In my previous post, I ran through some lessons I’ve learned about interactive read alouds and shared a few of my favorite books for K–1. And while many of those books can absolutely stretch up into grades 2–5, there are others that, because of their math content or overall reading complexity, are a better fit for this upper‑elementary grade band. So today, I’m sharing a set of book recommendations that support joyful exploration and productive mathematical discussions! These titles open space for noticing patterns, justifying ideas, engaging in debate, and connecting mathematical thinking to the world around them.
And if you’re working across multiple grade levels, you can always revisit my earlier K–1 read‑aloud list. Together, the two posts offer a collection of my favorite books that invite curiosity, support authentic access to the mathematics, and build a shared mathematical community from kindergarten all the way through fifth grade.
Concept
Book Suggestions
Number and Operations
Equal Shmequal by Virginia Kroll One Is a Snail, Ten Is a Crab by April Pulley Sayre and Jeff Sayre One Hundred Hungry Ants by Elinor J. Pinczes A Remainder of One by Elinor J. Pinczes How Much Is a Million? by David M. Schwartz 100 Mighty Dragons All Named Broccoli by Larochelle & Cho Dozens of Doughnuts by Carrie Finison Hello Numbers by Harriss and Hughes
Fractions
The Lion’s Share by Matthew McElligott Fry Bread by Kevin Noble Maillard Give Me Half! by Stuart J. Murphy The Doorbell Rang by Pat Hutchins How Many Ways Can You Cut a Pie? by Jane Belk Moncure
Measurement and Data
Spaghetti and Meatballs For All! by Marilyn Burns How Big Is a Foot? by Rolf Myller The Penny Pot by Stuart Murphy Curious Comparisons by Jorge Doneiger Coa Chong Weighs and Elephant by Songju Ma Daemicke Greater Estimations by Bruce Goldstone Actual Size by Steve Jenkins Which Would You Rather Be? by William Steig
Geometry
This is Not a Maths Book by Anna Weltman (not really a children’s book, but it is sooo good!) Which One Doesn’t Belong? by Christopher Danielson Shapes, Shapes, Shapes by Tana Hoban Grandfather Tang’s Story by Ann Tompert
I hope this collection gives you fresh inspiration for sparking mathematical curiosity in your classrooms. The best way to know whether a read aloud resonates with yourself and students is simply to try it. You can see how your students respond, notice the ideas they generate, and decide how to leverage their thinking toward the learning goal.
Give these titles a spin, and let me know what mathematical conversations they open up for you and your students. I’d love to hear what you try!
I had always been a fan of bringing stories into math class; however, as a fifth-grade teacher, it was hard to convince a group of almost–middle schoolers that a children’s book could be full of interesting, grade-level ideas to explore. More often than not, when I began reading aloud, I could tell right away that the vibe was off and that they saw it as too babyish.
At first, I assumed the problem was the book itself. Over time, though, as I learned more about mathematizing from Allison Hintz and Tony Smith, I came to realize that the issue wasn’t what I was reading, but how I was framing the experience. I was making the book’s concept the thing, rather than centering the story and the mathematical practices involved in mathematizing. And while the concept is extremely important when choosing a book, the facilitation really can make or break the experience.
Around that same time, I was incredibly fortunate to step into a role as a K–5 math specialist, where I had the opportunity to partner with an amazing reading specialist, Erin. Together, we tried out different books across grade levels and content areas, reflected on those experiences, and took up interactive read alouds as a way to blur the lines between content, habits of mind, and capacities. Through this work, I not only learned a great deal about teaching ELA, but also came to understand mathematizing as a way to invite students to see mathematics everywhere, including within stories that don’t appear to be mathy at all!
The Why
Mathematizing is a process of inquiring about, organizing, and constructing meaning with a mathematical lens (Fosnot & Dolk, 2001). Through mathematizing, students:
have access to mathematics
see math when and where it may not be obvious
see mathematics as a way to make sense of things
see math as a way of thinking, not solely a way of doing
focus on meaning-making
The What: Books
Choosing a Book
If you are a K-1 teacher, I am sure you already have an expansive book collection! As you look through your library, consider how different stories might invite mathematical thinking. Some books center math directly in the plot, others offer rich illustrations to examine, and some simply create situations that spark mathematical curiosity—even if the math lives quietly beneath the surface.
If you are having trouble choosing, I have highlighted some of my favorite books in the table below!
Concept
Book Suggestions
Counting and Cardinality
I Spy a Dinosaur’s Eye by Jean Marzollo Truman by Jean Reidy Grumpy Bird One Is a Snail, Ten Is a Crab by April Pulley Sayre & Jeff Sayre Ten Black Dots by Donald Crews Ten Flashing Fireflies by Philemon Sturges Anno’s Counting Book by Mitsumasa Anno Ten Ways to Hear Snow by Cathy Camper One Fox by Kate Read How Many? by Christopher Danielson
Geometry
This is a Ball by Beck & Matt Stanton City Shapes Square Cat by Elizabeth Schoonmaker The Shape of Things by Dayle Ann Dodds More-igami by Dori Kleber Inch by Inch by Leo Lionni
Addition and Subtraction
The Doorbell Rang by Pat Hutchins There Is a Bird on Your Head by Mo Willems Double Those Wheels by Nancy Raines Day
Composing Decomposing Comparing Numbers
Lia & Luis: Who Has More? by Ana Crespo Dozens of Doughnuts by Carrie Finison (also great for counting & add/subtraction) 12 Ways to Get to 11 by Eve Merriam
Measurement and Data
Curious Comparisons by Jorge Doneiger The Animals Would Not Sleep! by Sara Levine
Other Lovely Books!
Count on Me by Miguel Tanco The Look Book by Tana Hoban Playful Puzzles for Little Hands by Taro Gomi
The How: Interactive Read Aloud
Now that you have a book in mind, it is time to structure the read aloud experience. Don’t worry, there’s no single path for engaging students in a mathematical read aloud. Instead, think of it as a series of intentional moments that help students notice, wonder, build curiosity, and explore the mathematics within a story. Here are some helpful lessons learned that helped me improve in my interactive read alouds:
Let the story be the story. The first reading is an opportunity for students to listen, imagine, and enjoy. Without pausing to analyze, students can make sense of the narrative and build a shared experience around the text.
Listen closely to students’ thinking. After the reading, invite students to share what they noticed or wondered. Their ideas, mathematical or not, offer important windows into how they are making sense of the story. Recording these thoughts allows them to remain visible and valued.
Revisit the text. Returning to the story, or to particular pages, creates space to look more closely at the mathematics that emerged. This second look helps surface relationships, quantities, and structures that may have gone unnoticed the first time. Allison and Tony give great structures for planning this in their book and in the Supporting Materials section on that same page.
Honor students as question askers. While we should have an idea of the math we want students to engage in from the book, it is also extremely important to encourage students to pose their own mathematical questions inspired by the text. When students generate the questions, the mathematics feels purposeful and connected to their own thinking.
Create time to explore the math. Follow the read aloud with an activity that either emerges from students’ questions or deepens the ideas you want to highlight.
Conclusion
Ultimately, an interactive mathematical read aloud is less about following a script and more about encouraging sensemaking. The goal isn’t to squeeze math out of a book, but to create conditions where students naturally notice, question, and explore the mathematics already living in the pages. Over time, these intentional moments help students see math as something they do and make sense of, not just something that shows up in a textbook. Try one of these ideas in your next read aloud, and let me know how it goes, I’d love to hear what your students notice and wonder!
Examples
If you want to check out some examples before jumping right in, I have blogged about it a bit in these posts! Next time I will focus on some of my favorites in grades 2-5!
Whenever it snows, it feels so cozy inside that I just have the urge to read and write. And nothing inspires me more to write than student thinking. And there is no better place to see student thinking than in math journals!
When I was a classroom teacher, my fifth graders wrote in their math journals almost every single day. Sometimes they used them before a lesson to record estimates or predictions. Other times they wrote during class to capture their ideas as they worked through a problem. Often, they ended the lesson with a short prompt. No matter how the journals were used, they were always a safe, ungraded space for students to put their thinking on paper. And no matter the prompt, I learned something new every day about my students’ thinking simply by reading their entries.
Later, as a math specialist, I had the opportunity to see student writing in math classrooms across many grade levels, and it was so fascinating. I could see where it all begins in Kindergarten, when students are representing ideas with drawings and numbers, and how that thinking evolves through fifth grade as students’ written reasoning becomes lengthier and the prompts become more metacognitive. In every lesson I planned with teachers, we would build in a writing prompt. Those student responses, would always give us a new window into each student’s thinking.
For example, when I planned a lesson on arrays with a third-grade team, we intentionally designed an exit prompt that went beyond a simple right-or-wrong answer. The lesson began with a Dot Image, and students spent the rest of the time building arrays and writing equations to represent them. At the end of the lesson, we returned to one of the dot images from the launch. Instead of asking students to write an equation, we asked them to choose two mathematical expressions that had been shared during the Dot Image discussion and explain how those expressions were the equivalent using the image.
When we later looked through the student journal responses, they became the anchor for our reflective conversation. Each journal entry revealed something a little different: how students were making sense of multiplication, the connections they were noticing, and where their thinking was still emerging.
Math journals don’t just show us what students can do; they offer a window into how students are thinking. Let’s take a closer look at some of that student work based on broader mathematical understandings.
The Commutative Property
The majority of students chose two expressions demonstrating the commutative property of multiplication. Often students see that you can change the order of the numbers in a multiplication problem and the product remains the same, however in the journal entries, we were able to see student understanding of this property in a representation.
16 x 2 = 2 x 16
8 x 4 = 4 x 8
16 x 2 = 2 x 16 and 4 x 8 = 8 x 4
Changing the Number of Groups and Number in Each Group
A few students noticed that when they changed the number of groups and the number of dots in each group, the product remained the same. While these students are not yet articulating how the groups are changing, this work provides a great opportunity to plan future conversations around this idea.
Rearranging the Groups
This response is very similar to the previous responses, however this student is beginning to articulate how the groups are changing. Instead of having 10 groups of 3, the student explains he took some dots away and added them to another group to make 16 groups of 2.
Relating Operations
Some students related expressions based on what they understand about the operations and were able to represent these understandings in the dot image.
While the team and I heard and observed so much interesting student thinking during the Dot Image discussion itself, the journal prompt allowed us to look more closely at each student’s understanding and see the connections they were making. It served as a important formative assessment, one that extended beyond what we could learn through discussion alone.
Math journals have transformed the way I listen to students’ thinking. I love seeing math journaling used across grade levels, from students who are just beginning to represent their ideas to those who are refining written explanations. Journals give students who may not feel comfortable sharing aloud a space for their voices to be heard, while giving teachers invaluable insight into how students are making sense of the mathematics. I encourage all math teachers to incorporate math journals into their classrooms—not just to see how students arrived at an answer, but to uncover the connections, understandings, and confusions that shape their learning. That insight truly informed every planning decision I made in my classroom and deepened my understanding of the not only the mathematics, but how students build mathematical understanding.
Now, off to make some more coffee, grab a good book, and then follow up with some Fortnite or Zelda gaming time:) Happy snowy Sunday all!
Word problems have always been challenging for me as a teacher and as a coach supporting teachers. I think part of the reason is that you can’t really teach word problems in the traditional sense. Solving them depends on students making sense of a situation and the question they are being asked to answer, and there are many factors that influence that sensemaking.
One factor is the context itself. I know how important it is for students to apply their understanding in both familiar and novel situations; however, every context will be a mirror for some students and a window for others, and when a situation is completely unfamiliar, I have seen it significantly impact how students approach the problem. Another major factor is the language of the problem itself. Many word problems include vocabulary, sentence structures, verb tenses, and multiple steps that shape how students make sense of the situation. These features require them to draw on things like reading comprehension, syntax, semantics, and sequential thinking, not just mathematical understanding and procedural skill. All of these elements influence the mental model students build based on the context and ultimately affect how they attempt to solve the problem.
Because of these complexities, it is not surprising that many students quickly grab numbers from a word problem and compute or search for key words. These strategies often worked for them in earlier grades, with one-step problems, or within curriculum units focused on particular operations. As a result, they do not always read the context as something that should make sense. Instead, they read while thinking, “Which operation do I need to use to solve this problem?” This reminds me of times when I am reading a book with something else on my mind. Even though I am technically reading the words, I can finish an entire page, or even a chapter, and realize I cannot remember anything I just read. I think this is similar to what happens when students read a word problem while also trying to figure out how they are supposed to solve it.
Understanding these challenges gives us important insight into the kinds of instructional adaptations that best support students in sensemaking. When we pause and give students an opportunity to make sense of a context before jumping in to solve, we set them up for more productive problem solving. And, the more we provide these opportunities, the more metacognitive those ‘sense-making structures’ become for students. There are some great math language routines out there, such as Three Reads and Co-Craft Questions, that are productive in a whole-group setting, but can take a lot of class time, require preparation, and may not transfer easily to a new problem for students. Because we sometimes can’t predict the problems that will be most challenging, I also like to have a few back-pocket, in-the-moment adaptations that promote the same type of reasoning and sensemaking.
These adaptations are all about helping students make sense of a word problem before they jump into solving. By giving them time to notice, wonder, visualize, and pose questions, we make the problem more accessible and give students the chance to build a strong mental model. This approach draws on both math and language skills, helping students focus on understanding rather than just grabbing numbers or looking for key words. When we use these adaptations in the classroom, students are more likely to engage in deeper, more productive mathematical thinking and problem solving.
For more ideas and examples, you can check out some related blog posts:
Just like the launch of a lesson shapes how students access the mathematics, the structures we use during group work support what they do once they’re in it. In these moments, who talks, who listens, and whose ideas move the work forward can either widen or narrow the thinking that happens. Group time can be a place where rich, collaborative work happens, but it involves much more from the teacher than giving the directions ‘work with your group,’ ‘turn and talk,’ or ‘think pair share.’ While a curriculum can provide teachers with helpful suggestions, the uniqueness of each group of students places the responsibility heavily on the teacher, which makes sense. Only the teacher knows their students and the dynamics within each small group. Small shifts in how we organize students, position their ideas, and support their discussions can dramatically impact both the mathematical practices that students engage in as they work and the mathematical thinking that students bring to the whole group discussion afterward.
After exploring ways to launch a lesson, the next opportunity for quick, high-leverage adaptations comes when students begin working together. From the moment we ask students to collaborate to the moment when we circulate and listen, the structures we choose can either uncover students’ thinking or unintentionally limit it. Thoughtful approaches to group work can support collaboration, build mathematical habits of mind, and strengthen the sense of community we hope to see in our classrooms. In this post, we’ll look at quick, in-the-moment ways to support group work so every student has an opportunity to contribute and every idea has a chance to surface.
Alternate Ways to Work in Groups
Instructional Challenges: When students jump into group work without clear structures for talking and listening, it becomes easy for one person’s ideas to dominate while others disengage. Without intentional support, some students simply “go along” with the loudest or quickest thinker, and opportunities for deeper reasoning are lost. Additionally, when students stay in the same assigned seats, groups can become static. While this consistency can help early in the year as a community is forming, it can also limit the range of perspectives and mathematical ideas students encounter over time.
Each of these routines require students to articulate their ideas and listen to the ideas of others. One routine I love to support these practices is Talking Points. It didn’t really fit with the others in the table, but I wanted to mention it here as I close out the post. This routine includes giving each group a carefully crafted statement (for example, a prompt about multiplication or division), and asking students to respond by agreeing, disagreeing, or saying they’re unsure while explaining why. Because everyone at the table gets a chance to voice their ideas, and then they collectively wrestle with different perspectives, students often reconsider or deepen their understanding about the topic at hand. Finally, when the groups come back together for a whole-class reflection, all students benefit from a wide array of reasoning. I have a collection of blogs about Talking Points in the K-5 math classes here if you are interested in trying them out!
Try it!
In your next PLC or planning session, review the activities in an upcoming lesson. As you read through each problem, discuss:
What questions should I ask students to discuss in small groups that will move their thinking toward the mathematical goal of the day?
What structures can I use to ensure all students have the opportunity to share their ideas and have their ideas heard by others in the class?
Which of the four group work structures will you use to support students as they learn together?*
*If you’re planning with your grade-level team, each person can try a different structure and then compare the affordances of each. I’d love to see what you try! Share your ideas in the comments or on IG (@kgraymath)!
Next up will be routines for supporting student learning as they engage in problem contexts, in particularly word problem sense-making strategies.
As a math teacher and coach, I’ve always adapted curriculum materials. I am sure we all have—it’s part of knowing our students and wanting to provide them with access to the math in ways that make sense to them. And while we have the best of intentions, sometimes the choices we make can unintentionally take the math out of students’ hands, disconnect from what they already know, or even reflect assumptions and expectations we didn’t realize we were holding.
Adapting activities is challenging for many reasons: some decisions require extensive prep, others happen in the moment, and all of them must be balanced with the pressure to keep pace while still giving students the time and space to share the incredible ideas, experiences, and mathematical thinking they bring into the classroom. I also found it difficult at times to know exactly what to look or listen for mathematically, and I sometimes interpreted students’ responses through my own assumptions and expectations. While a high-quality curriculum can provide problems that elicit student thinking, the deeper work of examining what we notice, what we value, and how we interpret and leverage students’ ideas extends far beyond even the most comprehensive teacher guide.
While thoughtful planning ahead of time is ideal, we all know it isn’t always possible. Because so much can get in the way of planning and prep time, I think a lot about how to make quick, in-the-moment adaptations based on what students are saying and doing—and on what I still need to learn about their thinking. I often wonder: What if our adaptations did more than simply “fix” a lesson? What if they opened space for deeper student thinking, a stronger sense of community, and richer mathematical understanding?
These questions keep pushing me to look closely at the flow of a typical lesson and identify the moments where even small adaptations can make a big difference. I’ve found that there are five points in a lesson where a few quick, minor changes can have a big impact:
Launching an activity
Working in small groups
Engaging with contexts
Sharing ideas
Synthesizing the lesson.
While it would be wonderful to have a single “right” adaptation for each of these, the reality is that different classrooms and moments call for different routines or structures. In the next series of posts, we’ll take a closer look at each of these opportunities and explore practical, in-the-moment ways to adapt them.
Ultimately, the goal of these small adaptations isn’t just to adjust an activity or lesson; they’re meant to open up richer mathematical experiences for students and provide increased insight for teachers. My hope is that these structures encourage sense-making and increase access to the mathematics; provide every student opportunities to share the amazing ideas they bring; create space for questions about one’s own or others’ thinking; build a stronger mathematical community; and support teachers in reflecting on why they’re adapting an activity and how those choices best serve students.
In this post, I’ll outline a few alternate ways to launch a lesson, and in subsequent posts, I’ll explore each of the other lesson moments listed above.
Alternate Ways to Launch a Lesson
Instructional Goal: Provide students with an entry point into the activity.
When deciding which adaptation to use, it is helpful to think about what aspect(s) of the task might make it inaccessible to your students. For example:
If the context or representation of the problem might be unfamiliar to students, you could use the Tell a Story, What Questions, or Making Connections routine.
If the activity relies on a prior knowledge from earlier in the year or prior grade levels, it might make sense to use the What Questions or Making Connections routine.
If the language (including math language) adds a layer of complexity to the activity, you could use the Word Splash routine.
Try it!
There isn’t one “right” adaptation for any classroom scenario; each is an opportunity to learn more about your students. In the end, it’s not the routine itself that matters as much as what the routine affords—the ways it reveals student thinking, provides an entry point, and supports problem solving. The only way to figure out what works best is to try these structures out! Fortunately, they require little materials prep. All you need is a piece of chart paper or whiteboard and an image from the problem students will be solving.
In your next PLC or planning session, review the activities in an upcoming lesson. As you read through each problem, discuss:
What knowledge and experiences are your students bringing to the problem?
Is there an activity where students might not have access to the problem? If so, what specifically is inaccessible and why?
Which of the four routines will you use to launch the activity?*
*If you’re planning with your grade-level team, each person can try a different routine and then compare the affordances of each one. I’d love to see what you try! Share your ideas in the comments or on IG (@kgraymath)!
I have always loved, loved, loved Jenna’s Slow Reveal Graphs! They provide such an engaging structure that encourages students to explore and reason about data. If you don’t know what Slow Reveal Graphs are, it might be helpful to read up on them here before continuing.
Sense-making is so front and center in a slow reveal activity that I think it could be interesting and exciting to extend this activity with problem posing! I have been learned so much about problem posing from my friend Jinfa Cai. Problem posing is an instructional approach in which students generate and solve their own mathematical problems. In doing so, students share mathematical authority in the classroom and position students as creators of mathematical inquiry rather than solely recipients. I wrote a bit about it in the new NCTM book, Teaching Mathematics Through Problem Solving (pg 73-74).
The Mashup
To think about what a mashup could look like, let’s go check out one of Jenna’s recent slow reveal graphs, Average Song Length by Genre. Which is so appropriate because Jenna and I love to chat 90’s hip hop artists;) Oh, and did I mention that Jenna creates a slide deck with notes for each one of these?!?
Step 1: Display the graph and ask students what they notice and wonder. (Slide 1)
Step 2: Display the graph with new information, ask questions that encourage students to interpret the new information, and make predictions about what is still missing…luckily, these directions are in the slide notes of Jenna’s slides. (Slides 2-3)
Step 3: Display the final reveal of the graph. (Slide 4)
Step 4: Problem posing: Ask students to pose problems that can be answered by information in the graph and record them on a piece of chart paper.
Students might pose many different types of problems such as:
Which genre had the longest average song in 2019?
About how much longer was the average latin song than hip-hop song in 2023?
About how much did the average pop song duration decrease from 2019 to 2024?
Which genre had an increase in average song duration? Between what years?
If you listened to a dance song and alternative song in 2022, about how many seconds would you be listening? How many minutes?
This list could go on and on, but you get the point. I know it could be nerve-racking to use a graph like this because there are not definite values at every point, but I think that actually increases the reasoning element and could raise some cool points for argumentation!
Step 5: Ask students to solve their problems.
This is a choose your own adventure — you could ask students to solve all of the generated problems or you could focus attention on one or two problems aligned to that day’s learning goal. I always lean toward the latter so the activity isn’t just a one-off random activity, but instead connected to what they are learning. This also leaves me a bank of ‘if you get done early‘ problems, which was always one of the biggest differentiation challenges for me, and helps me focus the whole group discussion afterwards.
Step 6: Synthesize the learning.
This step is really dependent on the learning goal of the day. After solving, you could have students do a gallery walk to compare solutions and solution methods, representations, and reasoning. Or you could decide to have a whole group discussion based on the monitoring that you did as students worked.
I have always been such a fan of numberless word problems, notice/wonder, and the 3 Reads MLR. Problem-posing feels like it pulls the most purposeful parts of each of these routines into one. Layering problem posing on Jenna’s slow reveal structure puts such an fantastic focus on data while also supporting other areas of mathematical focus. It is also so adaptable by grade level, which makes it so flexible!
Final Problem Posing Thoughts
When students have the opportunity to pose their own mathematical problems based on a situation, they must make sense of the constraints and conditions from the given information to build connections between their existing understanding and a new understanding of related mathematical ideas (Cai, 2022). And when teachers have the opportunity to listen to connections students make, understand the math students see in situations, and make teaching decisions on how to elicit, share, and move forward with student thinking, it shares the mathematical authority in the classroom and leads to deeper, more powerful learning for all!
At the beginning and end of a curriculum unit, I find it valuable to learn what students already know and what questions they have, to help guide my planning and instruction. While pre- and post-assessments can provide useful information, they also tend to limit the range of students’ thinking, especially when students show minimal written work. Because of this, I began using a few key routines. One of these routines, which I call “Tell me everything you know and want to know about [topic],” invites students to share their ideas more openly.
For example, after a 3rd grade unit on multiplication and division, Katie (an amazing 3rd grade teacher) and I wanted to gain insight into what students learned, in their own words. We wanted to give them some individual think time first, so we gave them this simple sheet to record their ideas. We decided to leave the page unlined so students could freely draw any representations that made sense to them. Their ideas definitely did not disappoint! (Click on each thumbnail to see the full page)
We only got one question, but it was such an interesting one!
I think since it was the first time doing this routine at the end of a unit, we didn’t get as many questions as we had hoped. I do wonder how changes in wording such as “What new questions do you have about multiplication?” or “What wonderings do you have about multiplication?” would impact the amount of questions we would get next time.
After students had their independent think time, we shared their responses as a whole class and recorded their ideas on chart paper to stay up as an anchor chart we could refer back to throughout the year!
If you would like to try this routine before the holidays to see what students have learned, I blogged the directions here. In my next blog post, I will explore another routine I love to formatively assess student thinking! Until then, I would love to hear some of your favorites in the comments!
I am sure we have all seen it happen at one time or another in math class. We give a student a story problem to solve and after a quick skim, the student pulls the numbers from the problem, computes them, and writes down an answer.
If the answer is correct, we assume the student has a grasp of the concept. However, if it’s incorrect, we’re left with a laundry list of questions: Do they realize their answer doesn’t make sense? Did they not understand the context? Did they simply pull the numbers and operate to be finished or did they truly not know what to do with them? Most importantly, we ask ourselves, how can I help students make sense of what they are reading and think about the sensibility of their answer in the context of the problem?
If we’re lucky, we can identify a mathematical misconception and work with that. Oftentimes, though, the answer isn’t even reasonable. Then what do we do?
Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt.
The best way I’ve found to help students make sense of what a problem is asking is, ironically, to take the question out altogether. Inspired by the wonderful folks at The Math Forum, I do a lot of noticing and wondering with students in this fashion. Most recently, after reading Brian Bushart’s awesome blog post, I have started taking the numbers out as well! Instead of students thinking about how they’re going to solve the problem as they read, they are truly thinking about the situation itself. It’s been an amazing way to give every student entry into a problem and allow me to differentiate for all of the learners in the classroom, while at the same time provide insight into my students’ mathematical understandings.
Recently, I had the opportunity to work with a 3rd grade class. The class recently finished their multiplication and division unit and will soon be starting their work with fractions. In order for their teacher and I to see and hear how students apply the operations, make sense of contexts, and currently think about fractions, I thought it would be interesting to take a story problem from their Student Activity Book and take the question and numbers out.
The Planning
I chose the problem below and thought about what I would learn about a student’s mathematical understandings and sense-making after they answered the questions.
I was curious to observe how students make sense of problems based on the idea of removing the numbers and the question so I changed the problem to this simple statement:
“Webster has boxes of granola bars to share with his class.”
I anticipated the students would wonder about the missing mathematical pieces involved in an open-ended statement like this. I believed their wonderings could lead them to develop questions that could be answered based on the very information they were wondering about. I knew the mathematical ideas of multiplication, division, and/or fractional sharing would arise and that I would learn so much more about their thinking then if I had given them the original problem.
In The Classroom:
I launched the lesson by posting the sentence on the board and recorded things they noticed and wondered.
They noticed:
“We don’t know how many boxes of granola bars.”
“There is not enough information to know what is going on.”
“We don’t know if it is adding, subtracting, multiplying, or dividing.”
“There are multiple people in the class because it says boxes and share.”
They wondered:
“How many granola bars are in each box?”
“How many boxes he bought?”
“How many kids are in his class?”
“What kind of granola bars are they?”
Based on their noticings and wonderings, I felt everyone had a strong grasp of the context and sense of where this was going. Based on their noticing that there is not enough information to know what is going on, I asked what more they would want to know. They responded that they wanted the answers to the first three of their wonders: bars per box, number of boxes, and number of kids in the class.
I asked them what questions they could answer if I gave them those pieces of information and they responded:
How many bars does he have?
How many bars does each kid get?
How many does he have left?
At this point, I could have given them the information they wanted. However, I thought it would be so much cooler to allow them to choose that information for themselves. I was curious: how they would go about choosing their numbers! Would they strategize about the numbers to make it easier for themselves? Would they even think that far ahead? What would they do with the leftovers?
When I told them I was not giving them the information and that instead they were choosing their own numbers along with the question they wanted to answer, they were so excited!
Some partners chose their numbers very strategically to make it easier for themselves. To me, this demonstrated a lot of sense-making and forethought of what was going to happen in their solution path. And as an added bonus, while only asked to answer one question, the group answered all three questions! (Teacher note: if students chose numbers strategically and therefore finished quickly, I gave them extra bars to factor into their problem to see how they dealt with the leftovers.)
Other students chose the opposite route and strategically picked numbers to make it “harder for themselves.” Check out the way these two students showed strong reasoning and perseverance through division of numbers larger than any they’ve ever worked with.
Others chose numbers without much forethought and dealt with some amazing leftovers. This was a great way to formatively assess students’ thinking related to fractions before they began that unit.
And then there are always the surprises. Who would have thought third graders would reason about the leftovers in terms of percentages?
Reflecting on what the students would have been asked to make sense of and the work they would have had to do based on the original problem versus the reasoning and work they did related to this one simple sentence, I’m amazed by the difference. I learned so much more about what each of the students know beyond simply multiplying 5 and 6. Taking out the numbers and question allowed every student to think about the meaning of the sentence, the implied mathematical connections, and plan a solution pathway before jumping into a solution attempt.
I highly recommend everyone try this strategy with a word problem from your current text. It’s a wonderful way to give every student access to the math and freedom to think beyond just getting an answer.
If you know me or have ever read my blog, you know I could talk for days about student math work! You can visit my blog for a more detailed description of the work shown in this post as well as additional work captured from the lesson.
Routine 1: Tell Me Everything You Know About ____________
Directions
Choose a word or phrase that is the focus of your first curriculum unit. This could be something like: fractions, addition and subtraction, shapes, data, multiplication, etc. If students are introduced to that concept for the first time during the unit, such as volume in fifth grade, use a term like ‘measurement’ to elicit prior knowledge related to volume.
Write your chosen concept or topic at the top of a piece of chart paper.
Prompt students, “Tell me everything you know about [your chosen topic].”
Give students 1 minute of independent think time and then 1 minute to quickly tell a partner one thing they are going to share with the whole class.
As a whole group, record students’ ideas on the poster as they share.
When they are finished, ask if there are any ideas on the chart paper they have questions about. This is a good opportunity for students to ask clarifying questions of one another, revise their thinking, and agree or disagree with others’ ideas. You do not need to come to a final conclusion on each point of disagreement, especially if it is something they will learn in the unit. Simply just mark that idea with a question mark and revisit it later.
If there is time, you could start another poster with the prompt, “Tell me everything you wonder or have questions about [your chosen topic].” This communicates that sharing things they wonder and asking questions are part of learning. The information you’ll learn about student thinking will be extremely helpful going into the first unit.
As you move through the first unit, refer back to the poster frequently and ask students if they would like to add anything new or revise a previous idea.
Print a copy of the talking points for each group.
As a class, review how each round works. The first time you do this, it might be helpful to also model the process with a fun talking point such as, “A hot dog is a sandwich.”
ROUND 1 – Read the first talking point aloud. Take turns going around the group, with each person saying in turn whether they AGREE, DISAGREE, or are UNSURE about the statement and why. Even if you are unsure, you must state a reason why you are unsure. As each person shares, no one else comments. You are free to change your mind during Round 2 and/or Round 3.
ROUND 2 – Go around the group a second time, with each person saying whether they AGREE, DISAGREE, or are UNSURE about their own statement OR about someone else’s agreement, disagreement, or uncertainty from Round 1. As each person shares, no one else comments. You are free to change your mind again during Round 3.
ROUND 3 – Go around the group a third time to take a tally of AGREE / DISAGREE /UNSURE votes and record that number on your Talking Points sheet. Then, move on to the next talking point.
Sample Student Handout with Third Grade Talking Points
Talking Point
Agree
Disagree
Unsure
Fractions are always less than 1.
A fraction is a number.
We can locate fractions on a number line.
Fractions tell us a size.
One half is always greater than one third.
We can combine fractions.
Sample Math Mindset Prompts
Being good at math means being able to do math problems quickly.
A person is either good at math or bad at math.
I prefer to work on problems that challenge me rather than ones I find easy.
When working in a small group, if one person knows how to solve the problem, they should show the others in their group how to do it.
There is always one best way to do math.
Getting a problem wrong in math means you failed.
Drawing a picture is always helpful when doing math.
Sample Math Content Prompts
5 is the most important number.
The number 146 only has 4 tens.
Fractions are numbers.
When multiplying, the product is always greater than the factors.
Division of fractions is just like division of whole numbers.
The opposite of a number is always a negative number.
It is easier to work with decimals than with fractions.
For any equation with one variable, there is one best way to solve for the variable.
It is easier to work with degrees than with radians.
MathMinds 