Stanford math education professor Jo Boaler spends a lot of time worrying about how math education in the United States traumatizes kids. Recently, a colleague’s 7-year-old came home from school and announced he didn’t like math anymore. His mom asked why and he said, “math is too much answering and not enough learning.”
This story demonstrates how clearly kids understand that unlike their other courses, math is a performative subject, where their job is to come up with answers quickly. Boaler says that if this approach doesn’t change, the U.S. will always have weak math education.
“There’s a widespread myth that some people are math people and some people are not,” Boaler told a group of parents and educators gathered at the 2015 Innovative Learning Conference. “But it turns out there’s no such thing as a math brain.” Unfortunately, many parents, teachers and students believe this myth and it holds them up every day in their math learning.
“We live in a society with lots of kids who don’t believe they are good at math,” Boaler said at an Education Writers Association conference. “They’re put into low groups; they’re given low-level work and their pathway has been set.” But math education doesn’t have to look like this.
Neuroscience research is now showing a strong connection between the attitudes and beliefs students hold about themselves and their academic performance. That’s a departure from the long-held traditional view that academic success is based only on the quality of the teacher and curriculum. But researchers like Carol Dweck, Camille Farrington and David Yeager have shown repeatedly that small interventions to change attitudes about learning can have an outsized effect on performance.
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Neuroscientists now know that the brain has the ability to grow and shrink. This was demonstrated in a study of taxi drivers in London who must memorize all the streets and landmarks in downtown London to earn a license. On average it takes people 12 tries to pass the test. Researchers found that the hippocampus of drivers studying for the test grew tremendously. But when those drivers retired, the brain shrank. Before this, no one knew the brain could grow and shrink like that.
“We now know that when you make a mistake in math, your brain grows,” Boaler said. Neuroscientists did MRI scans of students taking math tests and saw that when a student made a mistake a synapse fired, even if the student wasn’t aware of the mistake. “Your brain grows when you make a mistake, even if you’re not aware of it, because it’s a time when your brain is struggling,” Boaler said. “It’s the most important time for our brains.”
A second synapse fires if the student recognizes his mistake. If that thought is revisited, the initial synapse firing can become a brain pathway, which is good for learning. If the thought isn’t revisited, that synapse will wash away.
A recent study of students with math learning disabilities found in a scan that their brains did behave differently from kids without the disability. “What they saw was the brain lighting up in lots of different areas while working on math,” Boaler said. The children were recruiting parts of the brain not normally involved in math reasoning.
The researchers tutored the group of students with math disabilities for eight weeks using the methods Boaler recommends like visualizing math, discussing problems and writing about math. At the end of the eight weeks, they scanned their brains again and found that the brains of the test group looked just like the kids who did not have math disabilities. This study shows that all kids can learn math when taught effectively. Boaler estimates that only 2 to 3 percent of people have such significant learning disabilities that they can’t learn math at the highest levels.
People who learned math the traditional way often push back against visual representations of math. That kind of thinking represents a deep misunderstanding of how the brain works. “When you think visually about anything, different brain pathways light up than when we think numerically,” Boaler said. The more brain pathways a student engages on the same problem, the stronger the learning.
An example of many ways to visually represent 18 x 5. (Jo Boaler/YouCubed)
GROWTH MINDSET AND MATH
Increasingly, educators are buying into the compelling research showing that what students believe about themselves affects how their brains approach learning. Growth mindset is probably the best known aspect of this research, and many school leaders are trying to figure out how to implement growth mindset programs in their classrooms.
“More kids have a fixed mindset about math than anything else,” Boaler said. And it’s no coincidence that they feel this way. Teachers often believe their students can’t achieve at the highest levels, and in turn, students believe that about themselves. Plus, the tasks themselves communicate a fixed mindset.
“It is very difficult to have a growth mindset and to believe that you can grow or learn if you are constantly given short, closed questions with a right or wrong answer,” Boaler said. Instead, she recommends giving visual problems that provoke discussion and have multiple ways they could be solved.
She also says kids should not be grouped by ability or tracked into “advanced” or “remedial” groups. That common practice sends fixed mindset messages to students, both the “advanced” ones and the “low-performing” ones. Kids considered to be “gifted” suffer from ability grouping the most because they develop the ultimate fixed mindset. They become terrified that if they struggle they’ll no longer be considered smart.
Instead, mixed ability grouping can work if the tasks are open-ended and what Boaler calls “low-floor/high-ceiling” tasks that allow every student to participate, while allowing lots of space within the task for students to grow in their thinking.
Boaler has lots of example tasks on her website, YouCubed, and on the NRICH website.
PUTTING IT INTO PRACTICE
During the summer of 2015, Boaler invited 81 seventh- and eighth-graders from a low-income district near Stanford to come to a summer math camp focused on algebra concepts. She gave the students a pre-test and found that their abilities ranged from very low (getting 0 answers correct) to fairly high. Then, for 18 days she taught them math well.
The instructional program focused on mindset messages, was full of inquiry-based, low-floor/high-ceiling tasks, was visual and used mixed achievement groups. At the end of 18 days, when Boaler gave them another test they had improved on average by 50 percent.
“They improved because they changed their beliefs that they were not a math person to believing they were a math person,” Boaler said. After the course, students said they looked forward to math and saw math as a creative subject.
Administrators from the district came to observe partway through the camp and couldn’t tell who was a low achiever and who was a high achiever in the class. Boaler also makes it clear to the students in the workshop what she expects from them, and speed is not something she’s evaluating. Instead, they do norm building so that everyone knows how to appropriately work in groups, help one another and be supportive.
“If we don’t pay attention to those kinds of interactions, and kids are dominating, or thinking they’re smarter, then we’re really in trouble,” Boaler said.
Removing the time pressure from math is another important issue for Boaler. Neuroscience research out of Sian Beilock’s lab at the University of Chicago has shown that time pressure often blocks the brain’s working memory from functioning. This is particularly bad for kids with test anxiety.
“The irony of this is mathematicians are not fast with numbers,” Boaler said. “We value speed in math classrooms, but I’ve talked with lots of mathematicians who say they’re not fast at all." But it is common for math teachers to call on the kids who get the answer quickly, reinforcing the idea for all students that rapidity is what matters.
COMMON PUSHBACK
Math education experts have been making the same case as Boaler for decades, and yet math education in the U.S. has not shifted much. Teachers often say they have to cover all the topics in the curriculum to prepare students for the tests they will be expected to pass, leaving them with no time for the kinds of open-ended, discussion-based math that Boaler advocates.
Boaler agrees with teachers that there is way too much to cover in the curriculum, especially because she finds much of it to be obsolete (don't get her started on the textbooks themselves). “The most important thing we can give kids is to think quantitatively about the world and apply a mathematical lens to different situations,” she said.
In addition to teaching students, Boaler trains teachers in her methods. Often they go back to their classrooms and apply these theories, which means they aren’t covering every topic in the textbook, and yet their students do better on the standardized tests anyway. Boaler is not a fan of all the tests American students must take, but she says teaching math the right way deepens kids' understanding of math in real ways that show up on tests, too.
Teachers and parents often push back against this kind of math. They wonder where memorization of math facts fits into the model, given the belief that kids must know their times tables to succeed in higher-level math. Boaler says that's unnecessary. She is a math education teacher and has risen to high levels of math learning without ever learning her math facts. She has number fluency, knows how to manipulate numbers and understands concepts, but she doesn’t have her math facts memorized.
The Programme for International Student Assessment test (PISA), which is often used to compare achievement across countries, has a section about attitudes and beliefs. Those surveys show that kids who approach math as memorization are the lowest achievers in the world. “America has more memorizers than almost any country in the world,” Boaler said. The highest achievers are those who think about the big ideas and make connections.
Likewise, repetition of math tasks is not helpful to deep learning. The same kind of problem with different numbers does not improve understanding, Boaler said. What students really need is “productive practice,” approaching the problem from different directions, applying the ideas and explaining reasoning.
Boaler is on a mission to “revolutionize” how math is taught in the U.S. She has written several books to help teachers learn to teach with her methods, offers a free online course, and even gives away curriculum for teachers, students and parents on her YouCubed website. During one week at the start of the 2015 school year Boaler gave away five free math lessons, encouraging teachers to try this approach. She’s pleased that 100,000 schools tried the lessons, and teachers could see the difference in their students. A survey of students found that after the lessons and the growth mindset videos, 96 percent believed they should keep trying after making a mistake in math.
Boaler said a big problem is that math teachers themselves are math-traumatized. They came through a system very similar to the one in which they work. Elementary school teachers in particular often feel insecure about math.
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“When they try math in these ways they get it, too,” Boaler said. “They can see this is much more valuable and enriching.”
