Every evening, after my family and I have cleaned up from dinner, I sit down to take on the New York Times game Strands. If you aren’t familiar with Strands, it involves a 6 x 8 array of seemingly random letters and a theme. The goal is to find words related to the theme by connecting letters in the grid until all the letters are used. Some days I figure out the theme right away, although that doesn’t necessarily help me find the actual words. Other days I find several words before figuring out the theme. And then there are the days when I stare and stare and stare at that screen and nothing emerges. At that point I’ll often put it down and come back to it on and off over the course of the night. The less familiar the theme is, the more likely I am to use a hint or two (or maybe more). If the theme is something I really enjoy, like Broadway shows or children’s books, I will refuse to use a clue, no matter how stuck I am. Regardless of what type of Strands day I am having, or how many hints I need, I will always—and I mean always—finish the puzzle, sometimes going to bed late to get it done.
My perseverance when solving Strands got me thinking about when I did and did not see such a level of commitment and engagement in my students. Where did I see my students dig in and persist in solving a problem? Where did I see them get frustrated and quickly give up? What were the activities they would keep coming back to, even after hitting an obstacle?
When I first started teaching, I taught sixth-grade science. I often had my students conduct observations or perform experiments in which they were challenged to uncover and make sense of the phenomena they were witnessing. Students loved these tasks and often suggested follow-up experiments to further test their ideas. Stopping them to wrap up the task was often met with groans. They were engaged. They wanted to keep going.
Now jump ahead to when I taught third grade. Unfortunately, we had a pretty standard math program: The teacher would introduce a skill. The class would do a guided practice. Students would go back to their seats to practice the skill with some application problems. Nobody was upset when I stopped them to wrap up those lessons.
It wasn’t until I started bringing more of my old science mentality to my math class that students got engaged. I started looking for problems that allowed students to deeply explore mathematical concepts, like open middle problems. I used rich problems of the week that were not rehashes of what we had just learned in class but that both allowed students to actively decide how to approach the problem and that pushed them to make connections between mathematical concepts. What I began to see was, when students were making sense of something, rather than just practicing a rote task, they were sticking to the task. Even if they got frustrated, they would take a break, but they would always come back later. They were engaged enough that they didn’t want to give up.
A word about words
As is often the case with education, a lot of different words get thrown around in reference to this type of engagement. You have likely heard of “productive struggle,” along with “persistence,” “perseverance,” and even “grit” in reference to “stick-to-itiveness” with mathematical problem-solving. While there is nothing wrong with any of these terms, I have come to like the term “perseverance” for two reasons. First, Common Core Mathematical Practice Standard 1 talks about having students, “Make sense of problems and persevere in solving them.” Regardless of whether your state or school uses the Common Core, the description of problem-solving in practice 1 nicely captures the active engagement I think all teachers hope to see in their students. Secondly, researcher Joseph DiNapoli’s description of perseverance in problem-solving seems like a perfect explanation of what I wanted my students to be able to do. In distinguishing between “grit,” “persistence,” and “perseverance,” he describes perseverance as “Initiating and sustaining, and re-initiating and re-sustaining, in-the-moment productive struggle in the face of one or more obstacles, setbacks, or discouragements.” He further clarifies that perseverance, as opposed to persistence, includes a willingness to re-evaluate and change one’s approach, which is critical.
Why is perseverance important?
Whatever name is used, the willingness to persevere in mathematical problem-solving has numerous benefits.
Multiple research studies have identified perseverance as one of several personality traits that act as “contributors to success in school and in life.” In terms of mathematics, the ability and willingness of students to persevere in productive struggle helps them think deeply about mathematical concepts and develop the ability to creatively solve problems. Or, as DiNapoli puts it, “Perseverance, or initiating and sustaining productive struggle in the face of obstacles, promotes making sense of mathematics.” He further asserts that when talking about “problem solving, a key process that supports the development of conceptual understanding is productive struggle.”
Indeed, educational researchers going back to John Dewey have found that productive struggle is necessary for constructing deep knowledge and understanding. Dewey further called out that the focus on quick answers, which was the hallmark of much traditional instruction, impedes opportunities to examine the conceptual aspects of a problem.
Perseverance and zone of proximal development
Creating opportunities for students to persevere in problem solving tells you that you’ve hit the sweet spot in terms of maximizing growth. When you see a student initially engage with a problem, find their way into it, get stuck, analyze and overcome setbacks, re-engage, and try a new approach, you know they are working in their zone of proximal development.
We know from the work of Lev Vygotsky that this edge of understanding—not too easy and not too hard—is where growth happens. Or as my colleague Brooke Mabry explains it, the “Zone of proximal development represents the metaphorical gap between what a learner can do and what they can’t do…yet.” If students race through a set of problems, they likely aren’t learning anything. If they can’t find an entry point to get started, the content is likely too difficult for them. Using pretests of relevant precursor content can help you ensure that students have enough background knowledge to wrestle with the problem while still having it be on the edge of their understanding.
Developing perseverance in math class
Helping students develop perseverance in mathematical problem-solving isn’t just about giving them high-quality problems, although that is an important component. You also need to create a culture that encourages perseverance, and you need to know how and when to provide support to avoid student frustration.
Let’s dig a little deeper into how to support perseverance in your classroom.
Make it worthwhile
Obviously, one of the key components of encouraging perseverance is giving students opportunities to wrestle with complex, interesting problems. There isn’t much incentive to persevere in completing ten nearly identical, rote problems. I like to think about it as the difference between painting a fence versus painting a mural. Yeah, the fence will look nice once it’s done, but other than picking the initial color, there aren’t a lot of big decisions to be made, and the task gets pretty repetitive pretty quickly.
Painting a mural is a far more complex task, with multiple decision points built into it, and it’s a lot more engaging and satisfying. I’ve written about rich mathematical problems before, but essentially, problems with multiple entry points, with nonobvious solution paths that demand active decision-making and the connection of mathematical ideas, and that allow for multiple solution methods are more engaging and promote deeper mathematical thinking and perseverance. Low-floor, high-ceiling tasks are great for this as they allow everyone entry but also can stretch high enough to encourage perseverance.
Be mindful of your mindset
I first got intrigued by the idea of perseverance in math over a decade ago after reading an article comparing how schools in the Western world and schools in the Eastern world approach the ideas of intellectual struggle and intelligence. Researcher James Stigler observed math classes in Japan and in the US and found fundamental differences in how each culture viewed intellectual struggle. In the US, Stigler noted that from a very early age we learn to associate struggle with low ability. In Eastern cultures, “it’s just assumed that struggle is a predictable part of the learning process. Everyone is expected to struggle in the process of learning, and so struggling becomes a chance to show that you, the student, have what it takes emotionally to resolve the problem by persisting through that struggle.” This comfort with struggle is reflected in deeper cultural attitudes. Jin Li, a researcher with Brown University found similar differences when observing conversations between American mothers and children and Taiwanese mothers and children. American mothers tended to connect academic achievement to intelligence, something innate and immutable. In contrast, Taiwanese mothers attributed success to hard work and persistence.
This difference really struck me as both a parent and an educator. The traditional US approach, which is intended to build students’ self-esteem, inadvertently fosters a “you’ve got it or you don’t” attitude rather than a growth mindset. Research has long supported the benefits of a growth mindset, in which you believe that knowledge can expand and develop. In contrast, a fixed mindset is one in which intelligence is viewed as a set trait that you either do or don’t possess.
Developing the growth mindset needed to persevere in problem-solving has a significant impact in the classroom. Stigler conducted another study in which he gave first-grade students in the US and in Japan an impossible math problem. The US students, who had more of a fixed mindset, averaged less than 30 seconds before they gave up, whereas the Japanese students had to be stopped after an hour of working on the problem.
Change minds
We’ve all had the student, or maybe have been the student, who looks at a problem and immediately says, “I can’t do that.” This type of knee-jerk reaction can mean a lot of things: That a student has internalized that they are “not a math person” because of repeated negative experiences. That they have learned that being “good” at math means getting the answer right—and getting it quickly. Or that they have internalized the, “If you don’t get it right away, you’re not smart” messaging. This is where the power of a students’ mindset comes into play. Luckily, there are several ways to help students develop a growth mindset.
One great way to start is to explicitly teach students that new experiences, including wrestling with tough problems, literally helps our brains expand. In a 2018 study, high school students completed online exercises that taught them how the brain can grow and change, the role of hard work in increasing neural connections, and how having a growth mindset can help you push through obstacles and achieve your goals. The study showed that changing students’ mindset positively improved both their perseverance and their mathematical performance.
Once students begin to understand how mental workouts build up their brains in the same way physical workouts build their muscles, change the language of your classroom to support this. Instead of telling a student they are smart, or that they are good at something, praise their process, their hard work, and their perseverance. Name specific steps that they took, what they did to get themselves unstuck, or how they engaged with other problem-solvers. Model using the language of “yet,” as in, “I don’t know how to do that, yet.” This is such a powerful way to reinforce that students are constantly growing in their knowledge and that they are all capable of learning.
Celebrating mistakes as steps to learning is also important to changing mindsets and encouraging perseverance. The fear of “getting it wrong” can really hold students back from engaging with a problem or sharing their ideas. Reframing mistakes as opportunities to learn and course-correct rather than as failures helps students grow their resilience.
Activities like “My Favorite No,” which highlight the learning that comes from making mistakes, are a great way to reinforce this idea. You can also tell students that research has shown that even failing to solve a problem correctly is productive. One study examined students who engaged in what the researchers termed “productive failure.” Students worked together on complex problems without support or scaffolding during the problem-solving process. Afterward, the teacher facilitated a discussion about their approaches, comparing them to a traditional solution approach. Even though these students had failed to solve the problem correctly, when given a post-test, they consistently outperformed students who had received direct instruction on the content.
Value the process
Math is often treated as a highly linear topic, generally for ease of developing a scope and sequence. And while topics and concepts do build upon each other, math concepts connect more in a web than in a single line. Problem-solving in math has also traditionally been treated as linear: follow a set of previously taught steps to find the answer. This type of follow-the-leader (aka, follow-the-teacher) thinking doesn’t promote perseverance or sensemaking in problem-solving.
What if we thought of mathematical problem-solving more like we think of writing, which is not so much a linear process but a recursive one? Much of what my colleague Julie Richardson posits in her article “5 reasons to give students feedback on their writing process” could be applied to mathematical problem-solving. She describes the process of writing as one where writers “constantly move between cognitive processes, such as planning (setting goals, generating ideas, organizing ideas), drafting (putting a writing plan into action), and reviewing (evaluating, editing, revising).” This is exactly what we want students to do when solving math problems: planning (thinking of relevant prior knowledge, listing possible solution paths, considering how to get started), drafting (testing different approaches), and reviewing (determining if an approach is productive, if there is a more efficient approach, and if any answers found are reasonable). Modeling mathematical problem-solving in this way supports perseverance and the idea that the process is as important as the answer. Also, just as with writing, we want students to collaborate, compare approaches, ask questions, and give each other feedback when solving mathematical problems.
Know when and how to step in
Speaking of feedback, knowing when and how to intervene to prevent frustration is critical. When a student is stuck or frustrated, it can be hard not to jump in and “rescue them” by giving them a next step to take. It can also be difficult to know just what the right thing to say might be. Thankfully, research points to a couple of ways to best support students.
Joseph DiNapoli and Emily Miller found that prompting students to conceptualize the mathematics involved in the problem before they started solving helped them to re-engage and persevere after hitting an obstacle. Students were given the prompt, “Before you start, what mathematical ideas or steps do you think might be important for solving this problem? Write down your ideas in detail.” When they got stuck in their solution process, prompting them to review their initial thinking helped students reconsider ideas and re-engage with the problem.
In general, asking open-ended questions designed to support reflection can help students either get started or get unstuck. Nancy Emerson Kress suggests these six questions:
- What do you notice?
- What additional information or clarification would be helpful?
- What can you do or figure out?
- How do you know that your work and/or answer are accurate?
- Is there another way you could approach this problem?
- What else can you say about the problem, and what else would you like to know?
Simple questions like these do a number of things for students. First, you are teaching them that expressing frustration doesn’t lead to the teacher telling you how to solve the problem. Second, by asking questions rather than providing information, you are affirming your confidence in them as capable problem-solvers. Third, this approach prevents teachers from reducing the complexity of the task by over-scaffolding it, a common practice. Finally, you are teaching them a set of questions that they can internalize to help themselves when they get stuck on future problems.
Another way to support students during productive struggle without rescuing them is to have them share their approach, ideas, or progress so far with other students. Hearing other students’ ideas may help students get “unstuck” and give them another direction or idea to pursue. The book 5 Practices for Orchestrating Productive Mathematics Discussions provides a great structure for both anticipating student challenges related to a given problem and for supporting high-quality classroom discussions of different approaches.
It takes time
Remember that seeing struggle as positive may be a new idea for students, and maybe for you, too, so it will take time to develop this muscle and mindset. The resources below can help you and your students grow comfortable with struggle and perseverance:
- Academic Tenacity: Mindsets and Skills that Promote Long-Term Learning. Researchers Carol Dweck, Gregory Walton, and Geoffrey Cohen created this free resource to help teachers develop mindsets and self-regulation skills that support perseverance.
- “Keep going! Building a culture of perseverance in math classrooms.” This PBS website includes videos and materials designed to help high school teachers support perseverance in mathematical problem-solving.
- “Khan Academy growth mindset.” Khan Academy provides resources for teachers as well as activities geared toward elementary, middle, and high school students to promote a growth mindset.
- “Teaching a growth mindset.” This site includes resources for teachers about understanding growth and teaching a growth mindset, including how to give feedback, celebrate mistakes, and select tasks that promote a growth mindset.
- Principles to Actions: Ensuring Mathematical Success for All. NCTM’s book on effective mathematics teaching and learning includes a chapter on posing purposeful questions and one on supporting productive struggle.
- “Promoting student learning and productive persistence in developmental mathematics: Research frameworks informing the Carnegie Pathways.” This resource from Carnegie Foundation for the Advancement of Teaching provides an in-depth look at promoting perseverance in math, including guiding questions and instructional moves teachers can use at various points in the problem-solving process.
- “Resources for teaching growth mindset.” This Edutopia site has 25 articles, videos, and other resources on developing a growth mindset, with specific content related to developing a growth mindset in math and improving classroom feedback.
