Productive Struggle in Math

When differentiation was part of my classroom instruction, creating a learning environment with clear expectations was very helpful. One expectation I held for myself was creating opportunities for students to practice "productive struggle" in math. While the level of scaffolding varied based on readiness, I made sure not to take away the challenge or integrity of a math task. Productive struggle wasn’t about not helping or leaving students to figure it out on their own. It was about helping them develop the ability to persevere, even when the math became difficult.

What is Productive Struggle? 

Productive struggle occurs when students encounter a learning challenge and persevere. It is the sweet spot of learning where students are challenged without becoming too frustrated and giving up. It involves experimenting with different strategies when an answer isn't quickly found or a solution isn't obvious. It requires students to understand that mistakes are part of the learning process and contribute to the footprints of thinking they leave when solving a problem. Rather than strictly relying on memorized procedures and algorithms, prior knowledge is applied to solve new problems and foster deeper mathematical understanding. 

This is what I wanted my students to have and experience. And over time, I saw productive struggle contribute to student-centered learning, strengthening my students' critical thinking and problem-solving skills.

How is Productive Struggle Tied to Math Stamina?

Math stamina plays a role in successfully navigating productive struggle. Let's face it, when we enter the struggle zone, it is human nature to want to give up. Instead of letting my students do that, I chose to teach them about the power of perseverance. In my classroom, I encouraged my students to see themselves as mathematicians who could push through challenges rather than give up. 

Building math stamina was part of the foundation. Students had to engage with math concepts, concentrate on the problem, and apply their knowledge when faced with productive struggle. To help them do this, I had to ensure they developed the math stamina to tackle and persist with challenging tasks. I wouldn't expect the teacher next door to run a marathon without training, and I wouldn't expect my students to have that math stamina without practice.

We worked on building math stamina together. I helped my students strengthen their math stamina by intentionally giving them tasks that helped to grow their willingness to grapple with math and persevere. It could be as simple as adding an extra word problem to extend their focus a few minutes longer, making the numbers a bit messier to work with, or providing open-ended tasks with multiple entry points to engage all learners. By being strategic about helping my students build stamina, I could see their progress unfold throughout the year. 

The goal was to help them maintain an "I Can" attitude and overcome hurdles when their understanding broke down. It was about taking a step back so students could tap into what they did know to help them figure out what they didn't know. When my students engaged in productive struggle, they strengthened their problem-solving and critical thinking skills and gained confidence in their mathematical reasoning abilities.

What Does Productive Struggle Look Like in the Classroom?

I encouraged productive struggle by giving my students non-routine problems with multiple entry points and more than one possible solution. Instead of looking for quick answers, my students learned to focus on how they approached a problem and the reasoning behind their choices. The goal was to help them see that grappling with math was just as important as getting the correct answer.

One of my favorite strategies to foster productive struggle was "Here's the Answer...What's the Question?" In this activity, I gave my students an answer and asked them to generate many varied responses to one answer. Students could see different approaches to completing the same task by displaying their responses. This strategy encouraged flexible thinking and reinforced the idea that there were many ways to think about numbers.

Closer Look at Here's the Answer...What's the Question?

My What’s the Question? Fractions Activity was one of my favorite ways to encourage deeper mathematical thinking. Instead of simply solving a problem, my students had to work backward from a given answer and generate as many possible questions as possible. This approach helped them build number sense, strengthen their reasoning skills, and develop flexibility in how they thought about math - in this case, fractions.

I introduced the activity by giving them a fraction as an answer. Then, I asked them to create different math problems that could result in that answer. Some of my students created real-world word problems, like figuring out how much of a pizza had been eaten, while others focused on numerical equations using division or multiplication. Seeing how creative they could get was exciting, especially when they started incorporating mixed numbers, equivalent fractions, and comparisons.

Sharing and discussing their work was where the real magic happened. As my students explained their thought processes, they began seeing how the same fraction could be represented in many ways. Their classmates often had completely different interpretations, which led to discussions about problem-solving strategies and mathematical reasoning. To keep the momentum going, I sometimes had students swap cards and create new problems for a different fraction or analyze their classmates' work to determine if it led to the given answer correctly.

This activity was an example of productive struggle in action. It wasn’t just about getting one right answer. It was about thinking critically and exploring multiple possibilities. By seeing themselves as problem solvers and embracing the process, students built the math stamina they needed to generate multiple questions that stretched their thinking.

Converting and Comparing Lengths Task Cards

My Convert and Compare Lengths Task Cards helped my students engage in productive struggle as they navigated the unfamiliarity of measurement. With its varied units and abstract conversions, measurement naturally offers opportunities for deeper thinking and productive struggle. Rather than simply memorizing conversion formulas that can be easily forgotten, the task cards required my students to grapple with various measurements and explore the relationships between different units.

Each task card presented a problem requiring my students to convert a given measurement or compare two different lengths. Some cards asked them to determine how many smaller units fit into a larger one, while others challenged them to decide which measurement was greater. The variety of questions ensured that my students had multiple opportunities to apply their knowledge in different ways to reinforce their understanding.

One of the best parts of using these task cards was the flexibility they provided. Sometimes, my students worked independently. They would use a dry-erase marker on laminated cards to check their own thinking before moving on. Other times, they worked in pairs, discussing their strategies and justifying their answers to a partner. When I wanted to add movement to the lesson, I turned the task cards into a scavenger hunt around the classroom, where they solved problems and recorded their responses on clipboards.

Working through these measurement problems, my students strengthened their math stamina. Instead of giving up when faced with unfamiliar units or forgotten conversions, they were willing to grapple with the math and reason through the problems. My students tapped into what they already knew and used that knowledge to make connections. They tested different strategies based on the context of the problems and used logic to determine if their conversions made sense. Seeing their confidence grow as they tackled these problems highlighted the power of perseverance.

Encouraging Productive Struggle With Open-Ended Scenarios

Using open-ended scenarios was another way I  encouraged productive struggle. These scenarios contained important mathematical information and data but did not provide the questions for students to answer. One example was the "beach balls in the warehouse" scenario. Instead of giving my students a ready-made problem to solve, I provided the data and challenged them to determine what information was important and how it could be used.

Students could approach this task in two different ways. Some chose the creative route, developing their own questions based on the information given. This required them to analyze the data, think critically, and ensure their question made sense mathematically. Others worked with the Four Question Task Cards, where they had to sift through the information and identify the relevant details to answer their assigned questions. Students could then share their thinking and notice the similarities and differences in the data required to answer each question. Both approaches reinforced the idea that math is about determining importance, making sense of numbers, and understanding how data connects to real-world situations.

By engaging in this process, my students developed their math stamina while strengthening their ability to independently work through a mathematical task. Rather than relying on me to guide them step by step, they learned to ask themselves, "What information do I need?" "How does this data connect?" "What strategies will help me find the answer?" These small shifts in thinking made a significant impact, helping my students build the confidence to navigate more complex problems on their own.

Inspire Productive Struggle

Helping my students to see themselves as mathematicians opened the door to their willingness to build stamina and work through productive struggle. They were more likely to persevere through challenges rather than give up at the first sign of difficulty. To reinforce this mindset, I displayed mathematician posters as a visual reminder in my classroom.

These posters were daily reminders that mathematicians don’t always get the correct answer on the first try. Sometimes math gets messy, and it may take multiple strategies before finding a solution. Each poster highlighted different aspects of mathematical thinking, from making mistakes and learning from them to using multiple strategies and explaining reasoning. When my students struggled with a concept, I referred back to the posters to remind them that challenges were a normal part of the learning process.

Over time, I noticed that my students started to internalize these ideas. Instead of feeling defeated when they hit a roadblock, they approached problems with more persistence. The language from the posters even made its way into our classroom discussions. My students would remind each other that mathematicians take risks, learn from errors, and keep trying. This simple shift in mindset helped transform the way they tackled math. This made them more confident and resilient learners.

Building Math Stamina Through Productive Struggle

Encouraging productive struggle in the classroom helped my students develop the confidence and perseverance to tackle challenging math problems. Using these activities and resources taught my students to think critically, explore multiple strategies, and embrace mistakes as part of the learning process. These tools reinforced the idea that mathematicians don’t give up. They push through challenges and strengthen their math stamina along the way. Seeing themselves as mathematicians, my students were more willing to engage deeply in math and accept productive struggle as a normal part of the learning process. 

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The Importance of Teaching Math Vocabulary

Math has its own language, and if our students don’t know the words, they will struggle with the concepts. When I was in the classroom, I knew I had to be intentional about teaching math vocabulary because it’s not something our students pick up naturally in everyday conversations. In fact, if we heard them using math terms walking down the hallway, we'd probably take a quick double-take! Keep reading to learn more about the importance of teaching math vocabulary. 

Explicit math vocabulary instruction is key because "math words" are not commonly used in day-to-day conversation. When was the last time you heard the word 'sum' or 'vertex' used outside of a math lesson? If we don’t give our students multiple opportunities to see, hear, and use math terms in context, their comprehension will take a hit.

Teaching Math Vocabulary Is More Than Copying Definitions

We’ve all seen the traditional approach: copy a definition and move on. That doesn’t always work for all of our learners. As a teacher, you've likely heard of Dr. Marzano and his theory of lesson development, which helps teachers improve their instruction. Well, this educational leader didn't stop there. He also offers valuable insights into effective vocabulary instruction. He stresses that vocabulary instruction has to be engaging, thought-provoking, and interactive. Math vocabulary doesn't have to feel dry or intimidating. When we make vocabulary playful and hands-on, our students actually retain it!

Marzano outlines eight research-backed strategies for effective vocabulary instruction. The ones below stood out to me the most as being perfect for math vocabulary:

• Don’t rely on definitions alone because our students need to interact with words.
• Use a mix of linguistic (verbal explanations) and nonlinguistic (visuals, gestures, or models) representations.
• Give our students multiple exposures to words over time.
• Break words down into meaningful parts (like milli- and centi-).
• Teach words in context. Math concepts make more sense when vocabulary is tied to real examples.
• Let our students talk about words in authentic ways.
• Incorporate games and wordplay. This approach helps the information stick!
• Focus on words that will actually help our students understand and apply math concepts.

Choosing the Right Math Vocabulary to Teach

There are so many math terms, and teaching them all is impossible. That’s why I focused on words that showed up in the standards. My students needed to know these words to build their math skills. When they understood the vocabulary, they could engage in math discussions, follow multi-step problems, and feel more confident in their reasoning.

I also found that my students needed repeated exposure in different settings. Whether it was math discussions, journaling, or even encouraging parents to use math terms at home, the more they heard and used the words, the better they understood them.

Make Math Vocabulary Meaningful Through These Strategies

Moving Beyond the Word Wall

For years, word walls were a staple in classrooms. As our understanding of literacy instruction has evolved, especially with the Science of Reading, we’ve moved toward more research-backed approaches, like sound walls for phonics instruction. While math vocabulary doesn’t follow phonetic patterns in the same way, we still need a system that helps our students connect words to meaning rather than just memorizing terms on a wall.

That’s why I took a different approach. Instead of treating math vocabulary as a static display, I made it a tool that evolved with our learning. The key was ensuring the words were directly tied to our learning content. I did not want just a list of terms they might never reference again.

I focused on relevance over quantity. I introduced words as they naturally came up in lessons and ensured they were words my students needed to understand the math. If a word wasn’t meaningful in context, it wasn't going to stick.

I also reinforced vocabulary in different manners, just like we do in our literacy instruction. Rather than relying on a list of words on the wall, we built understanding through discussion, writing, and visual representations. I encouraged my students to explain words in their own language, draw models, and use gestures to show what a term meant. The more ways they engaged with a word, the deeper their understanding became.

The goal was never to memorize words. It was to give my students a toolbox of academic vocabulary they could pull from when solving problems, explaining their reasoning, and making sense of math concepts. This shift made math vocabulary more accessible and valuable, which is exactly what I wanted!

Keep Vocabulary Visible and Interactive

I wanted my students to see and use math vocabulary regularly. Instead of having a traditional word wall that just took up space, I turned it into an evolving resource. It was a tool we actually used. The goal was to make math vocabulary something we used, not something that just existed on the wall.

I kept the vocabulary relevant to our learning, constantly adding and revisiting terms. When new words came up in math discussions, we’d pause to define them, give examples, and then add them to our vocabulary display. During small groups or math huddles, I’d refer to these words. I encouraged my students to do the same when explaining their thinking. It became a habit for them to look at the words as they worked through problems, reinforcing their understanding.

Use Graphic Organizers to Build Connections

Graphic organizers were a game-changer in helping my students make sense of math vocabulary. They gave my students a structured way to break down words and connect them to what they were learning. Instead of just memorizing definitions, they could visually see how a term related to different concepts.

One of my favorite ways to use graphic organizers was with Frayer models. I had students define a word in their own terms, draw a representation, list examples, and even write out non-examples. This really helped them develop a deeper understanding instead of just memorizing.

I also found that graphic organizers worked well when comparing terms. When we learned about area and perimeter, using a Venn diagram helped my students see the similarities and differences between the two. Over time, they became more confident in recognizing when to apply each concept.

I quickly learned that modeling was key. I always walked through a new organizer first, thinking out loud as I filled it in before gradually handing the responsibility over to them. This gradual release helped build their confidence. Before long, they were using the organizers independently as a reference tool.

Bring in Games and Word Play

Nothing gets our students more excited about vocabulary than games! Math words can feel abstract, but when students play with the vocabulary, they absorb it in a whole new way.

One of my favorite activities was using vocabulary task cards for quick, engaging practice. I’d print and cut out different word-based challenges, put them on a ring, and use them during transitions or as a warm-up. Sometimes, we’d pull a card with a math word. My students would then need to use it in a sentence related to what we were learning. Other times, they had to act it out or give a real-world example.

Another simple but effective strategy was math vocabulary charades. I’d write different math terms on slips of paper. Then, my students had to act them out while their classmates guessed the word. Seeing my students try to act out words like parallel or acute angle was always a highlight. It reinforced their understanding in a way that was fun and not forced. In the end, making math more approachable. 

I also incorporated math vocabulary games into small groups for my students who needed extra reinforcement. We’d do quick rounds of matching games or drawing clues. These quick, low-pressure activities helped build their confidence. They made vocabulary practice something they actually looked forward to.

Helping Students Build a Strong Math Vocabulary

Teaching math vocabulary isn’t about memorizing words. It’s about giving our students the tools to confidently understand and use math concepts. When vocabulary instruction is interactive, meaningful, and connected to authentic learning, it helps students think more critically and communicate their reasoning. Instead of relying on static word walls, we can create dynamic vocabulary experiences that grow alongside students’ learning. 

Focusing on relevant terms, encouraging discussion, and reinforcing words through multiple exposures, we help our students build a strong foundation in math language. When our students see math vocabulary as a tool rather than a hurdle, they become more confident problem solvers who can clearly explain their thinking.

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Alternate Assessment Examples for the Math Classroom

During my time in the classroom, one thing that transformed how I approached math instruction was shifting the focus of assessments. Traditional testing often felt rigid, leaving little room for my students to showcase their creativity or growth. That’s why I leaned into alternate assessment examples to create a more dynamic and student-centered approach to evaluating understanding.

This concept, inspired by the work of Jo Boaler in Mathematical Mindsets, emphasizes the power of assessments that celebrate learning and encourage growth. Today, we will look at what alternate assessments are, why they’re so effective, and how you can use them to empower learners in your classroom.

What Are Alternate Assessment Examples

Alternate assessments focus on the learning process rather than just the final product. Instead of simply marking answers as right or wrong, these assessments highlight student thinking, problem-solving strategies, and areas for improvement. For me, this approach meant using tools like self-assessments, reflection prompts, and exit tickets to better understand where my students were in their learning journeys.

These types of assessments allow our students to reflect on their mathematical thinking in a way that encourages ownership and growth. When my students considered questions like "What was the big idea we worked on today?" or "In what situations could I use the knowledge I learned today?" they started to see math as a set of skills they could apply in real-life scenarios. That kind of awareness doesn’t come from a simple quiz or test!

Why Are Alternate Assessment Examples Effective?

The power of alternate assessment examples lies in their ability to focus on learning rather than achievement alone. Traditional grading often paints an incomplete picture of a student's abilities. Alternate assessments provide actionable feedback and celebrate progress. This approach also helps identify where our students need support or where a misconception might exist.

One tool I relied on was exit tickets. At the end of a lesson, my students would complete a quick reflection on the key target understanding. They’d then drop their ticket into a red, yellow, or green basket to indicate their confidence level. This simple system helped me gauge not only their understanding but also their mindset. For example, if a student placed a ticket in the red basket despite demonstrating understanding, it signaled that they needed more encouragement and confidence-building.

Using Math Exit Tickets/Self-Reflection in Your Classroom

My Math Exit Tickets/Self-Reflection resource is a tool for helping your students engage in meaningful reflection while giving actionable insights into their learning. This resource includes a variety of prompts that guide your students to think critically about their progress, such as "What did I learn today?" or "What questions do I still have?" By encouraging this type of reflection, your students develop ownership of their learning and build a stronger connection to the material.

You can use these exit tickets as a way to assess understanding and as an opportunity to encourage a growth mindset. The flexibility of the resource allows you to bring these prompts to where they fit best in your classroom routines. Prompts can align with specific lessons or be used as a check-in at different points in a unit. These reflections help track progress and provide documentation for goal setting and parent-teacher conferences.

I saw the benefits of this approach and encourage you to try it in your classroom. Fill in the form below to receive 10 free Math Exit Tickets.  These exit tickets are perfect for engaging your students in meaningful reflection while giving you valuable insight into their progress.

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Shifting the Focus to Feedback With Alternative Assessment Examples

One of the biggest takeaways I’ve learned from using alternate assessment examples is the importance of feedback. Moving beyond traditional grading, I could focus on how my students grew as mathematicians. Grading became less about correctness and more about their strategies, the questions they asked, and how they justified their thinking.

Assessments should be about learning. Within the parameters set by district guidelines, there’s always room to prioritize constructive feedback over simple scores. This shift helps our students develop a positive relationship with math but also equips them with tools for success beyond the classroom.

The Value of Alternate Assessment Examples

It’s clear that alternate assessment examples were a game-changer for my teaching and students’ learning. These approaches encouraged growth, creativity, and math talk, moving beyond grades to focus on learning. Using tools like exit tickets and self-reflection prompts, I could meet my students where they were and help them take ownership of their progress.

If you’re ready to transform how you approach assessments, start by exploring alternate assessment strategies. Don’t forget to grab your free Math Exit Ticket (above) to kickstart the journey toward more meaningful and reflective assessments!

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