Productive Struggle in Mathematics: A Two-Part Series
Putting Effective Intervention into Action
In the first blog of this series, Paul examined the mindset and classroom culture that allow productive struggle to thrive. Here, he focuses on the instructional decisions that determine whether struggle becomes growth or frustration.
Practical Strategies for Supporting Productive Struggle
In my Precalculus class, students were learning to solve exponential and logarithmic equations. Knowing there were about six types of equations, I needed to think carefully about how I designed the lesson. Any lesson or task will best support a productive struggle environment when it is what we call “low floor, high ceiling.” Such was the case for this Precalculus lesson. I modeled each type before gradually releasing students to guided practice. Various students had to miss one or both lessons, so I made basic videos for each type of equation that allowed for multiple entry points. Throughout the lesson, there was strategic silence, ample wait time for students to identify relevant elements of an equation, think about the process, and strategize on how best to solve a given equation. When students were stuck, I used questions that extended their thinking rather than “rescue them.” For instance, one student said, “I am totally lost on station 4. Can you do that one on the board?” Rather than rescue the student, I asked two questions: “What do you recognize?” and “What category of equation would you classify this as?” Immediately, the student correctly identified the type of equation and glanced at an earlier modeled example. She asked, “So would I take the log of both sides?” Correct! Just like that, I had supported the student on her way to progress.
These kinds of targeted questions became my primary tool throughout the lesson. Naturally, I needed to assess students in real time to ascertain their level of understanding. I used various techniques to do this, including individual whiteboards and a station rotation activity. Initially, the focus of these formative assessments was not to see who could correctly solve an equation. Rather, I wanted to see who knew how to start each equation. These types of check-ins allowed me to scaffold the support I offered each student. When I felt it appropriate, I presented students with an exit ticket as a more formal assessment of their understanding. One fascinating encounter occurred when a pair of students came upon an equation that could be solved using two very different strategies. I listened in on their conversation as they debated which strategy was “the right one.”
Student A: “Do we rewrite the bases?”
Student B: “I don’t think so. The bases are different. I think you have to take the log of both sides.”
Student A: “Yeah, but look at the base of this side. It’s already a 6. Can’t we just turn 216 into a base of 6?”
Eventually, a third student jumped in, saying, “I think either way is right. It’s just which one you feel more comfortable with.”
Voilà! A classic example of what is possible when students are focused on the process rather than the outcome.
The Teacher’s Tightrope: When to Step In
Full disclosure: In the first part of my teaching career, I struggled in deciding how or when to engage with students when I saw them struggle. Only after years of experience in the classroom could I begin to develop a framework that supports students in the midst of their struggle, ultimately guiding them toward progress.
Here are three critical questions I ask myself before intervening:
- Are students making progress? Even if the progress seems slower than I expected, progress is always something to celebrate.
- Is frustration building or is focus maintained? Look for signs of persistence despite apparent frustration. Students can be frustrated with their own pace while still actively engaged. That’s productive. But if you see shutdown behaviors, i.e., head down, erasing everything, or staring blankly, that’s when frustration has crossed into unproductive territory. Positive encouragement or sharing your own struggles with learning can help redirect them.
- Have students exhausted their available strategies? The only way to do this is to engage with the student directly. Sure, you can observe their work, but many students erase work that did not seem to get them anywhere. This could be the issue! Probe what strategies the student has already tried. Ask them to explain their thinking. It is here you may uncover valuable insight to nudge them in the right direction.
Generally speaking, I give students 5 to 10 minutes of genuine struggle time before I intervene, depending on the class and topic. The power move, though, is developing a willingness to offer the correct intervention and then walk away. This powerfully communicates to students that you believe they can continue independently, fostering a sense of autonomy and increasing their resolve.
To support students while maintaining an appropriate level of cognitive demand, I tend to do one of three things.
- I will point students to a resource, rather than simply tell them what to do next. For instance, when my students struggled to choose an appropriate model to represent their data, I redirected them, asking, “Have you explored what happens when you try different models in Desmos?”
- Instead of making a statement, I will ask a question. For example, when one of my students recently struggled to factor a quadratic expression, I asked them if they had considered all the possible factors of a specific constant in the expression.
- If necessary, I offer a similar, simpler problem to reinforce that their feelings of uncertainty are a natural part of learning.
Even with the best intentions and frameworks, there are common pitfalls to avoid. By default, a teacher’s instinct is the desire to see their students succeed. First, I was guilty of creating an inequitable distribution of challenges, perhaps the biggest pitfall to avoid. Embarrassingly, I would often design a task for my AP Calculus students that I believed to offer just the right amount of rigor, when in fact it only served certain students who were equipped for that level of difficulty. After studying the Chain Rule for derivatives, I provided students with a blank table and instructed them to fill in the table such that a student could execute a problem with a specific answer. It turned out that a select number of students had gotten to a point where this task challenged them at an appropriate level. To the others, blank stares were followed by complete surrender, rendering the task irrelevant for the majority of the class. This underscored the need to be mindful of where each student is in their learning. Only by recognizing my missteps over time have I come to realize the benefits of offering more personalized tasks where possible.
Second, be careful not to “over-scaffold” by giving endless hints. Doing so often creates dependency rather than independence. Finally, learn how to distinguish between confusion and limitation. Too often, a student seems puzzled or confused, but upon further examination, there are significant limitations, perhaps with prerequisite skills, that hinder their ability to meet the demands of the task at hand. My daughter came home one night, exclaiming, “Dad, we have to lock in. I am so confused in math.” Her homework required her to graph transformations of functions. Initially, I thought she was struggling to see the patterns in each equation that would assist her in translating or reflecting graphs. It only took a minute or two to realize that she wasn’t confused at all. She understood those rules fine. However, she did not recognize the different parent functions she had been tasked with learning earlier in the course. Only once we tackled that deficiency could she progress further. What looked like confusion about transformations was actually a gap in foundational knowledge and a reminder that our diagnostic instincts matter as much as our intervention strategies.
Playing the Long Game
Vince had my attention the first week of school. He was enrolled in a class designed for students who had traditionally struggled in mathematics, yet he loved a challenge. He loved to think. He was happy to move ahead of the class, at his own pace, pushing himself further. At first, Vince thought he had to appease me by getting things done quickly. He would devour a task, only to feel defeated when I pointed out mistakes, even when I celebrated what he’d gotten right. Once I told him that I was more interested in bringing out his best, he became amenable to sitting with tasks longer. He embraced initial missteps, often trying to see how far he could progress in a given class period. When we tackled the Normal Distribution, he spent nearly two class periods fixated on applying the z-score to a probability, refusing to move on until he understood why his approach was not working.
Over time, Vince began to build the stamina to sustain excellent work over multiple class periods. He developed a disposition to mathematics, interested not just in getting the right answer, but in understanding the underpinnings of each concept. Toward the spring, I pulled Vince aside, saying “This is what it feels like to develop a mathematical identity. I don’t know if you have any interest in pursuing a career that involves mathematics or science, but you should be able to see now that you are capable!”
Vince’s reply caught me off guard. He said, “Actually, I’ve been meaning to ask you. My counselor was asking me what math I should take next year. Do you think I can do Precalculus?”
“Do you?” I replied.
“Well, I really like this class, and I love the challenge. I’m just nervous about whether it would be too much.”
“Vince, if it were too much, it would have been too much in this class. You’re ready.”
The smile on Vince’s face said it all. This is how productive struggle can have a positive impact on a student. Vince not only took Precalculus but thrived in it, proving that productive struggle doesn’t just build math skills, it builds the learners we hope all our students can become.
How Will You Respond?
Throughout this post, we’ve explored how productive struggle transforms students, from Stephanie’s breakthrough with polynomials to Vince’s journey toward mathematical identity. We’ve examined the frameworks, strategies, and pitfalls that determine whether struggle becomes frustration or growth.
The philosopher and psychologist William James once wrote: “Beyond the very extreme of fatigue and distress, we may find amounts of ease and power we never dreamed ourselves to own; sources of strength never taxed at all because we never push through the obstruction.”
This is what we’re really after when we talk about productive struggle in mathematics. Not suffering for suffering’s sake, but creating the conditions where students like Stephanie, like Vince, like my daughter, working on transformations, can discover capabilities they didn’t know they had. When we honor struggle without letting it become despair, we’re not just teaching math. We’re helping students find “sources of strength never taxed at all.”
Nearly 30 years ago, I became an amateur triathlete. Whether swimming laps, biking miles from home, or pounding the pavement, I’m constantly receiving feedback from my body, especially after a poor training session. That feedback is littered with invaluable information about my performance and the adjustments I need to make. Sound familiar? I often reflect by asking similar questions to those I would ask in my teaching practice: Where is there a need for harder effort? How am I responding when I seem stuck? Can I push just a bit longer than planned?
As teachers, we face our own productive struggles: learning new strategies, shifting our instincts, resisting the urge to rescue. We don’t have to worry about perfection. We just need to take the first step. This week, begin with reflection. Reflect on your own practice by asking:
- Where in my lesson plans is there space for struggle?
- How do I currently respond when students are stuck?
- What would it look like to wait 30 seconds longer before intervening?
If you already know the answers to these questions, then start with one strategy. Try it. Put it into practice.
Productive struggle is not about making mathematics harder. It is about making learning deeper and more durable. When teachers intentionally create space for thinking, resist the urge to rescue too quickly, and respond thoughtfully to student confusion, they help students build more than skills. They help students build confidence, resilience, and identity.
Explore more insights from Paul Battaglia and the authors of Math & YOU on building confident, capable mathematical thinkers.
