The Way We Teach Math Is Holding Women Back

Time

March 29, 2017

A Stanford math professor encourages a different teaching approach

First Daughter Ivanka Trump and Education Secretary Betsy DeVos toured the National Air and Space Museum with a group of middle school students Tuesday, encouraging girls to pursue careers in science, technology, engineering and mathematics — even while President Donald Trump’s administration put forth a budget proposal that suggests cutting funding for education and research. There is nothing more important than advancing the STEM fields — and those groups who are underrepresented within them.

One area in desperate need of examination is the way we teach mathematics. Many Americans suffer from misconceptions about math. They think people are either born with a “math brain” or not — an idea that has been disproven — and that mathematics is all numbers, procedures and speedy thinking. In reality, mathematicians spend most of their working lives thinking slowly and deeply, investigating complex patterns in multiple dimensions. We sacrifice many people — women and students of color, in particular — at the altar of these myths about math.

Math is a prerequisite for most STEM fields, and the reason many students abandon STEM careers. In higher levels of mathematics, gender imbalances persist: In 2015, about 76% of math doctorates were awarded to men. This figure should prompt alarm in mathematics departments across the country — and encourage focus on an area that is shockingly neglected in discussions of equity: teaching methods in classrooms.

At Stanford University, I teach some of the country’s highest achievers. But when they enter fast-paced lecture halls, even those who were successful in high school mathematics start to think they’re not good enough. One of my undergraduates described the panic she felt when trying to keep pace with a professor: “The material felt like it was flying over my head,” she wrote. “It was like I was watching a lecture at 2x or 3x speed and there was no way to pause or replay it.” She described her fear of failure as “crippling.” This student questioned her intelligence and started to rethink whether she belonged in the field of math at all.

Research tells us that lecturers typically speak at between 100 and 125 words a minute, but students can take note of only about 20 words a minute, often leaving them feeling frustrated and defeated. “I’ve essentially given up in my math class right now,” another student of mine wrote. “In such a fast-paced environment where information is constantly coming at you, there just isn’t time to think deeply about what you are learning.”

The irony of the widespread emphasis on speed in math classrooms, with damaging timed tests given to students from an early age, is that some of the world’s most successful mathematicians describe themselves as slow thinkers. In his autobiography, Laurent Schwartz, winner of the world’s highest award in mathematics, described feeling “stupid” in school because he was a slow thinker. “I was always deeply uncertain about my own intellectual capacity; I thought I was unintelligent,” he wrote. “And it is true that I was, and still am, rather slow. I need time to seize things because I always need to understand them fully.”

When students struggle in speed-driven math classes, they often believe the problem lies within themselves, not realizing that fast-paced lecturing is a faulty teaching method. The students most likely to internalize the problem are women and students of color. This is one of the main reasons that these students choose not to go forward in mathematics and other STEM subjects, and likely why a study found that in 2011, 74% of the STEM workforce was male and 71% was white.

Women are just as capable as men of working at high speed, of course, but I’ve found in my own research that they are more likely to reject subjects that do not give access to deep understanding. The deep understanding that women seek, and are often denied, is exactly what we need to encourage in students of mathematics. I have taught many deep, slow thinkers in mathematics classes over the years. Often, but not always, they are women, and many decide they cannot succeed in mathematics. But when the message about mathematics has changed to emphasize slower, deeper processing, I’ve seen many of these women go on to excel in STEM careers.

When mathematics classes become places where students explore ideas, more often than they watch procedures being rapidly demonstrated by a teacher or professor, we will start to liberate students from feelings of inadequacy. In a recent summer camp with 81 middle school students, we taught mathematics through open, creative lessons to demonstrate how mathematics is about thinking deeply, rather than calculating quickly. After 18 lessons, the students improved their mathematics achievement on standardized tests by an average of 50%, the equivalent of 1.6 years of school. If classrooms across the country would dispel the myths about math and teach differently, we would improve the lives of many students and enable the creation of a more diverse STEM workforce. It will take a generation of young, creative, adaptable and quantitative thinkers to tackle our society’s problems — thinkers that we are currently turning away from mathematics classrooms and lecture halls in droves.

Jo Boaler is a Stanford professor, co-founder of youcubed.org and author of best-selling book, Mathematical Mindsets: Unleashing Students’ Potential through Creative Math, Inspiring Messages and Innovative Teaching.

How to Help Students Develop a Growth Mindset

100 Percent Is Overrated

The Atlantic

People labeled “smart” at a young age don’t deal well with being wrong. Life grows stagnant.

Debra Hughes / Shutterstock
ASPEN, Colo.—At whatever age smart people develop the idea that they are smart, they also tend to develop vulnerability around relinquishing that label. So the difference between telling a kid “You did a great job” and “You are smart” isn’t subtle. That is, at least, according to one growing movement in education and parenting that advocates for retirement of “the S word.”

The idea is that when we praise kids for being smart, those kids think: Oh good, I’m smart. And then later, when those kids mess up, which they will, they think:Oh no, I’m not smart after all. People will think I’m not smart after all. And that’s the worst. That’s a risk to avoid, they learn.“Smart” kids stand to become especially averse to making mistakes, which are critical to learning and succeeding.

“Mistakes grow your brain,” as the professor of mathematics education at Stanford University Jo Boaler put it on Monday at the Aspen Ideas Festival, which is co-hosted by The Atlantic. I wondered why, then, my brain is not so distended that it spills out of my ears and nose. I should have to stuff it back inside like a sleeping bag, and I should have to carry Q-tips around during social events as stuffing implements. Boaler notes, more eloquently, that at least a small part of the forebrain called the thalamus can appreciably grow after periods of the sort of cognitive stimulation involved in mistake-making. What matters for improving performance is that a person is challenged, which requires a mindset that is receptive to being challenged—if not actively seeking out challenge and failure. And that may be the most important thing a teacher can impart.

People are born with some innate cognitive differences, but those differences are eclipsed by early achievement, Boaler argues. When people perform well (academically or otherwise) at early ages and are labeled smart or gifted, they become less likely to challenge themselves. They become less likely to make mistakes, because they stay in their comfortable comfort zone and stop growing. And their fixed mindset persists through adulthood. The simple and innocent praising of a smart kid feeds an insidious problem that some researchers track all the way up to gender inequality in STEM careers.

So ending the reign of the S word, as Boaler calls it, is a grand mission. “It’s imperative that we don’t praise kids by telling them they’re smart,” she argued in a Monday lecture to an audience that received her message with many knowing nods. “You can tell kids that they’ve done something fantastic, but don’t label them as smart.”

The idea of a fixed mindset, in which people are smart or not smart, stands in contrast to a growth mindset, in which people become intelligent and knowledgeable through practice. In her 2006 book The New Psychology of Success, psychologist Carol Dweck described the two: People with growth mindsets believe that the harder they work, the smarter they get. And the subtleties of the ways in which we praise kids are related to the mindsets those kids develop.

The group most damaged by fixed-mindset thinking is high-achieving girls, Boaler argues, because it’s girls who are told by society that they probably won’t be as good as boys at math and science. That means girls are only more likely to avoid challenging themselves in science and math, and that aversion to making mistakes leads to less learning and progress. The more that certain disciplines cling to ideas of giftedness, the fewer female Ph.D.s there are in those fields.

“When we give kids the message that mistakes are good, that successful people make mistakes, it can change their entire trajectory,” Boaler said. 100 percent is not an ideal score. When kids come home from school and announce that they got everything right on their school work, Dweck advises parents to offer some sympathy: Oh, I’m sorry you didn’t get the chance to learn.

Speaking of percentages, math is a good example of the importance of avoiding the fixed mindset. The idea of a “math person” or a math gene is a primary reason for so much math nihilism, math failure, and “math trauma,” as Boaler called it on Monday. When kids get the idea that they “aren’t math people,” they start a downward trajectory, and their career options shrink immediately and substantially. There is also the common idea of a wall in math: People learn math until they hit a wall where they just can’t keep up. That wall may be trigonometry, and it may be advanced calculus, and it may be calculating a tip. In no other discipline but math are people so given to thinking, instead of I need to practice, just Well, I’m not good.

“Big news,” Boaler said during her lecture, “there is no wall.”

With that, she advanced her Powerpoint and to a slide bearing a rendering of the Kool-Aid Man busting through a brick wall.

“I didn’t know who this was,” she said. “One of my teammates made this slide. I’ve learned that this is Kool-Aid Man.”

Memorizers are the lowest achievers and other Common Core math surprises

Hechinger Report

In this Feb. 12, 2015 photo, Yamarko Brown, age 12, works on math problems as part of a trial run of a new state assessment test at Annapolis Middle School in Annapolis, Md. The new test, which is scheduled to go into use March 2, 2015, is linked to the Common Core standards, which Maryland adopted in 2010 under the federal No Child Left Behind law, and serves as criteria for students in math and reading.

It’s time to debunk the myths about who is good in math, and Common Core state standards move us toward this worthy goal. Mathematics and technology leaders support the standards because they are rooted in the new brain and learning sciences.

All children are different in their thinking, strength and interests. Mathematics classes of the past decade have valued one type of math learner, one who can memorize well and calculate fast.

Yet data from the 13 million students who took PISA tests showed that the lowest achieving students worldwide were those who used a memorization strategy – those who thought of math as a set of methods to remember and who approached math by trying to memorize steps. The highest achieving students were those who thought of math as a set of connected, big ideas.

The U.S. has more memorizers than most other countries in the world. Perhaps not surprisingly as math teachers, driven by narrow state standards and tests, have valued those students over all others, communicating to many other students along the way – often girls – that they do not belong in math class.

The fact that we have valued one type of learner and given others the idea they cannot do math is part of the reason for the widespread math failure and dislike in the U.S.

Brain science tells us that the students who are better memorizers do not have more math “ability” or potential but we continue to value the faster memorizers over those who think slowly, deeply and creatively – the students we need for our scientific and technological future. The past decade has produced a generation of students who are procedurally competent but cannot think their way out of a box. This is a problem.

Mathematics is a broad and multidimensional subject. Real mathematics is about inquiry, communication, connections, and visual ideas. We don’t need students to calculate quickly in math. We need students who can ask good questions, map out pathways, reason about complex solutions, set up models and communicate in different forms. All of these ways of working are encouraged by the Common Core.

Technology leaders are publically arguing that calculation is not math, and that math is a much broader subject. Conrad Wolfram, one of the leaders of one of the world’s most important mathematics companies, Wolfram-Alpha, urges schools to stop emphasizing calculating and focus instead on problem solving, modeling, thinking, and reasoning as these are the mathematical abilities that students need in the workplace and their high tech lives. This broad, multidimensional mathematics is the math that engages many more learners and puts them on a pathway to life long success.

Related: Should we stop making kids memorize times tables?

Part of the problem in the U.S. is the desperation of many parents to advance their children in math, pushing them to higher levels of math faster and sooner, somehow believing that a resume packed with advanced math courses will guarantee their future. Bill Jacob, a mathematics professor at the University of California, Santa Barbara, speaks openly about the dangers of students being pushed to higher levels of mathematics too soon. “I know it is hard to persuade parents that their students shouldn’t race to get calculus, but I really wish they wouldn’t. Too much content and depth is left out when they do.” said Jacob, who is not alone in saying that he would rather have students in his university mathematics courses that have breadth in their mathematical experiences than any additional Advanced Placement courses. Experts in England are giving the same advice to parents of high achieving students. Geoff Smith, chairman of the British and International Math Olympiads warns that accelerating children through the system is a “disaster” and a “mistake”. He, like me, recommends that high achieving students explore the mathematics they are learning in depth, instead of rushing forward.

Mathematics is not a subject that requires fast thinking. Award winning mathematicians talk about their slow, deep thinking in math. Fields Medal winning mathematician Laurent Schwartz wrote in his autobiography that he felt stupid in school because he was one of the slowest thinkers in math. Eventually he realized that speed was not important – “What is important is to deeply understand things and their relations to each other.  This is where intelligence lies. The fact of being quick or slow isn’t really relevant.”

Related: Testing gives third-graders upset stomachs, tears, even fevers

Some school districts, such as San Francisco Unified, are trying to slow down the math experience, requiring that advanced students go deeper rather than faster. Students still reach calculus but the pathway to calculus consists of deep understanding rather than procedures and memorization. This is an important move. There is no harm in students being introduced to higher-level mathematics earlier, as long as the mathematics is enjoyable and ideas can be explored deeply. Third graders can be fascinated by the notion of infinity, or the fourth dimension, but they do not need a race through procedural presentations of mathematics.

New brain science tells us that no one is born with a math gift or a math brain and that all students can achieve in math with the right teaching and messages. The classrooms that produce high achieving students are those in which students work on deep, rich mathematics through tasks that they can take to any level they want. No one is told what level they can reach and no one is held back by narrow questions that limit students’ mathematical development and creativity.

Many people across the U.S. have gross misconceptions about mathematics learning, thinking that mathematics is a narrow subject of memorization and speed and that people who do not calculate fast or memorize well are not ‘math people’. We need to change the conversations about mathematics, communicating to all children that they can learn. We also need to change the way math is taught, valuing the different ways of thinking that are so important to the subject. Mathematics, itself, needs this and although change is hard, it should be embraced.

We need to broaden mathematics and open the doors of mathematics to all students. When we do this we will see many more creative, energized young people equipped to think quantitatively about our ever-changing world. We all need this.

Jo Boaler is a professor of mathematics education at Stanford University, co-founder of www.youcubed.org, and author of What’s Math Got To Do With It: How Teachers and Parents can Transform Mathematics Learning and Inspire Success. (Penguin, 2015)

This story was produced by The Hechinger Report, a nonprofit, independent news organization focused on inequality and innovation in education. Read more about Common Core.

Researchers: How memorization can inhibit math fluency

Students most effectively learn math working on problems that they enjoy, not drills or exercises

Physics.org

January 30th, 2015 by Clifton B. Parker in Other Sciences / Social Sciences

Students most effectively math working on problems that they enjoy, not drills or exercises
Stanford Professor Jo Boaler finds that children who excel in math learn to develop “number sense,” which is much different from the memorization that is often stressed in school.

Stanford Professor Jo Boaler finds that children who excel in math learn to develop “number sense,” which is much different from the memorization that is often stressed in school.

Students learn math best when they approach the subject as something they enjoy, according to a Stanford education expert. Speed pressure, timed testing and blind memorization pose high hurdles in the youthful pursuit of math.

“There is a common and damaging misconception in – the idea that strong are fast ,” said Jo Boaler, a Stanford professor of mathematics education and the lead author on a new working paper. Boaler’s co-authors are Cathy Williams, cofounder of Stanford’s YouCubed, and Amanda Confer, a Stanford graduate student in education.

Curriculum timely

Fortunately, said Boaler, the new national curriculum standards known as the Common Core Standards for K-12 schools de-emphasize the rote memorization of math facts. Maths facts are fundamental assumptions about math, such as the times tables (2 x 2 = 4), for example. Still, the expectation of rote memorization continues in classrooms and households across the United States.

While research shows that knowledge of math facts is important, Boaler said the best way for students to know math facts is by using them regularly and developing understanding of numerical relations. Memorization, speed and test pressure can be damaging, she added.

On the other hand, people with “number sense” are those who can use numbers flexibly, she said. For example, when asked to solve the problem of 7 x 8, someone with number sense may have memorized 56, but they would also be able to use a strategy such as working out 10 x 7 and subtracting two 7s (70-14).

“They would not have to rely on a distant memory,” Boaler wrote.

In fact, in one research project the investigators found that the high-achieving students actually used number sense, rather than rote memory, and the low-achieving students did not.

The conclusion was that the low achievers are often low achievers not because they know less but because they don’t use numbers flexibly.

“They have been set on the wrong path, often from an early age, of trying to memorize methods instead of interacting with numbers flexibly,” she wrote. Number sense is the foundation for all higher-level mathematics, she noted.

Role of the brain

Boaler said that some students will be slower when memorizing, but still possess exceptional mathematics potential.

“Math facts are a very small part of mathematics, but unfortunately students who don’t memorize math facts well often come to believe that they can never be successful with math and turn away from the subject,” she said.

Prior research found that students who memorized more easily were not higher achieving – in fact, they did not have what the researchers described as more “math ability” or higher IQ scores. Using an MRI scanner, the only brain differences the researchers found were in a brain region called the hippocampus, which is the area in the brain responsible for memorizing facts – the working memory section.

But according to Boaler, when students are stressed – such as when they are solving math questions under time pressure – the working memory becomes blocked and the students cannot as easily recall the math facts they had previously studied. This particularly occurs among higher achieving students and female students, she said.

Some estimates suggest that at least a third of students experience extreme stress or “” when they take a timed test, no matter their level of achievement. “When we put students through this anxiety-provoking experience, we lose students from mathematics,” she said.

Boaler contrasts the common approach to teaching math with that of teaching English. In English, a student reads and understands novels or poetry, without needing to memorize the meanings of words through testing. They learn words by using them in many different situations – talking, reading and writing.

“No English student would say or think that learning about English is about the fast memorization and fast recall of words,” she added.

Strategies, activities

In her paper, “Fluency without Fear,” Boaler provides activities for teachers and parents that help students learn math facts at the same time as developing . These include number talks, addition and multiplication activities, and math cards.

Importantly, she said, these activities include a focus on the visual representation of number facts. When students connect visual and symbolic representations of numbers, they are using different pathways in the brain, which deepens their learning, as shown by recent brain research.

“Math fluency” is often misinterpreted, with an over-emphasis on speed and memorization, she said. “I work with a lot of mathematicians, and one thing I notice about them is that they are not particularly fast with numbers; in fact some of them are rather slow. This is not a bad thing; they are slow because they think deeply and carefully about mathematics.”

She refers to the famous French mathematician, Laurent Schwartz, who wrote in his autobiography that he often felt stupid in school, as he was one of the slowest math thinkers in class.

Math anxiety and fear play a big role in students dropping out of mathematics, said Boaler.

“When we emphasize memorization and testing in the name of fluency we are harming children, we are risking the future of our ever-quantitative society and we are threatening the discipline of mathematics. We have the research knowledge we need to change this and to enable all children to be powerful mathematics learners. Now is the time to use it,” she said.

More information: “Fluency Without Fear: Research Evidence on the Best Ways to Learn Math Facts”: youcubed.stanford.edu/fluency-without-fear/

Provided by Stanford University