Game Play, Math, Math Discourse, Problem Solving, Teaching and Learning
Serious Play, Part 2
We’ve all heard that “Every teacher is a reading teacher.” However, since learning is a social endeavor, “Every teacher is a communication teacher” makes more sense to me. Explicitly teaching children listening and speaking skills in the context of ones’ own discipline promotes the interactions that facilitate learning. In math class, math games present the perfect scenario for teaching children to talk math.
Below, I have listed four ways in which the speaking and listening standards interact with the standards for mathematical practice and connect to game play. For consistency, I’ve used in the previous post that referenced multiple variations of War as the context for each.
1) Communicate precisely – complete sentences.
I once had a middle school teacher say to me, “If you teach kids to speak in complete sentences, they’ll more easily learn to write in complete sentences.” That really spoke to me! I used that principle in oh-so-many ways. In the context of game play, I made sure that students communicated their thinking in complete sentences in each round.
For example, when students played Addition War, I asked them to use a comparison statement before taking the cards. It sounded something like this: “14 is greater than 8.” By listening to my students as they practiced their skills, I could learn more about their thinking and precision.
2) Communicate precisely – vocabulary.
By asking students to verbalize thinking while playing math games, I ensured that they also rehearsed using mathematical vocabulary. Math-specific vocabulary is rarely used outside of math class. So asking students to integrate these words into their game play increases their usage.
Listening carefully to students’ conversations and use of mathematical vocabulary pays off. When my students were playing Double War (addition, subtraction, or multiplication), I heard Travis proudly state, “14 is gooder than 8.” He was so confident and excited that he had won the round. His statement, however, revealed something I had not anticipated. He had interpreted the mathematical term “greater than” as “better than” rather than “more than.” Because students were required to verbally state their comparisons, I detected Travis’s misconception. I then wove this idea into my instruction just a few minutes later.
3) Construct viable arguments.
Another way to embed language use into the classroom includes teaching students a simple process for explaining their solutions. My favorite structure comes from a writing structure I used: “First – Next – Then – Last.” Students create a three- to four-sentence explanation that may sound something like the following.
“First, I wondered how many more I needed to add to 8 to make a 10 (it was 2). Next, I took 2 from the 6 and added it to the 8 to make a 10. Then, I noticed that there were 4 left from the 6. Last, I added the 10 and the 4 to get 14.”
By giving students a sentence frame or a language structure to use when playing math games, you increase the likelihood that their learning will be deeper and longer-lasting.
4) Critiquing the reasoning of others.
For children to learn to be good listeners , we must teach them what good listening looks and “sounds” like. In my classroom, we talk about looking at the speaker without fidgeting, about the inner “listener dialogue.” We learn how to respond when you don’t agree. We talk about what it means to be a respectful listener. We discuss that deep listening is more than just being quiet. It’s about thinking about what you’re hearing and asking questions inside your head. And, most importantly, it’s about listening to learn and to understand. A reply should not be formulated until you’ve listened and processed what you heard.
Stephen R. Covey said, “Most people do not listen with the intent to understand; they listen with the intent to reply.” Learning to listen serves our children well, and math games afford them time to practice. As Bryan H. McGill said, “One of the most sincere forms of respect is actually listening to what another has to say.” Wouldn’t our world be a wonderful place if all the adults learned to construct arguments worthy of being listened to, and then, in turn, listened with the intention to understand and to learn?
For now, I’m just happy if my students use their emerging speaking and listening skills to make math game-play richer in my classroom.
How about you? What do you think about this idea that speaking and listening can be taught in math class? Please share your thoughts in the comments box below.
Kimberly Rimbey, Ph.D., works with teachers and leaders to develop system-wide change in mathematics teaching and learning.
Hands-On Math, Problem Solving, Seeing The Math You Teach, Teaching and Learning
Have You Ever Really Seen the Math?
Did the way you learned math suit you? Were the concepts connected, or were they more about following steps and rules without really understanding why they worked?
For me, dividing fractions was always a mystery. I could recite the steps: flip the second fraction and multiply. But I had no clue why it worked. And why, when I divided two fractions, did the answer sometimes get larger? Isn’t dividing supposed to make things smaller?
If you’ve ever felt this way, you’re not alone. Many of us were taught to do math, but not necessarily to see it.
And that brings me to Ted.
Have You Met Ted?
I met Ted a couple years ago at a workshop I led in his district. Ted had been teaching fourth grade for 15 years. He was confident, experienced, and deeply committed to his students. But recently, changes to his state’s standards had shaken his confidence.
One shift in particular? Multi-digit multiplication.
Instead of simply teaching students to multiply using the standard algorithm, the new standards asked Ted to teach multiplication “using strategies based on place value, properties of operations, and the relationship among operations.”
Ted was puzzled. How do you teach strategies you’ve never seen? And even if your curriculum includes those strategies, how do you use them effectively if they weren’t part of how you learned math?
This led Ted to a moment of self-reflection: his students could follow steps, sure. But did they really understand what those steps meant?
Seeing the Math You Teach
Rather than jumping into “show and tell” mode, I invited Ted and his colleagues to engage with a two-page spread from a visual math resource I was working on with my co-authors – Seeing the Math You Teach. We explored multiple strategies for multi-digit multiplication. No algorithms. No formulas. Just visuals and reasoning.
Rimbey, K., Basham, K, Berda, C. (2025). Seeing the Math You Teach. Corwin: Thousand Oaks, CA.
Take a moment to imagine what Ted saw on these two pages. Think: base-ten blocks, area models, number lines, and symbolic representations all working together to build meaning.
What do you notice? What are you curious about?
As you examine visuals like these, you begin to uncover the connections:
- Place value models that show how tens and hundreds grow
- Arrays and area diagrams that highlight the structure of multiplication
- Number bonds that reveal flexible thinking about numbers and their relationships
- Symbols that begin to make sense after the concepts take shape visually
And just like that, the math starts to come alive.
Beyond Elementary: A Universal Invitation
While the examples in this post come from elementary classrooms, the ideas are universal. Whether you’re helping first graders explore base ten or guiding high schoolers through slope-intercept form, visual models support conceptual understanding.
They help students connect the why behind the how. They turn math from mystery into meaning.
So let’s explore together.
Visual Strategies in Action
Let’s look at how some common tools and representations connect concepts across K–12.
- Number Bonds. Great for developing part-part-whole reasoning and flexible number sense. Later, they laid the groundwork for decomposing expressions and functions.
- Base-Ten Manipulatives. Perfect for understanding place value, regrouping, and decimal operations. With the right scaffolding, they even support Algebra Tiles!
- Area & Array Models. Help students visualize multiplication as repeated addition and structured grouping. Also helpful for algebraic expressions and factoring in secondary grades.
- Number Lines. Useful for addition, subtraction, fractions, decimals, and integer operations. Use Linking Cubes as physical number lines in early grades. Extend into coordinate graphing and slope visualization in later grades.
- Symbols. A powerful tool, especially when paired with visuals. They offer precision and efficiency, but only after meaning is established – whether you’re connecting quantities to numerals, building understanding of comparison symbols, or establishing ways to record exponential growth.
Whatever your grade level or content area, visual representations offer a bridge to understanding.
Seeing the Math You Teach – Part 1
By Kimberly Rimbey, Ph.D., NBCT
Have You Ever Really Seen the Math?
Did the way you learned math suit you? Were the concepts connected, or were they more about following steps and rules without really understanding why they worked?
For me, dividing fractions was always a mystery. I could recite the steps: flip the second fraction and multiply. But I had no clue why it worked. And why, when I divided two fractions, did the answer sometimes get larger? Isn’t dividing supposed to make things smaller?
If you’ve ever felt this way, you’re not alone. Many of us were taught to do math, but not necessarily to see it.
And that brings me to Ted.
Have You Met Ted?
I met Ted a couple years ago at a workshop I led in his district. Ted had been teaching fourth grade for 15 years. He was confident, experienced, and deeply committed to his students. But recently, changes to his state’s standards had shaken his confidence.
One shift in particular? Multi-digit multiplication.
Instead of simply teaching students to multiply using the standard algorithm, the new standards asked Ted to teach multiplication “using strategies based on place value, properties of operations, and the relationship among operations.”
Ted was puzzled. How do you teach strategies you’ve never seen? And even if your curriculum includes those strategies, how do you use them effectively if they weren’t part of how you learned math?
This led Ted to a moment of self-reflection: his students could follow steps, sure. But did they really understand what those steps meant?
Seeing the Math You Teach
Rather than jumping into “show and tell” mode, I invited Ted and his colleagues to engage with a two-page spread from a visual math resource I was working on with my co-authors – Seeing the Math You Teach. We explored multiple strategies for multi-digit multiplication. No algorithms. No formulas. Just visuals and reasoning.
Rimbey, K., Basham, K, Berda, C. (2025). Seeing the Math You Teach. Corwin: Thousand Oaks, CA.
Take a moment to imagine what Ted saw on these two pages. Think: base-ten blocks, area models, number lines, and symbolic representations all working together to build meaning.
What do you notice? What are you curious about?
As you examine visuals like these, you begin to uncover the connections:
- Place value models that show how tens and hundreds grow
- Arrays and area diagrams that highlight the structure of multiplication
- Number bonds that reveal flexible thinking about numbers and their relationships
- Symbols that begin to make sense after the concepts take shape visually
And just like that, the math starts to come alive.
Beyond Elementary: A Universal Invitation
While the examples in this post come from elementary classrooms, the ideas are universal. Whether you’re helping first graders explore base ten or guiding high schoolers through slope-intercept form, visual models support conceptual understanding.
They help students connect the why behind the how. They turn math from mystery into meaning.
So let’s explore together.
Visual Strategies in Action
Let’s look at how some common tools and representations connect concepts across K–12.
- Number Bonds. Great for developing part-part-whole reasoning and flexible number sense. Later, they laid the groundwork for decomposing expressions and functions.
- Base-Ten Manipulatives. Perfect for understanding place value, regrouping, and decimal operations. With the right scaffolding, they even support Algebra Tiles!
- Area & Array Models. Help students visualize multiplication as repeated addition and structured grouping. Also helpful for algebraic expressions and factoring in secondary grades.
- Number Lines. Useful for addition, subtraction, fractions, decimals, and integer operations. Use Linking Cubes as physical number lines in early grades. Extend into coordinate graphing and slope visualization in later grades.
- Symbols. A powerful tool, especially when paired with visuals. They offer precision and efficiency, but only after meaning is established – whether you’re connecting quantities to numerals, building understanding of comparison symbols, or establishing ways to record exponential growth.
Whatever your grade level or content area, visual representations offer a bridge to understanding.
Seeing = Understanding
Too often, in mathematics, we are in a rush to move to the abstract. In doing so, we overlook the visual.
-Peter Liljedahl
Seeing the math you teach means making ideas visible.
It means sketching. Moving things around. Building understanding.
And when we see it, our students can too.
Next Steps: What Could You See More Clearly?
Teachers: What’s one concept you teach that students struggle to grasp? Could a visual representation help uncover the why behind the steps?
Leaders: How might you use ideas like those above to spark the conversation with your teacher teams? How might you subtly use these questions to help your teachers see the math in new ways?
Everyone: The book is available now, both from Corwin and Amazon!
In the next post, we’ll explore ways to use Seeing the Math You Teach to support both teachers and students to really see the math.
Until then, keep seeing the math—and helping your students see it too.
Kimberly Rimbey, Ph.D., works with teachers and leaders to develop system-wide change in mathematics teaching and learning.
Hands-On Math, Problem Solving, Professional Learning, Seeing The Math You Teach, Teaching and Learning
When You See the Math, They Will See It Too
by Chryste Berda
If math has ever made you feel small, you’re not alone. Many of us have internalized the message, intentionally or not, that math is a fixed ability: you either have it or you don’t. But what if making math visible could be the very thing that transforms that feeling of smallness into strength?
In my classroom, we lived by just two rules:
- Math makes sense.
- Everyone will be good at math.
I picked up these rules at a math conference years ago, and they became my mantra from that point forward. But recently, a moment reminded me that these rules are deeply connected—and that one cannot exist without the other.
When Rule #2 Is Broken
I was screening number sense for a colleague when a 9-year-old student caught my attention. She got the problem 14-9 wrong and then sighed in frustration.
“No, that’s not right, but I don’t know why this is so hard for me! I’m just not good at math.”
In that one sentence, I heard it: Rule #2 had already been broken for her.
But here’s what I’ve learned: Everyone will be good at math only WHEN we teach it in ways that make sense. That’s Rule #1. And when we break Rule #1, we can’t keep Rule #2.
Making Math Visible Is the Key
Just the day before, I had been working with our district’s special education teaching team. The focus? Making math accessible by making it visible.
Making math visible doesn’t just mean using manipulatives or diagrams – it means illuminating the thinking, patterns, and connections behind the math. It means helping learners see what’s going on under the surface.
One teacher pulled me aside during that workshop. In a hushed voice, she said:
“I’m not a math person, but I want to be—for my students. The way you’re showing us how to make the math visible? I’m actually learning the math for myself!”
She followed up this comment by asking if we could continue to collaborate as she begins this with her students. You see, math becomes visible not just through clear explanations, but through shared exploration – with colleagues who remind us we’re not in this alone because in Rule #2, “everyone” doesn’t just mean all students, it literally means everyone – colleagues, parents, administrators, students, – everyone!
From Professional Learning to Classroom Confidence
This philosophy is at the heart of our book, Seeing the Math You Teach. We’ve seen how making math visible doesn’t just support student understanding–it builds teacher confidence, too.
At a recent 6th-grade professional learning community meeting, teachers tackled one of the most misunderstood standards: Mean Absolute Deviation (MAD). Before it appeared in standards, many educators had not even heard of it. And early resources? Practically nonexistent!
So the PLC team used page 195 from our book to understand MAD together. In those quiet moments of collaboration – around a whiteboard, over a problem, inside a question – something shifted. Confidence took root. You could feel the shift in the conversation as the process was illuminated and the math became visible—for them.
Rimbey, K., Basham, K, Berda, C. (2025). Seeing the Math You Teach. Corwin: Thousand Oaks, CA.
And when they brought that same resource into their classrooms, it became an anchor chart — a living tool for students. The color coding and step-by-step clarity in the resource helped illuminate the process behind MAD and made it visible to every student, much in the same way that it had become visible for the teachers. That’s the power of making math visible. It builds what I like to call mathematical swagger—for teachers and students.
Everyone Means Everyone
Let’s stop pretending there’s such a thing as a “math person.” Confidence in math doesn’t come from talent—it comes from clarity. From ideas being made visible. From sense-making being prioritized. And it doesn’t happen alone. It happens in community, through collaboration and exploration, with people who remind us we’re not in this alone. Because when you see the math, they will see it too.
Make Math Visible for Someone This Week
Take one small step this week to make math more visible—for a student, a colleague, or even yourself. Use a model. Ask a clarifying question. Share your own learning journey. Pull out that anchor chart one more time.
And if you’re ready to dig deeper, invite a colleague to co-plan with you or explore a page from Seeing the Math You Teach together. Learning is meant to be visible—and shared.
Math makes sense. And everyone will be good at math–because of you.
Chryste Berda - Chryste is energized by sharing her passion for learning with her colleagues as the district math coordinator and as a Regional VP for the Arizona Association of Teachers of Mathematics. She is intensely curious about students’ thinking and spends much of her time listening to students explain their ideas.
Hands-On Math, KP® Ten-Frame Tiles, Math, Problem Solving, Professional Learning, Teaching and Learning
Because we have been talking a lot about fluency in the past few blog posts, it suddenly occurred to me the other day that I can see a surprising connection between our efforts to help students develop fluency and my experiences becoming a successful high school sprinter. Let me explain.
When I was in high school, I loved running the quarter mile, and I made it to the state track meet every year. The mile relay was my absolute favorite event on the track – four young women working tirelessly to perfect our individual races in order to combine our skills to claim the final prize of the meet. We each played an important part in creating that final success:
- We worked on our form and rigorously built our individual skills (using the starting blocks, perfecting our just-right paces, engaging in distance running, sprinting shorter distances, etc.), all the while receiving immediate and continuous feedback from our coach.
- We honed and customized our individual races by putting together our own well-practiced skills.
- And then we took that last, important step — we combined our efforts into a longer, collaborative race, blending the traits of our individual races into a more complex and challenging final event.
So here’s the connection to math fluency…Just as my coaches helped me and my teammates achieve success through a prescribed sequence of skill-building activities, we, as teachers and coaches, can use the same strategies to help our students build the skills necessary to achieve fluency.
- I equate students’ development of number sense and math-fact fluency with the smaller, individual skills we sprinters developed (our sprints, form drills, and the like).
- Then, students’ fluency with number sense and math facts become the building blocks to fluency with multi-digit operations, just as we sprinters individually put together our running skills to design our full races.
- Eventually, operational fluency supports problem solving (the “more complex and challenging event” that is the goal of fluency) by enabling students to invest their thinking in the problem-solving process without getting stuck in the mire of operations.
So here we are, having developed number sense and math fact fluency with our students, laying the groundwork for them to perform more complex operations with multi-digit base-ten numbers (and fractions). Ensuring that students achieve operational fluency is critical…and how we get there can be a challenge. Below are the five steps I have found to be most important in helping students gain operational fluency.
- Continue to reinforce conceptual understanding. Using tools such as KP Ten-Frame Tiles to reinforce and describe what’s happening in the procedures will continue to support procedural understanding (Akin & Rimbey, 2017).
- Understand that speed and fluency are not synonymous. Fluency combines accuracy, efficiency, flexibility, and appropriateness. An over-emphasis on speed alone increases math anxiety (Boaler, 2018).
- Support fluency by giving relevant, immediate feedback. If you simply give students arithmetic worksheets with problems to practice over and over, they will not thrive because there is no opportunity for them to receive necessary and meaningful feedback.
- Build fluency in a fun and motivating atmosphere (Boaler, 2018). Provide opportunities for students to work on their skills through interactive games and online activities where meaningful feedback is possible. Something as simple as having students work side-by-side with models and with white boards, one “acting out” while the other records the operation, can make these practice sessions more meaningful.
- Make math fluency meaningful. All along the way, provide students with authentic opportunities to experience application of their math skills so they can see first-hand how fluency is an asset (Akin & Rimbey, 2017).
Had my mile-relay team neglected to rigorously build our individual skills, had we not had the opportunity to practice together toward a cohesive outcome, had we not focused on our common goal, we would not have been successful in seeing our vision become reality.
So it is with math fluency – students must actively build their competence with number sense and math fact fluency, put these skills together as they practice toward cohesive strategies for operations, and apply these skills as they successfully engage in deep and meaningful problem solving.
What are your thoughts about building math fluency for operations? What strategies have you found successful? Please share your thoughts in the comments box below.
Kimberly Rimbey, Ph.D., works with teachers and leaders to develop system-wide change in mathematics teaching and learning.
Game Play, Math, Problem Solving, Teaching and Learning
As the holidays approach, and with them an extended break between semesters, motivation wanes, not only among students but also within the teaching ranks. Classroom teaching can be a challenge – trying to keep students on-target with their learning goals while acknowledging the distracting festivities all around.
As a classroom teacher, I sometimes found myself frustrated by the ways in which my students got caught up in the holiday cheer. After a few years, I realized that it wasn’t only the students who were pulled away from the learning at hand; I suddenly realized that I was contributing to the tension between learning and celebrating. I was eager to get out the door to get more shopping done, to get ready for yet another party, to be released from the pressures of work so I could engage in more holiday play. I was contributing to the frenzied atmosphere in my classroom because I was distracted by my own desire to be on vacation. I was tired, overwhelmed, and a bit burned out — and my students knew it.
At that moment, I realized I needed to do two things: have fun with teaching during the holiday season and attend to some self-care.
Having Fun with Teaching and Learning
First, I made a list of the standards to be covered during the weeks between Thanksgiving Break and Winter Break. I looked at the tasks and problems included in my district-adopted textbook. And then I contemplated ways I could redesign the tasks to include holiday flair while meeting the objective.
- Shopping Task: Given that shopping is a big activity this time of year, I like to set students up with a math activity that involves buying gifts for others. I give them a budget, a list of people, and a time limit for purchasing gifts for each person. This activity may be as open-ended as letting them shop online or as limited as giving them a specific scenario such as this one from NCTM. You might even take your students on a field trip to the mall. When I did this, I made arrangements ahead of time with 5 stores, all of whom agreed to ring up the purchases and give my students mock receipts (no money changed hands). The students worked in groups to buy gifts for a family of 5. They had to plan ahead as there was a minimum for each family member in order to qualify (they couldn’t just buy a deck of cards for the last person because they had cut it too close). The team that came closest to $100 without going over “won.” Afterwards, each team received $25 (earned from a math-related fund raiser we did before Thanksgiving) to shop for a child on the Angel Tree at the mall.
- Coordinate Geometry: My 5th-graders are usually working on coordinate geometry around December, so I like to replace some of the textbook tasks with graphing tasks that result in creating holiday pictures on the coordinate grid. The students love this so much that they ended up creating their own coordinate grid puzzles for one another.
- Game Play: Designating a set period of time for game play provides a fun way to review concepts already covered during the year. I like to set out some marshmallows and chocolate chips for the kids to munch on as they play games that have become favorites throughout the semester such as One, Two, Switcheroo or Go Fish.
There are so many ways to connect holiday fun to the math standards you’re currently teaching. Give it a try…and let us know how it goes!
Attending to Self-Care
I’m saving this one for next week. In the meantime, take some time for yourself!!!
We wish you the best as you navigate these weeks between Thanksgiving and Christmas. Focus on making things fun for your students – and you’ll have more fun, too! And be sure to take care of yourself in the process.
We would love to hear from you. What are you going to do to make math fun for your students during the holidays?
Kimberly Rimbey, Ph.D., works with teachers and leaders to develop system-wide change in mathematics teaching and learning.
Problem Solving, Professional Learning, Teaching and Learning
Last week, I shared ways to shift homework practices to maximize student learning and understanding. This week, I turn my attention to grading practices. Just as with homework, when students receive their projects or tests back with grades written at the top, they simply look at the grade, make a judgement about whether or not they are good at math or if their teachers like them, and pretty much disregard any other comments on the paper.
We, as teachers, know that reflecting on work and learning from mistakes is where real learning takes place. However, even when we ask students to reflect on how they did, that score at the top of the page supersedes anything else on the page that could lead to a learning experience.
So…what can we do to push students to focus on the learning rather than on the grades? Here are a few ideas you might try…
- Grading Practices 1: Give students the opportunity to share everything they know. As teachers, we hold the power to choose which facts and details we deem most important to include in an assignment or test. However, our students have so much more knowledge to share than we have anticipated with our questions. Why not include a blank page at the end that asks students to tell us everything they know about the topic at hand that wasn’t asked? The example below is from a science class, and is easily transferrable to math class!
- Grading Practices 2: Highlight mistakes, but don’t give a grade. When grading an assignment or test, simply highlight the mistakes each student made. Do not assign point values, and do not put a score at the top of the page. At first, your students will ask, “What did I get?” or “How much is this one worth?” Resist, resist, resist! Do not go there.
- Grading Practices 3: Give students time to reflect upon and discuss mistakes. Rather than going over the assignment or test as a class, ask students to reflect on their on their own work. They may discuss and compare answers with their friends, but in the end, everyone must learn from his/her own mistakes if learning is to take place. As teacher Leah Alcala says, “Learning from mistakes is really what learning is.” Click here to see Leah and her students in action.
- Grading Practices 4: Share “my favorite mistakes” before passing out papers. After highlighting mistakes as described above, go through the papers again to look for trends and teachable opportunities. Select 3-5 of your favorite mistakes to discuss with the whole class prior to passing out their papers. This helps to glorify mistakes, communicating that mistakes are among the best learning opportunities we have.
I’m hoping that between last week’s homework post and this week’s grading post, you’re seeing ways in which teachers might shift their practice to maximize student learning without adding to the plethora of things they already do every single day. Simple tweaks in our practices can bring about tremendous shifts in student learning.
Please let us know how it goes by sharing your thoughts and stories in the comments box below.
Kimberly Rimbey, Ph.D., works with teachers and leaders to develop system-wide change in mathematics teaching and learning.
