Episode #10: Mathematical Mindsets: An Interview With Dr. Jo Boaler
Our conversation begins with Jo sharing some of her memorable moments from math class followed by a rich discussion about the importance of thinking deeply to communicate mathematical understanding, ideas for differentiation in the math classroom, teaching through task, and much more!
- How math is visual and by teaching with visual resources we can strengthen our students understanding.
- Why math class needs to be social and collaborative instead of isolated.
- How to differentiate your math lessons with intentionality.
- Why we need more than just memorization of math facts and procedures.
- How to meet your standards while teaching through tasks.
- Making Math Moments That Matter Online Workshop https://makemathmoments.com/onlineworkshop
- YouCubed www.YouCubed.org
- Mathematical Mindsets [Book]
- What’s Math Got To Do With It? [Book]
- Mindset Mathematics [Books]
- Math Is Visual www.mathisvisual.com
- Math Is Like… http://mrorr-isageek.com/math-is-like-first-day/
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Jon Orr: Hey, hey, hey, Math Moment Makers, do we have a treat for you? We’ve got Dr. Jo Boaler on the podcast today.
Kyle Pearce: Our conversation begins with Jo sharing some of her memorable math moments from her math class, followed by a rich discussion about the importance of thinking deeply to communicate mathematical understanding, ideas for differentiation in the math classroom, teaching through task and much more.
Jon Orr: This is Episode 10, Mathematical Mindsets, an interview with Dr. Jo Boaler. Get ready to grow your brain with a variety of amazing mathematical ideas for your classroom. Start the intro music.
Kyle Pearce: Welcome to the Making Math Moments That Matter Podcast, I’m Kyle Pierce.
Jon Orr: And I’m Jon Orr. We are two math teachers who together-
Kyle Pearce: With you the community of educators worldwide want to build and deliver math lessons that spark engagement, fuel learning and ignite teacher action. Are you ready for this excellent episode, Jon?
Jon Orr: For sure, for sure. We want to welcome Dr. Jo Boaler, Professor of Mathematics Education at Stanford University, author of the popular book Mathematical Mindsets, co-creator of youcubed.org.
Kyle Pearce: Before we get into the interview, we want to give you a heads up that registration for our spring six module online workshop closes this Friday, February 8, 2019, and will not reopen until the fall. For details and to register before it’s too late, head to makemathmoments.com/onlineworkshop. Again, that’s makemathmoments.com/onlineworkshop. Without further ado, here’s our chat with Jo. Good evening, Dr. Boaler, how are you doing today?
Jo Boaler: I’m doing well, thank you. But you have to call me Jo.
Kyle Pearce: [crosstalk 00:02:06]. So awesome to have you on the show with us today. We are really excited and we can’t wait to dive into some math. So we’re wondering, who is Dr. Jo Boaler? for those who don’t know. I’m not sure how it’s possible but for those who don’t know, tell us a little bit about yourself.
Jo Boaler: Who is Dr. Jo Boaler? That’s a deep question. Okay. I am a professor at Stanford, Professor of Maths Education, co-founder of YouCubed. So few years ago, my buddy Cathy and I decided to make YouCubed, and it’s now a center at Stanford and also a website that I think we passed 31 million views just recently. So it’s lots of people like it and keep coming back to get more of the free nice things that are on the website.
Kyle Pearce: That’s beautiful. We will definitely have some more questions for you, YouCubed. I’m sure those who are listening whether they realize it or not, have likely been to youcubed.org because I find so many fantastic resources on that website, and it’s such a great tool I share with so many of our teachers. So thank you so much for that work.
Jo Boaler: It’s wonderful. Happy to hear that.
Jon Orr: One question we ask all of our guests is, what is your memorable or most memorable math moment from your math class experience? So this could be stretching way back or not too far way back but [crosstalk 00:03:25]-
Jo Boaler: … too insulted but carry on.
Jon Orr: … stretching back to your history as a student in math class, what stands out to you, something that kind of sticks with you even today?
Jo Boaler: I have a couple of memories that really stick out for me as a student in math class, and both of them when I was, I don’t know, probably 16 to 18. And one was bad memory, one was a great memory. And the bad memory was I was in my class, the teacher was the head of maths in the school. And I remember very clearly him saying, “Does anybody have any questions?” And I put my hand up and I said, “Yes, I have a question.” And I told him my question and he said, “You don’t understand that?” And I thought, “Okay, no more questions for this teacher.” And I remember very strongly that feeling. I can still remember where I was sitting, obviously it was pretty traumatizing for me.
Jo Boaler: But I also have a great memory which is, really math had always been just follow these methods, repeat the methods that I do and get answers, until I was doing A level which is this kind of advanced older level maths in England. I had a really different teacher, her name was Mrs. Marshall. She used to run into the room at the start of lessons, shut the door and say, “I got it. I just escaped seeing the head teacher,” because she used to wear dangly earrings, which wasn’t allowed amongst the teachers. And she was funny. She used to chat with us and she got us to talk about math. We would actually turn and talk to each other.
Jo Boaler: We were learning calculus, and we’d sit there and have chats about what it meant, and which methods we were using and why. So that just changed everything for me. From that point on, I saw maths differently. So yeah, that was a pretty big moment for me that went on for two years. She was my teacher for two years, which was great.
Kyle Pearce: That’s really interesting. I’m sure many listening can relate to, hopefully with the latter, but I would guess that some have experiences similar to that. First, we all tend to have that memory that sort of etched in our brain, you can go back there in your mind and picture the whole scenario. And something that I felt was interesting to compare and contrast, that negative experience and that positive experience, was really just this idea of that non-threatening classroom environment. And even just the mathematical discourse, it sounds like it was rich in that second experience, and probably quite lacking in that first.
Jo Boaler: Exactly, yes. And when we sat and chatted about maths, I mean, whenever anybody talks and kids get to talk to each other about it, you just go deeper. It becomes more interesting, you realize there were all these different perspectives, it’s not the black and white subject it’s portrayed as. So yeah, everything changed for me from that point.
Kyle Pearce: I definitely relate. At the time of this recording, our semester has just ended today. And even today, at the end of my semesters I ask my students to write down some big takeaways from the year. And I’ll ask one big takeaway as, “What’s a big math takeaway? Like, what math lessons are sticking with you to go on?” And another big takeaway I say is like, “What’s a non math takeaway from this year? I hope I’ve instilled some lessons here that are not necessarily math related.” And even today just reading them, it really stuck out to me that almost half of them had commented on the value that everyone in my class was providing them, about having them work together, and talk about math and at the boards problem solving, and they had that comment of, “Never really before had they learned that there was so much value to learn from each other on problem solving.” I really appreciate their comments, but it helps verify some of the thinking and work that was going into these lessons.
Jo Boaler: Definitely. I mean, I work with lots of teachers nowadays. And I’ve worked with some very traditional teachers, who have only sort of ever given lectures and have kids work through problems. And I remember one of them saying to me recently, “I just changed one thing, I started asking the kids what they thought, and if anybody had different ideas,” and it changed everything. And she said, “Now I’ve opened up to students perspectives on maths and them sharing these different ideas with each other so that I’m never going back. It’s opened this whole different world to me.” I do think that’s a really key moment when teachers start to realize they can ask kids, “What do you think?” And talk to each other about this and share your different ideas. It’s not a big change, but it changes everything for kids.
Kyle Pearce: Oh, absolutely. This was something I know I was not comfortable with for a good chunk of my career. And part of me coming from the secondary side, I don’t know if it may have been the fear of maybe losing control of the students or something along those lines. But it’s just going back to the relationship piece and that conversation when you’re having students talking and really empowering every student to be a part of the math lesson, and not having them fearful of asking a question. I mentioned in a previous episode that I used to want to plan my lessons as if it was done so well that no students asked any questions. Now it’s quite the opposite. It’s like, “What can I do to get kids to ask more questions, so we can dig deeper here?” It’s almost like trickery in a way.
Jo Boaler: Well, and I always think that we could probably all agree that mathematics really is all about reasoning. Scientists prove things by finding more cases but mathematicians prove things by reasoning, by making clear statements and making links between different statements. And so it’s very hard to reason mathematically if you’re not talking. When we give kids these silent maths experiences, they’re missing out on that super important part. I remember actually on that, I also do a lot of work with parents and one of the pushbacks I get from parents, particularly parents of high achieving kids is, “Why should my student write about their thinking? Why should they communicate their thinking when they can just get the answer? That’s a big waste of time for them. They can just speed to getting answers.”
Jo Boaler: I had a conversation with Conrad Wolfram, who is head of WolframAlpha, a great maths website. He said to me, “People who can’t communicate their mathematical thinking will have no use to me as employees, because it’s dangerous to me if people could just get answers. You don’t know if it could be incorrect. People can’t branch off from their thinking and go off in another pathway.” So I thought that was really interesting also, like what use are you really if you can’t communicate mathematically when you’re out in the workplace?
Kyle Pearce: I think that’s a really interesting segue into a question I had intended to ask a little bit later, but I think it really fits nicely here. It comes down to this idea. It sounds like getting students to communicate and represent their thinking, and really it kind of brings me to this idea of trying to differentiate for students. Something in your Mathematical Mindsets book that I really appreciated were some of the strategies for differentiating a math class that weren’t necessarily what I had envisioned earlier in my career, what I was taught differentiation was, and I’m sure some people listening could probably relate.
Kyle Pearce: They’re thinking, “If I have a group of students…” We all know that even if you’re in a system where they’re still doing streaming, or tracking or any of those types of organizational structures, which I’m sure we can get into later. But the thought was that you would differentiate for your students by essentially giving everybody something completely different. And in your Mindset book, I felt like your strategies were much more accessible for teachers to be able to use but much different than what maybe we were taught in our pre-service or what we were hearing from administrators or even other consultants. So do you mind elaborating on that a little bit? How do we differentiate for students and relate it to these tools?
Jo Boaler: Well, there’s different ways of differentiating. One, as you say, is having different work available. If you’re going to go down that route, what I always say is, “Please let the kids choose which they do.” Don’t say, “Oh. Here, you do this, you do this.” But there’s a whole different way of differentiating which I prefer, which is to give much more open tasks. So my favorite task to those that I say a low flow and high ceiling, everybody can access them, but they go to high levels. They’re really a great friend for teachers. Because as you say, whatever class you’re teaching, kids are probably all over the place and what they know, and when we give these narrow little tasks out, some kids can do them, some kids can’t, different status issues come up. But if you give these low floor high ceiling tasks that we have many of in our website, everybody can access them, but different kids take them to different places.
Jo Boaler: And so my favorite sort of differentiating tools when I’m teaching with these open-ended tasks is if somebody finishes one question I often ask is, “Can you now think about this problem in a different way? Can you go back in and think about it, solve it differently?” Another one I like to ask is, “Okay, you’ve finished this problem. Can you now write a problem that’s as challenging or maybe more challenging than this?” The kids love to do this. They love to write their own problems. So those are a couple of my favorite sort of differentiation strategies. I may have talked about others in the book though, did I?
Kyle Pearce: You know what? Another one that I really loved as well, it’s actually inspired me to go on a little bit of my own journey and has even just this idea of making it visual like bringing that floor down by making it more visual. The first time I had an opportunity to hear you speak live, you did the patterning problem, the quadratic patterning problem where, I’m going to lose what the name of it is, but you had all kinds of different tricks for how you could visualize, not tricks, names for how you could visualize it. That completely blew my mind. Do you mind elaborating on maybe that task or something like that visual task for us?
Jo Boaler: I love tasks like this. I mean, I worked with a neuroscientist at Stanford, and one of the things that we now know, which we just didn’t know five years ago even, is that every time the brain processes maths, there are two visual pathways involved in the brain. And so your brain really wants to see things visually, and it’s very important to see things visually. When we see something visually and we also see it with numbers or algorithms that causes your brain to make connections between different brain areas. And that’s very important also. So just always giving kids problems as numbers doesn’t allow that kind of rich, connected brain to develop.
Jo Boaler: But yeah, so the problem that you’re talking about is one of my favorites. Lots of problems we can do in this way. It is a pattern like growing cases where first of all, you see a collection of squares. And then in the next case, there’s more squares. And in the next case, there’s more squares and it follows a sort of pattern sequence. These are often used in algebra lessons and usually the question is, “How many would there be little squares with the B maybe in the hundredth case or in the nth case? It’s used as a way of developing algebraic thinking. But I like to ask the question, “Where do you see the extra squares? Just show me where you see them.”
Jo Boaler: Now, when I ask that question it’s always amazing that in any room of people, there’s like eight or nine or 10 different ways of seeing it. I love that. I ask people to give names to the different ways. So on that task, for example, people will say to me, “I see like raindrops falling down that make the extra squares,” and other people will say, “Oh, I see like a bowling alley with a row of Skittles coming down in the bottom of the bowling alley.” And so they have all these really amazing ways of seeing it and ways of describing it. But of course, every visual way they see it, you could also describe algebraically and in fact if you asked everyone to describe their visual algebraically you would have a whole room full of different algebraic expressions that are all equivalent to each other. So that’s a lovely lesson in algebraic equivalence.
Jo Boaler: But you can give that task to kids and ask them the number question and some kids will be able to do it, some won’t. They’ll be status differences. Or you can ask them the easier question, “How do you see it?” They’ll share these amazing different ways they see it and have great conversations. But also they will understand that function so much better than if we just get them to push the numbers around and symbols around.
Kyle Pearce: Well, it’s so funny because that task and many others now that I’m, like you said, it’s your favorites, one of my favorite as well, but now looking at any pattern, it’s like you’re forced to see it. It’s so beautiful because I work with high school teachers and Jon works with high school teachers as well, and anytime we take a visual pattern, no matter what it is, it could be quadratic, it could be linear. When we start talking about how it’s growing or how it’s shrinking, or these very open questions where people are able to speak and almost share it because it is unique, it’s like they’re empowered to do so instead of everybody kind of converging on the same answer or the same value. For me, that’s like a big aha moment. I remember sitting there watching that pattern, and I saw it like the stairway to heaven [crosstalk 00:16:18].
Jo Boaler: Oh, yeah.
Kyle Pearce: … which happened to relate to me because I really liked that song when I was growing up. But yeah, what a fantastic problem for sure.
Jo Boaler: And I have, as you say, also different visual problems that are similar to that one. And there’s another one I keep using at the moment, and I use it with teachers in workshops, and regularly teachers cry when they’re working on it which is interesting. And I’ve come to realize that these different teachers who cried, have cried because they’ve been able to see the pattern growth and come up with a quadratic expression through the visual, so they can understand how the shape is growing and then they can name that algebraically. And one woman came up to me, she had tears running down her face and she had formed this quadratic expression. And when I talked to her, she said she’d sat through hundreds of hours of algebra lessons, never thinking she could do quadratic, solve a quadratic equation let alone sort of come up with her own expression. And so it was like this liberating moment for her and I keep having this happen. I should worry about the number of people I make cry. It’s kind of like tears of-
Kyle Pearce: … tears of joy.
Jo Boaler: Yeah, joy and release. These people, even teachers have spent their whole life walking around with the idea that they can’t do higher level maths. And when you let go of that idea, it is kind of freeing emotional.
Kyle Pearce: I was at a PD session with my district last year, and our consultant was using some of these resources and sharing a quadratic function pattern. And he posed the question to the group of high school teachers, and he said, “Does anybody see how they could use this in the grade 10 applied course?” And in Ontario, the grade 10 applied course is not the high functioning academic kind of math course. It’s more geared towards students going to workplace or college, but still studying quadratic expressions and patterning. And no one was putting their hand up. And some couple teachers eventually said, “Yeah, we could see that in the academic course,” so the course that’s the higher level courses, “but not what the applied kids. They wouldn’t catch on to that expression that fast.”
Kyle Pearce: And I had to stop everybody because I said that this is exactly where that is great for it, and my students when I’ve introduced quadratic functions through the patterning, they’ve grasp it so much deeper and I’ve radically changed how I start all the function types in my class. It always starts with pattern and each of those, my grade nine course and my grade 10 course, I’ll do a patterning problem just to warm up the class couple times a week. And by the time it’s like, “Oh, we’re ready for the quadratics unit,” we’ve already captured everything we need to know about quadratic patterns and linear patterns. It’s been mind blowing for me for sure.
Jo Boaler: I would also say that all of those students in that course, are capable of learning higher level maths if they wanted to. Whatever has happened in their life or they’ve been tracked [inaudible 00:19:04] system, but it’s unfortunately we get these ideas about kids, “Oh, they won’t be able to do this,” because of course they can.
Kyle Pearce: And I had almost the exact kind of opposite situation happened in that same room, that there was higher level teachers who are teaching the academic courses and they’re saying that, “You don’t want to use the algebra tiles or the visual models with … We have to have this abstract focus.” If that’s for the kids that are operating at a lower mathematical thinking, and I know you have some thoughts on this, could you elaborate a little bit on that?
Jo Boaler: Well, we really try and push back on that message. It’s another one of the damaging myths that’s out there in education, that visuals, manipulatives are for lower level kids or younger kids. I use visuals with my undergrads at Stanford all the time, but more importantly, many very important visual mathematicians. And some of the most important things now in computer science and all sorts of fields is visual. People are realizing that visual thinking is actually very powerful. So there’s no hierarchy of, “It’s better to work with visual and then go to abstract maths.” You could go from abstract maths to a higher place with visual maths. And that’s another one of those myths that we work hard to try and bash and get rid of.
Kyle Pearce: Yeah. And it’s really damaging as you mentioned. And earlier, you referenced some of the parents of students who can come up with the answer. They’re likely the students who know all their math facts, and they’ve typically been getting high marks, and I’m using little bunny ears here, because what are we assessing? Like, what do these students know? Do they just know answers? Or do they actually understand the mathematics? And that brings me to this idea of, when I have this discussion with teachers or with parents, it’s about this idea of tools for thinking and representing thinking.
Kyle Pearce: So if I am dealing with a parent, with a student who can come up with the answer very quickly, the key for them in my mind is to be able to represent it concretely and visually. And if they cannot, then they actually don’t have a high level of proficiency. And I would say the same could be true for our students who are working in the concrete stage, and they’re struggling to show it in other ways as well. So really, for all of our students, we want them all to be able to represent it in different ways, and really have multiple options and opportunities to try to convince us, or maybe offer us a different perspective in seeing the thinking, I think that idea alliance for sure.
Jo Boaler: When a person I work closely with, Cathy Williams who founded YouCubed, and then she was a district coordinator in charge of math for the districts in San Diego. And she would get parents bring their kids to her say, “My child should be doing calculus in third grade,” and all the things that parents want. And she would do this assessment with the kids, and sure they could divide one divided by two thirds and come up with an answer, but then she would ask like, “Okay, can you give me a visual proof for why that works, or even tell me why it works.” And nothing. Of course, we can train kids to flip, and multiply and do these procedures, but if they don’t understand why things work or how to think about it visually, that is a big red flag for me.
Kyle Pearce: Absolutely. That actually … This new world I was exposed to as a student, I was that kid. I was the kid that a teacher would say, “Yes, you’re a good math student.” I was good at math. And I was put into that group simply because I had a good memory. Like, I just lucked out. I could memorize and I didn’t need the connections to come up with an answer, but I’ll tell you, I had no idea what it was that I was doing or why I was doing it. And eventually, the wheels fall off for all of us if we don’t have those connections.
Kyle Pearce: So for me, it’s been kind of a journey I’ve been on. I’ve created a website called mathisvisual.com. And whenever I get a free moment, I’m trying to put up visuals and trying to really almost for my own thinking, so I can visualize some of the ideas that I never really grasped when I was a student, why multiplying two negatives is a positive? And how could I show that? How could I articulate that? And it really makes learning these ideas, such a more rich experience and more worthwhile experience.
Jo Boaler: And I love that you’ve gone on that journey. I think that all of us have been on that journey at some point. I mean, I also grew up just quickly memorizing and spewing out answers, but I realized at some point in my life, I can’t remember when exactly, that there’s another mathematics really that’s visual, and deep and creative. And I meet teachers who are at just at the beginning of that journey or maybe they’ve never thought visually about anything before, and it’s scary for them. And some of them are quite worried about it and I always say to them, “You’re really lucky you’re at the beginning of this really wonderful journey, where you can start seeing things differently and seeing maths differently.” And it is as you say, you never go back after you’ve made this transition, and you start to see maths differently.
Kyle Pearce: Yeah. So exciting.
Jon Orr: So it was definitely a transition for me, and something that didn’t happen like Kyle until I became a teacher. I’m wondering is, what got you on that visual journey? Is there anyone that you looked up to as you began your journey, to fight the equity in math or the visual in math education? If so, who might that be? And why is that the person you looked up to?
Jo Boaler: When I was learning to be a teacher, it was in the late 80s and 90s. And it was the start of what’s known now as the reform movement. So it was right at the beginning of my teaching credential at London University, when I was introduced to maths that was visual, and we could talk about it and discuss it. So my very first teaching job in London, I went in having learned these ideas in my year of my teaching credential that nobody else in the department was using them, and they were catering sets as we call them in England like tracks. My very first class was the bottom set. The kids had just been put into the bottom set. And the first words, the first kid I taught were, “What’s the point? [inaudible 00:25:10] try.”
Jo Boaler: So I worked during that year to detrack the school. We actually ended up getting rid of sets that year. But I also brought in these ideas like, “My kids are going to talk about maths.” And I can remember in one of my lessons, the traditional second in department came to my door and said, “Your classroom is too noisy, make them be quiet.” And I said, “Well, come in and tell me which of these kids are not talking about math. Just show me any of these students who aren’t doing something really productive.” And he came in frowning, and he walked all around the room, and he came up to me and he said, “No, you’re right.” And he left the room. It was pretty early for me. It was before even my first teaching, that I learned these ideas from math educators in England. Now if you go into schools in England, you’ll see people sitting in rows memorizing, but there was a brief good time and that was probably when I was learning, fortunately.
Jon Orr: Then you left there. Interesting.
Jo Boaler: Yes. And then I went back, and then I left again.
Kyle Pearce: I’m wondering, because I know that especially in North America and in many places, including in Ontario, we in Ontario call it pathways. Students are in different pathways and really the way it’s framed from a ministry perspective, from the Ministry of Education, is more or less trying to put students on a track to help them get to the place that they’re hoping to get to. But unfortunately, the way it gets interpreted or the way it gets delivered, sounds an awful lot like the streaming, or sets or tracks like you’ve articulated. So I’m wondering if there was a teacher listening, who is maybe getting a little nervous going like, “I’m having a hard time teaching in my academic course or in my workplace course,” or whatever that course might be. What are some benefits that you saw when you eliminated sets, during your short time over in the British system?
Jo Boaler: It wasn’t that short but [crosstalk 00:27:02]. And I’ve worked in different places now where we’ve detrack classrooms and immediately you see huge benefits, one of which is, so I can tell you another anecdote where I was working with other schools to detrack in Brighton. This wasn’t so long ago when I was in my last stint in England, and they got rid of the sets. And on the first lesson where they taught the kids, one of the maths teachers rushed up to me and said, “Oh my gosh, the student who solved the problem, was one of the kids who would have gone into the bottom set.” And that to me is what it’s all about. We don’t know what kids can do. The neuroscience is showing this very clearly, anybody can grow or develop those mathematical pathways anytime.
Jo Boaler: So when we put kids into tracks, and I know we do it with the best intentions to give kids the most appropriate work, but what we do is we communicate to them that they are of a certain level, and we also give them work that is of that level. And of course, once we do that we’re putting kids into boxes, and those boxes will limit kids. So it’s harder to teach those mixed achievement groups. I know it’s harder. And I don’t think teachers can make that shift without a lot of training and help on how to do it. But if we want to give kids this limitless experience of maths, we really need to do that.
Kyle Pearce: It’s interesting, because I definitely agree especially if we are teaching through a traditional format of, “Take up the homework. I’m going to give you a note or a lecture on the new topic, and then I’m going to give you examples and set you on your way.” I don’t see that going very well. But I wonder, in Ontario we’ve been working for quite a long time. And sadly early on, I don’t think there was enough professional development around this idea, but really the idea of really teaching through task and consolidating the learning later. And I see that through your work a very, very similar, whether you want to call it a three part lesson, there’s all kinds of different ways we can name this type of teaching.
Kyle Pearce: But I see that teaching through task is how we can actually bring students from all different levels, into a classroom and have those productive discussions in mathematics, as well as do some pretty interesting things. It allows them to really show off what they know, and what they have in their back pocket when we don’t pre-teach all the steps and procedures, and we sort of give them that space for playing. I like in your book how you use the word “play” a lot. It’s like time to play with this idea. And we’re not going to play forever. There’s a very specific learning objective here that we’re going to pull out of this, but we’re giving everybody free space and I think that’s so powerful. Do you mind speaking to that idea of teaching through tests?
Jo Boaler: Yeah. And I am more and more a fan of using words in maths that people just don’t put alongside math. So yeah, we can play with math so it’s a really great thing to do. And mathematical freedom is another concept we’ve been putting on our website and talking to people about, what does it mean to feel mathematically free? Now for many people, they can’t even imagine those two words side by side. Mathematics is all about following someone else’s rules. There’s no freedom anywhere. So what does that mean for people to feel mathematically free? It’s a lovely actual feeling. And while we taught at summer school a couple of years ago, we invited 84 middle school students to campus. And we taught them. They were a totally heterogeneous group. So we had kids who scored zero on the pre-tests that we gave them, and they had statements of special need.
Jo Boaler: And then we had these super high achieving kids that were fast and high achieving, and then we had everybody in between, but we gave the kids open, rich, creative tasks. And what would happen is, as the kids worked on these tasks, if some students really needed extra support, they didn’t know things, we would go around and help them with that and teach things to them. And some of the kids were powering on, and seeing things differently and taking them in different directions. But there was plenty of opportunity to give the right kind of support to different kids. And they were with us for only 18 lessons, but they improved on standardized test measures, the equivalent of 2.7 years during 18 period.
Jo Boaler: Now, I know that a lot of that improvement for the kids was there was no tracking, and we gave them constantly these messages that, “There’s no such thing as a math person, everybody can achieve in maths. And maths isn’t about just speed and procedures. It’s about ideas and creativity.” And so the kids changed during that short period of time, and they really improved in their mathematical work. So I am a big believer in having diverse groups. I actually would rather have a very diverse set of students in my classes. I find that kids who attract often think in the same way and [crosstalk 00:31:56].
Kyle Pearce: … conforming-
Jo Boaler: … conversations. They’re very tracked. So I like diversity. I like diversity in maths and in students also.
Kyle Pearce: I would definitely agree. It’s almost as if the students in these different courses, they sort of conform. The thinking becomes one big group think, and oftentimes, again, is the way we are teaching the mathematics and doing it very direct instruction, and not very … There isn’t lack of investigation, and curiosity and all of those things. Now, when you think back to some of the experiences you’ve had with these diverse groups, what role do you think understanding the content from a conceptual and a developmental standpoint, how important is that for the teacher to be able to kind of zoom out, and be able to see the content like how it’s growing in order to really be able to lead some lesson, and knowing that students are all in different places along their learning journey, how we can sort of help bump them? Because that I think something we need to really understand too, is that we’re not expecting all students are going to be at the same point by the end of the lesson. But hopefully every student’s moved forward in their journey. So I’m wondering if you have thoughts on that.
Jo Boaler: And I think it’s hard if we do expect students to all do the same thing in a lesson. And I always know that there’s no creativity in a math class, when you walk in and all the students’ work looks the same, that you want students to be thinking differently, doing things differently. And we wouldn’t expect them to get the same end place. I think that’s right. You provide a rich environment for them, and then you support them in moving forward wherever they are.
Jo Boaler: But for teachers, well, we are about to find out how hard that is for teachers across the country, because we are embarking on a new project this summer where the camp that we did over the summer, we now have about 20 different groups around the country who are going to do the same thing this summer. And we have kids in Alaska, kids in Hawaii and all the states in between, and we have kids in other countries. We have some kids in Glasgow doing it too. But in all those cases, we’re going to help those teachers try and do the same thing that we did, and be there to support them and see what happens, and find out from the teachers and from the kids what they were able to do.
Jon Orr: It’s amazing that the support that you’re giving the teachers and trying to make that difference, and I can definitely relate to that. It was actually through your book, What’s Math Got to Do with It?, where I felt validated for the first time teaching through task. And I was the radical teacher in my building and most likely in my whole district, where I was not teaching kind of that standard lesson example, homework kind of flow. And when I switched to teach through tasks and listen to what the kids had to say, and try to fill in gaps and make consolidations after, I picked up your book and started reading it, and I was like … I’m yelling out “Yes.”
Jon Orr: You had the research to back up everything that I was doing, and it really gave me the confidence to just keep going. You’re in a world where everyone else is doing something different and you’re wondering, “Am I doing the right thing?” And when I read your book and you’re saying, “Yes, I have the stats and I have the data to prove it,” and I’m like, “Yes, this is it.” I definitely want to thank you for that. And I know that there’s lots of teachers out there who would definitely want to thank you.
Jo Boaler: That’s great. That’s all I hope for, is to be able to support teachers and help them. That’s what I really hope for. But I know … I mean, you’re in Ontario, there’s quite a bit of pushback up in your region from people who think this isn’t the right way to go. And some of them have quite aggressive stances towards people like us.
Kyle Pearce: I wasn’t sure if we would go down this path. But since you’ve mentioned it, I thought that we probably should. And I’m sure you’ve heard we have a new premier elected to the Ontario government. So there’s lots of … And unfortunately, a lot of misinformation that is out there. And I mean, that’s pretty typical, right? It seems like you’re either yes or no, right or wrong. There’s no balance in between. So I’m wondering, when someone listens to this conversation, if let’s say they happen to be in maybe one of these groups that’s still not seeing it here, do you mind sharing a little bit of your perspective around the idea of number fluency? I know that you know it’s important, you believe it’s important, but you have some pretty important messages about how we want to help students get there. I’m wondering if you can share some of your thoughts on that.
Jo Boaler: That is always contentious while on the memorization of math facts and times table tests, and certainly I am misrepresented often actually in that subject area. But what I strongly believe is that number sense is what is really important for kids. And number sense means knowing numbers flexibly, conceptually. Yes, it can be useful to memorize math facts, but there’s different ways of memorizing math facts. One is what I think of as blind memorization. You’re given a whole list of them and said, “Memorize them.” There’s no meaning, there’s no context, there’s no understanding. The other way is to use them in different situations and to look at a multiplication table, find patterns and understand what’s going on when numbers are multiplied together.
Jo Boaler: You still develop memories of math facts, but you develop a deeper conceptual understanding at the same time. So it’s very, very sad to me the probably millions of people who’ve turned away from maths, because of speed tests they did in second or third grade. And I teach undergrads at Stanford and the class is called How to Learn Math. And I have many math traumatized students who come to me. And they’re at Stanford, they’re very high achieving, but they’re really traumatized by maths. And I say to them, “When did this happen? What happened?” And I can tell you that in virtually every case, they will recall for me the math fact tests they did in second or third grade. And for some of them, they did well in them but at that point they thought, “Well, math is this shallow subject of fast recall,” and they just turned away from it. Others are pretty traumatized by them. And that started a whole path of math anxiety, but it’s just so sad because there are so many better ways to help kids know math facts.
Kyle Pearce: Absolutely. And when Jon and I go and do workshops, whether it’s in our district or beyond, we often ask … Well, we every time ask that question about our most memorable math moment for people, and we have them share and then we share it out to the group. And then for us it’s like a nice icebreaker. But how many times teachers will share that negative memory, and that negative memory is related to Mad Minutes, or Around the World or whatever it might be, where people felt that pressure. And as that student who traditionally was called the “decent” or “good at math,” I knew my times tables. But even in those situations, I always remembered me feeling super anxious because I didn’t want to be wrong, and it really put me in such an awkward spot.
Kyle Pearce: And only later I realized that while I knew my 12 times tables, if my mom or dad was doing flashcards and they just slipped in at 12 times 13, I wouldn’t have known the answer. And how sad is that, that I didn’t even understand that I could have added an additional group. So for us, that’s so important. And when you talk about blind memorization as well as being flexible with numbers, I often use the words memorization versus automaticity, like coming up with this … I memorize phone numbers because it’s difficult to make a connection. There’s no real reason why Jon’s number is what it is. But yet, when it comes to math, there’s so many connections we can make. And of course, they will become automatic the more we use them, but let’s use them for a purpose. Let’s not save the interesting math problems until we know the math facts. Let’s learn the math facts as we’re solving interesting problems.
Jo Boaler: I had somebody write to me today who said her son was in tears, because they do these Mad Minutes and his teacher said if anybody doesn’t get them all correct, they will have detention for the whole year. And this kid’s in tears every night at home and won’t do anything. And so, where do people think that this is a good idea?
Kyle Pearce: I know. It just shuts you down.
Jon Orr: You know what? I have a story because when I first started teaching, I did Mad Minutes, and I was that teacher because that was what I was put through. Mad Minutes is a thing. It’s everywhere. Like, I went through Mad Minute calculations. And I remember someone showing me a resource that had just Mad Minute sheets, like galore. And I remember thinking, this is back when I first started teaching, “Oh, great. I’m going to give one of these to my class every day at the beginning of class in three minutes.” When I’m thinking back, I don’t know why I picked three minutes. It was just, “Here’s 100 multiplication facts in three minutes,” seemed reasonable. And if you didn’t get it, if you didn’t get them all, you had to do more. That was it. It was like, “We’re going to just do more.”
Kyle Pearce: Those poor children.
Jon Orr: Yeah, I know. It wasn’t until I want to go back and say sorry to all of them. Probably I was the teacher that I’m sure kept this negative math feeling for all these kids that I taught for a few years anyway, before I started to open my eyes.
Jo Boaler: I really feel for teachers who just didn’t know, they were taught that way. And that’s the way they’re taught. And particularly the ideas I put my Mindset book and what we’ve learned from the brain science in recent years, who knew? Nobody knew. I hope that people don’t feel bad about doing things that they were taught to do. But yeah, I think there’s a lot of people who are in that boat. That’s what they had in school. That’s what they have been told is important. That’s why we really try hard to get different ideas out as widely as we can.
Kyle Pearce: Yeah. I think it’s so important as long as we keep that open learning stance, and when you know better, you do better. And that’s really the big, big takeaway. I hope that people are hearing out of this.
Jon Orr: There’s a question that we get a lot, because during our workshops and when we talk with teachers, we talk about teaching through task, and talks with our students and having them work together. What advice could you give to a teacher who’s trying their darndest to do all this stuff and try to do what’s best for students, but they keep saying or they keep feeling that the curriculum or the standards are just getting, “I have to cover the curriculum. I can’t do both”? We have to address that question. I wonder what advice you could give to teachers who have that same question.
Jo Boaler: Of course, many teachers are in that boat. They want to give kids this rich deep experience, creative visual maths, but they’ve got this whole list of standards to get through and district tests coming along. Not a good situation to be in. I’m a little wary of curriculum standards, even better sets of standards. And we could talk about which ones are better and worse, but in any event, what the people who write curriculum standards do is, I mean, if you ask any mathematician what they love about maths, it’s very common for them to say, “Well, maths is the subject of a few really big ideas and these really rich connections between them.” And those connections are really important.
Jo Boaler: But what the standard writers do, is they take this interconnected web of ideas and they chop it up into like 200 pieces, and then they give teachers these many pieces and say, “Here, teach this.” Their connections have gone. Math teachers don’t see the connections and of course their kids don’t either. And we need to change that. We need to change the way we’re communicating to teachers. But my advice really, if you’re in a school where you just feel like you’re under pressure to get through lots of content, is open up any textbook page and there may be 10, 20 questions in there, choose the three that you like the most and then have kids engage with them in a multi dimensional way. Ask them to talk about the problems together, but also to draw them and to represent them visually, and go deeper in a smaller number of problems instead of filling it, after half hours very monotonous repetition of the same kind of question over and over again.
Kyle Pearce: Absolutely. I think in the U.S. especially with the Common Core, I think that was a great step in the right direction. But as you mentioned, there’s so much there and oftentimes I feel like as educators, especially elementary educators that may be teaching multiple subject areas, that could be really, really challenging for them. You almost get to this point, like if I’m looking at it as a checklist, no wonder we’re sort of just skimming the surface. And for me, the part that I guess finally I gave myself permission, to be a little bit more fluid and flexible working within my curriculum here in Ontario. But what I did one year was I actually taught a grade nine course, second semester, and then the same group in grade 10, first semester.
Kyle Pearce: And when I saw the lack of retention that those students had coming into that grade 10 course, and they’re telling me, “We’ve never seen this or that,” or whatever it was, for me, that was the huge Aha. I’m like, “I’m doing all this work, thinking I’m checking off all these ideas, but at the end of the day, it’s not doing anything.” So that was when I had to … I stopped and I started reflecting, and thinking more deeply about, clearly we’re not engaging in this deep enough. And I really think that your advice is such a great first step, for people to maybe strip away all of the multitude or that huge magnitude of questions that you have for every day, and let’s go deeper with fewer.
Jo Boaler: And I feel … I mean, the Common Core set of standards I think helped in the elementary years a little bit, pruning out some content so people could go a bit deeper, but totally bottled out in the high school years which looked exactly like other high school sets of standards. And unfortunately, just a lot of antiquated content that kids will never use. I think we can all be clear that they will never use a lot of these methods again.
Kyle Pearce: Yeah, so why not go deeper with some of them, even the ones you’re not going to use at least let’s do interesting things with them, so we can engage in some deeper thinking?
Jon Orr: We only have a couple of questions left. But one that I’m really interested in hearing your take on or your thoughts on is, there’s an activity that I do at the start of every semester, but at the end of every semester. And again, I did it today because today was the last day of the semester. But at the beginning of every semester, one of our first activities when the kids come in, I don’t know them and they’re all bringing in different experiences from all the different places of their math experiences. So one thing I always have them do is I have them complete the phrase, “Math is like …” And then I always just write … And so some kids will say like, “Math is like a swamp,” and then they elaborate on why. It’s like, “Oh, it stinks and no one likes it.”
Jon Orr: Or a kid will say, “Math is like exercise because it grows your brain.” So I love this activity at the beginning of every semester, because it gets everybody’s ideas on their past experiences out into the open, and we can have that conversation about their beliefs and what this class will look like. So it’s a nice flow to set a stage for the semester. And then we end the semester the same way and we have them answer the question again, and maybe it changed and that’s what I’m looking for. So I’m wondering if, what does math look like for you? What would you say to complete the phrase, “Math is like …”?
Jo Boaler: Math is like a rainbow.
Jon Orr: Why would it be like a rainbow?
Jo Boaler: Just because I see it as very multicolored literally like rainbow, with a lot of difference and I like the many different sort of textures and colors inside maths.
Kyle Pearce: Right. Vibrant. I’m like picturing as soon as you said that, I’m going, “Yeah, it’s so interesting to look at and if we do it visually.” That’s a great idea for what math is like for sure.
Jon Orr: And one of our last [crosstalk 00:47:15]. Sorry, go ahead.
Jo Boaler: I was just going to say, I really like to ask my kids that I teach, whether they’re adults or children, to write a math history for me before they come into the first class. “Telling me about your relationship with maths.” And they write about big things that have happened to them that have changed their relationship. I always find it so hopeful to the teacher.
Jon Orr: That’s a good idea. Make it a note for that too [crosstalk 00:47:40].
Kyle Pearce: Yeah. It’s interesting because if we better understand where students are coming from, then at least it makes sense when they are acting or maybe not wanting to participate, and really trying to find a way to help bring them in and sort of welcome them to the math club versus feeling like they’ve always been on the outside. So that’s a great idea. I appreciate that.
Jon Orr: One of our last questions here is, what math content areas, or maybe it’s pedagogical, are on your mind lately? I’m sure you’re always up to new ideas and new thinking. What’s going on in your mind right now? What’s sort of peaking your interest when you have a few seconds to do some tinkering in the math space?
Jo Boaler: Well, we did a little playing recently, actually with exponents, which you might be surprised by. But I was working with a school and a little bit horrified I have to say, to see that they were doing flipped classrooms and kids were watching videos, and I got to watch the videos. And it listed all these rules for exponents, and then said, “Memorize these and then answer these questions with which rule has been used.” And I was like, “Oh my gosh, this is just awful.” And it inspired me to make a little worksheet for the school. Often we do these really rich, deep creative tasks which I love, but we made what was really a worksheet but it just got the kids to work out why each of these rules were working, and get it out and share it.
Jo Boaler: So not a huge deal at all. But it made me think, “Is that what’s happening in classrooms, people are being asked to memorize the rules of exponents?” So my next move in this arena, is I’m going to put it on Twitter this little task we made. It’s not amazing. It’s not deep. But it’s just a chance for kids to think about why exponents are working, and I plan to just ask people whether this is helpful or not. Or maybe they’re doing something better than this. Or maybe they’re not, maybe they’re asking kids to memorize them and see what happens. So that’s my latest little maths adventure.
Kyle Pearce: It’s great to hear that you have so many amazing tasks that are so interesting, and create so many amazing discussions, and that it doesn’t always have to look and sound exactly like that. And that’s a conversation we tend to have with a lot of teachers, and you may have seen some of the things that Jon I do for sparking curiosity. We like playing with animations, and videos and those types of things. But the message we try to ensure teachers see is that not every single class needs to look and sound like that. And if it did, it might get pretty boring after a while. It’s all about finding the beauty in the activity that you’re doing. And sometimes it’s just about figuring out how this stuff works. And as long as we’re doing that, that is so much better than just that straight memorization without meaning. So that’s great to hear you articulate it.
Jo Boaler: “Why does it make sense?” is a great question to ask kids. “Why does that make sense?” And [crosstalk 00:50:36].
Kyle Pearce: And that’s kind of the whole point, why we’re in the class in the first place, right? And we have calculators and we have Google that could find answers to so many different questions. We don’t need to be able to just calculate. We want to be able to actually think things through and sort of debate, and convince and I love kids making conjectures, those types of things.
Jo Boaler: Me too. And we need kids to be able to do more than their calculators can do, because we don’t need them to do what the calculators can do. That reasoning, making conjectures, coming up with their own questions, those are really important for kids.
Jon Orr: Our last question for you would be, where would people learn more about Jo Boaler and your work? Do you have a link that they could go to? Do you want to share that out?
Jo Boaler: Yes. I think you could learn more about me and our approach if you go to youcubed.org, which is Y-O-U and C-U-B-E-D. And on there, you’ll find lots of different ways of changing kids’ math relationships, videos, tasks to use in lessons, ideas and papers, all sorts of different things. And I have a book that’s called Mathematical Mindsets that also has a lot of those big ideas. And so those are probably my two main things at the moment, although I’m also on Twitter, Instagram, all those different places. I have a new book coming out in the fall that I’ve just finished that’s called Limitless, and the subtitle is Learn, Lead and Live Without Barriers. And this new book really has come about because so many people who read Mathematical Mindset said to me, “You have to get this out to other teachers, not just maths teachers, but you have to get it out to other teachers, and parents and anybody really.” So this book is attempting to do that, to get the ideas [inaudible 00:52:20] sort of live differently to anybody.
Kyle Pearce: Well, that’s excellent. Mathematical Mindsets has shifted my thinking, and I know so many other educators that I’ve come across when I asked them favorite math books and so on, that title comes up so often. What’s Math Got to Do with It? as Jon referenced earlier, had a huge impact on his teaching practice. For those listening, we also have a book list of about 60 titles currently, but it’s actually growing every day because some more suggestions are coming my way, and I can’t believe I forgot them. But also on that list are the mindset mathematics books which are grade specific books.
Kyle Pearce: So just for those listening, definitely check out those links we will put in the show notes, and we can’t wait to check out this new book coming out in the fall it sounds like, sometime later in the year. Well, thank you so much for the time tonight. Jo, it’s been a true pleasure to have you on the show. I know that you are very busy, and to take some time and carve it out for the math community, the Math Moment Makers, it means a ton to us. And we can’t wait to see you at our next conference, which will probably be NCTM I’m thinking.
Jo Boaler: I can’t wait either. It probably will be NCTM. Tomorrow morning I set off anchorage, where I’m going to be talking to 1,100 maths teachers in the Alaska. But yeah, we will probably get together again at NCTM. Yeah.
Jon Orr: Awesome. Thank you very much.
Kyle Pearce: Yes, we appreciate your time. And I hope you have a great night.
Jo Boaler: Thank you.
Jon Orr: What a great math infused discussion with Jo Boaler. Kyle, what was your big takeaway from this conversation?
Kyle Pearce: The biggest takeaway for me was how Jo articulated how she feels when she’s mentoring a teacher, who’s yet to embark on the journey of approaching mathematics differently than how we were taught, seeing mathematics as exciting, vibrant and a visual endeavor that all students and adults can engage in and enjoy. While many coaches, consultants and administrators might see shifting the beliefs of an educator who teaches mathematics traditionally as a challenge, she welcomes this work and is excited to help one more teacher see mathematics differently than how they were taught. How about you, Jon, what resonated with you?
Jon Orr: Mine was on a similar path, but hearing how dedicated she is to helping teachers teach math better. She’s a researcher, she’s an author, and she still gets into those classrooms to work with teachers to give them ideas, tasks and structures. She’s created a whole website dedicated to helping us do this better. She writes books with the main goal of just changing the way math should be taught, so that children can have memorable math moments that aren’t terrible. Her sharing has made an impact on my teaching, and so many more too. I really appreciate her.
Kyle Pearce: Awesome stuff Jon. Thanks for sharing your reflections. So how about you at home, what’s your big takeaway from this episode? Share it with a friend, a colleague, or send us a message on social media @MakeMathMoments on Twitter, Instagram or Facebook.
Jon Orr: In order to ensure you don’t miss out on our new episodes as they come out each week, be sure to subscribe on iTunes or your favorite podcast platform.
Kyle Pearce: Also, if you’re liking what you’re hearing, please share the podcast with a colleague and help us reach a wider audience by leaving us a review on iTunes, and tweeting us @MakeMathMoments on Twitter. Show notes and links to resources from this episode can be found at makemathmoments.com/episode10. Again, that’s makemathmoments.com/episode10.
Jon Orr: We release a new episode every Monday morning. Keep an eye open for our next episode, play is the engine of learning, an interview with Dan Finkel, where we’ll dive into Dan’s story about how he fell in love with mathematics and in particular teaching with games in the math classroom.
Kyle Pearce: Well, until next time, I’m Kyle Pierce.
Jon Orr: And I’m Jo Orr. High fives for us and high fives for you.
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