Mathematical Mindsets

Unleashing Students' Potential through Creative Mathematics, Inspiring Messages and Innovative Teaching

Jo Boaler

• Publisher: Jossey-Bass
• Year: 2022
• ISBN:
  9781119823070
• Edition: 2
• Topic: Education
• Subtopic: Teaching Mathematics

Reverse mathematics trauma and find a universal blueprint for math success

In Mathematical Mindsets: Unleashing Students' Potential through Creative Math, Inspiring Messages and Innovative Teaching mathematics education expert and best-selling author Jo Boaler delivers a blueprint to banishing math anxiety and laying a foundation for mathematics success that anyone can build on.

Perfect for students who have been convinced they are naturally "bad at math, " the author offers a demonstration of how to turn self-doubt into self-confidence by relying on the "mindset" framework.

Mathematical Mindsets is based on thousands of hours of in-depth study and research into the most effective—and ineffective—ways to teach math to young people. This new edition also includes:

• Brand-new research from the last five years that sheds brighter light on how to turn a fear of math into an enthusiastic desire to learn
• Developed ideas about ways to bring about equitable grouping in classrooms
• New initiatives to bring 21st century mathematics to K-12 classrooms

Mathematical Mindsets is ideal for K-12 math educators. It also belongs on the bookshelves of the parents interested in helping their K-12 children with their math education, as well as school administrators and educators-in-training.

CHAPTER 1
The Brain and Mathematics Learning

In the last few decades we have seen the emergence of technologies that have given researchers new access into the workings of the mind and brain. Now scientists can study children and adults working on math and watch their brain activity; they can look at brain growth and brain degeneration, and they can see the impact of different emotional conditions upon brain activity. One area that has emerged in recent years and stunned scientists concerns “brain plasticity.” It used to be believed that the brains people were born with couldn't really be changed, but this idea has now been resoundingly disproved. Study after study has shown the incredible capacity of brains to grow and change within a really short period (Abiola & Dhindsa, 2011; Maguire, Woollett, & Spiers, 2006; Woollett & Maguire, 2011).

When we learn a new idea, one of three things happens in the brain (see Figure 1.1). The first possibility is that you start a new brain pathway. The more deeply you learn, the stronger the pathway becomes. The second possibility is that you strengthen a pathway you already had, and the third possibility is that you make connections between pathways. This brain development is taking place all the time, and the pathways you build, strengthen, or connect were not in your brain at birth; they are created by your learning experiences.

I wish all students knew this—when you are teaching them math, you are changing their brains! Neuroscientist Norman Doidge (2007) likes to share with his audiences that every day you wake up, your brain is different from the day before—that is the extent of brain growth and change that occur every day. If you learn something deeply, you form lasting brain pathways that you can revisit and use, but if you visit an idea only once or in a superficial way, the pathway can “wash away” like a path made in the sand. These brain connections form when learning takes place, but learning does not occur only in classrooms or when reading books; as we all know, we are forming brain connections when we have conversations, play games, or build with toys, and in the course of many, many other experiences.

The first research on what became known as neuroplasticity, which shocked the scientific world, came from studies of “Black Cab” drivers in London. I am from England, and I have traveled in taxicabs in London many times. I still have fond memories of the exciting day trips my family and I took to London when I was a child, from our home a few hours away. As an adult I studied and worked at King's College, London University, and had many more opportunities for trips around London in taxis. A number of different taxis work in the London area, but the queen bee of taxis in London is the Black Cab (see Figure 1.2).

FIGURE 1.1 Brains form, develop, and connect pathways.

FIGURE 1.2 The Black Cab of London

Source: Peter Fuchs/Shutterstock.

For most of my rides through London in a Black Cab, I had no idea how highly qualified the drivers were. It turns out that to become a Black Cab driver in London, applicants need to study for four years or more, and during that time learn routes around 25,000 streets and 20,000 landmarks within a 25‐mile radius of Charing Cross in London. Learning your way around the city of London is considerably more challenging than learning your way around most American cities, as London is not built on a grid structure and comprises thousands of interweaving, interconnected streets (see Figure 1.3).

At the end of their training period the Black Cab drivers take a test that is simply and elegantly called “The Knowledge.” If you ride in a London Black Cab and ask your driver about “The Knowledge,” they are usually happy to regale you with stories of the difficulty of the test and their training period. The Knowledge is known to be one of the world's most demanding courses, and applicants take the test an average of 12 times before passing.

In the early 2000s scientists chose to study London Black Cab drivers to look for brain changes as the drivers took years of complex spatial training, but the scientists were not expecting such dramatic results. Researchers found that at the end of the training period the hippocampus in the taxi drivers’ brains had grown significantly (Maguire et al., 2006; Woollett & Maguire, 2011). The hippocampus is the brain area specialized in acquiring and using spatial information (see Figure 1.4).

In other studies, scientists compared the brain growth of Black Cab drivers to that of London bus drivers. Bus drivers learn only simple and singular routes, and the studies showed that they did not experience the same brain growth (Maguire et al., 2006). This confirmed the scientists’ conclusion that the Black Cab drivers’ unusually complex training was the reason for their dramatic brain growth. In a further study, scientists found that after Black Cab drivers retired, their hippocampus shrank back down again (Woollett & Maguire, 2011). This was not because of age but because of lack of using the brain pathways.

FIGURE 1.3 Map of London

Source: jason cox/Shutterstock.

The studies conducted with Black Cab drivers, of which there have now been many (Maguire et al., 2006; Woollett & Maguire, 2011), showed a degree of brain flexibility, or plasticity, that stunned scientists. They had not previously thought that the extent of brain growth they measured was possible. This led to a shift in the scientific world in thinking about learning and “ability” and the possibility of the brain to change and grow.

FIGURE 1.4 The hippocampus

Source: decade3d/Shutterstock.

Around the time that the Black Cab studies were emerging, something happened that would further rock the scientific world. A nine‐year old girl, Cameron Mott, had been having seizures that the doctors could not control. Her physician, Dr. George Jello, proposed something radical. He decided he should remove half of her brain, the entire left hemisphere. The operation was revolutionary—and ultimately successful. In the days following her operation, Cameron was paralyzed. Doctors expected her to be disabled for many years, as the left side of the brain controls physical movements. But as weeks and months passed, she stunned doctors by recovering function and movement that could mean only one thing—the right side of her brain was developing the connections it needed to perform the functions of the left side of the brain. Doctors attributed this to the incredible plasticity of the brain and could only conclude that the brain had, in effect, “regrown.” The new brain growth had occurred faster than doctors imagined possible (http://www.today.com/id/36032653/ns/today-today_health/t/meet-girl-half-brain/#.UeGbixbfvCE).

This operation has now been performed on many different people. Christina Santhouse was eight when she underwent the operation to have half of her brain removed. Christina went on to many notable achievements, including making the honor roll at high school, earning a master's degree, and becoming a speech pathologist.

The new findings that brains can grow, adapt, and change shocked the scientific world and spawned new studies of the brain and learning, making use of ever‐developing new technologies and brain scanning equipment. In one study, researchers at the National Institute for Mental Health gave people a 10‐minute exercise to work on each day for three weeks. The researchers compared the brains of those receiving the training with those who did not. The results showed that the people who worked on an exercise for a few minutes each day experienced structural brain changes. The participants’ brains “rewired” and grew in response to a 10‐minute mental task performed daily over 15 weekdays (Karni et al., 1998).

In another study on mathematics learning in particular, Teresa Iuculano and her colleagues at Stanford's school of medicine uncovered critically important information. They brought into their labs two groups of students. One group had been diagnosed in schools as having mathematics learning disabilities; the other group comprised “regular” performers. The researchers looked at the students’ brains as they worked on mathematics, using MRI scans. They found fascinating differences—the students identified as having learning disabilities had more brain regions lighting up when they worked on a mathematics question. This is counterintuitive for many people, who think that students with learning disabilities have less going on in their brains, not more. The researchers point out that success does not always come from more brain activity, but focused activity in certain areas. The research then became even more interesting. Both sets of students were given eight weeks of one‐to‐one tutoring. At the end of the eight‐week period, the two sets of students had not only the same achievement but the exact same brain areas lighting up (Iuculano et al., 2015).

These and other results should prompt educators to abandon the traditional fixed ideas of the brain and learning that currently permeate schools—ideas that children are smart or dumb, quick or ...