to swim by literally being thrown in the deep end and being told, figure it out. And maybe a parent or an older family member thought that was funny and thought that was a perfectly good way to learn. And ultimately, we want kids to be able to swim in the deep end. We want kids to be able to swim if they get pushed or fall in the deep end. But we're saying we don't want to start there because that's not the best way to get to that destination.

[19:56]

Is that right?

[19:57] SPEAKER_01:

Yes. And how do very intelligent, committed people in education make the mistake, make this mistake? Like you would never say we want kids to swim in deep water without any support from an adult. That doesn't mean we throw them in there. We know if you coach them and teach them the strokes and start in shallow water, why do we make that mistake in mathematics or in other subjects? It's like we've hypnotized ourselves in education to completely ignore the research.

[20:24]

that's so obvious to a coach or so obvious to me, because I've taught thousands of kids, including kids. The very first student in JUMP was in grade six, back when JUMP started as a tutoring club before it became a class program. She was in grade six, she was assessed as MID, which means they thought her IQ was 70. She was testing at a grade one level, couldn't even skip count by twos in grade six. Three years later, after three years of tutoring, she went into academic grade nine math. Then she skipped a year and finished academic grade 10.

[20:57]

So she was writing tests. I've got copies of her tests that she was writing with multi-step ratio problems using algebra, decimals, and so on. This is a girl who couldn't skip count by twos three years earlier. It's just because I took her on that path, you know? I didn't throw her off the deep end. I taught her how to swim.

[21:13]

And a lot of it was like once she could actually pay attention and was starting to develop some foundational concepts, a lot of it was just Socratic questions. I let her figure out as much as possible. So I write about that in the book. I argue that we've neglected one of the deepest causes of inequality in our society. We've had trouble developing sustainable economies or fair economies because we've overlooked a deep root cause, which I call intellectual inequality. It's very hard to eradicate a material inequality, but we've done studies like the one I mentioned with the kids who wrote the Pythagoras competition that show you can eradicate intellectual inequality in a couple years.

[21:51]

with no more expense than what you're going to spend on regular math programs. You can eradicate intellectual poverty in a few years. You can create a whole classroom of kids who enjoy math and think it's worth engaging in and that they can do it. So imagine the impact on our society if we invested in that, or even just allowed teachers to follow the science of learning and to see examples of this happening so they know it's possible. So I'm going to make a crazy claim. I'm sure I sound like a snake oil salesman, but I took a girl who they thought her IQ was 70 from a grade one level to grade nine level, and she did academic grade 10 in the same year in an hour of tutoring week in three years.

[22:34]

The New York Times article, it's called A Better Way to Teach Math, shows similar results across a whole class. This is possible, and we've been seeing this data for 10 years now, more than 10 years, but it's heartbreaking because It's still a massive battle to get people to actually teach in a way that's going to elicit that potential in every child. Because we're so confused by these arguments, like the equivalent of throwing kids off the deep end, because we want them to swim in the deep ocean. That's not how you start.

[23:10] SPEAKER_00:

And there's this kind of aesthetic argument for the opposite of what you're talking about, right? Like we have all these, what I really think are aesthetically motivated arguments about how math should be taught in a way that feels nicer or feels more modern or feels, I don't know. I have to conclude that that is people's motivation to do something that feels better rather without really much attention to the question of whether it actually works better. And I don't want to name names too much here, but I mean, there are people who essentially advocate for the opposite of what you advocate and often do so on aesthetic grounds and without evidence. What's your take about why that's an appealing thing for people to do?

[23:48] SPEAKER_01:

I'm a very progressive person, but I think that Somehow very well, meaning people have created a generation, particularly in the U S it's very anti-scientific, very anti-math. I think partly because they didn't take enough responsibility for getting some evidence for doing studies before they promoted these progressive ideas. And I donated 10 years of my life to jump. We made it a not-for-profit on purpose so that we couldn't, you know, be forced to do the wrong thing by shareholders. So I'm very progressive. I believe in an equitable, a truly equitable society where kids just don't have equal opportunities to jobs and things, but actually could choose to do any subject they want.

[24:32]

So I'm not arguing for a traditional society. I'm arguing for a radically different society, but you can't keep mistaking the end where you want to get kids to for the means to get them there. And you can't keep ignoring the science. which is very clear on the means, like the 85% zone, deliberate practice, those things. You can't keep ignoring the science because your theory sounds good. JUMP has been involved in two randomized control trials.

[24:54]

You have to test your ideas at scale. And multiple other studies, you have to be willing to do that work. It's very easy to be seduced by theories that sound good. And the research says that an incremental approach, a more guided approach is good for novice learners. As they become expert learners, they'll be exactly what you wanted them to become. They'll be 21st century problem solvers, but they're not gonna start there.

[25:20]

and you're gonna have to guide them very carefully, and you're gonna have to take responsibility that they're actually mastering and understanding what you're teaching. If you assume that every child can learn math at a deep level, almost without exception, like I don't even, like that girl I taught was, they thought her IQ was 70. If you assume that, then you have to keep refining your theories and your methods based on that idea. And if you're not, with entire classrooms, if you're not with entire classrooms producing that result, then you've got to go back and check your theories.

[25:52] SPEAKER_00:

So John, if people want to learn more about Jump Math and learn about your other books, as well as All Things Being Equal, where are some of the best places for them to go online?

[26:01] SPEAKER_01:

Well, we have a website, jumpmath.org. And a lot of our lesson plans are there. We realized teachers didn't, particularly if they were math phobic or didn't have a great background in math, don't have a lot of time to learn it. So we wrote lessons in a way that they could learn the math as they teach it. So those lesson plans are free on our website if they want to look at them.

[26:24]

There's lots of materials there. And my book, I think, is the most complete argument I've made for things I'm talking about today, all things being equal. But the website is jumpmath.org. Plus, if you Google Jump, you'll find articles. The New York Times article is a better way to teach math.

[26:41]

I wrote an article for Scientific American Mind called For the Love of Math. So I guess maybe start there.

[26:48] SPEAKER_00:

Well, John, thank you so much for joining me on Principal Center Radio. It's been a pleasure. Thank you so much.

[26:53] Announcer:

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