A Low-Floor/High-Ceiling Math Problem for Multiple Grade Levels

            In this Mathematics Teacher article, Stephen Caviness (University of Virginia) shares what he calls a quintessential low-floor/high-ceiling math task – one that is accessible to every student and has multiple, creative solutions to explore. “Such tasks,” he says, “support students in embracing uncertainty, engaging in productive struggle, and expressing creativity in doing mathematics.”

            He’s had success using this task – Close to 1,000 – with students from second grade to college level: 

 Using the digits 1 to 9 exactly once each, place a digit in each box to make the sum as close to 1,000 as possible. 

Caviness encourages students to start filling the boxes without a strategy to see what happens – perhaps 123 + 456 + 789. Common mistakes are using a digit more than once and using zero, which is not an option. “From here,” he says, “students will have to think about place value as they make small iterations, narrowing in on sums closer and closer to 1,000. Ultimately, the closest that one can get to 1,000 is 999, but there are many ways to get there.” 

            As students find correct answers, Caviness suggests a number of ways the task can be extended to get students thinking about the math in deeper ways:

• Why do we regroup? What does it mean to “carry the one”?
• Which sum is furthest from 1,000? What would be the “wrongest” answer?
• What if you were allowed to use a zero?
• Are there any values as close to 1,000 as 999?
• Why can’t we add to 1,001?
• What patterns do you notice about our closest solutions?
• How many ways are there to add to 999? (There are 180.) 

“Close to 1,000” by Stephen Caviness in Mathematics Teacher, July 2026 (Vol. 119, #7, pp. 520-523); Caviness can be reached at slc9th@virginia.edu. For an extensive collection of non-routine math problems, see Stella’s Stunners, compiled and organized by Ohio teacher Rudd Crawford.

Please Note: This summary is reprinted with permission from issue #1148 of The Marshall Memo, an excellent resource for educators.

How a Children's Book Can Build Empathy and Understanding

            In this Council Chronicle article, teacher/author Priten Soundar-Shah says she reads the same children’s picturebook, Fish Is Fish by Leo Lionni, to every class she teaches – middle school, summer camp, college, adult learners. She reads the book with sing-song expression, holding up the illustrations page by page. “Some people look at me like I’ve completely lost my mind,” she says. “But some are intrigued. They want to know where I’m going with this.” 

            Here’s the book’s story. A minnow and tadpole become best friends. The tadpole grows into a frog and leaves the pond to explore the world. When he returns, he tells the minnow about the amazing things he’s seen – birds, cows, people. But his friend can only imagine these unknown things as fish-shaped versions. Birds are fish with wings, cows are fish with udders, people are fish walking upright. “The fish’s imagination is constrained entirely by what it already knows,” says Soundar-Shah. “It imagines these things, but it’s limited by its own experiences. So, it imagines things that are slightly incorrect.”

            The frog’s perception is also distorted. “He’s excited,” says Soundar-Shah. “He genuinely wants to communicate. And yet, because he doesn’t know what he doesn’t know, because he has his own lived experiences that he’s drawing from, he can’t even conceptualize what might not make sense to the fish. He doesn’t know what the gaps are.”

            This becomes a theme as Soundar-Shah and her students explore new material. “We’re going to read something,” she says. “Our intuition might be to immediately disagree. But remember, we might be fish in this scenario. We really want to listen carefully to make sure we’re understanding the whole context.” When students are inclined to think an idea is completely ridiculous, she asks, “Are we being the frog? Are we being the fish? Have we conveyed things as accurately and clearly as possible to someone who doesn’t have our lived experiences?”

            Fish Is Fish becomes a way to approach difficult conversations. “It’s a reminder that understanding across difference requires both humility about what we might be missing and generosity toward what others are trying to communicate,” says Soundar-Shah. “In a moment when so many of us are struggling with how to hold space for genuine dialogue in our classrooms, a five-minute children’s book might be exactly the kind of disarming, humble starting point we need.” 

 “Why I Teach the Same Picturebook to Every Class I Teach” by Priten Soundar-Shah in Council Chronicle, Spring 2026 (Vol. 35, #3, pp. 27-28)

Please Note: This summary is reprinted with permission from issue #1148 of The Marshall Memo, an excellent resource for educators.

Helping Insructional Coaches Do Their Best Work

            In this article in The Learning Professional, Kimberly Richardson (Hampton City Schools, Virginia) says that instructional coaches “are often expected to navigate complex leadership responsibilities with limited preparation, unclear expectations, and inconsistent support.” To address these common issues, Richardson suggests that districts:

• Clarify the purpose of coaching. For coaches to maximize the time they spend working directly with teachers on systemwide priorities, it needs to be clear that they should not be doing substitute coverage, coordinating testing, monitoring duties, providing logistical support, or playing an indirect role in teacher evaluation. “Teachers are more likely to engage meaningfully with coaching,” says Richardon, “when leaders consistently communicate that coaches are support partners rather than evaluators or managers.”

• Develop human-centered leadership skills. Just because coaches were outstanding teachers and skillfully implemented their curriculum doesn’t mean they know how to support adult learning, navigate resistance, manage difficult conversations, or deal with colleagues’ stress and burnout. There needs to be intentional training and support around building trust with teachers, communicating clearly, navigating conflict, and creating psychological safety so teachers will address shortcomings and grow professionally. New coaches should get thoughtful instruction in the role, observe effective coaching, and work with a mentor.

• Orchestrate the conditions for effective coaching. Districts need to carve out time for coaches to reflect, plan, collaborate, and problem-solve with each other, while protecting them from being asked to do work that is extraneous to their mission. “Supporting coaches is leadership work,” Richardson concludes. “If schools expect coaches to cultivate growth in others, leaders must create systems that cultivate growth in coaches as well.” 

 “Setting Coaches Up for Success: How Leaders Can Help Coaches Thrive” by Kimberly Richardson in The Learning Professional, June 2026 (Vol. 47, #3, pp. 28-31)

Please Note: This summary is reprinted with permission from issue #1146 of The Marshall Memo, an excellent resource for educators.

Can Real-Time Teacher Coaching Be Effective?

            In this article in The Learning Professional, consultant/author Jacobé Bell says that for instructional coaches to have a positive impact on teaching and learning, they need to:

• Build a trusting relationship with each teacher;
• Have a shared commitment to student success;
• Create low-stakes spaces to practice;
• Provide timely support when teachers are trying something new. 

Bell believes that under the right conditions, coaching can sometimes happen while teachers are working with students. “The coach’s intent is not to take over, but to make teaching practices visible and more transferable during live instruction,” she says. “When done well, active coaching creates a scaffold that supports teachers as they try on new practices, building their confidence and competence until they can execute those moves independently.” She describes two approaches:

• Preplanned in-the-moment coaching – In a planning meeting, the coach and teacher identify possible challenges in a lesson and plan how they’ll collaborate – perhaps tag-team teaching, a short demonstration or segment taught by the coach, prompts at a particular point in the lesson, the coach monitoring 4-5 students’ work and giving the teacher real-time feedback.

• Spontaneous in-the-moment coaching – These coaching interventions are responsive, based on what the coach sees during the lesson, says Bell, and “are often subtle, quick, and highly situational” – for example, the coach makes a suggestion (“This might be a good time for students to turn and talk”), uses an agreed-upon signal (a circling gesture to suggest that the teacher should circulate and look at students’ work), modeling an instructional move (when a student gives a vague answer, saying, “What in the text makes you say that?”), and redirecting students (“I see that Table 3 is using whisper voices”). 

            Moments like these can make teachers feel like they have a supportive thought partner as they build their teaching repertoire in real time, says Bell. “Jumping in is most effective when the coach asks the teacher ahead of time how they prefer the coach to do so. Without this clarity, in-the-moment coaching can feel intrusive or undermining, and not all teachers are comfortable with support that is clearly visible to students.”

            In that vein, Bell suggests that coaches ask themselves three questions to decide when and how to get involved during a class:

• Is the need urgent? “If yes,” she says, “I use whisper, signal, or reset strategies. If no, I let it play out.”
• Is the skill easily teachable? “If yes, I model the skill quickly or script the next line for the teacher to say. If no, I let it play out.”
• Will it stall the lesson? Should it be saved for the after-lesson debrief? “If yes, I take notes and let it play out. If no, I whisper, signal, or use strategies.” 

[See my 2015 Kappan article, “Should Supervisors Intervene During Classroom Visits?” summarized in Memo 606. K.M.] 

 “Coaching in the Moment: Real-Time Practices That Accelerate Learning” by Jacobé Bell in The Learning Professional, June 2026 (Vol. 47, #3, pp. 18-23)

Please Note: This summary is reprinted with permission from issue #1146 of The Marshall Memo, an excellent resource for educators.