Mixing Up Math: Why Memorization Isn't Enough
The author of 'Mathematical Mindsets' discusses how math should be taught
In her new book, Stanford University math education professor Jo Boaler shares a story of three boys in a summer math class she taught. The curriculum focused on algebra but encouraged students to complete problems visually, rather than with a traditional "solve for x" strategy. The boys, who had been disruptive and doing poorly at the start of class, grabbed hold of a problem that used multilink cubes. Together, they worked studiously, so engaged that they didn’t even respond when fellow classmates tried to distract them.
"[They] zigzagged around, moving closer to a solution, then further away, then back toward it again," Boaler writes. Eventually, the boys had workable solutions and, Boaler says, something much more important than a completed math worksheet: They'd learned a way to work together to uncover answers.
This, Boaler says, is how all math should be taught. Not the classic procedural short questions again and again, but open-ended, task-based visual math.
Only nerds do math. You must have a math brain to do math. You must be good with numbers. Boys are better at math. All these prejudices [...] foster math anxiety, which leads to poor math performance.
Bringing 'real' math to the classroom
Math taught in a high school classroom has little to do with anything in the real world, says Boaler.
"It is not what mathematicians work on," she says. Their math is broad and multidimensional. It's not only about numbers or equations but is a subject enriched by different perspectives and visual images.
Boaler wants to see more of that creative uncertainty in the classroom rather than the times tables, tests and rote memorization so commonly used.
"A lot of teachers just teach as they were taught," says Boaler. "Now they're reproducing what they learned growing up."
They'd prefer something different, though. "When I present to teachers, they are so excited and inspired. They get it," says Boaler. "But so many tell me that they can't teach this way because they have to teach to the textbook and have to do all of these tests."
Memorization or numbers sense?
Science backs up what Boaler already knows: New doesn't mean less effective. In her paper, "Fluency Without Fear: Research Evidence on the Best Ways to Learn Math Facts," Boaler reviews data from 13 million 15-year-olds from around the world. Results show that the students who do most poorly in math often focus on memorization. Sure, Boaler writes, it's useful to commit some facts to memory, but what students really need to develop is their "numbers sense."
Numbers sense, Boaler explains, is "the foundation for all higher-level mathematics." When students have it, they use numbers more flexibly, interacting with them conceptually. Students who have no numbers sense tend to adhere to standard memorized procedures. The difference: Students with numbers sense can think outside the box to solve problems.
But does that mean then that students with numbers sense are fluent in math? The idea of "math fluency" is itself fluid with Common Core math standards defining subject fluency as the following:
Procedural skills and fluency: The standards call for speed and accuracy in calculation. Students must practice core functions, such as single-digit multiplication, in order to have access to more complex concepts and procedures. Fluency must be addressed in the classroom or through supporting materials, as some students might require more practice than others. (Officers, 2016)
It's good to have those skills in math, among other subjects, but Boaler takes issue with "equating fluency with blind memorization." The best way to be fluent, she says, is to be comfortable with numbersno matter what.
Fighting math prejudices
One of the many roadblocks preventing change in math education is how we see math in our society. Only nerds do math. You must have a math brain to do math. You must be good with numbers. Boys are better at math. All these biases are perpetuated in our society, says Boaler, and they foster anxiety about math, which leads to poor performance. They also influence how we design and teach math curriculums.
"Some people don't want to believe that they can be good at math," said Boaler. What's needed, she says, are strong role models for students and ongoing encouragement from teachers.
Of course, says Boaler, it takes time to make any change, but she thinks we're moving in the right direction.
"What the brain scientists are saying fits with what the research is says," she says. "This fits with what mathematicians and Silicon Valley leaders are saying. So when the head of Google Recruitment says that they will not look at test scores anymore, we do have signs of change."