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

(This is the first of two of our most popular Blog posts that we repeat for the month of July. )

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.*

Greetings from India.

Thank you. You made my life easier. As a maths teacher I often see teachers and parents making discouraging observations and comments about their children/students. I can now quote you, to prove how they are all wrong.

Thank you. This gives me hope that learning math is possible if I put my mind to it.