How Does a Growth Mindset in Mathematics Transform Struggle into STEM Success?

Why Do High-Achieving Students Hate Math and How Can Visual Thinking Fix It?

Discover how Math-ish by Jo Boaler challenges narrow teaching methods to solve math anxiety. Explore strategies like metacognition and visual learning to foster creativity, diversity, and genuine equity in education today.

Dive into the full review to apply specific metacognitive strategies that turn mathematical struggle into a powerful tool for lifelong learning and creative problem-solving.

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High schools in the United States typically teach mathematics in a narrow, mechanical way. There’s only one way to solve a mathematical problem — and only one answer. Students are obsessively tested and placed in math education streams, ensuring widespread inequalities. As a result, even high-achieving students often hate math, and avoid university majors that involve mathematics. Encouraging a better relationship to mathematics should involve playful, creative thinking and ideas, and collaboration with a diverse group of people. What needs to be taught isn’t exactly mathematics, but “math-ish.”

Take-Aways

• Educators must replace today’s narrow approach to mathematics with a more flexible, diverse approach.
• “Metacognition” practices form the foundation of better math education.
• Learning math involves making mistakes — and embracing the struggle.
• Math-ish education involves diverse approaches to numbers, shapes, and data.
• Math can be an exciting and interactive visual experience.
• Mathematics isn’t just a set of rules — it’s a way of thinking.
• A flexible, diverse approach to mathematics should be incorporated into all teaching.
• Changing the status quo in mathematical education could help address inequalities.

Summary

Educators must replace today’s narrow approach to mathematics with a more flexible, diverse approach.

Most people learn mathematics in a “narrow” way. In the narrow form of mathematics taught to children and young adults for over 100 years, there’s only one valid approach to a mathematical problem, and only one solution. Narrow mathematics is a stolid, plodding enterprise. It doesn’t transform numbers into something more tangible, like images or physical objects, and does not involve being creative or the free play of ideas. Students are constantly scrutinized and evaluated based on tests. The narrow mathematics approach and the performance-oriented culture it engenders has spoiled mathematics for literally millions of students. Some 60% of university students drop their science majors after taking introductory math courses — even talented, high-achieving students.

“Narrow mathematics ends the hopes and dreams of millions of college students. Not only is this a problem for the students, but it also causes problems for US society, threatening the future of the economy, [and] the development of science, technology, medicine, and the arts.”

Narrow mathematics denies people the sheer joy of mathematical ideas and what they can reveal — to the detriment of society at large. Fortunately, there’s an alternative to narrow mathematics, rooted in two core ideas: “Mathematical diversity” refers to a diversity of people learning and discussing mathematics and a parallel diversity of approaches to mathematics; “math-ish” means de-emphasizing contextless, numeric problem-solving in favor of a focus on the host of creative, interesting ways people use math in real-world settings.

“Metacognition” practices form the foundation of better math education.

People who do unusually well in mathematics aren’t necessarily more naturally able or talented than others. Most simply learn how to approach math in ways that aren’t taught in the classroom. The key to better math education starts with a concept formulated by Stanford University psychology professor John Flavell in the late 1970s: metacognition. Metacognition is effectively thinking about thinking, or put another way, “learning to learn.” The effects of metacognition practices go beyond math itself. Individuals who practice metacognitive strategies thrive in every aspect of their lives: They are better at solving problems, communicating with others, sustaining relationships — and performing at their jobs. People who appreciate and put into practice metacognitive approaches are passionate about learning and curious about diverse points of view.

“Flavell describes metacognition as including knowledge of ourselves, knowledge of the task at hand, and knowledge of strategies, so it is no surprise that it boosts problem-solving, enables mathematical diversity, and enhances work performance.”

There are several metacognitive strategies you can deploy to approach and solve a mathematical problem. First, take a step back and reflect upon what kind of problem you’re facing and what it’s demanding of you. Since neurologists have discovered that mathematics employs visual areas of the brain, you can continue by creating a visual representation of the problem — and seeking a new way to solve it. Ask why a particular procedure, such as multiplication, actually works, distill the problem into simpler elements, and then speculate about how to solve it. When you’re working with others, one of you should raise skeptical questions and arguments about your speculations, and if push comes to shove, you might turn the problem into something smaller and simpler to solve. In general, it’s important to cultivate a reflective or metacognitive mindset, and to learn how to problem-solve with others in a way that respects their ideas and perspectives.

Learning math involves making mistakes — and embracing the struggle.

Extensive research shows that people who adopt a “growth mindset” are better learners and problem-solvers. A growth mindset doesn’t mean that you will invariably succeed at whatever you try, no matter what. Rather, it means that when you make mistakes, you believe you can learn from them. And when you face obstacles, you persist in your efforts because you know setbacks are opportunities to course-correct and find new paths forward.

“Neuroscientific studies have shown that whereas people with a fixed mindset view mistakes as evidence of their own weakness, people with a growth mindset view mistakes as opportunities to learn.”

Scientific studies show that people with a growth mindset are more responsive to critical feedback. And unlike people with a fixed mindset, people with a growth mindset don’t get overly anxious and upset — or self-recriminating — when things don’t go according to plan. Failures strengthen them.

Math teachers should encourage their students to “struggle” with math problems — that is, to work to try to understand the nature of the problem in question and come to a deeper understanding of the concepts involved. Struggling with a problem, rather than being handed a preset procedure for solving it, is more likely to spark creative ideas and inspire new and novel ways to solve the problem.

Teachers can encourage their students to value struggling with problems in a variety of ways. They can present their students with problems that invite diverse approaches, using visual or physical aids. They can propose problems that students can solve in multiple ways and invite conversation and collaboration. The teacher can also praise students’ struggles, even when they lead to mistakes, and reward students for the work they invest in a problem rather than just getting an answer right. And of course the teacher can always recount legendary mistakes. After all, great scientists make mistakes, sometimes enormous ones.

Math-ish education involves diverse approaches to numbers, shapes, and data.

Math-ish education, with an emphasis on the “ish,” involves understanding math as it manifests itself in ordinary life in a population of people with diverse identities and backgrounds — racial, national, religious, gender, and sexual. Mathematical diversity includes both people’s personal differences and the different ways they perceive and think about mathematics. Student diversity encourages mathematical diversity, but student diversity is important in other ways, too. Research shows that racial segregation results in lower achievement, beginning as early as the third grade.

“Diversity enhances life, and it enhances mathematics.”

The question of what mathematical concepts to focus on is another important one. The fact is, K-12 teachers can’t cover everything, especially if they aim to teach in a mathematically diverse way. According to a College Board survey of higher education professors, there are three critical areas in mathematics that students need a feeling for to thrive at the university level. The first is arithmetic or numbers, the second is data analysis, and the third is linear equations that describe how the world’s elements relate to one another. People can acquire a feeling for numbers by noting the way they identify patterns — think of the way a child arranges and rearranges blocks — and drawing patterned shapes that mirror numbers. When students improvise number patterns and shapes, they begin to relate to numbers in a new way. And, of course, numbers are everywhere in real life.

In an online world dominated by social media, understanding data is a critical life skill. Children should begin early on to learn how to understand and not be deceived by data, and slightly older students should be introduced to statistics and the nature of probability. Students can apply an important concept from data science and statistics to understanding linear equations and relationships: The difference between correlation and causation. Just because one phenomenon consistently occurs in conjunction with another doesn’t mean that they have a causal relationship. In this era of the chronic dissemination of misinformation, people should always ask where data comes from, whether all the relevant data is being presented, whether the data is being skewed to emphasize a particular perspective, and whether the relationships presented are correlated or causal.

Math can be an exciting and interactive visual experience.

When mathematics is only approached through numbers, which are in and of themselves abstract, opportunities for surprising connections and ideas, and deeper forms of engagement, are relinquished. In mathematics and elsewhere, visual representations open the mind to new ideas and approaches. For instance, educators can teach algebraic reasoning to children by dipping a large cube divided up into smaller cubes in a certain color of paint. The question is which and how many of the smaller cubes would have sides colored by the paint.

“The physical building of the larger cube meant that students were activating different brain pathways, developing the areas of their brains responsible for numerical, visual, and physical processing.”

Mental representations aren’t created just by thinking about something. You have to build them carefully, the way you would build a cube or draw an image. Mental representations can serve as models. And cognitive scientists and neuroscientists have shown the way mental models play a central role in learning. The brain’s frontal cortex, for instance, is central to problem-solving, and it does this in part by creating models of the world. Research suggests that the parts of the brain most active for mathematicians aren’t the parts responsible for language processing, but the parts of the brain responsible for visual processing. In a way, people literally see numbers — at least when they are represented as groups of dots. Math can be a visual experience.

Mathematics isn’t just a set of rules — it’s a way of thinking.

Many people think of mathematics as little more than a set of inflexible rules or procedures that they must memorize — like how they learned their multiplication tables. Research suggests that students who don’t do well in math tend to conceptualize math in this way. In fact, mathematics is a set of interconnected concepts that people must approach in a flexible manner. Students who do well in math understand these truths. A student who really grasps the concept of numbers, for instance, might find different solutions to adding to numbers: They might divide the numbers into a set of smaller units that are easier to add, for example.

“Learning concepts involves thinking deeply, considering, for example, ‘What is a number?’ ‘How can it be broken apart in different ways to make other numbers?’ ‘How can it be represented visually?’ ‘Where do we see numbers in the world?”

If mathematics is rooted in a few fundamental concepts or ideas, it’s fueled by the ways those concepts and ideas connect to one another. Unfortunately, the standard way schools teach math involves breaking it into small units, making concepts seem disconnected. People who research expertise suggest that humans develop expertise through conceptual mastery, mental representations, and the network of interconnections between mental representations. Teachers can promote this sort of connected understanding of mathematics in their students by suggesting they make “sketchnotes”: blending notes with images to help explain a topic or idea.

A flexible, diverse approach to mathematics should be incorporated into all teaching.

A diverse approach to mathematics — in the learning process and in day-to-day life — should involve thinking about the connections between core concepts, and metacognitive reflections that insist on asking “why” at every stage and at every level of problem-solving. The crucial issue is how to put this into practice. According to Swiss psychologist Anders Ericsson, developing expertise in an area like mathematics demands “deliberate practice.”

“Ericsson defines this as a practice that is purposeful and leads to specialized mental representations, with clear feedback loops on ways to improve.”

Traditionally, math students learned mechanically and by rote, and the only type of feedback they got was the score and grade they got on an exam. Deliberate practice requires you to encounter substantial concepts and ideas — mathematical or otherwise — and create mental images and models that include feedback loops you can actually use to improve your performance. Effective math practice protocols will include problems that require new approaches, an emphasis on ambitious concepts and ideas, models with visual or physical features, and connections between different mathematical ideas and the real world.

Changing the status quo in mathematical education could help address inequalities.

There’s no getting around the fact that traditional math education is highly inequitable. In this system, just a few students move forward to more advanced math education, and those who get left behind tend to be Black and Brown. Those who never learn advanced math in school are then excluded from science and technology professions after graduation.

“Students are more interested and more successful when the content they are learning allows them to engage in different ways.”

The purpose of a new model of math education, reflected in the 2023 California Mathematics Framework supported by the California State Board of Education, is to simultaneously promote the development of mathematical expertise and reduce inequalities. It does this in part by changing the way math education progresses, starting with a more general discussion of the problems at hand, and insisting on the value of diversity in the perspectives people bring to mathematics and the possible approaches to mathematical problems.

About the Author

Dr. Jo Boaler is the Nomellini-Olivier Professor of Education at Stanford University. She is the author of the first Massive Open Online Course (MOOC) on mathematics and of the bestseller Limitless Mind.