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.

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How Kids Benefit From Learning To Explain Their Math Thinking

MindShift

same-different-jh

Math teachers of older students sometimes struggle to get students to explain their thinking with evidence. It’s hard to get kids in the habit of talking about how they are thinking about a problem when they’ve had many years of instruction that focused on getting the “right answer.” That’s why educators are now trying to get students in the habit of explaining their thinking at a young age. The Teaching Channel captured kindergarten and first grade teachers pushing students to give evidence for their answers in situations where there are several ways to think about a problem.

Pattern recognition is a fundamental part of mathematics and kindergarteners are not too young to notice, compare and describe simple patterns. In this video, kindergarten teacher Donella Oleston describes how she had to back up and explain to these young learners what it means to “explain your thinking,” because at first students would only answer, “My brain told me so.” With practice, she says students have gotten to deeper levels of noticing, moving past the obvious and picking out more abstract similarities and differences between two pattern sets.

 

Want Your Kids To Do Better In Math? Have Them Trace Math Problems With A Finger

By Angela Laguipo, Tech Times | January 31, 2016

Not all kids are fond of math, and for them, solving math problems can be a tedious task. A new study suggests that students who trace certain math problems using their fingers are able to solve them more quickly and easily.

Researchers from the University of Sydney found that students who used a technique called finger tracing were able to solve math problems with more ease than others.

The researchers said that students who used their fingers to trace over examples while at the same time reading arithmetic and geometry material were able to perform better by completing tasks more easily and quickly than those who did not apply the technique.

Tracing involves using the index finger to physically trace and touch the angles of a triangle in geometry, for example. The research team believes this may help reduce the load on working memory and enhance the ability to retain complex information.

“Our findings have a range of implications for teachers and students alike. They show math learning by young students may be enhanced substantially with the simple addition of instructions to finger-trace elements of math problems,”says corresponding author Dr. Paul Ginns.

In the study published in the journal Learning and Instruction and Applied Cognitive Psychology, the researchers recruited 275 children from ages 9 to 13 years old. They discovered that tracing over math elements while reading them enhanced the children’s understanding of problems in algebra and geometry. Previous studies have also confirmed that finger tracing helps kids recognize shapes and letters.

Dr. Ginns says this simple and zero-cost teaching technique can help teachers assist students by giving them specific instructions to “trace over” important elements in mathematical textbooks.

The researchers are now looking for more ways to use the technique on more complex and harder math problems that require higher levels of cognitive ability.

They add that they are confident that the new technique can be used in the classroom setting and even in subjects other than math. Further research is needed to explore the technique.

Teaching Math With Modular Origami

Scholastic.com

By Alycia Zimmerman on January 22, 2016

  • Grades: 1–2, 3–5, 6–8, 9–12

Several years ago, I had the good fortune to attend a workshop by Rachel McAnallen (aka Ms. Math) about teaching geometry with a fun and tactile method: origami! Since then, introducing my students to modular geometric origami is one of my favorite teaching moments each year. Origami math gives my tactile and spatially gifted students a chance to shine, it helps students with sequencing and direction following, and it’s a fun way to introduce a wide range of geometry terms and concepts.

I had NEVER created origami before the abovementioned two-hour workshop. You absolutely do not need to be a talented origami artist to pull off these lessons with your students. With the straight-forward tips below and a few minutes of practice, you’ll be ready to guide your students through an origami math experience that will have them clamoring for more. (That’s when you can hand them an origami book and challenge them to figure it out!)

 

I was bursting with pride upon making my first “skeletal octahedron.” Students feel a similar sense of accomplishment when completing their origami structures.

 

What is Modular Origami?

Modular origami is the fancy name for geometric origami that is made up of many repeating “units” that are then assembled to create a more complex geometric form. Unlike traditional origami that uses a single sheet of paper to fold a figure, modular origami uses many sheets of paper that are folded into basic modules or units. Once you learn how to make the basic unit for a design, you repeat the process to make enough copies of the unit to assemble your final form. (For a look at some modular origami projects, check out my Pinterest board.)

Although the process of making the units is repetitive, I find that many students enjoy it as a calming, almost meditative process. I often introduce this activity right before standardized tests, because the repetitive folding soothes some students and gives them a purposeful active for jittery hands. I always have a few students who find folding the units to be a chore. I team these students up to divide and conquer the unit folding work and then assemble a joint final product. I’ve had so many students become nearly obsessed with folding units — they bring origami paper to lunch and recess (especially on rainy days) to get in extra folding time.

A student shows off his first modular origami creations: sonobe cubes. (See the video tutorial below to make these simple cubes.)

 

What Supplies Will We Need?

I buy very inexpensive origami paper for my students since we go through a fair amount of it, like this 500-sheet pack of 6”x6” paper. I keep a pack or two of fancier paper on hand for special projects that individual students tackle. Colored copier paper cut into squares also works well.

The only other supplies you’ll need are a Popsicle stick and a Ziploc bag for each student. The students use the Popsicle sticks to press “crispy creases” into the paper, and the bag to hold all of their units before they assemble the modules into the final design.

 

How Do I Incorporate Math Into Origami?

The math comes entirely through the discussion as you guide the students through making a module/unit. I sit all of my students down and VERY slowly go through the stepwise process of making the first module for a design. I model the process using the document camera, and I have a couple of student experts circulate to help other students who get stuck. (I pre-teach the folding process to these student experts so they are available to be my assistants.)

Before, during and after each fold, we discuss the shapes that we are pressing into the paper, we classify the angles, and I invite the students to name each step to help them remember the sequence of paper folding. This way, students can remember that “the large trapezoid comes after the double horizontal rectangle step.” By folding while discussing geometry, students are also more likely to memorize vocabulary-heavy geometry; they create kinesthetic associations to go along with the geometry terms.

Two of my student experts show off their icosahedrons. Empowering these guys to assist their peers not only helps to build their confidence, it also means that struggling students get timely hands-on support.

 

How Do I Get Started With Modular Origami?

The sonobe cube uses a very simple modular origami unit: the aptly named Sonobe unit. As a cube with six faces, this design requires six units. That makes for a pretty short project. Students can get the feel for modular origami without having to create dozens of units for a single project. And Sonobe cubes are so much fun to assemble! You can find plenty of online tutorials about the Sonobe cube, or follow along with my video below. Plus, once your students have mastered the Sonobe cube, they can use the same units to make octahedrons and icosahedrons.

 

What Do We Do After Our First Project?

After you teach your students how to make the Sonobe cube (and possibly the other Sonobe shapes), you might like to help them through one other project. An octagon-star is another favorite because it is a transforming shape — the final design transforms from a star to an octagon and back. For the second project, I provide written directions, but I still walk them through the process step by step. I have the students refer to the written directions (and diagrams) so they can learn to follow origami directions independently.

Students who caught the modular origami bug will be so motivated after learning the first two projects, that they will likely want to figure out other origami designs. I provide a basket of modular origami books and printouts that they can peruse to choose other projects. At that point I step back and let my students become the expert origami crafters — their skills soon surpass my basic ones, and I am very happy to take on the role of appreciative spectator.

  

Beyond Working Hard: What Growth Mindset Teaches Us About Our Brains

Mind/Shift

(iStock)

Growth mindset has become a pervasive theme in education discussions in part because of convincing research by Stanford professor Carol Dweck and others that relatively low-impact interventions on how a student thinks about himself as a learner can have big impacts on learning. The growth mindset research is part of a growing understanding and acknowledgement that many non-cognitive factors are important to academic learning.

While it’s a positive sign that educators see value in the growth mindset research and believe they can implement it in their classrooms, the deceptively simple idea has led to some confusion and misperceptions about what a growth mindset really is and how teachers can support it in the classroom. It’s easy to lump growth mindset in with other education catchphrases, like “resiliency” or “having high expectations,” but growth mindset actually has a much more concrete definition. As Eduardo Briceño wrote in a recent post for MindShift, “It is the belief that qualities can change and that we can develop our intelligence and abilities.”

This simple idea can lead to big changes in learners, but it has been commonly misinterpreted to mean that if teachers praise students for working hard, they will develop a growth mindset. In many cases that isn’t true and students will feel that praise is disingenuous. Briceño explains it this way: “Students often haven’t learned that working hard involves thinking hard, which involves reflecting on and changing our strategies so we become more and more effective learners over time, and we need to guide them to come to understand this.”

To foster growth mindsets in students, teachers can coach students to try different learning strategies that make the brain work smarter. Educator praise can be used to acknowledge specific strategies students have tried and can push students to reflect on themselves as learners. This process is more complex than it looks and ultimately should help lead students to become more independent thinkers.

Growth mindset is also not a panacea for low achievement or education inequality, although the fervor with which some districts have adopted the idea might lead one to believe that. Critics like Alfie Kohn point out that no individual attitude shift is going to overcome the very real structural inequalities that exist in schools. He worries that focusing on mindsets will not only mask those bigger problems, but could undermine the imperative to provide compelling learning experiences that lead students to discover an innate love of learning.

Another common way of boiling down the mindset research is to tell students that “mistakes are good; we learn from mistakes.” While that can be true, not all mistakes are worth pursuing. Some mistakes are just sloppy and others are made in such a high-stakes environment. Reflecting on these kinds of mistakes can improve performance next time, but they aren’t necessarily the most fruitful kinds of mistakes.* Mistakes that lead to the most learning are the ones made when students are stretching outside their comfort zones to grasp an idea that’s just out of reach. Or, when someone has an “aha” moment after doing something she thought was right but then realized was a mistake based on new information. Reflecting on these mistakes, and formulating a new plan of action based on them, is what makes them powerful.

PUTTING IT INTO PRACTICE

It’s exciting that this research has been around long enough and has reached enough educators that many districts and schools are already trying to put the research into practice. Their successes and failures are important to share as educators work to figure out how to implement it.

Many schools quickly realized that growth mindsets are not only important to students, they are crucial for educators trying to make change. And helping educators to develop their own growth mindsets hinges on positive working environments and trust at school. Educators have a hard time taking risks in their teaching practice if they believe the outcome must be perfect the first time. And yet, one of the most important ways to instill a growth mindset in students is to model the disposition as teachers, making it even more crucial that district and school leaders create a climate conducive to growth mindsets in adults.

Some high schools are weaving explicit instruction around growth mindset into workshops and classes for incoming freshmen. Educators hope that if students get the same messages about stretching to learn and improving based on those mistakes from day one of high school and from every subject-area teacher, that a growth mindset will become part of school culture.

Other schools focus on normalizing struggle in the classroom by honoring students who are honest about their difficulties and making thinking transparent to everyone. In this Teaching Channel video produced in partnership with PERTS, second-grade teacher Maricela Montoy-Wilson models for other educators what it looks like to praise specific strategies. She celebrates the public mistakes her students make in math and makes them feel proud of how their brains grow in those moments.

GROWTH MINDSET AND MATH

Approaching the world with a growth mindset can be very liberating. It gives educators and students freedom to try new approaches, reflect on the positives and negatives, and then try again. But somehow this process is easier for students and teachers to believe in subjects like English or science. Even students who understand that their brain can grow and change with effort, and believe that to be true in some areas of their life, persist in a fixed mindset about math.

Many find math to be the most difficult and hated subject in school. In some ways that’s not so surprising, since many math classes are set up to value speed over careful reasoning and often offer closed questions requiring one right answer. When a student struggles in that type of classroom structure, it becomes difficult to believe she can grow or change her abilities. The questions asked and skills valued are projecting the opposite message.

The common experience of math trauma

The Brilliant Blog

By Annie Murphy Paul

A note to Brilliant readers: I’m continuing my confessional streak here (last week I wrote about my experiences of belonging in college). In the piece below, I’ve chosen to share a memory from my own life because I think it is likely to be similar to memories you have as well.

In writing about “math trauma,” I don’t in any way mean to trivialize trauma or its devastating effects. But I think mathematics expert Jo Boaler is right that the humiliation and shame that many of us have experienced in regard to school math does constitute a kind of trauma, one that often produces a lifelong aversion to and avoidance of the subject.

As always, I’d love to hear your perspective.—Annie

There was a math genius in my first-grade class. His name was Hank, and we all knew he had a gift for numbers and we did not. When we filled out our daily timed math quizzes, he was always done first, after which he idly tapped the eraser end of his pencil on his desk and whistled under his breath, waiting for the rest of us to be done. When the teacher called him up to the front of the classroom to demonstrate how he would solve a series of addition or subtraction problems, the chalk in his hand became a white blur, moving faster than we could follow it.

Early one Friday afternoon our teacher introduced a new activity: speed math competitions, in which pairs of students would vie to be the first to answer correctly all the math problems on one’s own half of the blackboard. My stomach tightened and my heart beat faster at the prospect of it; I shrunk down in my seat, trying to make myself invisible, but as if in a monstrously foreordained nightmare I heard the teacher call out the first two contestants: Hank, and me.

I slowly approached the blackboard and with a trembling hand picked up the chalk. The rows of problems rippled before my eyes: I couldn’t see or think about them clearly, even though I was perfectly capable of answering such questions when left alone to work at my desk. Just then I heard a knock on the window overlooking the playground; my mother and sister stood outside, smiling and waving. For a moment my heart lifted with the idea that they’d come to take me home; then I remembered that my sister, younger than me by a year, had a half-day of kindergarten on Fridays. They weren’t going to help me escape, and in fact were going to be additional witnesses to my humiliation.

“And ready . . . set . . . go!” Hank got down to work, his chalk clacking furiously against the blackboard. He moved as smoothly as a typewriter carriage, working his way through the problems left to right, left to right. Meanwhile I stood frozen, only turning my head to glance at my mother and sister, still smiling encouragingly, and then to look at the board again, still swimming with incomprehensible symbols.

“And . . . stop!” The teacher patted the shoulder of Hank, who had finished his final problem with seconds to spare. I tried to hide my tears from my mother and sister, who mercifully slipped out of sight.

Jo Boaler has heard many, many stories like this one. She is a professor of education at Stanford University and the author of a new book, Mathematical Mindsets. I heard Jo speak at Stanford last week and was so impressed, and even moved, by what she had to say about the destructive way we teach math and the harm it wreaks on students.

Here, I highlight several of my favorite passages from the book’s early chapters. I can’t recommend Mathematical Mindsets highly enough; read it, and tell others about it! They likely experienced math trauma too.

• “A high level of intensity of negative emotion around mathematics is not uncommon. Mathematics, more than any other subject, has the power to crush students’ spirits, and many adults do not move on from mathematics experiences in school if they are negative. When students get the idea they cannot do math, they often maintain a negative relationship with mathematics throughout the rest of their lives.”

• “[The negative experiences that many people have with math flow] from one idea, which is very strong, permeates many societies, and is at the root of math failure and underachievement: that only some people can be good at math. That single belief—that math is a “gift” some people have and others don’t—is responsible for much of the widespread math failure in the world.”

• “Math is special in this way, and people have ideas about math that they don’t have about any other subject. Many people will say that math is different because it is a subject of right and wrong answers, but this is incorrect, and part of the change we need to see in mathematics is acknowledgement of the creative and interpretive nature of mathematics.”

• “Mathematics is a very broad and multidimensional subject that requires reasoning, creativity, connection making, and interpretation of methods; it is a set of ideas that helps illuminate the world; and it is constantly changing. Math problems should encourage and acknowledge the different ways in which people see mathematics and the different pathways they take to solve problems. When these changes happen, students engage with math more deeply and well.”

• “Another misconception about mathematics that is pervasive and damaging—and wrong—is the idea that people who can do math are the smartest or cleverest people. This makes math failure particularly crushing for students, as they interpret it as meaning that they are not smart. We need to dispel this myth. The combined weight of all the different wrong ideas about math that prevail in society is devastating for many children—they believe that mathematics ability is a sign of intelligence and that math is a gift, and if they don’t have that gift then they are not only bad at math but they are unintelligent and unlikely ever to do well in life.”

• “My work [on the growth mindset, originally developed by Boaler’s Stanford colleague Carol Dweck] over recent years has helped me develop a deep appreciation of the need to teach students about mindset inside mathematics, rather than in general. Students have such strong and often negative ideas about math that they can develop a growth mindset about everything else in their life but still believe that you can either achieve highly in math or you can’t. To change these damaging beliefs, students need to develop mathematical mindsets, and this book will teach you ways to encourage them.”

• “Growth mindset ideas [can be] infused through all of mathematics. Teachers of mathematics, and parents working with their students at home, can transform students’ ideas, experiences, and life chances through a growth mindset approach to math. General mindset interventions can be helpful for shifting students’ mindsets, but if students return to mathematics classrooms and math work at home working in the same ways they always have, that growth mindset about math slowly erodes away. The ideas that I share with teachers and parents and set out in this book include paying attention to the math questions and tasks that students work on, the ways teachers and parents encourage or grade students, the forms of grouping used in classrooms, the ways mistakes are dealt with, the norms developed in classrooms, the math messages we can give to students, and the strategies they learn to approach math—really, the whole of the mathematics teaching and learning experience.”

Mathematical Mindsets includes many more such insights, as well as fun, hands-on exercises you can do with your child or students. If you read it, please share the ways in which it changes your thinking and your practices on my blog, here.

And send questions and comments to me at annie@anniemurphypaul.com—I look forward to hearing from you!

Teachers Nurture Growth Mindsets in Math

EdWeek

A blend of family attitudes, cultural ideas, and frustration often lead students to believe that math ability is a fixed trait like eye color, teachers say. They believe they are either born with the skills necessary to succeed in math class or they’re not.

Those pervasive ideas and the way math has traditionally been taught can make it exceptionally difficult for math teachers to nurture growth mindsets in their students, they say.

“There’s a cultural mystique in mathematics and sort of salient, counterproductive conceptions about what it is, that it’s somehow harder than art, which of course is crazy,” said Philip Uri Treisman, a mathematics professor and director of the University of Texas’ Charles A. Dana Center, which focuses on math and science education.

“It has sort of cultural baggage with it that is not helpful to the field,” he said.

The concept of growth mindsets has gained a foothold in many schools, where teachers emphasize that the brain can grow and change and that students don’t enter school with a set of unchangeable strengths and weaknesses. In general, that means praising effort over personal traits and encouraging students to learn from mistakes by developing new strategies to approach problems.

As more schools buy into the research that shows that student mindsets and persistence are linked to academic success, researchers are working to develop more specific strategies for nurturing positive learning attitudes in areas like mathematics.

The key, they say, is changing both the student’s ideas about learning and the way teachers approach math content.

Open Math Problems

Traditional math problems often encourage students to quickly work toward one solution, but “open tasks” can teach the same content while giving students more opportunities to struggle and interact with math concepts on a deeper level. This encourages students to develop a growth mindset, or a recognition that math can be a learned skill, rather than a fixed trait, researchers say.

The Dana Center’s Academic Youth Development program, for example, blends mindsets research with math concepts through both professional development for teachers and summer programs for students as they prepare to enter 9th grade, which is often the year students take Algebra 1. Researchers are studying the effects of the program, which is being used in 1,250 middle and high schools around the country.

‘My Favorite No’

Some teachers are also making efforts on their own to learn about the mindset concept. Stanford University’s Project for Education Research That Scales, or PERTS, released a series of online courses about mindsets for parents and teachers last month. It included just one subject-specific course: a series of videos, exercises, and sample lesson plans tailored for math teachers.

That course includes guidance on how to “normalize failure” by encouraging students to ask questions that they may be afraid to ask and to share incorrect answers with their peers.

“Sometimes it’s important to simply tell students that you love mistakes because that’s how students learn,” one sample discussion plan says. “Start the class with a lesson on why you like mistakes and what students can learn from them.”

Classroom teachers say many of their students approach math with the expectation that they have failed if they can’t quickly solve a problem using a prescribed algorithm.

With a mindset approach, teachers help students focus on learning from that failure and trying the problem from a different angle so that students can understand the underlying concepts.

The Stanford course includes a video of a teacher doing a daily exercise called “my favorite no.” At the beginning of class, she has every student solve a problem on an index card, sorting the resulting answers into a pile of correct and incorrect answers. She then copies an incorrect answer onto the board and asks students to identify all of the correct elements before honing in on what part of their classmate’s solution led them to an incorrect answer.

These sorts of exercises allow even the students who initially solved the problem correctly to learn from their peer’s errors, said Mari Montoy-Wilson, who teachers 3rd grade at a charter school in East Palo Alto, Calif., a small city that is mostly Latino and much less affluent than neighboring communities in the surrounding Silicon Valley.

“People aren’t equipped with this idea that if something feels hard, that’s your sweet spot and you need to persevere and unpack that,” she said.

Merely challenging students to change their mindsets without also changing the way math is taught can be “dangerous,” Treisman said. Without a grasp on math skills and opportunities to apply those skills and develop strategies, students will receive the message that even effort can’t help them improve, he said.

That’s why math teachers who emphasize mindsets advocate for teaching through “open problems,” which challenge students to explain a concept rather than quickly identify one solution. This gives them a chance to explore strategies for solving a problem and recognize there is often more than one way to make sense of it rather than judging their own math skills by whether or not they get the initial answer correct.

Stanford University math education professor Jo Boaler explains the concept in a video included in the course: In a traditional problem, a teacher may give students the dimensions of a rectangle and ask them to find its perimeter. In an open problem, a teacher may ask students to draw three rectangles with a certain perimeter and explain their work.

Exploring Concepts

Mariel Triggs, who teaches math at a private high school in San Francisco, said her students unpack open math problems step-by-step to explore a concept.

For example, she will ask them how many baseballs it would take to fill a room, and then allow them to determine all of the information they need to solve the problem. After they arrive at the answer, she will ask them how the problem would change if they expanded the dimensions of the imaginary room.

“I get these students and they will say, ‘I am not good at math,’ and I began to realize that what they were really saying was, ‘I don’t know how to do the problem in front of me,’ ” she said. “I frame it like a fun puzzle.”

Teachers said those strategies dovetail nicely with the Common Core State Standards’ emphasis on sense-making, abstract reasoning, developing strategies to use math concepts, and critiquing the reasoning of others.

And open problems allow students to understand how math concepts relate to each other, rather than merely understanding how to use an algorithm the teacher prescribes, they said. It’s not that getting the correct answer doesn’t matter, Montoy-Wilson said. But open problems emphasize that the process of arriving at the answer matters too, she said.

“It’s pretty scary in terms of what we want for our future to think of kids who only know the algorithm and not why it works,” Montoy-Wilson said. “When you just focus on getting to the answer, you really rob kids of grappling and working on that sweet spot. You don’t want to scaffold or carry the load too heavily for your kids.”

Across subjects, researchers have found that a teacher’s own orientation to learning can affect whether their students have a fixed mindset or a growth mindset about their own abilities.

Views on Math

But a teacher’s views on math also matter, said Kathy Liu Sun, an assistant professor at Santa Clara University in Santa Clara, Calif., and a former public high school math teacher.

Sun used surveys to gauge the mindsets of about 3,400 students and 40 teachers. She also assessed teachers’ approaches to math, whether they valued speed and memorization or “multi-dimensional” problems that allow for multiple strategies and sense making.

Through post surveys and classroom observations, Sun determined that teachers with a “multi-dimensional” view of math were more likely to have students with a growth mindset at the end of a course.

“I think that teaching is really really important,” she said. “It’s not just about changing kids’ beliefs, it’s about giving them opportunities to experience it.”

Sun also identified strategies that can help boost students’ confidence in the math classroom.

Teachers should encourage multiple attempts at problem solving, they shouldn’t offer unsolicited help to students, and they should provide opportunities for students to resubmit their work, she said.

Some mathematicians have said professional development for math teachers should prioritize content knowledge over pedagogy. Treisman said many math teachers have the math knowledge to teach in a more open format that encourages growth mindsets, and they just need to strengthen the skills necessary to do so.

Many of today’s math teachers were taught in very traditional classrooms, and many have not explored the subject in this way on their own, he said.

That’s why teachers need to practice their own sense-making and model it for their students, Treisman said. If math were music, mastering the basic concepts would be like learning scales and leading students through discussions of open problems would be like playing songs, he said.

“Teachers love the idea of mindsets as almost a panacea,” Treisman said, “but they themselves have very fixed ideas of their own learning.

Vol. 35, Issue 03, Pages 1,10-11