Jo Boaler transcripts Shifts in Math Education >> We've had a very traditional model of mathematics teaching for a long time. In fact, if you go into a lot of math classrooms they don't look that different from Victorian days. The teacher is still at the front telling methods. And the kids are sitting, reproducing them. Even though we have moved forward so much in our understanding of learning, it hasn't really gone into classrooms. So, we know that in the traditional model of math teaching, the teacher stands there and tells you. Some kids do okay with that. Many of us have had that model. But there are actually the students who are actively making connections themselves beyond that. And in fact, a lot of people who are mathematically successful, a lot of people, will tell you about something that has given them that success that was nothing to do with the math teaching they experienced. But for most kids in those classrooms there it's a failing model. So, what we need to do is we need to broaden mathematics. We need to teach problem solving and inquiry. One sort of fundamental thing that I think is interesting is we know from experimental studies that compared teachers teaching some math and then having kids work on problems. That's the sort of traditional way of doing it with kids working on problems where they don't know how to do it, and then kids, and then teachers teach them the math. And they find that that second approach kids do massively better. And the reason for that is if you give kids problems that they can't solve and they have to think about them and talk about them and then later you teach them a method, their brain is primed to kind of receive that method. They're interested. They want to know how to solve it. So, that's like the complete switch in what happens in math classrooms. So, what we should be doing is giving kids rich and complex problems, having them work on them. Maybe some kid knows how to solve it and other kids might learn it from them, but whatever happens you have them engaged in thee rich problems, and then teachers can teach a formal method if it helps them go forward in the problem. So, we've really got the balance wrong right now. It's a lot of skill procedure. Somebody I like to quote from Conrad Wolfram, he has a campaign where he's going around the world saying we've got to stop math being calculating. Right now, so he would describe the importance of working mathematics. Then he says that working mathematics there's four components. First of all, you have a problem, well, first you ask a question. So, the biggest growing job in the world now is a data analyst. All companies have huge amounts of data. So, the first thing you'd have to do is ask a question of the data. So, he says working mathematically you have to ask a question, then you set up a model, then you run a calculation, and then you have to go back to that model and interpret it. And schools spend 85% of the time on the calculation step. And that's the one step we don't need people to do anymore. And in fact, you know, they never get people to do calculations. They have computers and calculators for that. But they do need people to set up models and understand. And this is a really important shift if we want people capable in the twenty-first century. We don't need them sitting in classrooms learning a mathematics that is actually 400 years old. Keith Devlin is a mathematician at Stanford. And he's really interesting. And he talks about most math classrooms are learning mathematics that's at least 400 years old if not older. We need to shift that and be teaching the math that people actually need.

Mindsets and Mistakes >> I found that we can talk to kids about mindset and having a growth mindset. And they'll sort of, you know, get some of that, but sometimes they'll remember it and sometimes they don't. But what's really important is to give them very specific examples. So, one of the things I think is very powerful that we now know from evidence is we now know that when you make a mistake in math your brain grows. The synapses fire, and there's brain growth that happens. But when you get work right in math there's no brain growth. So, this has huge implications. It means we want kids making mistakes in math. We want them working on hard, challenging work. So, actually working with kids to reposition mistakes is, has a big impact on their learning. Good teachers have always said, "Oh, mistakes are helpful. They help you learn." But this is a different way of communicating to kids. Mistakes are more than helpful. We need kids making mistakes. That's when your brain grows. When kids get that idea, that changes everything for them because up to that point they often think I make a mistake. I'm not a math person. And they just give up. So, we need to teach them that seeing, having a growth mindset comes out when you have, when you face mistakes and when you face hard challenges. And that's when we see what kind of mindset you have, your approach to hard work. Kids really need to know the evidence about the brain growth. That's a really important part of mindset intervention for them to know the brain is like a muscle. The more you exercise it, the bigger it gets. You know, it's all about everybody can learn in this way. But mindset is very linked with approaches to math and strategies are very important for mindsets. So, actually the latest PISA data, which is done with 30 million 15 year olds, shows that the lowest achievers in the world are those with a fixed mindset. The kids with the growth mindset are much, much higher. But the kids who are the highest achievers in the world are those with a growth mindset and use good math strategies. So, we know that those two are very compatible. And the failing math strategy, the lowest achievers in the world are the kids who use what is called a memorization strategy. So, they go into math and think you have to remember all of this. I have to, and they say that they approach for tests, they approach tests by trying to memorize all the steps. The high achievers in the world are those who say when they approach math they look for the big ideas. They think about the connection between math and the world. So, a lot of what I try to teach is math strategies that are helpful for kids, thinking about what problems to how are you drawing them, it's really helpful talking about them.

Mindsets and the Learning of Math >> A top research at Stanford, Carol Dweck, has shown through decades of research that people have one of two mindsets. If you have a growth mindset you believe that you get smarter the more work you do. If you have a fixed mindset, you believe that you're smart or you're not. And there's not a lot you can do to change that. So, it turns out these different mindsets have huge implications for the learning of math and huge, the, they change kids achievement in math pretty drastically. So, we know that the lowest achievers both in the U.S. and in the world, we're just getting new data from PISA on this. We know that the lowest achievers are the kids with the fixed mindset massively, by massive differences they achieve differently to kids who have a growth

mindset. And that's because growth mindset behaviors are really helpful. So, the kids with the growth mindset if they fail because they don't think it means they're not smart, they'll keep going and learn from that. They're more persistent. They're more able to work through hard things. So, we really need kids to have a growth mindset. So, there are interventions. Fortunately people's mindsets can change. There are online interventions, sort of on general mindset that kids go through. And the data shows that they can be achieving at one level. They do a mindset intervention, and then they're going on a whole different trajectory. So, I really believe that kids need mindset interventions in math. Like math is the subject with the most fixed mindset thinking. So, in fact, this online student course I've designed is designed to change kids' mindsets in math. It's a really important thing to do. And we know it has great potential.

Math Perceptions: Dispelling the Myth >> We have many teachers coming who've only ever received traditional math instruction themselves and then they just repeat it in the classroom. I think the most important thing to do with new teachers and I teach new teachers at Stanford, is to have them engage in math differently. So one of the first things we do and what we constantly do is have the teachers do math in the ways we want students to do. When I do that, actually I have teachers crying in those sessions because they work on math problems and for the first time, they understand the math and they see math as this connected subject, which they had always thought of as lists of rules so I think having teachers learn math themselves, relearn math really in the ways we want them to teach it, is pretty important. But we have many adults, including teachers who have had a very bad experience math themselves, some of them are really traumatized by math actually and it affects them on a daily level in their lives and so a lot of adults in our population believe that only some people can be good at math and if they had bad experiences in math, they've interpreted that as meaning they're not a math person. There's a very strong belief that there's math is like a gift that you have or you don't so it's really important to challenge those ideas. I'm finding -- I'm doing of work now with parents and teachers and getting these ideas out to them and finding generally, people are very receptive. Once they actually see the brain evidence and what we know from research, it's very hard to not understand it. That was a misconception that they may have had their whole lives so I think getting change to happen, getting more people to understand that anyone can learn math is really about getting the research ideas out to them in a really good, well -- a good form that's easy to understand so the only challenge really, I think, we're finding comes from people who don't want to dispel that myth. There are people who have their whole life, particularly if they're successful in math, believe that they're kind of special in some well or that their kids are special in some way, so those people are somewhat resistant to this new knowledge that actually everybody can learn math and it’s not that you’re some pre-genetically designed person that was able to do that. But generally I find little resistance, once people get the evidence. Unfortunately a lot of research evidence has sat in academic journals for a long time and universities don't have good systems for getting it out to the people that need it often, so what I have been doing a lot of my work on recently is taking research evidence that actually in journal articles that are pretty difficult to read and turning it into really sort of practical forms for teachers and for parents, good ways to understand it, good ways to implement

it. ^E00:02:58

Learning Beliefs in Mathematics >> And so we used to think from research that being good at math was all about this knowledge you developed; we now know that just as importantly, possibly more importantly, that the beliefs you hold about yourself and math and learning and the messages you receive. So I would say the beliefs students have about their own learning and probably more important than the math knowledge that they've learnt when they come -- need to do well in life and in tests and exams. So this is really new evidence. We really had no idea until recently of the power of the messages kids have and the ways they feel about themselves and the importance of that in their learning. Though one of the studies that really shocked me that showed this more than anything for me was a really big experiment they did with large numbers of high schools kid that you had to -- they wrote an English essay and then they all got diagnostic kind of challenging comments, diagnostic comments and half of them got one sentence added at the end and the other half didn't. And the half that got the sentence added -- and that was the only thing different in their whole year -- did significantly better in exams and tests a year later from once sentence, I mean it's pretty shocking but the one sentence was, and this was put at the end of the teacher's feedback, I'm giving you this feedback because I believe in you and the kids who got that, I mean it was just incredible. Nothing else changed, the teacher didn't know who got that sentence so I share that with teachers not so that they go around writing that but just to show the power, partly of teacher's words but also the importance of kids knowing that their teacher believes in them. It turns out, kids are processing this all the time, every day in class; does my teacher believe in me, are they telling me this because they don't think I can do it, are they giving me this work because they don't think I can do it and so we -- these ideas and messages kids pick up are just as important or more important than what they learn cognitively. And we didn't know that really five years ago, but there's a lot of evidence coming out now that we need to pay attention to non-cognitive aspects like how kids feel about their learning, what they think they can do, who they think they are in relation to mathematics. These are really important so I would want to see in a classroom, a lot of just messages up, messages on the walls about yet, the word yet is very powerful. I haven't learnt this yet so some teachers have bit yets on the wall and the messages about math grows your brain and struggle, you know, if you're not struggling, it's not helpful, mistakes are good. So messages are good and then we want to see kids talking about math, I mean that's just a really important part of math, teaching for understanding. I see the mindset kids have very clearly because if you give say, a more open piece of work out to kids, you'll find that probably 40% of kids' first response to that is to say I can't do this, call the teacher over, will you help me, I don't know what to do; that's a very classic growth -- fixed mindset approach. Whereas the kids with the growth mindset are ones who will, you know, be prepared to struggle for a bit without getting anywhere, without giving up and saying I can't do this. ^E00:03:16

STEM Implications >> So I spoke last week at the White House actually to the Council of Women and Girls about why too few girls are choosing STEM subjects. So we know that girls and boys can be very successful in STEM subjects: science, technology, engineering, math- but they choose out of it. There's a huge, leaky pipeline at the university, and kids that don't even get that far. So, they do that because of the teaching of those subjects. And what I spoke to them at the White House about was the critical role-play by mathematics in that. And a lot of people decide STEM isn't for them because of the experiences they have in math class from an early age. So, I also, I show them a video of an undergrad. Unfortunately when people have STEM aspirations when they come to the university, they're often dashed in their first math class. And we heard from a student who had wanted to be a math major her whole life. Her father is a math professor. Came to Stanford. Within two weeks of taking the required math class she dropped the class and decided to be a Russian literature major. And this is what's the reason we don't have enough students in STEM. It's a lot to do with mathematics. And it's a lot to do with the teaching of mathematics where students think they can't be successful. Students think it's a very route kind of subject that's incompatible for them with the kind of identities they want for themselves as thinkers and doers. So, that particularly hits women and girls. But so in encouraging people in STEM, the subjects can be integrated. Projectbased math, science, putting ideas together has been shown to be successful. They can also be taught separate subjects, but they have to be taught well. So, when we're teaching students that they can't do math, which is what a lot of teaching does for them, then they drop STEM aspirations. So, we really critically have to change that not just in schools but in the universities as well. I have some sympathy with that. I mean, one thing I don't like is where we teach science and math together, and we design these lovely science projects. But the math use is very sort of procedural. It's like, you know, math is just a service to this project. I actually find more creativity and engaging math when it's taught as math. So, I believe that there are people who can integrate it and teach it really well. I do believe that. But I think it has to be really done carefully so that the math isn't just becoming a service to it or the fit isn't very unreal, like getting kids to go off and do a bit of it just to rehearse that aspect of math or whatever. So, the good approaches I've seen have been ones where students sort of choose starting points. And then they take them in their own directions. And teachers are kind of flexible. So, they might, they may be a designer of a project, students take it on like they want to design something new for the city where it's more environmentally friendly, or, and they go in the directions that the project takes them in and teachers then are there to teach them that math they need as they need it to go forward at that point. So, it's less about designing these contrived situations and more about letting kids engage in inquiry.

Mathematical Success: A Conceptual Approach >> I can talk a little bit about algebra. I think algebra exemplifies the problems we have and what a great approach is. So an algebra class is typically across many, in many places they start with solving for X. You open a math textbook, it teaches you how to solve for X in algebra. So that's finding a particular value of X in an equation. That's actually a very narrow piece of algebra and the key idea of algebra is that it, this X thing represents a variable which isn't captured in this solving for X. So we lead kids down the

wrong path early on and they solve for X, solve for X, solve for X and then they're told, X can actually be anything. It can be a variable. And that causes a huge conceptual block for them. And that's one of the reasons many kids fail algebra. So a good algebra task, one I use a lot, is we show a growing pattern and ask students how the pattern is growing. So, I could ask students, look at the, look at what you have in the first case, the second case or the third case and tell me how many are in the hundredth case in which case they'd draw up a table, come up with a number, write an algebraic expression. So I don't do that. I ask them how do you see this growing? So what happens when we do that is kids see mathematics in many different ways. So, first of all they'll talk about their different ways of seeing it. They've probably got, you know, usually kids have different ways of seeing it and they can talk about why they see the growth in different ways but it still works. So we open up tasks to be more visual, to have different, kids see, use mathematics and bring their own perspectives to it. To give opportunities to talk about different ways of solving. And eventually they'll probably all come to the same answer, but valuing their thinking and the different methods and the different ways you can see things mathematically is a really good task. So I have a task framework where we decide which are good tasks and which aren't, and the good ones are ones that are more open where they have different ways of seeing the math, different pathways through the problems valued, different representations of valued. They involve sense-making instead of just, you know, calculating without any thought. So those are the important things. I teach a class at Stanford to freshmen which is a class called How to Learn Math, and they have to write a letter, write an application to get in, an essay, a very short thing, and it's really interesting. The course is meant to be small, for 15 students. And every year I get 100 applications from Stanford students and they all read the same. They say, “I used to love math, I used to be good at math until, dot dot dot.” So I have this class of math traumatized Stanford students every year and I work with them to change that. And they come out and they'll tell you, they come out with math being totally different for them. But some of the key things is people have to understand that any math trauma or anxiety that's set up has come from the math experiences they've had. It's a key shift. It's not you, it's some things that have, you know, been done to you or happened with you. That's really important. And then I show them the evidence on the brain and on how we learn that shows them that anybody can do good, do well at math. That's really important. We have to take the emphasis off speed. So a good two thirds of my Stanford freshman tell me that they developed anxiety when timed tests were used in schools. It's amazing to me that we give these things to kids. In the U.S. they, in my daughter's school they give, from first grade upwards, 50 questions to do in three minutes. So the point of that is to go fast. Now we now know from very recent research and brain imaging that time tests are the onset of early math anxiety for kids. About a third of kids will develop math anxiety around that sort of speed pressure. And it doesn't have to be a timed test, it can be any sort of speed, public humiliation. All sorts of things can set up math anxiety but it's perfectly a, you know we can take that away. So I'm teaching an online class for students which is just only at Stanford which is designed really for kids who've had a bad experience with math, or adults where it gives them these ideas and helps them see math differently and see themselves as capable. Students with special needs I think fall into different categories. There are students with very particular needs that may have come from, you know, brain differences or something that developed early on. And then there are a lot

of students who are actually math traumatized and have been sort of sent down a track where they become low achievers. And I'm actually speaking at a conference in England on special needs students and my, the title of it is "How We Create Low Achievers in Math Classrooms". So oftentimes when students aren't doing well in math, they get the opposite of what they actually need which is they'll get more rote instruction and more attempts to drill them in particular things. What we know separates low achievers from high achievers in math is not how much they know, it's how willing they are to engage with numbers flexibly. That completely separates kids into two achieving groups. So we know that in mathematics it's really important to engage with numbers conceptually. That helps with brain compression and other important things. So what happens is soon as a kid starts not doing well they often get a very bad form of instruction which is more rote, they don't engage with things. Kids with special needs actually are often more able to engage in math creatively. They, they are a lot of particular needs, processing disorders and others where kids really have difficulty with remembering facts. But actually if you engaged them in a problem solving task and get them to use their thinking, they can be better, they can do better than kids who are traditionally high achievers. So the opening of mathematics is really important for kids with special needs. It's, you know, one of the most important things we can do for those students.

Timed Learning and Math Anxiety >> One of the things that I'm finding very interesting is a finding from neuroscience that math should never be associated with speed so this is a very interesting research finding. They can now scan brains and see how they work and one of the things that [inaudible] and others have found is it turns out that you hold -- you -- math facts are held in a part of your brain that's got the working memory and when people are anxious, their working memory is blocked. You can't access it so what they find is you can be made anxious by math and in all sorts of ways. [Inaudible] and I've experienced this where you're like working at a table doing math publicly with people watching and you suddenly feel like oh, I just can't think; that's the impact of stress blocking the working memory. Whenever we give kids stressful situations, timing, exams, they actually can't function. For kids, you know, they develop this anxiety and they cannot produce math and what happens then is they start to believe they can't do mathematics so I think the research that we're getting on speed from neuroscience is really interesting and important. ^E00:01:09 Mental Math: Thinking Flexibility with Numbers >> Practice is important. We want kids to have some practice where they learn something and are able to use it in different situations. It doesn't need to be, and probably shouldn't be, repeat this method 20 times with different numbers. That's not the kind of practice we need. So, but we do want kids to have experience using the methods they learn. But you know I often, I say to teachers and parents, we're really out of balance now. Kids are practising all the time in math class and practice, one of the best ways to practice actually on math is through an exciting math app or a video game.

They give kids loads of opportunities to practice and they're enjoying it at the same time. So, I, you know, say to particularly elementary teachers, get rid of those boring worksheets that turn kids off math and get them working on this app where they're using the same math facts and numbers but they're really engaged and enjoying it. So, we know the most important thing for kids is number sense. This is what separates high and low achievers. People often say oh algebra's so hard, algebra's the difficulty for kids. That's, the reason kids fail algebra is because they don't have number sense. And number sense is needed through all higher level mathematics; calculus and beyond. So, we have to have kids develop number sense. That means they're flexible with numbers, they think creatively with numbers, they can work on problems in different ways. So, one of the great strategies which I love for teaching kids number sense is something called a number talk. And number talks are great because they take about five minutes and it's one of the only strategies I know that teaches kids mental math skills and it teaches them conceptually at the same time. So, in a number talk the teacher just puts up a standard math problem, like 8 times 15 is one of my favorites and the idea is they, so they say to kids solve 8 times 15. And they ask that when the students have a solution they just put up what's called a silent thumb rather than saying I've done it, I've done it, which puts pressure on other kids. So then the teacher collects in, usually there's one or two answers. And then the teacher finds the different ways students have solved it. So, I've used that problem with CEOs, with Stanford undergrads, with 7th graders. Whenever I do it there's at least five different ways of solving that problems. So, in a number talk the teacher will collect these different methods and then the kids can look like oh that method worked and that method works. So, they have to do it mentally in their head. So, if you give the problem the kids have to mentally, no pen and paper, work out the answer and then it's all these different creative methods come out. So, number talks are great. They help kids learn the math conceptually. They help make them fluent with mental math but maybe for me the biggest thing is when I've taught number talks to kids, it has changed their views of what math is. For most people, seeing the 8 times 15 can be worked out in these five really different but great methods, blows their mind. You know just the idea that any number calculation can be seen in so many different ways. So, I think they're an amazing strategy to use in classrooms.