### You’ve likely seen it a hundred times. A student is working diligently on a math problem. They’re leaning in, scribbling possibilities frantically. They pause to look up toward the ceiling for a moment. Then back to their paper with renewed gusto.

### And then it happens. The pencil is forcefully dropped on the desk. The student leans back, sighs, and their head drops in defeat. Their productive struggle has turned UN-productive.

**The Research**

**What is Productive Struggle? **

Productive struggle is often referred to in math as struggle that advances student thinking and deepens their mathematical understanding (Warshauer, H. K. (2015). It takes time. And it requires students to be moving toward a resolution (Hiebert & Grouws, 2007). Without that movement, students can slip into unproductive struggle.

Unproductive struggle is dangerous for student’s experience with mathematics. Why? The answer lies in understanding what is happening beneath the surface: in your student’s brain.

**Your Brain on Productive Struggle**

Cognitive scientists would rebrand “productive struggle” as being in a “positive neurological state”. A state is a mind-body response that often involves the interplay of emotions, thoughts, and bodily feelings (Oosterwijk et al., 2012). Examples of states include frustration, confidence, cynicism, and optimism.

What determines whether a state is a “positive state”? Positive states trigger a chain reaction in the brain that enhances learning. For example, a student in a state of curiosity will experience a burst of dopamine release (Lloyd et al., 2015). Dopamine is the neurotransmitter most associated with motivation. It also increases activity in the hippocampus, a critical brain region for encoding new memories (Gruber et al., 2012).

Similar, but unique, neurobiological processes occur for other positive (or productive) states. They include anticipation, curiosity, confidence, and confusion. (More on why confusion is productive in a moment.)

__Signs of Productive Struggle__

Learn to recognize the signs of students in productive struggle. When you see it, smile, and let them continue.

*A pattern of fast movement combined with stretches of deep reflection (confident)**What if I changed this? I wonder what might happen then? (curiosity)**I think I’m getting close (hopeful)*

**Productive struggle is only productive if accompanied by a ****brain-state that supports student learning.**

**Your Brain on Unproductive Struggle**

Negative states have the opposite effect on the brain and learning. They trigger a counterproductive effect on learning in the brain. For example, a student in a state of fear or anxiety will experience a surge of cortisol. This stress-related hormone can impair (or halt) cognitive function (McGarry et al., 2016). Other negative states for learning include frustration, hopelessness, boredom, and anxiety.

__Signs of Unproductive Struggle__

Learn to recognize the signs of a student in unproductive struggle. Then keep reading to learn powerful interventions to support your students.

*Sighs; moans (frustration)**Leaning back in chair; shoulders slumped (despair)**I can’t do this; This is too hard. (hopelessness)*

**Why it’s Hard to Stay in Productive Struggle?**

States last for seconds, maybe minutes. And then the brain shifts into a different state. You feel rage when someone cuts you off on the freeway. Soon your rage downshifts to anger, then annoyance, and hopefully back to calm or perhaps even empathy. Similarly, students shift in and out of states throughout their math class (and beyond). Being confused is a positive state for learning. The brain seeks out meaning and resolutions to the unknown. When some degree of resolution is found, confusion can shift to confidence. Without any progress toward a resolution, confusion can quickly shift into frustration – a negative state for learning.

**3 Reasons Students Slip into Unproductive Struggle**

There are many reasons why a student may slip from a productive state to an unproductive state. Let’s take a look at three of the most prevalent in mathematics.

**Student Self-Efficacy**

Self-efficacy is a student’s belief in their capabilities (Bandura, 1989). A student who believes they are capable of attaining progress will be in a more positive state. Thus, better equipped to engage in productive struggle. Students with low levels of self-efficacy are more prone to slip into states of hopelessness or despair. These are not helpful states when engaging in challenging mathematical thinking.

**Prior Knowledge**

Students who lack sufficient prior knowledge will more easily shift into an unproductive state for learning. Mastery of prior knowledge shapes a person’s present self-efficacy. How? We rely on our past success and mastery of tasks to determine our current efficacy (Lin et al., 2018). A student with limited fact fluency, for example, will engage in less productive struggle with algebraic thinking.

**Excessive Cognitive Demand (too challenging)**

Challenging tasks can be highly motivating to students (Baumeister, 2016). However, too much challenge can lead students to slip into negatives states such as overwhelm, anxiety, and despair. The optimal level of engagement and motivation (often called *flow*) is found in the balance of challenge and skill level (Csikszentmihalyi et al., 2014).

Students are constantly sliding along the continuum of brain states. Each one impacts the students’ ability to engage in mathematical thinking. Highly effective teachers help guide students toward positive states that facilitate productive struggle.

**Practical Application**

From the three common roots of unproductive struggle above, here are tools to build student capacity for productive struggle.

**Foster Greater Math Confidence and Identity**

The goal here is to build greater student efficacy. All strategies below are designed to increase the frequency of dopamine release in the brain. This will nudge students toward more positive states for learning.

**Affirmations:**Start your math class with a class-wide affirmation or creed. “I am a math master. I have the power to think critically and improve every day. I take my time to notice patterns and wonder about the beauty of mathematics. I am a math master.” Positive self-talk is a well-researched tool to boost effort and achievement (Tod et al., 2011).**Develop language patterns of productive struggle**: When a student says, “I don’t get it” help them rephrase it as “I don’t get it YET.” The addition of yet helps students avoid slipping into hopelessness. It also implies forward movement as they engage in the struggle (Warshauer, 2015).

**Meet Them Where They Are**

Here are two tools (one proactive; one reactive) to support student’s need for sufficient prior knowledge.

**Get in the know:**The appropriate level of struggle for a math learner is related to their depth of prior knowledge. Develop teaching habits to pre-assess students so you enter each unit aware of students’ knowledge base. Brain dumps are one way to gather data. Before beginning a new unit, have students “dump” everything they know about the topic on paper. This type of retrieval exercise can provide valuable information when planning for productive struggle (Karpicke, 2012).**Use analogies and metaphors:**When a student is slipping into an unproductive state for learning, analogies and metaphors can help them bounce back into a positive state for learning. How? Connecting the current learning task to something the student is already familiar with reduces the cognitive pressure by utilizing other neural networks (Richland et al., 2004). Bonus brain points for analogies that connect to something of interest to your students.

**Engage in More Social Learning**

It can be tempting to lower the rigor of mathematical thinking when students slip into an unproductive state. Before making any modifications, consider boosting options for social learning.

**More collaborative learning:**This is a strategy to disperse the cognitive pressure amongst group members (Paas, & Sweller, 2011). Repeatedly ask yourself, “Is this something that students MUST complete independently? Or could they combine their mathematical thinking with a peer?”**Create systems of social support:**When students sense their struggle is slipping from productive to unproductive, do they know the class-wide protocol? Perhaps they first ask a neighbor, then reference their notes, and then approach the teacher. Teaching students to self-regulate through different experiences of mathematical struggle is a life-long skill.

The roots of productive mathematical struggle lie in students maintaining positive brain states. These tools can help your students maintain momentum as they engage in productive struggle – an important element of teaching math with the brain in mind.