Rethinking the way we teach math: Beyond right and wrong

Jo Boaler (00:00):

The language the world is written in is mathematics. And when we learn maths, we start to see and understand the patterns in the world in many situations. And maths gives you a way of thinking that allows you to see more.

Denise Pope (00:19):

Welcome to School's In, your go-to podcast for cutting edge insights and learning. From early education to lifelong development, we dive into trends, innovations, and challenges facing learners of all ages. I'm Denise Pope, senior lecturer at Stanford's Graduate School of Education and co-founder of Challenge Success.

Dan Schwartz (00:42):

And I'm Dan Schwartz. I'm the Dean of the Graduate School of Education and the faculty director of the Stanford Accelerator for Learning.

Denise Pope (00:52):

Together, we bring you expert perspectives and conversations to help you stay curious, inspired, and informed. Hi, Dan.

Dan Schwartz (01:03):

Hi, Denise. How are you?

Denise Pope (01:05):

I am good.

Dan Schwartz (01:06):

Okay. Well, good, because I'm going to ask you a really hard question.

Denise Pope (01:10):

Uh-oh. Okay.

Dan Schwartz (01:11):

You ready?

Denise Pope (01:12):

I think.

Dan Schwartz (01:14):

Denise, what's the purpose of learning math?

Denise Pope (01:17):

Well, some people might say it's to torture you, but I'm going to go with the positive, which is I think it's to learn how to do very practical things. I will give you an example, Dan. I am making a recipe for cupcakes this weekend, and we're having many more people than I thought over. And I need to figure out how to double or maybe even triple the recipe. That takes math, right?

Dan Schwartz (01:45):

Or it takes AI.

Denise Pope (01:47):

Oh, okay. I didn't think about that. Okay. How about this? There's a 3% chance of rain tomorrow. I have to figure out, should I take an umbrella? I guess I could ask AI, should I take an umbrella? But it's going to get tedious to ask AI about every little thing when math comes up.

Dan Schwartz (02:04):

So, I have a different answer.

Denise Pope (02:06):

Okay. What's your answer?

Dan Schwartz (02:07):

So, your answer is the skills answer. I have two answers. One is cynical. It's so you can get into college.

Denise Pope (02:13):

Oh, boo.

Dan Schwartz (02:14):

Yeah. I told you that. I said that was cynical. The other one is more sincere, which is I think math enables you to appreciate the world more deeply. Oh. So, it's not just to be able to compute and solve problems, but to see structures that you couldn't see before. There's a lot of beauty in it.

Denise Pope (02:31):

Say more about that. What kind of structures are you noticing that you couldn't notice before without the math?

Dan Schwartz (02:36):

Proportionality. To be able to see things and say, oh, these are nice pleasing ratios, for example, in art might be an obscure example. We've done studies with kids learning science. And in one condition, we say explain this. The other condition when you say use math to explain this. And the kids who use math to explain it end up understanding the physical phenomenon much more deeply. They can reason more flexibly about it. They can think about what if questions. So, not all of education is just so you get practical skills. Some of it's so you can appreciate the world that you're in. But we need to ask someone who actually knows the answer and-

Denise Pope (03:16):

Whose lifelong study has been the answer.

Dan Schwartz (03:20):

That is true. The challenge is, can you come up with an answer that serves everybody else's answer? All these different reasons people think you should learn math. So, our guest today is Jo Boaler. Jo is a beloved professor at the Graduate School of Education and the founder of YouCubed, the research center at Stanford that designs math curriculum and professional learning. I think it's reached about eight million downloads at this point. And Jo is the great communicator. She's recently written a new book called Math-ish: The Groundbreaking Guide to Finding Joy and Understanding in Mathematics. So, welcome, Jo. Thanks for joining us.

Jo Boaler (03:58):

Thanks for having me.

Dan Schwartz (03:59):

So, Denise and I debated two questions to start with. So, I'm going to give you the easy one first and then the one that's relevant to our conversation. So, when does one use maths versus math?

Jo Boaler (04:10):

Oh, we should use maths all the time.

Dan Schwartz (04:14):

Is this British?

Jo Boaler (04:16):

It is British. I mean, everybody in the UK says maths, but I also prefer it because maths is short for mathematics. And mathematics is a plurals noun that's supposed to denote all the different ways of being mathematical. So, math sounds so singular and narrow compared to maths.

Denise Pope (04:41):

I like it.

Dan Schwartz (04:43):

Okay. So, Jo, what is the purpose for learning math?

Denise Pope (04:48):

No, maths. What is the purpose for learning maths?

Dan Schwartz (04:49):

Maths. Oh God, I failed that quickly.

Denise Pope (04:51):

You've already blown it, Dan.

Jo Boaler (04:51):

That didn't take long. Well, I'm going to say the language the world is written in is mathematics. And when we learn maths, we start to see and understand the patterns in the world in many situations, in sports, in finance, in health, in art. And maths gives you a way of thinking that allows you to see more, to know when we're being duped by AI or financial companies. So, it has many values for students. Of course, it helps them in education and will give them access to interesting courses and higher paying jobs. But for me, I'm most excited about the way it enhances our everyday lives.

Dan Schwartz (05:41):

Is it possible to make a curriculum that sort of satisfies the practical skills, the appreciation of the world, the do well on the tests so you get into college?

Jo Boaler (05:53):

Definitely. I think you can create a curriculum that's based on all the ways maths is beautiful and in the world. And it would help kids with tests. I mean, we see that in teachers who teach in those kind of ways that their kids still do really well on tests. It would be a lot easier if the standards were changed. I think the standards we have in maths could definitely do with a reboot.

Dan Schwartz (06:19):

So, as Dean, I talk to a lot of people and I suggest education could be improved. And the response I get is, I was taught with drill and kill and I turned out just fine.

Denise Pope (06:32):

Mm-hmm. In case people don't know what drill and kill is, it's very boring. Do 80 problems on this worksheet, come back, take a test. Yeah?

Dan Schwartz (06:43):

But if it's on the computer every time you get a right answer, there's an explosion on the screen. What could be more fun?

Denise Pope (06:49):

It's literally kill. Okay.

Dan Schwartz (06:52):

Yes. Anyway, Jo, what can I say back to people who say I was taught that way and I turned out just fine?

Jo Boaler (06:58):

I think most people do not turn out fine. And even if you do turn out okay from just having a drill and kill approach, you miss out on seeing the beauty in maths. You miss out on joy and creativity. And we turn off so many students with that drill and kill approach. And we see that in the data now with only what, 20% of students in this country proficient in maths. So, I don't really see the reason we have to put people through that.

Denise Pope (07:31):

And it definitely turns people off. I am one of them. In fact, I would love a really sort of concrete example of how you might teach in a different way.

Jo Boaler (07:40):

Sure. So, instead of putting up methods and rules on the board and having kids follow them with none of their agency as human beings, you can give them a rich problem to tackle in their maths class where they're collaborating, thinking about how they can move forward. Teachers can teach them new methods while they're working on those problems. And so, students are really excited to learn those new methods. And when teachers actually make a shift from just standing there and telling kids methods to having the students engage in problem solving and reasoning, they enjoy it more too.

Dan Schwartz (08:24):

I believe that. Where the students actually get to be creative and sort of generate solutions. My guess is if I put up a list of adjectives like creative and so forth, and then a bunch of disciplines, I don't think people would draw the line matching creative to math. Is my intuition right on this one?

Jo Boaler (08:44):

Yeah. But it turns out when you give students the opportunity to think creatively about maths, they see it as a whole different subject. We've had that. We run summer camps through our YouCubed center. And one of the things kids say is I used to think maths was just right or wrong answers. Now, I see it as creative and interesting and about ideas. It's a lot more inspiring.

Dan Schwartz (09:12):

So, can you give an example of this, the kind of tasks you give with the work you were doing with the Stanford women's basketball team?

Jo Boaler (09:20):

Yeah, that was a fun project. So, we know that as well as students needing to learn the traditional maths curriculum, these days, they really need to develop data literacy. It's an important equity imperative in my view. If we don't teach our young people to be able to separate fact from fiction and to be able to understand whether data visuals they're being shown in social media are misleading, then we're leaving them quite vulnerable. So, I approached the coach of the women's basketball team here at Stanford. I had a season ticket, so I chose them. But we asked them if they would help us develop a data unit on basketball.

(10:03):

And so, the endpoint of this unit we developed was kids going into a data set and picking their own team and justifying why they chose different players. But we realized some kids won't know anything about basketball. A lot of them do. So, we actually got two players from the team, taught the kids basketball. And then one of the activities in the unit is kids make a basketball court on their desk and have a ping pong ball and try out different moves. Oh, and then they get a data card and they get to collect data on a game. So, they watch a little bit of a Stanford game and they're recording the data on their chart.

Dan Schwartz (10:44):

What's on a data card?

Jo Boaler (10:46):

So, it was a little card we designed for kids as sports analysts do during a game. Record like three points, two points, assist.

Dan Schwartz (10:55):

And then what's an example of a task they would do once they've recorded the data?

Jo Boaler (11:00):

Well, they did a number of smaller tasks. They looked at averages, which are a big part of the middle school curriculum, measures of center and spread in player statistics. But then the big task really was them going into a data set, choosing their own team, analyzing the different players, using statistics, making graphs to justify why they chose particular players.

Dan Schwartz (11:25):

Oh, interesting.

Jo Boaler (11:26):

So, just lots of stats and-

Denise Pope (11:28):

It's like fantasy football. It's like choosing your fantasy football team, right?

Dan Schwartz (11:32):

If you tracked them longitudinally, like would more of these kids be playing fantasy sports?

Jo Boaler (11:36):

Maybe. I mean, one of the things we were hoping for in this unit is to show kids that you can be involved in sports even if you're not an athlete. There are so many different sports analysts now, and that's a very cool job to have.

Denise Pope (11:51):

What I love about that, even if they don't want to be sports analysts, you came up with a way to show them a real life example of math. Ideally interesting to many kids, right? Very different from go home and practice these stats. And it's also closer to real life than the word problems that we used to do were so pathetic. If Bob has six apples and Sheila has two and a half and you're just like, "Who cares about Bob and Sheila?"

Dan Schwartz (12:21):

I want to hear you finish the word problem, Denise.

Jo Boaler (12:23):

Who cares? Yeah. I go into maths classrooms a lot. And I often sit there and I'm watching kids solving two simultaneous equations and in no context at all. And I just think to myself, why should these kids care about the answer to this question?

Denise Pope (12:41):

You don't just think about, they think about it. And this is why they just go home and throw it all into AI or they cheat or they copy or they get super frustrated and they say, "Forget it. I'm not a math person."

Jo Boaler (12:52):

Exactly right, Denise.

Dan Schwartz (12:53):

Wow.

Denise Pope (12:53):

It's true.

Dan Schwartz (12:55):

Boy, was that a rapid downward spiral?

Denise Pope (12:57):

Well, no, but it's true. I mean, school is so mind-numbing at times. Why are we adding to that?

Jo Boaler (13:07):

And it's not like this is not true anymore. Denise, you, like me and you too, Dan, have had children go through high school math in the not too distant past.

Denise Pope (13:20):

Yeah, yeah. No.

Dan Schwartz (13:21):

More distant for some of us than others. But yes, I remember.

Denise Pope (13:25):

The majority of the people do not teach the Jo Boaler way. Although there are a bunch of people who come up to me and say, "You teach at Stanford School of Education. Do you know Jo Boaler?" So, you are kind of a movie star in your own right, Jo, and you are definitely having an impact.

Dan Schwartz (13:45):

So, Jo, I'm a parent. I was taught the old school way. So, I have huge anxiety about math. You got any tips so I don't communicate this to my kid? Same for teachers. They may have math anxiety.

Jo Boaler (13:59):

One thing I would say to teachers or parents is we have a little online class on YouCubed. It's free. It's designed for students, but many, many districts have given it to parents as well. And it's like six 15 minute sessions that shares about mindsets and shows maths as a different kind of subject.

Dan Schwartz (14:20):

Learning more about it can bring down my anxiety.

Jo Boaler (14:23):

For districts, I wanted to tell you a little story of a district we worked with for 10 years, Healdsburg. And when we started working with them, 16% of fifth graders were proficient in maths. In their last round of state testing, 75% of the fifth graders were proficient in maths.

Denise Pope (14:42):

Wow.

Jo Boaler (14:43):

And what happened in Healdsburg is they went all in on a mindset culture and all their teachers came to our workshops. But another thing they did was a parent evening. I did the parent evening with them. And we presented on why math should be visual and creative and conceptual. And the parents who were critical, skeptic, became huge advocates. And they were saying, "Why didn't I get maths like this?"

Dan Schwartz (15:16):

Congratulations.

Jo Boaler (15:17):

They started going out in the community and getting others to learn about it. So, I do think we have to help parents understand. We have to bring parents in and teach them about these better ways of learning maths. We have a lot of resources on our website for parents and for teachers. We have something called the Week of Inspirational Maths, and it's the whole first week of the school year planned for you with creative open tasks and mindset videos for kids. About two thirds of schools in the US use that week of inspirational maths. But we also have some really nice lesson starters for teachers that are visual ways of seeing things.

Denise Pope (16:01):

Can you give an example of a lesson starter?

Jo Boaler (16:03):

So, we have data talks, number talks, visual art talks. So, in a data talk, you see a cool data representation. Might be a basketball court or it might be polar bears or we partnered with NASA to produce some that show the earth heating up. And you just ask kids, what do you notice and what do you wonder about it? And so, there were no right and wrong answers. Kids can share their thinking. They learn to unpack data visuals and understand what they're telling them.

Dan Schwartz (16:39):

Is this one of the reasons you wrote the book? Who is the Math-ish book for?

Jo Boaler (16:44):

Oh, Math-ish. I have to tell you about Math-ish. I'm very excited about it. So, Math-ish is for anybody, but I introduce an idea in Math-ish. I make the argument that in classrooms, maths is pretty much always very precise. And that is often what scares kids about it. They're afraid of being wrong. It's right and wrong. But in the real world, we nearly always use what I call ish numbers. So, we could be talking about how much rain fell last night or how many people are in the room. I mean, I could give you a million examples. And I make the argument that, okay, we need both in maths, but we're way too much on the precise end and could bring a bit more ish thinking in.

(17:29):

And I know, we've always known that estimation is important. We can cite all the estimation research, but kids hate to estimate. They will often do a calculation and round it off to look like an estimate. But I suggested to teachers that before kids ever do a calculation, we should ask them, what's your ish number? And the kids love coming up with ish numbers. And what the teachers are saying is, I used to have a class discussion and maybe a third of the kids would be involved. When I ask them for an ish number, the whole class puts up their hand. They're released from this fear of it being wrong.

(18:13):

But it also, like if I was to ask you to divide 22 by nine, that's not a lovely calculation. Yeah. Denise is wrinkling her nose.

Dan Schwartz (18:26):

Two and a third.

Denise Pope (18:26):

I would say two-ish.

Jo Boaler (18:27):

Two-ish, right?

Denise Pope (18:27):

Right?

Dan Schwartz (18:27):

I'd say two and a third.

Jo Boaler (18:29):

And if you come up with an ish number, two and a third, two, then you have a sense of those numbers, what we call number sense. But most kids don't have that. But if we keep getting them to come up with ish numbers, they have to think conceptually. They have to not get lost in all the details and actually think of the big picture. You can do it with everything in maths. You can ask them kids before they do like a detailed graph in algebra, before they're plotting functions. You could ask them, "What do you think that's going to look like? What would your ish graph look like?" So, then kids have to pull back from all that detail.

Denise Pope (19:07):

Jo, I love that so much because if you say 22 divided by nine, I immediately went, "Ugh." But when you say ish, I'm like, "Well, I know nine plus nine is 18. That's close to 21. I'm going to say two-ish." That literally felt very freeing to me.

Dan Schwartz (19:24):

Can I give you a problem I had once?

Jo Boaler (19:27):

Sure.

Dan Schwartz (19:28):

So, we were trying to get people to understand big numbers. And we showed like a little yellow square, a quarter of inch on each side.

Jo Boaler (19:36):

Okay.

Dan Schwartz (19:36):

And I'd say, "How big would 10 of these be?" And then I'd say, "How big would a hundred be?" And then I'd say, "How big would a million be?" And people were off. A million isn't that much. They would say it would fill a stadium. So, they could keep guessing all they want. And I don't think it's going to help them to sort of get a sense of big size.

Jo Boaler (19:59):

You're talking about something that humans have a big problem with, which is grasping those very, very big numbers. Often, I think that people are way off when they think about those really large numbers. But well, fractions, think about fractions. If we ask most people to divide a fraction, most people will tell you, teachers included, the only thing they know to do is to flip and multiply. Yep. But if you ask them, what would an ish answer to divide by-

Dan Schwartz (20:30):

Sorry everybody. Denise just raised her hand.

Denise Pope (20:33):

I did. I flip and multiply.

Dan Schwartz (20:35):

And I have a confession. Driving in today for this, I was trying to remember, do you flip and multiply when it's division? Or do you flip when it's multiplication?

Denise Pope (20:44):

Okay. Don't confuse me. I know it's a division that you flip and multiply.

Jo Boaler (20:47):

You had that burned into your brain, right, Denise?

Denise Pope (20:51):

I don't know why. And I couldn't give you an ish. I couldn't give you an ish for that, Jo.

Jo Boaler (20:55):

You could give me an ish, Denise. If we were to sit down for half an hour, I'm confident you can.

Denise Pope (21:00):

Okay, yeah. Okay. So, I would need to draw it out. I can't in my head come up with an ish, but I would need to draw it out and kind of like think about it.

Jo Boaler (21:07):

Exactly. Drawing it is great. And this is part of what we are always trying to get is kids to do more visualizing with maths.

Denise Pope (21:16):

Well, okay. On that same note, in some paper I was reading, they were trying to explain to people what a trillion dollars would be. And they said it would be if you laid dollars down back to back, it's kind of like your square, Dan.

Dan Schwartz (21:33):

It'd take you to the moon and back like 16 times.

Denise Pope (21:36):

That doesn't help me. No, that didn't help me. That's not what they said. They said this much is about the size of Los Angeles. This much is the size of California. This much is the size of the United States. Those are things I can relate to. To the moon and back, I couldn't relate to, but that was kind of cool. And they actually showed visuals. Okay. It's okay for me to draw a visual.

Jo Boaler (21:55):

It's better than okay.

Denise Pope (21:58):

What about this counting on fingers? Are you cool with all of that?

Jo Boaler (22:01):

Yeah. The research on fingers is wild. They've actually found that we have an area of our brain that sees fingers. So, if you hide your hand under a book or a table and someone touches your fingertips, it's how well you know the feeling of each of the different fingers. It's called finger perception. And how well kids know their fingers in grade two is a better predictor of how well they do in maths in the following year than test scores.

Denise Pope (22:31):

Okay. Wait, what is up with that? What explains that?

Jo Boaler (22:34):

So, it's really important. I mean, we literally have this visual number line, which are our fingers. And this is why they now think that musicians tend to be good at maths. That is in part because they have very finely developed finger perception. I did a project with Allison Okamura in engineering, where we made little haptic devices for kids where they answered number questions by using their fingers. So, they were associating numbers and fingers and that improved their maths achievement significantly in a short space of time.

Denise Pope (23:09):

It can also be that that was just more fun than drill and kill. And so, that was going to improve their... I'm poking holes in it already, Jo.

Jo Boaler (23:17):

But yeah, I mean, the finger research is part of a broader group of research that shows that visual thinking is really important. We actually have two pathways in the back of the brain that are visual pathways. We want to activate those when we're learning maths. It's super important. So, pretty much everything we do throughout YouCubed site is visual or invites kids to think visually. Thinking visually about fractions is a lot better than just blindly following rules with fractions.

Denise Pope (23:48):

Okay. I have to ask this because in the title of Math-ish, you have the word diversity. Talk to us about that.

Jo Boaler (23:55):

So, what I talk about is mathematical diversity is just valuing all the different ways people see and think about maths. One of the other problems of maths education is kids are taught there's one way to do it. When of course, there were always many different ways to think about mathematical problems. And when we open up maths to value the different ways people think about it, that brings about much more enjoyment from kids and understanding. So, celebrating mathematical diversity, celebrating the different ways we see it, the different ways we think about it, the different ways we problem solve is part of that message.

Denise Pope (24:34):

Okay. So, I don't want a teacher or kids or anyone to walk away with the wrong message here. I could see someone saying, "Are you telling me that there's no such thing as a wrong answer in math?"

Jo Boaler (24:46):

No, I'm not telling you that. But there's always different ways to get to that answer. So, one of my favorite problems is asking people to think about 18 times five and how they would work out 18 times five. And so, some people think about 20 times five and take off two fives. Some people think about 10 times five and eight times five. Some people think 18 times five is the same as nine times 10.

Denise Pope (25:19):

That's pretty good. I was with you with the 20 because that's how I do it. I do 20 times five. I was going to give you an ish.

Jo Boaler (25:25):

I could probably give you 12 different ways of thinking about 18 times five.

Denise Pope (25:27):

Wow.

Jo Boaler (25:28):

And when we do this with kids and we show them all the different ways and we visualize all the different ways, many students are mind blown by that because that's not how they think about maths. But super valuable to show that maths is a subject that can be thought about differently.

Denise Pope (25:47):

So, Dan?

Dan Schwartz (25:48):

Yes, Denise.

Denise Pope (25:48):

As always, I like to put you on the spot. What did you learn today from our podcast?

Dan Schwartz (25:56):

Opening the gate so students can try different approaches. There might be a right answer, but there's a lot of different pathways and different pathways show you different patterns. But I think the thing that I learned that we didn't quite get to talk about is what an amazing curriculum designer Jo and her team are, right? These are great ideas. I don't know how you come up with them.

Denise Pope (26:17):

And two thirds of schools are using that opening week. I mean, that's amazing.

Jo Boaler (26:21):

My team is the answer to that. I have a small but mighty team.

Dan Schwartz (26:26):

Yeah. There has to be some strategy to help you think of these. But Denise, what are your takeaways?

Denise Pope (26:33):

I love the ish. I'm so excited to read the Math-ish book because it just calms me down and it makes me think, it doesn't have to be precise, but it helps me conceptualize it. I hadn't really thought about that.

Jo Boaler (26:46):

Great. I'm glad.

Denise Pope (26:46):

Yeah. So, Jo, thank you. And thank all of you for listening to this episode of School's In. Be sure to subscribe to the show on Spotify, Apple, or wherever you tune in. I'm Denise Pope.

Dan Schwartz (26:59):

I think I'm Dan-ish. I'm sort of close to Dan, but not quite Dan.

Denise Pope (27:04):

I like it. Dan-ish.