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Low Floor, High Ceiling, Wide Walls: Using Tasks to Elicit and Leverage Ideas

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The answer is 1/2. What is the question? 

Pause for a moment to think about how you would respond to this prompt. After you have a question in mind, reflect on the thinking you did to come up with that question. What topics did you think about? Did they include mathematical representations, a real-world context, or calculations? Something else? 

Last week, we had our second book study session for NCTM’s Teaching Mathematics Through Problem Solving K–8. The focus of the session were chapters four and five around instructional tasks and teaching signposts that support students in learning through problem solving. We wanted to open the session with a good math prompt as everyone was arriving, a prompt that gave them something interesting to think about as we waited a couple of minutes and also set the tone for the learning we were going to do together. Since the session was focused on instructional tasks, we wanted to begin with one that reflected what we value in classrooms—tasks that invite everyone in and elicit a wide array of student thinking.

After a few seconds, the chat was buzzing. Here is a sampling of the amazing responses we got.

The variance in responses was so fun and got me thinking about when, where, and with whom a prompt like this might be especially powerful. I often use tasks like this to launch a unit, lesson, or activity so I can learn how students are thinking about the mathematical concept we are about to explore.

Launching a Lesson

A former colleague, Jenn, used this prompt to launch a 3rd‑grade lesson on comparing fractions. Using the task at this point in a lesson not only provided insight into students’ thinking, but also supported differentiation. Students who finished early had access to a bank of questions to evaluate, compare, and justify. Even better, those questions came directly from the students themselves.

Launching and Wrapping Up a Unit 

Building on this idea, what might it look like to use a prompt like this not just to launch a lesson, but to bookend an entire curriculum unit?

Before beginning a unit on fraction multiplication and division in 5th grade, we might pose the question, “The answer is 3/4. What is the question?” At the start of the unit, we would likely see fractional diagrams, addition expressions such as 1/4+1/4+1/4, or multiplication expressions like 3 ×1/4​. All of these responses are incredibly valuable for eliciting what students currently understand about fractions.

By the end of the unit, however, students have developed new understandings about fraction multiplication and division. Revisiting the same prompt invites them to apply those new understandings in more sophisticated ways. Building on the original list of student‑generated questions at the end of the unit could serve as a powerful formative assessment and, just as importantly, a rich anchor chart that documents students’ learning over time.

Professional Learning 

This idea also extends naturally to professional learning settings, particularly those that bring together educators across a wide range of grade levels. For example, in professional learning focused on the fraction learning progression, the list we generated at the beginning of our book study session would be invaluable.

A rough draft PD flow could look something like this: 

  1. Pose the prompt and ask teachers to write their responses on index cards. 
  2. In small groups, teachers share their questions and work together to align them to grade‑level standards.
  3. In larger groups, teachers collaborate to form a learning sequence, ordering the cards from the earliest fraction understandings to the latest. 
  4. Teachers move into grade‑level groups to identify their curriculum unit in which they could use a prompt like this. For K–2 teachers, this would mean adjusting the prompt to use the words “one‑half” instead of a numerical representation (a variation I used in 1st grade).
  5. In grade‑level groups, teachers anticipate what students might say and plan how they will leverage student understanding and use student‑generated questions.
  6. The whole group shares ideas. 

Reflection

Coming back to the book study, I found this book quote reflective of prompts like these: 

Activities that provide access and extension are often referred to as having a low-floor and high-ceiling. Meaning, the problems invite all students to engage, while also providing space for deeper exploration. These types of tasks provide accessible entry points without lowering or limiting the cognitive demand of the mathematics. Mitch Resnick (2020) takes this idea even further with the concept of “wide walls,” which reminds us that learning shouldn’t just move from easy to hard, but should also give students space to explore ideas in different ways and directions.

Coming Up

If this prompt has you thinking about your own classroom or professional learning spaces, I’d love for you to continue the conversation with us. Join me on February 9 for a free webinar with Ashley Powell and Shawn Wigg as we explore instructional tasks that invite a wide range of thinking. We will relate the ideas in the NCTM book to practical applications of tasks in the classroom. Participants will receive an excerpt from the NCTM book, and we’ll raffle off a free digital copy at the end. Hope to see you there!

Adapting Lessons Part 2: Structuring Group Work

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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.

Adapting Lessons Part 1: Launching an Activity

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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)!

Part 2: Supporting student learning as they work in small groups

Two Math Routines to Learn About Student Thinking

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Routine 1: Tell Me Everything You Know About ____________

Directions

  1. 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.
  2. Write your chosen concept or topic at the top of a piece of chart paper.
  3. Prompt students, “Tell me everything you know about [your chosen topic].”
  4. 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.
  5. As a whole group, record students’ ideas on the poster as they share.
  6. 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.
  7. 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.
  8. 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.

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Routine 2: Talking Points

Directions

  1. Arrange students in groups of 3 or 4. 
  2. Print a copy of the talking points for each group. 
  3. 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 PointAgreeDisagreeUnsure
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.

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