At some point, every math student will struggle. Teachers know this. They expect it. They plan for it. Naturally, they worry that challenging lessons will turn into impasses where students are overwhelmed by their frustrations. But teachers also know that these moments of struggle can be fruitful learning opportunities. Indeed, they can be the moments where students make the greatest strides, develop confidence, and mature into independent thinkers.

The deliberate process of grappling with mathematical concepts in an effort to understand them is known as productive struggle.

As rote learning continues to come under scrutiny, productive struggle offers an alternative method that is creative, constructive, and meaningful.

## Interactive vs. Passive Learning

When students learn by rote, they tend to think of math strictly as a series of steps toward getting the right answer. When they forget a step or don’t know how to apply one, they turn to the teacher for answers. Seeing that a student is stuck, the teacher will naturally provide the next step or demonstrate how to apply it. After all, this is what teachers do. The problem, however, arises when students turn to the teacher the moment they can’t figure something out on their own. Over time, many students come to depend on the teacher for the right answer. “Just show me how to do it. Just give me this part of the formula, and I can solve the problem,” students will often say. This habit betrays passive learning instead of interactive learning. In the eyes of a passive learner, the role of teachers is simply to transfer knowledge from themselves to their students. Active learners, on the other hand, involve themselves in their own education in various ways, constructing meaning, asking questions, and trying to figure things out on their own before turning to the teacher for help.

Productive struggle is a form of active learning that produces grit—the determination to solve problems on one’s own. Students develop grit when they feel stuck, thinking they have exhausted their knowledge, but choose to persevere. By doing so, each student discovers methods that help them to solve problems on their own. This might mean retracing their steps and reviewing their process to ensure that they didn’t forget something or miscalculate. Or they might look back at their notes. Perhaps they reflexively identify the source of their confusion by asking themselves questions about why they are stumped. Regardless of their method, they are not memorizing specific formulas for specific problems, but rather developing habits for solving problems in the classroom and in life.

## What Productive Struggle is Not

Now, to be clear: productive struggle does not entail teachers leaving students to flounder. As many education experts point out, there is a fine line between struggle and frustration. Struggling students may get stalled and take extra time to work through a problem, but they remain focused on their task. Frustrated students, on the other hand, feel overwhelmed, then shut down and give up. They are the ones who ask for help without trying. The former student not only learns the mathematical concept, but also practices controlling negative emotions; the latter gives in to them and learns nothing.

## Productive Struggle: Time Consuming

Facilitating productive struggle can be intimidating to the teacher, because it is time consuming. Productive struggle requires real thinking by students rather than recitation of facts. Moreover, it’s not a test-prep method, so you may wonder, “Why even bother?” For starters, evidence suggests that productive struggle correlates to higher test scores. In one study, students who learned through productive struggle performed 10 percent higher than students who learned through rote memorization. Students who can persevere through challenges demonstrate more intellectual dexterity than those who simply memorize formulas.

Students who can persevere through challenges demonstrate more intellectual dexterity than those who simply memorize formulas. Even though productive struggle may require more time and effort from the teacher, the rewards are well worth the effort.

Even though productive struggle may require more time and effort from the teacher, the rewards are well worth the effort.

## Intentional Discourse

Well enough, but how does a teacher encourage students to push through challenges? One idea is to engage students in intentional discourse. Imagine, for example, that you have assigned your class a problem that requires students to exercise skills and knowledge from a recent lesson on adding and subtracting fractions. You begin class one day by posing the following scenario: “The first night your family ate four-tenths of pasta. The second night they ate three-tenths of pasta, but by the third night everyone is tired of spaghetti. Your mother says she won’t cook anything else until it’s all gone. How much more spaghetti do they need to eat?” You instruct students to raise their hands and suggest how to solve the problem, but get nothing but blank stares. Obviously, you can’t let the silence go on forever, so you ask: “Is this addition or subtraction?” One of the students answers addition. “Good. Now, what are we adding?” Slowly students begin to offer answers and comments. These kinds of targeted questions promote productive struggle by helping students arrive at the answer on their own.

## High-Achieving Students

Productive struggle may be overwhelming even for high-achieving students. The success of straight-A students often depends on their ability to memorize information, so assigning non-routine or open-ended problems may move them out of their comfort zone. Many high-achieving students are accustomed to reaping easy results, so these problems will likely frustrate them. They may for once even question their own abilities. Such challenges, though, are ripe with opportunity for intellectual and emotional growth. If a student’s struggle turns into a hard stop, remind them what you as a teacher are trying to do for them—help them become capable of solving any problem, no matter how impossible it seems.

Productive struggle emphasizes the creative process of solving mathematical problems. However, for many students math has always been about “getting the right answer.” Consequently, when students struggle during the process of solving a problem, their immediate reaction is to think that they are failing and bad at math. After all, if producing the right answer is the whole purpose of math, any delay in achieving this goal means that the student is missing the mark. Yet this is not how mathematicians think or work. Instead, they apply their knowledge to problems that can take several years to solve. They fail and experience intellectual and professional growing pains along the way. In short, getting the right answer involves a sometimes tedious yet rewarding and creative process. Why should it be any different for students? Even students who have no interest in pursuing mathematical careers still need to learn that achieving goals requires a process of struggle.

We should, therefore, encourage students to see math as a process. To do so, we need to change the way we grade. Instead of marking answers right or wrong, try to provide informative feedback. Some studies found that “right and wrong” feedback hurts student performance. Moreover, informative feedback reveals the thought process behind math and disabuses students of the notion that math is a mechanical means to the right answer. Consider, for example, a student who’s having a tough time creating equivalent fractions. Despite the challenge, he works hard, and one day, he gets an A- on a quiz. Instead of simply check-marking the correct answers, his teacher writes a short note at the bottom: “I’m so impressed! Your hard work has paid off. You’re now creating equivalent fractions because you’re identifying the similarities and differences between the fractions.” The feedback here is specific and concise. The teacher reinforced what the student is doing right, so he knows exactly what works.

## Math-People Myth

One of the other unfortunate results of emphasizing the answer over the process is the “math-people myth.” This myth divides the world into two groups: Those who are inherently good at math, and those who are not. According to this view, math is not a skill students develop: It’s a trait they are born with. As with everything else, students have varying aptitudes for math, but research indicates that everyone can do math. Sadly, the math-person myth has embedded itself in the minds of many students, and sometimes it becomes an excuse. Teachers have often heard the same students complain that “they just aren’t a math person” when they don’t immediately understand something. Productive struggle not only debunks the math-people myth, it sets the expectation that everyone struggles at some point. Productive struggle not only debunks the math-people myth, it sets the expectation that everyone struggles at some point.

Productive struggle not only debunks the math-people myth, it sets the expectation that everyone struggles at some point.

## CRA

Teachers can incorporate alternative teaching methods to help students realize that they too can do math well. Consider, for example, a class that is learning decimal place values. Although most of the class is catching on, a few students are falling behind. Up to this point, the teacher has taught most of the lessons on the board and assigned worksheets. Feeling overwhelmed, one of the students pushes her worksheet aside and lays her head on the desk. The teacher turns to the Concrete-Representational-Abstract (CRA) method to help the student emerge from her shutdown. She brings out various small plastic shapes. “Certain pieces represent certain place values,” the teacher explains. “These cubes represent the hundredth place, the cylinders represent the tenth place, the squares represent the ones, and the circles represent the tens place. We will use these shapes to visually construct the decimal numbers.” She still struggles, but the visual aids help her stay focused and push through her irritation. Teachers may also try the 5E method to facilitate productive struggle.

## Conclusion

Productive struggle is a difficult intellectual and emotional space for students to occupy. Nobody enjoys feeling stuck. Yet the beautiful thing about productive struggle is that students emerge on the other side of their struggles feeling empowered. Indeed, productive struggle is where knowledge and confidence crystalize. As educators, it is our responsibility to give students just enough help for them to discover this for themselves.