*By Jo Boaler, Professor of Mathematics Education, Stanford University, and co-founder of youcubed.org*

2018 was an important year for the Letchford family – for two related reasons. First it was the year that Lois Letchford published her book: *Reversed: A Memoir*.^{[1]} In the book she tells the story of her son, Nicholas, who grew up in Australia. In the first years of school Lois was told that Nicholas was learning disabled, that he had a very low IQ, and that he was the “worst child” teachers had met in 20 years. 2018 was also significant because it was the year that Nicholas graduated from Oxford University with a doctorate in applied mathematics.

Nicholas’s journey, from the boy with special needs to an Oxford doctorate, is inspiring and important but his transformation is far from unique. The world is filled with people who were unsuccessful early learners and who received negative messages from schools but went on to become some of the most significant mathematicians, scientists, and other high achievers, in our society – including Albert Einstein. Some people dismiss the significance of these cases, thinking they are rare exceptions but the neuroscientific evidence that has emerged over recent years gives a different and more important explanation. The knowledge we now have about the working of the brain is so significant it should bring about a shift in the ways we teach, give messages to students, parent our children, and run schools and colleges. This article will summarize three of the most important areas of neuroscience that directly apply to the teaching and learning of mathematics. For more detail on these findings, and others, visit youcubed.org or read Boaler (2016).^{[2]}

The first important area of knowledge, which has been emerging over the last several decades, shows that our brains have enormous capacity to grow and change at any stage of life. Some of the most surprising evidence that highlighted this came from studies of black cab drivers in London. People in London are only allowed to own and drive these iconic cars if they successfully undergo extensive and complex spatial training, over many years, learning all of the roads within a 20-mile radius of Charing Cross, in central London, and every connection between them. At the end of their training they take a test called “The Knowledge” – the average number of times it takes people to pass The Knowledge is twelve. Neuroscientists decided to study the brains of the cab drivers and found that the spatial training caused areas of the hippocampus to significantly increase.^{[3]} They also found that when the drivers retired, and were not using the spatial pathways in their brains, the hippocampus shrank back down again.^{[4]} The black cab studies are significant for many reasons. First, they were conducted with adults of a range of ages and they all showed significant brain growth and change. Second, the area of the brain that grew – the hippocampus – is important for all forms of spatial, and mathematical thinking. The degree of plasticity found by the scientists shocked the scientific world. Brains were growing new connections and pathways as the adults studied and learned, and when the spatial pathways were no longer needed they faded away. Further evidence of significant brain growth, with people of all ages, often in an 8-week intervention, has continued to be produced over the last few decades, calling into question any practices of grouping and messaging to students that communicate that they cannot learn a particular level of mathematics.^{[5]} Nobody knows what any one student is capable of learning, and the schooling practices that place limits on students’ learning need to be radically rethought.

Prior to the emergence of the London data most people had believed either that brains were fixed from birth, or from adolescence. Now studies have even shown extensive brain change in retired adults.^{[6]} Because of the extent of fixed brain thinking that has pervaded our society for generations, particularly in relation to mathematics, there is a compelling need to change the messages we give to students – and their teachers – across the entire education system. The undergraduates I teach at Stanford are some of the highest achieving school students in the nation, but when they struggle in their first math class many decide they are just “not a math person” and give up. For the last several years I have been working to dispel these ideas with students by teaching a class called How to Learn Math, in which I share the evidence of brain growth and change, and other new ideas about learning. My experience of teaching this class has shown me the vulnerability of young people, who too readily come to believe they don’t belong in STEM subjects. Unfortunately, those most likely to believe they do not belong are women and people of color.^{[7]} It is not hard to understand why these groups are more vulnerable than white men. The stereotypes that pervade our society based on gender and color run deep and communicate that women and people of color are not suited to STEM subjects.

The second area of neuroscience that I find to be transformative concerns the positive impact of struggle. Scientists now know that the best times for brain growth and change are when people are working on challenging content, making mistakes, correcting them, moving on, making more mistakes, always working in areas of high challenge.^{[8, 9]} Teachers across the education system have been given the idea that their students should be correct all of the time, and when students struggle teachers often jump in and save them, breaking questions into smaller parts and reducing or removing the cognitive demand. Comparisons of teaching in Japan and the US have shown that students in Japan spend 44% of their time “inventing, thinking and struggling with underlying concepts” but students in the U.S. engage in this behavior only 1% of the time.^{[10]} We need to change our classroom approaches so that we give students more opportunity to struggle; but students will only be comfortable doing so if they have learned the importance and value of struggle, and if they and their teachers have rejected the idea that struggle is a sign of weakness. When classroom environments have been developed in which students feel safe being wrong, and when they have been valued for sharing even incorrect ideas, then students will start to embrace struggle, which will unlock their learning pathways.

The third important area of neuroscience is the new evidence showing that when we work on a mathematics problem, five different pathways in the brain are involved, including two that are visual.^{[11, 12]} When students can make connections between these brain regions, seeing, for example, a mathematical idea in numbers and in a picture, more productive and powerful brain connections develop. Researchers at the Marcus Institute of Integrative Healthhave studied the brains of people they regard to be “trailblazers” in their fields, and compared them to people who have not achieved huge distinction in their work. The difference they find in the brains of the two groups of people is important. The brains of the “trailblazers” show more connections between different brain areas, and more flexibility in their thinking.^{[13]} Working through closed questions, repeating procedures, as we commonly do in math classes, is not an approach that leads to enhanced connection making. In mathematics education we have done our students a disservice by making so much of our teaching one-dimensional. One of the most beautiful aspects of mathematics is the multi-dimensionality of the subject, as ideas can always be represented and encountered in many ways, such as with numbers, algorithms, visuals, tables, models, movement, and more.^{[14, 15]} When we invite people to gesture, draw, visualize, or build with numbers, for example, we create opportunities for important brain connections that are not made when they only encounter numbers in symbolic forms.

One of the implications of this important new science is we should all stop using fixed ability language and celebrating students by saying that they have a “gift” or a “math brain” or that they are “smart.” This is an important change for teachers, professors, parents, administrators – anyone who works with learners. When people hear such praise they feel good, at first, but when they later struggle with something they start to question their ability. If you believe you have a “gift” or a “math brain” or another indication of fixed intelligence, and then you struggle, that struggle is devastating. I was reminded of this while sharing the research on brain growth and the damage of fixed labels with my teacher students at Stanford last summer when Susannah raised her hand and said: “You are describing my life.” Susannah went on to recall her childhood when she was a top student in mathematics classes. She had attended a gifted program and she had been told frequently that she had a “math brain,” and a special talent. She enrolled as a mathematics major at UCLA but in the second year of the program she took a class that was challenging and that caused her to struggle. At that time, she decided she did not have a “math brain” after all, and she dropped out of her math major. What Susannah did not know is that struggle is really important for brain growth and that she could develop the pathways she needed to learn more mathematics. If she had known that, and not been given the fixed message that she had a “math brain,” Susannah would probably have persisted and graduated with a mathematics major. The idea that you have a “math brain” or not is at the root of the math anxiety that pervades the nation, and is often the reason that students give up on learning mathematics at the first experiences of struggle. Susannah was a high achieving student who suffered from the labeling she received; it is hard to estimate the numbers of students who were not as high achieving in school and were given the idea that they could never do well in math. Fixed brain messages have contributed to our nation’s fear and dislike of mathematics.^{[16]}

We are all learning all of the time and our lives are filled with opportunities to connect differently, with content and with people, and to enhance our brains. My aim in communicating neuroscience widely is to help teachers share the important knowledge of brain growth and connectivity, and to teach mathematics as a creative and multi-dimensional subject that engages all learners. For it is only when we combine positive growth messages with a multi-dimensional approach to teaching, learning, and thinking, that we will liberate our students from fixed ideas, and from math anxiety, and set them free to learn and enjoy mathematics.

*This blog contains extracts from Jo’s forthcoming book*: Limitless: Learn, Lead and Live without Barriers, *published by Harper Collins.*

[1] Letchford, L. (2018) *Reversed: A Memoir*. Acorn Publishing.

[2] Boaler, J (2016) *Mathematical Mindsets: Unleashing Students’ Potential through Creative Math, Inspiring Messages and Innovative Teaching*. Jossey-Bass/Wiley: Chappaqua, NY.

[3] Maguire, E. A., Gadian, D. G., Johnsrude, I. S., Good, C. D., Ashburner, J., Frackowiak, R. S., & Frith, C. D. (2000). Navigation-related structural change in the hippocampi of taxi drivers. *Proceedings of the National Academy of Sciences*, 97(8), 4398-4403.

[4] Woollett, K., & Maguire, E. A. (2011). Acquiring “The Knowledge” of London’s layout drives structural brain changes. *Current **b**iology**:CB*, 21(24), 2109–2114.

[5] Doidge, N. (2007). *The Brain That Changes Itself*. New York: Penguin Books,

[6] Park, D. C., Lodi-Smith, J., Drew, L., Haber, S., Hebrank, A., Bischof, G. N., & Aamodt, W. (2013). The impact of sustained engagement on cognitive function in older adults: the Synapse Project. *Psychological science*, 25(1), 103-12.

[7] Leslie, S.-J., Cimpian, A., Meyer, M., & Freeland, E. (2015). Expectations of brilliance underlie gender distributions across academic disciplines. *Science*, 347, 262-265.

[8] Coyle, D. (2009). *The Talent Code: Greatness Isn’t Born, It’s Grown, Here’s How*. New York: Bantam Books;

[9] Moser, J., Schroder, H. S., Heeter, C., Moran, T. P., & Lee, Y. H. (2011). Mind your errors: Evidence for a neural mechanism linking growth mindset to adaptive post error adjustments. *Psychological science*, 22, 1484–1489.

[10] Stigler, J., & Hiebert, J. (1999). *The teaching gap: Best ideas from the world’s teachers for improving education in the classroom*. New York: Free Press.

[11] Menon, V. (2015) Salience Network. In: Arthur W. Toga, editor. *Brain Mapping: An Encyclopedic Reference*, vol. 2, pp. 597-611. Academic Press: Elsevier;

[12] Boaler, J., Chen, L., Williams, C., & Cordero, M. (2016). Seeing as Understanding: The Importance of Visual Mathematics for our Brain and Learning*. Journal of Applied & Computational Mathematics*, 5(5), DOI: 10.4172/2168-9679.1000325

[13] Kalb, C. (2017). What makes a genius? *National Geographic*, 231(5), 30-55.

[14] https://www.youcubed.org/tasks/

[15] Boaler, J. (2016) *Mathematical Mindsets: Unleashing Students’ Potential through Creative Math, Inspiring Messages and Innovative Teaching*. Jossey-Bass/Wiley: Chappaqua, NY.

[16] Boaler, J. (2019). *Limitless: **Learn, Lead and Live without Barriers.*

Minor point, and irrelevant to the message of the post, but “Einstein flunked math” is a myth. Apparently the origin of the myth is that the grading system changed while he was in school. It went from 1 to 5 (one being the best) to 5 to 1 (five being the best). Einstein went from being the best to still being the best.

Hmmm, is this true? Curious of your source on Einstein.

Not quite. In Germany, it was (and is) 1-5 (1 best). In Switzerland, where he moved, it was (and is) 1-6 (6 best).

The author did not claim that “Einstein flunked math”. She included him among “people who were unsuccessful early learners and who received negative messages from schools.” And there is evidence of that. From the Einstein Archives at the Hebrew University of Jerusalem:

“After one year of homeschooling, Albert was sent to primary school, entering second grade already at age 6 1/2. He may not easily have accommodated himself to the school’s expected mindless obedience and discipline aimed at instilling authoritarian civic virtues. Unable – or unwilling – to provide quick automatic responses, the boy was considered only moderately talented by his teachers.”

http://www.albert-einstein.org/article_handicap.html

I had a struggle similar to Susannah’s in my undergraduate education. I believe the class was Abstract Algebra. The first test did not go well. The second was terrible so the professor offered everyone a retake. I scored worse on it. The third test firmly cemented me in the low Cs. As I sat down to the legal sized study sheet of concepts and proofs that could be on the final, I was looking through the university’s course guide, trying to find a new major that would not set me back too many credits. Great way to choose one’s profession, right? When a solution didn’t present itself, I decided I was going to prove and memorize every proof on the sheet. Once I started, I realized there were only two types of proofs and once I knew which one a problem required, it followed a pattern, just as the professor had been telling us all along. I scored a 188/200 on the final and an A for the course. I share this story with my students today and remind them that when they encounter something that is difficult if they work at it, they can figure it out.

Sure. The brain is a remarkable organ, capable of change at almost every level.

My concern with this view–which I mostly agree with–is that it might lead us to ignore the needs of children whose brains are ready for more mathematics, for more rapid learning.

Whatever ‘gifted’ means–and we can talk about that–it’s difficult to deny that some children learn more quickly than others. And to deny them access to mathematics which challenges them, which makes them struggle, which engages their minds, is to deny them the education they need. Just as much as writing off the minds of other children as ‘not math minds’ denies them the education they need.

And so I have a problem with the characterization of the word ‘gifted’ above. There is no contradiction between this notion of ‘gifted’ and the notion that the brain can change and adapt.

Msaul, I don’t think that anyone is disputing the fact that some students seem to pick up the learning faster than others but we mistakenly assume this makes them ready for more content instead of deeper content. Not only that, my question would be, why are they learning “quicker”? Is it because they are regurgitating information thrown at them and mimicking mathematics or are they truly, authentically grappling with rich tasks that make them think and wonder? It’s the difference between acting like a mathematician and actually becoming one. We want our students to become thinkers and challenge their surroundings by wondering “what if”? We want our students to have mathematical conversations that allow them to defend their ideas and solidify their own thinking. And, don’t forget, we want our students to be tolerant of all types of people – especially people that think differently. I often wonder if the kids we are labeling as “smart” are actually the kids that are compliant and do what their told. Don’t get me wrong, I realize there are students who just seem to make connections faster than others. My own son is one of them. He absorbs math, loves it and takes as many math classes as he can. But, I often wonder, what would have happened if he had been asked to think deeper about topics? Would he see himself as more of a creator of mathematics rather than a consumer?

No, I am talking about students who reason proficiently about mathematics. Not those who are acting. I’ve spent decades working with such students, and it’s easy to tell them apart from those you are describing. Yes, they have mathematical conversations. Yes, they challenge their surroundings. Yes, they think deeper about these topics–not just faster.

I don’t know what your son’s experience is, but mine, with gifted students, is that they learn faster and deeper at the same time.

As for tolerance: that’s another story completely. But I would not want to sacrifice mathematical learning for learning of tolerance. And I don’t think we have to. That’s another story completely.

In what way or how do you think this view could lead educators to ignore the needs of rapid learners of mathematics?

Good question. I am not sure. The author leaves it ambiguous whether she denies that some kids have a gift for mathematics or whether she denies the utility of labeling them ‘gifted’. Likewise there is no indication here, or acknowledgment, that some students–on both ends of whatever spectrum we are talking about–have special learning needs. (And she does acknowledge that there are differences in how people learn mathematics, including–I think–the rate at which they learn mathematics.)

But if you don’t identify students who are rapid learners of mathematics in some way, you cannot meet their special needs. And because of budgets and expedience, will tend to let them languish. I see this most often in rural, minority, and working-class populations.

Thinking about it, with the example of ‘Suzannah’ the problem was a dual issue of being labelled as ‘gifted’, whilst NOT being stretched. If she’d been actually challenged on a regular basis, or not been told she was something special, she might not have crashed-out when she did.

Everyone who is not intellectually handicapped, can learn mathematics. One does not need to be “gifted” or “clever” to learn mathematics, only to create/discover new mathematics. Many very clever mathematicians have, over the centuries, invented/created/discovered mathematics and presented it in such a way that much lesser mortals with limited mathematical ability can learn and use it.

Mathematics is both the simplest and most difficult subject I have studied, the principal difficulty being to find/master the approach/perspective that lays bare the underlying simplicity. If you understand what your are doing in mathematics, you do not need to rely on memory – you can work it out. If I forget the German word for a table, I’m stuck. But when I forgot the quadratic formula in an exam (in high school) I was able to work it out, because I understood it.

Wait. I am certainly not denying that anyone can learn mathematics to high levels. That is a different claim from the claim that there is no such thing as a ‘gift’ for mathematics. That is also a different claim from the claim that the label ‘gifted’ (for students who learn quickly and reason well) is pernicious. These are three different statements. And I think the author confounds them.

Gifted and talented people are usually aware that they are “good at” what they are gifted and talented at, and do not need to be told so, unless they have been subjected to treatment that has led them to doubt themselves.

I think it’s best to let students develop, offering guidance and assistance, only discussing whether they are gifted, or handicap, if it is absolutely necessary.

Well, but when gifted or handicapped people require services that supplement or replace those we offer to others, we have to somehow explain what we are doing. As Imi notes, the explanation is usually not news to the person involved. And, I would add, not usually news to his or her peers or colleagues.

Indeed. For a “gifted” student, I would suggest something like: “Here is some extra (mathematics) you might find interesting.” For the struggling student, it’s more delicate, depending on the source/cause of his/her difficulties.

I’m not sure what your experience has been, but I’ve taught both struggling and promising students for decades.

On one level, I can add to your suggestion: for struggling students, something like “Here is a new way to think about things that you might find interesting.” The point is that the worst thing you can do for such students is the same thing over again, even if slower.

But both these responses make light of a serious educational problem: differentiation of instruction. The reason I find it important to identify outliers (gifted, struggling) is that it is difficult–more difficult than usually portrayed in the literature–to provide for them. And research, as usual, does not address the task of teaching adequately: most research is about learning, which is a different process.

I have extensive experience of teaching, both gifted and struggling students.

I have always told students that there is a VERY simple reason they find mathematics hard, namely, mathematics IS hard! There would be something wrong if they did NOT find mathematics hard. However, mathematics is also the simplest subject they’ll ever study, for if you understand what you are doing, and forget a formula or result, you can work it out. But if you do NOT remember the German word for a table, you are truly stuck in an exam, or if you forget the details of some law and you are sitting for a law exam, you’re in trouble.

You don’t need to be clever to learn and use mathematics, only to create it. Because extremely clever and capable people, including genuine geniuses, have developed mathematics painstakingly over centuries, even lesser mortals, like us, can learn, understand and apply mathematics.

The hardest part of mathematics is to find the perspective that lays bare the underlying simplicity.

I am really disturbed by the number of ad hoc tricks used to “explain” things in mathematics that actually mask what is really going on, such as the “explanation” so often used for the arithmetic equality -(-a) = a. Typically, -a is “explained” as turning to face in the opposite direction, so doing it twice has you facing in the original direction. This is a trick that works only in this case. It cannot be used to show why 1/(1/a) = a, or why the inverse of the inverse of a function is the original function. Yet there is a simple underlying universal argument that explains all of these (and any similar!) statements. This universal argument is conceptual and requires more care to introduce and explain, but not inordinately so, and it makes many things much easier later.

The author did not define “gifted.” She did note that *labeling* people as “gifted” can later cause problems: “When people hear such praise they feel good, at first, but when they later struggle with something they start to question their ability. If you believe you have a ‘gift’ or a ‘math brain’ or another indication of fixed intelligence, and then you struggle, that struggle is devastating.”

Her website also has a short film interviewing students about the difficulties they faced as a result of being labeled gifted.

https://www.youcubed.org/rethinking-giftedness-film/

It has never been clear to me what the objection is. Sometimes a label of ‘gifted’ can cause problems, but in my experience not as often as implied here. The claim does not seem to be research based, although I am sure that there are anecdotes both for and against the label. In fact, kids who learn quickly but don’t get fed the mathematics they need–whether or not labeled ‘gifted’–often frustrate when they run into mathematics with which they must struggle. Worse, they will get bored, drop mathematics, and never reach the level of struggle. In short, the claim quoted above rings hollow in my decades of experience.

And it is hard to believe that all the students interviewed in that film did a dance of happiness without being prompted in some way.

Are the difficulties “gifted/talented face greater than those who are challenged by maths at many levels and forced to show what they can’t get? Same happens in PE with some children, same in music, all subjects and all walks of life. Can the term disposition be used? Some parts of life are of more interest to me so I am prepared to struggle to learn in these areas. What are the problems that a fixed mindset creates?

If you provide an ‘open’ flexible task it will cater for all students. The problem with a closed question is, as you suggested, it only challenges a limited number of students at an appropriate level because there is a desired response. When the task is open to provide opportunities for different responses, pathways or representations, and there is a classroom climate of embracing challenges and taking risks, all students are catered for. Those ‘gifted’ students will probably teach you something when their ceiling has been removed and so will the rest of the students.

I appreciate the “dream” of mathematical talent for everybody. The catch, however, is that the learning can only be done in the correct context

You implied that statement in your post. I laud you for that.

You also implied that the United-States culture discourages open thinking to make the process of doing math more palpable and achievable.

Keep on communicating your voice of universal math learning and teaching. This will create a better future for everyone.

Maybe it is not general knowledge what Maria Montessori was writing in the Introduction of her Psicogeometria. It is exactly what Boaler is saying.