Mr. Maier, a terrific geometry teacher that Robert (one of the coauthors) had in high school, once told his class, “Math is not a spectator sport.” His students understood that he wanted them to be active participants in their learning—thinking critically, asking questions, and engaging fully in the material. This expectation was reinforced by his dynamic teaching style and strong belief in every learner’s ability to succeed.
Despite Mr. Maier’s enthusiasm, in too many classrooms, math becomes something students watch rather than engage in. The teacher demonstrates a skill, students copy it, and their focus quickly narrows to finding the “right” answer (NCTM, 2014). This traditional model leaves little space for sense-making or student voice, especially for struggling learners who need time and structure to access content (Boaler, 2016).
As math educators, we believe that engaging struggling learners in mathematics begins with structured observation. When teachers explicitly teach students how to notice, they transform from passive participants into active investigators. Observation creates a powerful entry point into mathematics, offering struggling learners a way to engage before procedures begin.
Why Observation Matters
Observation in mathematics extends beyond simply looking at what appears on the board. It involves attending to details, recognizing patterns, and interpreting what those patterns reveal. For example, a student might begin a Do Now problem that resembles one they solved the previous day. Instead of only noticing similar words or context, they notice the mathematical structure. As they sketch out the situation, certain numbers repeat across their representations—20, 40, 60. Soon they realize they are not simply counting or matching terms; they are identifying a pattern of common multiples. This shift in noticing transforms the task from surface-level reading to deeper mathematical thinking.
Without developed observation skills, many students default to replication, copying steps without truly understanding them. When observation is intentionally taught and practiced, students pause, notice, and make sense of relationships—a shift that raises engagement and cultivates agency (Boaler, 2016).
This routine aligns with well-established practices that promote productive discourse and conceptual connections, including launching lessons with cognitively rich tasks and prompting explanations before procedures (NCTM, 2014). Observation is not an add-on; it is the opening move that prepares scholars to think critically and reason. In an Algebra I class, for example, when students encounter an image of a graph with four translated lines, Elly (one of the coauthors) and her coteacher invite students to describe what they notice, using mathematical language to articulate shifts, slopes, and relationships. Students then shift from seeking answers to making sense of the problem, positioning observation as a powerful entry point into conceptual understanding.
The Four-Step Observation Process
For students who struggle with math, observation prompts can serve as sentence frames. They lower cognitive load while allowing for different ways of thinking. Struggling learners benefit most when they can enter lessons as noticers and questioners, rather than simply answer-getters (NCTM, 2024).
Struggling learners stop asking, “What step do I copy?” and begin asking, “What do I notice?”
As founder of The Structured Writing and Thinking Program, Robert (coauthor) studied decades of research and practice to conclude that observation isn’t passive noticing but a deliberate cognitive move that strengthens reasoning and understanding (Pauker, 1999). Structured observation is a necessary mathematical habit, preparing students to make sense of concepts before engaging in procedures or solutions. Students practice responding to four simple prompts:
This problem shows . . .
One important idea is . . .
Two details are . . .
One question I am curious about is . . .
Over time, these prompts become a habit of mind, defined by Costa and Kallick (2008) as intellectual behaviors that characterize efficient and effective problem solving and learning. Habits of mind represent a composite of skills, attitudes, and dispositions that enable students to persist when solutions are not immediately apparent. Instead of waiting passively, students begin lessons by noticing, predicting, and questioning.
Let’s explore how these prompts can be effective in the classroom.
Demystifying Fractions in Elementary School
In a 5th grade class, students are learning to add fractions with unlike denominators. Rather than beginning with computational procedures, the teacher creates an opportunity for structured observation. Two circles are projected on the whiteboard, one divided into sixths and the second into fourths. The class quietly studies the representation before discussing what they notice. Together, students articulate observations such as:
“This picture shows two circles, each split into fractional parts.”
“One important idea is that, as denominators increase, the pieces get smaller.”
“Two details I notice are that I see unit fractions, and 1/4 is larger than 1/6.”
“One question I am curious about is: How do we add fractions that aren’t the same size?”
Only after this foundation-building does the teacher present the problem: Liam ate 1/4 of a pizza and Sarah ate 1/6. How much did they eat together? Because the students have already noticed the size relationships between fractions and named how denominators affect the size of each piece, they enter the problem with a conceptual anchor. They are not just adding numbers; they are reasoning about parts of a whole and building a meaningful pathway into the procedure.
For struggling learners, this routine accomplishes more than procedural preparation. It validates their capacity for mathematical thinking. Instead of approaching fractions with anxiety, they enter the lesson having made meaningful mathematical observations. The teacher then explicitly connects those student-generated insights to strategies for finding a common denominator and solving for the answer, strengthening both conceptual understanding and procedural fluency (NCTM, 2014).
Making Sense of Equations in Middle School
In a 7th grade classroom, the teacher prepares to introduce the concept of multistep equation solving. Before demonstrating solution methods, she writes the equation on the board and asks students to observe:
Solve for x: 5x + 6 = 3x + 12
The teacher gives students individual thinking time, then facilitates pair discussions before whole-group sharing. Students might respond to the observation prompts like so:
“This problem shows variables on both sides of the equal sign.”
“One important idea is that I’ll need to move terms to one side.”
“Two details are that both sides involve x, and the constants differ.”
“One question I am curious about is: Do I combine like terms before isolating x?”
For students who become overwhelmed when solving multistep equations, this routine provides essential scaffolding. They can identify structural elements, variables, constants, and the equal sign before being asked to manipulate them mathematically.
With time to study the visual and discuss what she sees, one student who is often hesitant to participate shares her observations with the group and remains present in the conversation. When the class moves toward computation, she is still with them, now anchored by ideas she helped generate. The routine shifts the experience for the student from “I can’t start” to “I already understand something here,” offering an entry point that supports sustained participation.
Pointing to the Right Answer in High School Geometry
In a 10th grade geometry class, students are exploring applications of coordinate geometry. Before introducing any formulas, the teacher displays a coordinate grid with three points labeled A (1, 2), B (7, 5), and C (7, 2). He then asks students to take a moment to observe what they see. After quiet reflection, the class shares individual observations to create a collective response:
“This problem shows three points on a coordinate plane.”
“One important idea is that I can draw a right triangle between them by drawing lines from A to B to C to A.”
“Two details are that there is a right angle at point C and the distance from point A to C is 6 units along the x-axis.”
“One question I am curious about is: Can I always connect coordinate points in this way?”
During the observation period, one student hesitates, unsure how to begin. The teacher quietly works alongside her, tracing the horizontal and vertical distances with a finger on the screen and asking, “What do you notice about how far A is from C along the x-axis?” The student counts the units, pauses, and then says, “It moves 6 across and 3 up. That makes a right triangle.” With that anchor, she remains engaged as the class transitions to formalizing the distance formula. Instead of memorizing a rule, students build understanding through reasoning.
Connecting Observation to Assessment
In addition to helping students formulate their thoughts and be active participants in their learning, observation skills directly support both formative and summative assessment goals. Teachers can use observation responses as daily comprehension checks to identify misconceptions before they solidify. The Standards for Mathematical Practice call for students to make sense of problems and look for structure. Observation routines directly support these expectations by slowing students down and providing them with space to examine relationships before taking action. When students develop the habit of systematic noticing, they strengthen the reasoning strategies required for complex assessment tasks and high-stakes tests—not through test preparation but through thoughtful mathematical thinking.
Building Agency and Mathematical Identity
What makes the observation routine transformative extends beyond improved computational skills to fundamental shifts in mathematical identity. Struggling learners stop asking, “What step do I copy?” and begin asking, “What do I notice?” By observing patterns, identifying key ideas, and generating critical questions, all students—not only the quick and confident learners—can find meaningful entry points into mathematical thinking.
Making Mathematical Observation Work for Every Student
Introducing observation routines into your mathematics classroom can sometimes surface challenges—from time constraints to student resistance. Use these evidence-based strategies to anticipate problems and respond effectively in your classroom.
1. Time Constraints and Pacing Pressures
Teachers worry about “losing time” in standards-heavy environments. However, frontloading observation accelerates learning. To overcome this concern, try:
Replacing one guided practice problem with a three-minute observation routine.
Transforming “bell ringer” activities into mathematical observation opportunities, such as a “See, Think, Wonder” task or “Which One Doesn’t Belong?”
2. Student Resistance to Slower Pace
Students conditioned to rapid-fire problem solving may initially resist slowing down. Gentle shifts in routine can help them settle into reflective observation. Consider strategies like:
Framing observation as a powerful mathematical move that gives everyone an entry point into the work.
Turning noticing into a low-stakes challenge by setting a timer for 20 seconds and asking, “Who can spot something interesting before the timer goes off?”
Spotlighting thoughtful observations during share-outs to show that careful thinking earns recognition.
The more the classroom culture honors depth over speed, the more students begin to see observation as a skill worth practicing rather than a pause before the “real math.”
3. Engaging Students Who Avoid Writing
Some students might carry resistance to writing from past school experiences. Asking them to immediately capture observations in writing can feel like a barrier rather than an entry point. Helpful supports include:
Pairing students to talk first, then jot ideas together to lessen the pressure of having to write alone.
Offering choice in how thinking is recorded—sketches, manipulatives, labels, or short audio notes.
Providing tools like voice-to-text to keep attention on ideas.
When students realize that writing is a means of representing their thoughts rather than a test of worth or correctness, resistance softens and their voices find space on the page.
4. Differentiation for Diverse Learners
Your students have unique needs, and you might have to diversify the way you approach teaching the observation method. Some ideas:
For English learners: Begin by offering translated prompts when needed and anchor discussions in visuals, gestures, and labeled diagrams.
For students with learning differences: Break prompts into smaller steps and provide options for showing thinking, such as drawing, selecting from sentence starters, or speaking ideas aloud.
For advanced learners: Add extension nudges, such as “What would happen if we changed one coordinate?” or “How might this look in three dimensions?”
The aim of all these strategies is not to simplify or speed ahead, but to open multiple pathways for struggling students into the same core mathematical idea. This way, observation becomes a shared doorway rather than a single gate.
