I'm Giving You This Feedback Because I Believe in You

Lex Fridman asks Jo Boaler, a Stanford mathematics educator and co-founder of youcubed.org, what is beautiful about mathematics. She loves the part many people do not even count as math: creative work where you look at a problem in different ways, visualize it, and find different solutions. A lot of people think math is one method and one answer. What she loves is the opposite — multiple methods, multiple ways of seeing, and sometimes multiple solutions. The sad part, she says, is that many people never get shown that version.

She tells teachers to hand over their driest, most boring content and she will turn it into a visual, interesting, open problem. It turns out you can do that with any area of math. Kids have been done a disservice by a diet of numbers and only numbers. Neuroscience, as she reads it, says that working a math problem involves five brain pathways, and the highest-achieving people in the world are the ones with more connections among those pathways. Seeing a problem with numbers and also visually forges a connection. Writing about it in words forges another. Building it with something physical forges a third. She wants what she calls a multi-dimensional experience of math.

The popular classroom split — you are a visual learner, so here are pictures; you are not, so here are formulas — is, she says, the wrong lesson from the science. If you think you are not a visual learner, it is probably more important that you get a visual approach so that part of the brain can develop. Physicists have been ahead of schools here. Einstein did not want to go anywhere with a problem without thinking visually; the train-and-light-speed story is intuition first. Mathematicians, when interviewed, talk about intuition constantly, but it is rarely invited into class. Sebastian Thrun once told her about robots built for the Smithsonian that were picking up white noise. He intuitively knew what the problem was. Proving it mathematically took three weeks.

Color is an underrated tool in the same family. If an algebraic expression has an *x*, write that *x* in a color and keep using the color wherever *x* appears, so the relationship is visible. In workshops with elementary teachers, Boaler has watched people come to tears when they first see ideas making sense visually. They had spent their whole lives not realizing you could really understand things that way.

Fridman asks whether math should be easy or hard. Boaler thinks challenge is great, with a condition. The key to dealing with hard math is knowing you can do it. Education’s most damaging idea, in her view, is that you are either born with a math brain or you are not. When kids first struggle they think they do not have that brain, and they literally switch off. From that point things go downhill. Struggle becomes usable when you know you have to go through it to get there, and that you are able to. The belief has to come first.

That is also why teaching is so hard and so under-appreciated: thirty students who think in different ways, some of whom dislike hard things and some of whom genuinely need a different route. A one-way class — here are numbers, here is a method, watch me, reproduce it — sorts the room into kids who like that game and kids who do not. Open the subject visually, with numbers, with words, and many more kids can get in. Different wiring stops being a verdict.

Fridman grew up in the Soviet Union and says he was forced into math early and fell in love through force. Two features stuck. One is a cult of excellence: success means being really good in that class, not merely better than yesterday, closer to how Americans treat sports stardom. The other is the belief that everybody is capable of being great, and if you are not, that is on you — work harder. He even remembers a half-delusional childhood plan to invent a time machine for a Nobel Prize. When you struggle, that belief that you could be great is what keeps you moving.

Boaler finds the contrast useful. In the US, excellence is also prized, but kids are told in many ways that they are either people who can or people who cannot. A large share decide they are in the second group, so there is no point trying. Switching that idea would be huge. The downside of “anybody can get there if they work hard,” which she names openly, is that people who do not get there carry blame they should not. She would add a clause: anybody can get there, but they need to work hard *and* they need good teaching. Plenty of people never meet a teacher who can make the subject make sense. As for more hours of math, she is for it if the math is interesting, open, and multi-dimensional. Stanford summer camps run three-hour classes. Teachers planning them asked whether kids could stay excited that long. The kids did not want to go to break or lunch.

Kids start giving up on themselves and on math around fifth grade, Boaler says, and the middle-school years are pivotal. Early elementary still lets them explore. Fifth-grade teachers often announce that it is time to prepare for middle school, and the grades and tests arrive. That is when kids start to feel badly about themselves. In those years many decide whether STEM is for them. Every year matters and every year can switch a pathway, but the lever is longest there. Her camps are built for middle-schoolers for that reason.

Does a teacher need a one-on-one “I believe in you, Johnny” speech? A study in high-school English classrooms had every student write an essay. All of them got diagnostic feedback. For half, researchers added one extra sentence at the bottom. Those students did significantly better in English a whole year later. The sentence was: “I'm giving you this feedback because I believe in you.” When Boaler shares this with teachers she says she is not suggesting they write it on every paper. One teacher asked about a stamp. Do not put it on a stamp. The power is that it is specific, and that the teacher means it. Kids sit in class thinking, what does my teacher think of me? Can I do this? The words are not given enough, and they cannot be faked.

Fridman admits he is an extreme perfectionist. When a student is clearly not close to good, the temptation is to give up on them mentally. History is full of people who sucked for a long time, sometimes late. You have to keep believing they can be great, and also say they have to work hard. Boaler points at Silicon Valley: it is filled with school dropouts and kids labeled with special needs who went on to do remarkable creative work. Her explanation is direct. The school system is set up to value good memorizers who can reproduce what a teacher shows, and to push away creative, often slower, deep thinkers. Those people get the “you cannot” signal early. Many of them are the ones who later do amazing things.

Maryam Mirzakhani, the first woman to win the Fields Medal, did work that was entirely visual. Her daughter thought she was an artist. Boaler chaired a PhD defense for one of Mirzakhani’s students and sat through two hours that were so visual she does not think she saw any numbers. She remembers thinking she could have brought a thirteen-year-old into the room and the child would not have recognized it as math. Other mathematicians said Mirzakhani had connected previously unconnected areas. When she was about thirteen, in Iran, a teacher told her she could not do math. All it takes, Fridman says, is one teacher. Boaler tells teachers that even inside a broken system of districts and textbooks, a single person can change a kid’s relationship with the subject forever.

A study found that the amount of math anxiety parents had predicted their child’s achievement — but only if they helped with homework. You can see how it works. An anxious parent sits down and says, I was terrible at this in school, and that is what gets passed on. One implication Boaler draws, half joking, is that if you hate math, just do not do math with your kids. youcubed.org has a sheet for parents on how to interact around the subject instead. When she talks to parents she tells them that even if they hate math they need to fake it: be endlessly optimistic and happy about doing it.

Fridman wonders whether it bruises the ego to want a child to surpass you. Boaler has not seen parents worry that their kids will be better than them. What she has seen is a bad relationship with math, not knowing how to help, and pulling away. Another study showed that when mothers say to their daughters they were bad at math in school, the daughter’s achievement goes down. Kids pick up the messages. Even Stanford undergraduates, people who have always done well, arrive thinking math is a set of methods to memorize and reproduce. Until parents have actually experienced the other math, it is hard for them to shift their kids, because in their eyes the subject just looks like that.

Boaler is one of five writers of a new mathematics framework for California, recommending what they call teaching to big ideas. Current standards, she says, get translated by textbooks into narrow questions, and success is then measured by delivering on those questions. Third grade, in one example she reads aloud, has three different standards about unit squares. One of them defines a square of side length one as a unit square with one square unit of area, usable to measure area. Math, for her, is a network map of a few big ideas and the connections among them. Standards take that map, chop it into lots of little pieces, and deliver the pieces to schools. Teachers do not see the connections. Neither do the kids.

The new framework sets out mathematics from kindergarten through grade 10 as big ideas and connections. Grade three is no longer sixty standards. It is a small set of concepts — square tiles, measuring, fraction, shape, time — and how they join. A teacher whose big idea this term is measuring does not read five procedural statements. She asks what rich, deep activity will actually teach measuring. Kids work on it for a few days and a lot of other math walks in. Teach to the big ideas, she says, not to a stack of short questions.

She is not a fan of grades and tests as daily instruments. A grade at the end of a course is a reasonable summative measure. The American math classroom uses grades every week or every day. Technology made it worse for her own kids in high school: everything was graded, visible in an online gradebook, updating as they walked into the next class. She calls that a performance culture — you are there to perform, someone is measuring you, you see your score — and it is not conducive to deep learning. A better mid-course tool is a rubric that shows the mathematical journey of a unit. Kids self-assess; teachers mark what they can do and write notes about what to try next. Instead of being an A kid or a B kid, they see where they are mathematically. At year’s end, sure, give a grade. During the year, give information.

Some of Fridman’s best classes were ones where a professor noticed a particular interest and gave him something extra, once with reverse psychology: I do not think this can be done. He poured everything into doing it. That kind of special treatment is precious and, he admits, a lot to expect when a room is full. Boaler pulls it back to something teachers can actually do. The important stance is that you do not know who will be excellent at something before you hand out the activity. In camp, when students finished early they were asked to write a question like this but different, or to extend it. One boy, after a lesson on borders of squares of different sizes, said he had been thinking about a pentagon. She had not predicted that question and had not reserved an extension task for him. She told him it was fantastic, and to go find out.

The same line lands differently on different kids. Fridman lights up when someone says he cannot do it. Another child might break. Teaching is difficult for exactly that reason. What she does when she first meets students — there is a video on the youcubed site — is show a picture of seven dots for a few seconds and take it away before they can count. Tell me how many, she says, but do not count them. Group them. Then she goes around the room. In a class of thirty there are perhaps eighteen ways of seeing the same seven dots: an outside hole with a center, stripes of lines, segments. She writes them all on the board. Look at this, she says. We are a class of thirty and we saw these seven dots in eighteen different ways.

The skill has a name: groupitizing. How well you groupitize predicts how well you do in math. She does not think you are born with it; some kids have developed it and you can learn it. That, to her, is part of how wrong school math has the assessment. We time kids on multiplication facts to decide who is good. Watching how they group dots might be a better signal. You can do the same task with young children or with graduate students. What the lesson really teaches, besides the dots, is that in this room the way you see a problem will be taken seriously.

What the 21st century needs, Boaler argues in *Limitless Mind*, is a flexible mind. School should not be about particular methods so much as approaching problems with flexibility. People do not put the words math and creativity together; that pairing is what she loves. She likes Leonard Mlodinow’s book *Elastic*, about elastic minds, the same claim. Schools do very little to develop that mind and a great deal to develop the kind a computer now handles: memorization and procedures. When kids leave school a computer will do those things better. Creative, flexible thinking is still near ground zero for machines. The human brain is perfectly suited for it. She would like the balance in schools to shift.

Other humans are part of that shift. In 2019 she taught calculus to a hundred Stanford undergraduates, just before Covid, and taught them to collaborate. Most came in against it. Their only experiences were competitive individual work, or group work where one person did everything. They also walked in loaded with social comparison: the person next to me is better, faster, thought of something I missed. The team spent four weeks training that out. For a pre- and post-test they refused a timed individual exam. They gave an applied problem to pairs, each person a different colored pen, no switching. On the first pass the papers showed a red section and a green section, separate work, and they did not do well, even though middle- or high-schoolers could have started the problem.

Four weeks later, same problem, same colors. This time they were actually collaborating, and the ones who collaborated did better. Students wrote that they had learned to value other people’s thinking on a problem. Fridman offers Daniel Kahneman as a model. In their first meeting Kahneman knew almost nothing about autonomous vehicles and asked question after question out of genuine curiosity; ten minutes in they were jointly trying to solve driving. Collaboration is a skill. You have to put ego and your own worldview aside and learn how the other person sees it. Boaler adds the other half: if you sit there thinking that was an amazing idea, he is so much better than I am, you will not take on the value of the idea. Stanford students were stuck on that. Fridman has also found, once a little more public, that celebrating others pays off far more than proving you are the smartest person in a small circle by refusing to admit anyone else is.

Fridman defends physical textbooks: pretty pictures, a smell, a way of opening. High school, for him, was among the happiest learning years of his life, a whirlwind of calculus, biology, chemistry, physics, anatomy, Shakespeare, computer science, with nobody yet telling you what to do with your life. Boaler does not argue with that memory. She has loved some books too. A lot of math textbooks, though, are not good experiences. In a youcubed camp video a student says that in math you have to follow the textbook, which is kind of like the Bible. Monday is 2.3.2, Tuesday is 2.3.3, and you never go off that. What the kids loved about camp was that there were no books. They still got sheets of paper. They felt free because they were not trotting through exercises.

Fridman grants the 2.3.2 structure a real gift: a sense of progress, a hard problem that finally yields, a chapter that closes. Boaler accepts that you can have a good experience inside a textbook. What matters is what is in it. She grew up in England, where math is not split into a year of algebra and a year of geometry — she does not think any country besides the US does that — and she looks at kids doing algebra for a year and thinks she would have been pretty bored. Cutting the subject into algebra, geometry, and then into 2.3.2 is the same move as chopping the map of big ideas into standards. The fragments are easy to schedule and test. The learner never sees the whole picture.

Why do the British say *maths*? Mathematics, she says, was always the different maths — geometry, probability, a multi-dimensional subject — and the plural survived the shortening. American English made it singular, and the common phrase “do the math” basically means do a calculation, an expression she dislikes because math does not have to be calculation. *Maths*, for her, keeps the broader feel: many subdisciplines, many approaches, many ways of seeing.

Boaler caught the online-teaching bug from Sebastian Thrun’s first MOOC and put out a free summer course in math education. She did not know if anyone would take it. Thirty thousand math teachers did, talking and sharing, and the course took off. That is what pushed her and co-founder Kathy Williams to create youcubed: people finished and asked, what now? About five years in, the site had more than 52 million visitors. The mission is to share good ideas for teaching math with teachers, students, and parents, with a subgoal of erasing math anxiety, and to show the subject as beautiful and creative.

The academy, she says, already knows a great deal about how to teach math well — journals, research, write-ups — but teachers do not read it. It sits behind paywalls and is written in inaccessible ways. A whole industry finds out how to teach well and does not tell the people who are teaching. youcubed’s move is translation: research into lessons, videos for kids, tips for parents. Inspired by Hour of Code they built a Week of Inspirational Maths and asked teachers to try one week. It has been downloaded millions of times, often as the first week of the school year. Teachers report that the kids’ lights were on. Then the week finished, they opened the textbooks, and the lights went out.

She would love to extend that week into a year. The Stanford team is small and cannot produce creative visual tasks for every year group for every day. They did spend five exhausting years making a K–8 set of books organized around big ideas, rich activities, and visuals, just finished when they spoke. Some teachers use them as a supplement to a boring textbook; some treat the book as the year. Her personal goal is that every math teacher knows youcubed exists. A lot who find it are grateful. A lot still do not know it is there. On pedagogy she learned a hard Udacity rule: never talk more than about five minutes, then make people do something. Half-hour videos are when viewers switch off. MIT OpenCourseWare-style lectures are good as access; she thinks the Udacity rhythm is more effective. The early MOOC dream of democratizing education did not fully arrive, because the people who take MOOCs tend to be the more privileged. There is still work to do to reach everyone else.

From the student’s side, Boaler’s first study habit is to stop reviewing by rereading. People look through a teacher’s explanation, it makes sense, they think they have it, and they move on. It is not until they try a problem that they realize they did not understand it. Neuroscientists, she says, recommend keeping giving yourself questions. Testing yourself is a better way to know what you actually know. Homework, in the same spirit, should not be twenty more questions. Send kids home to reflect: what was the big idea today, what was difficult, what did you struggle with, what was exciting? Plenty of students go through years of math getting questions right without knowing what the topic is, why it matters, or where it lives in real life.

Fridman pushes the same habit up a level. Children ask layers of why; adults stop. Boaler cites research she finds sad: the number of questions kids ask in school goes down in a roughly linear slide. In the early years you cannot stop them. They learn not to ask. One of her own memories is vivid enough that she remembers where she was sitting. First class with a new high-school math teacher, who had said to ask if you have questions. She put her hand up. He said something like, “that's your question.” She asked no more questions in that class. He was chair of the math department. She guesses he had no idea of the impact. Teaching is hard because even tone carries weight; a bad day and one sentence can last decades. youcubed’s counter-move is to put kids’ questions on the wall and celebrate them.

The flip side of that power is a single teacher who keeps someone in the subject. For Boaler it was Mrs. Marshall, her A-level math teacher in England, where at sixteen you drop most subjects and go deep on three or four, closer to a US master’s in specialization. For the first time in her math life, Marshall gave them problems and told them to talk to each other. At seventeen Boaler sat with friends figuring out how to solve something. That was the whole change. She started to hear other people’s ways of seeing, and they came up with solutions together. It was not a private pep talk. It was a teaching approach. A teacher can change a child’s relationship with math without saying a word of encouragement, just by changing how the period is run.

Fridman brings up Cal Newport’s *Deep Work* and a world of distractions. Boaler’s answer is to give kids genuinely deep problems, even if only occasionally. When she trains teachers she hands out a problem and says she will check in after an hour. They are shocked. An hour later they ask how an hour could already have gone. Most kids go through years of math without once working on a problem that way. Her Stanford undergraduate class does it every session. Students leave saying they never want to go back to a relationship with math that was only about quick answers. Teachers do not have to do it every day. They do have to give kids some experience of working slowly, going deeper, and not being told they have five minutes to finish twenty questions.

Sitting still in focus is itself a skill, Fridman says, a muscle you can build from five minutes to ten to thirty to an hour. Without it you never reach the addictive part of solving. A survey asked schoolkids how long they would work on a math problem before giving up and deciding it was not possible. The average across the kids was two minutes. That is less a verdict on intelligence than on a system that never let them stay.

Visualization at scale has a public face in Grant Sanderson, 3Blue1Brown, whose programmatic animations Boaler loves. She jokes that she should send him visuals and let him make them astonishing. The craft is storytelling as much as pictures: a small example, then a turn, a surprise, and you fall in love with the concept. Sanderson has said he does not see himself teaching the core curriculum so much as lighting a spark, and she agrees you cannot ask one person to convert the whole syllabus at that quality. He open-sourced Manim, a Python library for animating equations and shapes. The code is not perfectly documented, but releasing it produced a crowd of people making math videos, many with a handful of views. For those makers, Fridman says, the process of teaching is the transformation — the old Feynman point that the best way to learn is to teach. Hotels, Boaler notices, are full of patterned carpets and walls. Humans love patterns and broken symmetries. Math is that subject, if you let kids be pattern-seekers. Multiplying 18 by 5 by thinking 9 times 10 is a pattern that always works. Once that is your job, the work gets exciting.

She also has a short free class for kids, *How to Learn Math*: six sessions of fifteen minutes, showing math as visual and creative plus some mindset and brain science. In a randomized control trial, middle-schoolers who took it — taught by the same teachers as a control group who did not — ended up 68 percent more engaged in math class and did better at the end of the year. Ninety minutes of video. Not a huge change to the education system. It changed a relationship.

Asked for advice to young people, Boaler knows it will sound corny: always believe in yourself and know that you can achieve. The reason it is worth saying is that millions of kids in school have been told they cannot, and adults too. They chose a job because those other things they could never have done. When someone says maybe you could, even adults go back, relearn, and change careers. Scientists keep looking for a limit to how much a person can learn and come away unable to find one. People go further and further. That includes people born with parts of the brain that do not function well, labeled with special needs, some of whom also go on to do amazing things. She wants people to know this deeply, not just say it.

Fridman adds that the same trap appears after success. People who love you celebrate what you already did and want you to keep doing it, rather than believing you can do the riskier thing your heart wants. He realized late how much it matters, when you are younger, to seek out mentors and surround yourself with people who will believe in you. You do not just land on an advisor. The people around you define the trajectory, so select them, and get away from people who do not believe you. Parents can love you and still load you with limiting beliefs from their own experience. If you are interested in mathematics and your parents are not, Boaler says, do not listen to them on that one dimension. In *Limitless Mind* she interviewed people told they could not do math by parents or teachers who later found another message and discovered a whole world of mathematical thinking was open. Find people who believe in you, however hard that is.

Is she optimistic about school in fifty or a hundred years? She has hope. Change in education is usually microscopically slow, but she sees movement. Data science can now be taken in high school instead of Algebra 2. The content of American high-school math, she says, was set out in 1892 and had not changed. A course developed in 2014; kids took it, nothing bad happened, they went to college. Then the University of California system emailed 50,000 high schools saying data science was a legitimate college pathway in place of Algebra 2 — a green light. When she last looked, twelve states allowed it; she expected that to double the next year. Data science does not sit neatly inside math, computer science, or statistics, which is part of why it got in: it forces the question of who teaches it and where it lives. A hundred years from now she imagines subject boundaries gone, programming integrated the way reading is, kids creating and finding things as they learn. Today’s factory-model school was never designed to inspire creativity. People may look back and find it hilarious.