{"id":24441,"date":"2024-12-19T05:00:00","date_gmt":"2024-12-19T13:00:00","guid":{"rendered":"https:\/\/www.nwea.org\/blog\/?p=24441"},"modified":"2024-12-20T09:37:09","modified_gmt":"2024-12-20T17:37:09","slug":"sticking-with-it-helping-your-students-build-perseverance-in-math","status":"publish","type":"post","link":"https:\/\/www.nwea.org\/blog\/2024\/sticking-with-it-helping-your-students-build-perseverance-in-math\/","title":{"rendered":"Sticking with it! Helping your students build perseverance in math"},"content":{"rendered":"
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Every evening, after my family and I have cleaned up from dinner, I sit down to take on the New York Times<\/em> game Strands. If you aren\u2019t 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\u2019t 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\u2019ll 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\u2019s 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\u2014and I mean always\u2014finish the puzzle, sometimes going to bed late to get it done.<\/p>\n\n\n\n

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?<\/p>\n\n\n\n

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.<\/p>\n\n\n\n

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.<\/p>\n\n\n\n

It wasn\u2019t 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<\/a>. 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\u2019t want to give up.<\/p>\n\n\n\n

A word about words<\/h2>\n\n\n\n

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 \u201cproductive struggle,\u201d<\/a> along with \u201cpersistence,\u201d \u201cperseverance,\u201d and even \u201cgrit\u201d in reference to \u201cstick-to-itiveness\u201d with mathematical problem-solving. While there is nothing wrong with any of these terms, I have come to like the term \u201cperseverance\u201d for two reasons. First, Common Core Mathematical Practice Standard 1 talks about having students, \u201cMake sense of problems and persevere in solving them.\u201d Regardless of whether your state or school uses the Common Core, the description of problem-solving in practice 1<\/a> nicely captures the active engagement I think all teachers hope to see in their students. Secondly, researcher Joseph DiNapoli\u2019s 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 \u201cgrit,\u201d \u201cpersistence,\u201d and \u201cperseverance,\u201d<\/a> he describes perseverance as \u201cInitiating and sustaining, and re-initiating and re-sustaining, in-the-moment productive struggle in the face of one or more obstacles, setbacks, or discouragements.\u201d He further clarifies that perseverance, as opposed to persistence, includes a willingness to re-evaluate and change one\u2019s approach, which is critical.<\/p>\n\n\n\n

Why is perseverance important?<\/h2>\n\n\n\n

Whatever name is used, the willingness to persevere in mathematical problem-solving has numerous benefits.<\/p>\n\n\n\n

Multiple research studies<\/a> have identified perseverance as one of several personality traits that act as \u201ccontributors to success in school and in life.\u201d 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<\/a>, \u201cPerseverance, or initiating and sustaining productive struggle in the face of obstacles, promotes making sense of mathematics.\u201d He further asserts<\/a> that when talking about \u201cproblem solving, a key process that supports the development of conceptual understanding is productive struggle.\u201d<\/p>\n\n\n\n

Indeed, educational researchers going back to John Dewey<\/a> 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.<\/p>\n\n\n\n

Perseverance and zone of proximal development<\/h2>\n\n\n\n

Creating opportunities for students to persevere in problem solving tells you that you\u2019ve 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.<\/p>\n\n\n\n

We know from the work of Lev Vygotsky<\/a> that this edge of understanding\u2014not too easy and not too hard\u2014is where growth happens. Or as my colleague Brooke Mabry<\/a> explains it, the \u201cZone of proximal development represents the metaphorical gap between what a learner can<\/em> do and what they can\u2019t <\/em>do\u2026yet.\u201d If students race through a set of problems, they likely aren\u2019t learning anything. If they can\u2019t 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.<\/p>\n\n\n\n

Developing perseverance in math class<\/h2>\n\n\n\n

Helping students develop perseverance in mathematical problem-solving isn\u2019t 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.<\/p>\n\n\n\n

Let\u2019s dig a little deeper into how to support perseverance in your classroom.<\/p>\n\n\n\n

Make it<\/strong> wort<\/strong>hwhile<\/strong><\/p>\n\n\n\n

Obviously, one of the key components of encouraging perseverance is giving students opportunities to wrestle with complex, interesting problems. There isn\u2019t 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\u2019s done, but other than picking the initial color, there aren\u2019t a lot of big decisions to be made, and the task gets pretty repetitive pretty quickly.<\/p>\n\n\n\n

Painting a mural is a far more complex task, with multiple decision points built into it, and it\u2019s a lot more engaging and satisfying. I\u2019ve written about rich mathematical problems<\/a> 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<\/a> are great for this as they allow everyone entry but also can stretch high enough to encourage perseverance.<\/p>\n\n\n\n

Be mindful of your mindset<\/strong><\/p>\n\n\n\n

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<\/a> 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, \u201cit\u2019s 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.\u201d 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.<\/p>\n\n\n\n

This difference really struck me as both a parent and an educator. The traditional US approach, which is intended to build students\u2019 self-esteem, inadvertently fosters a \u201cyou\u2019ve got it or you don\u2019t\u201d attitude rather than a growth mindset<\/a>. Research has long supported the benefits of a growth mindset<\/a>, 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\u2019t possess.<\/p>\n\n\n\n

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.<\/p>\n\n\n\n

Change minds<\/strong><\/p>\n\n\n\n

We\u2019ve all had the student, or maybe have been<\/em> the student, who looks at a problem and immediately says, \u201cI can\u2019t do that.\u201d This type of knee-jerk reaction can mean a lot of things: That a student has internalized that they are \u201cnot a math person<\/a>\u201d because of repeated negative experiences. That they have learned that being \u201cgood\u201d at math means getting the answer right\u2014and getting it quickly. Or that they have internalized the, \u201cIf you don\u2019t get it right away, you\u2019re not smart\u201d messaging. This is where the power of a students\u2019 mindset comes into play. Luckily, there are several ways to help students develop a growth mindset.<\/p>\n\n\n\n

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<\/a>, 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\u2019 mindset positively improved both their perseverance and their mathematical performance.<\/p>\n\n\n\n

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 \u201cyet,\u201d as in, \u201cI don\u2019t know how to do that, yet<\/em>.\u201d This is such a powerful way to reinforce that students are constantly growing in their knowledge and that they are all capable of learning.<\/p>\n\n\n\n

Celebrating mistakes as steps to learning<\/a> is also important to changing mindsets and encouraging perseverance. The fear of \u201cgetting it wrong\u201d 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.<\/p>\n\n\n\n

Activities like \u201cMy Favorite No,\u201d<\/a> 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<\/a> examined students who engaged in what the researchers termed \u201cproductive failure.\u201d 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.<\/p>\n\n\n\n

Value the process <\/strong><\/p>\n\n\n\n

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\u2019t promote perseverance or sensemaking in problem-solving.<\/p>\n\n\n\n

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 \u201c5 reasons to give students feedback on their writing process\u201d<\/a> could be applied to mathematical problem-solving. She describes the process of writing as one where writers \u201cconstantly 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).\u201d 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.<\/p>\n\n\n\n

Know when and how to step in<\/strong><\/p>\n\n\n\n

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 \u201crescue them\u201d 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.<\/p>\n\n\n\n

Joseph DiNapoli and Emily Miller found that prompting students to conceptualize the mathematics<\/a> involved in the problem before they started solving helped them to re-engage and persevere after hitting an obstacle. Students were given the prompt, \u201cBefore you start, what mathematical ideas or steps do you think might be important for solving this problem? Write down your ideas in detail.\u201d 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.<\/p>\n\n\n\n

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<\/a>:<\/p>\n\n\n\n