Math Instructional Strategies: Using Checks for Understanding

Effective mathematics instruction depends on more than presenting a strong task and listening to a few students explain their answers. We also need ways to notice what the entire class understands while learning is still taking place. Among the most useful math instructional strategies are checks for understanding that make student thinking visible and give us information we can act on right away.

A check for understanding is not simply a pause to ask, “Does everyone understand?” It is an intentional opportunity to gather evidence about what students are thinking, interpret that evidence, and decide what should happen next. When used well, checks for understanding help us respond to students without interrupting the sense making at the center of the lesson.

What Are Checks for Understanding in Mathematics?

Checks for understanding are brief opportunities built into instruction to reveal how students are making sense of a mathematical idea. They may occur while students are beginning a task, as they develop a strategy, during a discussion, or near the end of a lesson. Their purpose is not to produce a grade. Their purpose is to inform the next instructional decision.

An effective check for understanding is a responsive cycle: gather evidence of student thinking, interpret what the evidence reveals, and use it to decide what should happen next. Each instructional response creates a new opportunity to gather evidence and continue the cycle.

Correct Answers Can Hide Incorrect Reasoning

That distinction matters. A completed page may show whether answers are correct, but it may not reveal the reasoning that produced them. A student can give a correct answer using faulty reasoning, just as a student can make an error while demonstrating productive mathematical thinking. If we look only at correctness, we can easily miss both.

Consider a common example involving exponents. When asked to evaluate 2², a student says 4. Because the answer is correct, it would be easy to move on. But when the student is asked to evaluate 3², they say 6. After being told the answer is incorrect, the student responds, “But I did the same thing. I multiplied the big number by the little number.”

The first correct answer did not demonstrate an understanding of exponents. It concealed a misconception because multiplying 2 by 2 happened to produce the same result as 2². A check for understanding must help us uncover the reasoning behind an answer before a convenient set of numbers makes that reasoning appear correct.

Incorrect Answers Can Reveal Productive Thinking

Incorrect answers can be equally revealing. Recently, I was practicing mentally adding 10 with my seven-year-old niece in the car. We had worked through several teen numbers, and I had emphasized that when we add 10, the number of tens increases by one while the number of ones remains the same. When I asked her to solve 57 + 10, she answered 27.

Her answer initially seemed to come out of nowhere. When I asked how she was thinking, however, she explained, “Because 10 + 10 = 20.” She appeared to be treating 57 as though it were 10 + 7, reasoning that 10 + 10 = 20 and then preserving the 7. Her answer was incorrect, but it was not random. She had generalized reasoning that worked with teen numbers without yet attending to the five tens in 57.

That explanation gave me far more useful information than the answer alone. Useful math instructional strategies help us uncover misconceptions hiding behind correct answers and recognize productive thinking within incorrect ones.

Math Instructional Strategies Must Include More Than a Few Voices

In many classrooms, the students who volunteer most quickly become the unofficial measure of whether the class is ready to move on. If two or three students provide correct answers, it can appear that the idea is secure. Meanwhile, quieter students—and students who need more processing time—may remain invisible.

One of the most important features of effective checks for understanding is that they allow every student to respond. This does not mean every response needs to be shared publicly. It means we need a way to see or hear enough thinking to understand patterns across the room.

Use Individual Whiteboards During Mental-Math Warm-Ups

In my work with teachers and students, individual whiteboards are one of my favorite math instructional strategies for making student thinking visible during a warm-up, number talk, or other short mental-math routine. Every student can record an answer, sketch a representation, write an equation, or jot down part of a strategy. When students display their boards at the same time, we can quickly read the room instead of relying on the few students who raise their hands.

Although mental-math routines are intended to develop reasoning and flexible computation, “mental” does not have to mean students are prohibited from writing anything down. The important distinction is between recording thinking and using a written procedure to replace the intended mental reasoning. A student might record an answer, capture a few quantities they want to remember, or sketch how they decomposed a number. For some students, this small amount of recording reduces the working-memory demands of the task and makes it easier to participate. The whiteboard should support the thinking, not become a requirement that every student produce a polished written solution before speaking.

Individual whiteboards provide a quick way to gather a response from every student and scan for strategies, representations, misconceptions, and questions before beginning a whole-class discussion.

Individual dry-erase boards from EAI Education are one option for incorporating this routine into warm-ups, number talks, and other formative checks.

Use Vertical Non-Permanent Surfaces During a Task

Vertical non-permanent surfaces serve a different purpose. Rather than collecting a brief response from every student during a warm-up, they give pairs or small groups space to develop, revise, and share their reasoning as they work on a task. Students can add to their work, reorganize their ideas, or try another representation as their thinking changes. As their work is visible across the room, we can monitor how ideas develop and decide which strategies to press, compare, or connect.

In rooms where wall space is limited, tabletop magnetic dry-erase easels can provide a flexible alternative. I especially like the magnetic option for card sorts because cards can remain visible and movable as students organize, revise, and explain mathematical relationships. These tabletop easels from Amazon are one option.

During a Grade 4 demonstration lesson, students worked in groups at vertical non-permanent surfaces to reason about 6 × 13 in different ways. One group represented six groups of 13 and tracked partial totals. Another decomposed 13 into 10 and 3, reasoning that 6 × 10 = 60 and 6 × 3 = 18. Both approaches led to 78, but the boards revealed different ways of thinking about the factors. Although these photographs feature Grade 4 students, vertical non-permanent surfaces can make developing mathematical thinking visible across grade levels.

 During a Grade 4 demonstration lesson, groups use vertical non-permanent surfaces to develop and share different ways of reasoning about 6 × 13. Although these examples come from Grade 4, working at vertical non-permanent surfaces can make students’ developing strategies visible across grade levels.

Gather a Quick Response from Everyone

Not every check for understanding requires students to write. A quick poll is one of the simplest formative math instructional strategies for gaining an immediate snapshot of how the class is thinking. Students might show thumbs up for agree, thumbs down for disagree, or a thumb to the side for unsure or not yet convinced.

The “not sure” option matters. It communicates that uncertainty is useful information rather than something students need to hide. A poll can help us decide whether to continue, pause, invite contrasting explanations, or give students time to reconsider. However, a thumb position does not reveal the reasoning behind it. Follow-up prompts such as “What influenced your decision?” or “What would you need to know to feel more certain?” turn the poll into a more meaningful check.

A three-position thumb poll—agree, disagree, or not sure yet—provides a quick snapshot of the room. Follow-up questions are still needed to uncover the reasoning behind each response.

Use Turn and Talk to Hear Developing Reasoning

Turn and talk can be especially valuable with younger students who may communicate more of their thinking orally than they are ready to record independently. Students can first consider a visual, quantity, or mathematical claim and then tell a partner what they notice, how they know, or whether they agree.

As students talk, we can listen for the ways they are counting, the relationships they notice, the language they use, and the ideas they may be ready to share with the class. These brief conversations can reveal understandings and misconceptions that may not yet be visible in students’ written work. When we listen with a specific mathematical purpose, turn and talk becomes more than a participation structure; it becomes a useful check for understanding.

During a demonstration lesson early in Grade 2, students use turn and talk to describe what they notice and explain their developing mathematical ideas. Although this photograph features younger learners, purposeful partner talk can reveal student reasoning across grade levels when we listen with a specific mathematical goal.

Use Multiple-Choice Questions to Reveal More Than an Answer

Strategically designed multiple-choice questions are another example of how math instructional strategies can reveal far more than whether students selected the correct answer. When the choices reflect meaningful mathematical relationships or likely ways of thinking, each response gives us information we can use. The amount of time students spend responding can also offer a clue about whether they are reasoning from a known relationship or stopping to calculate.

During a Grade 5 demonstration lesson, I used Plickers to pose a question connected to multiplication as scaling. Students were asked whether (9/16) × (2/3) would be smaller than, larger than, or equal to 2/3.

Note: Online content management tools don't always handle math notation well. If a fraction shows up with a slash, just remember that in the classroom, we want to format them as shown in the graphic below, since that helps students really see the numerator and denominator.

This Grade 5 multiplication-as-scaling question can reveal whether students reason about the effect of multiplication or apply a computational procedure. Response time provides one piece of evidence, while students’ explanations help us understand the reasoning behind their choices.

A student who understood the effect of multiplying by a factor less than 1 could answer almost immediately: because 9/16 is less than 1, the product must be less than 2/3. No multiplication algorithm was needed.

Yet many students immediately began multiplying the numerators and denominators and then comparing the product with 2/3. Some eventually selected the correct response, but the time they spent and the work they recorded suggested that they were calculating rather than reasoning about the effect of the multiplication. That did not tell me everything they understood, but it told me what I needed to investigate next.

Plickers is one of many response tools that can gather a multiple-choice answer from every student quickly. I like it for this purpose because students can respond individually with reusable cards, allowing me to quickly scan the room and collect response data from the entire class. In this example, the results showed that only 8 of 18 students selected “smaller than 2/3,” while 10 students selected “larger than” or “equal to” 2/3. That distribution immediately showed me where I needed to probe, but it still did not explain why students chose each response. The selected choice provided one piece of evidence and response time provided another. Follow-up questions such as “How did you know?” helped me distinguish between a correct answer produced through computation and a quick response grounded in an understanding of the Grade 5 scaling standard.

Plickers provides a quick way to gather and scan individual response data. In this Grade 5 demonstration lesson, the distribution identified an idea that needed further investigation, while follow-up questions helped uncover the reasoning behind each choice.

Plan Questions That Reveal Mathematical Thinking

The quality of math instructional strategies such as checks for understanding depends on what we ask students to do. Questions such as “Do you understand?” or “Are there any questions?” invite a yes-or-no response rather than mathematical evidence.

And let’s be honest: Unless we have intentionally cultivated a classroom culture in which uncertainty is expected and valued, many students will not feel secure enough to admit that they do not understand. Others may not yet have the metacognitive awareness to identify what is confusing or formulate a question. As a result, a room can remain completely silent even when many students are unsure of what to do or why a mathematical idea works.

Instead, we can plan prompts that focus attention on a specific idea. We might ask students to choose which equation matches a representation, explain why two strategies are equivalent, identify an error, or decide which solution method is more efficient for a particular set of numbers. We can also follow an answer with “How were you thinking?” before confirming whether it is correct.

Anticipate What Student Responses Might Mean

As formative math instructional strategies, checks for understanding are most useful when we anticipate possible responses before the lesson. If students select an incorrect representation, what misunderstanding might that choice reveal? If they use a particular strategy, what does it suggest they understand about the numbers? If several approaches are likely, which connections will be important to make visible?

Anticipating does not mean predicting every response. It means preparing to recognize a few likely strategies, partial understandings, and misconceptions. This preparation helps us listen for meaning instead of making an immediate judgment about whether an answer is right or wrong.

This is where checks for understanding connect closely to Margaret S. Smith and Mary Kay Stein’s 5 Practices for Orchestrating Productive Mathematics Discussions. Anticipating prepares us to recognize student thinking. Monitoring helps us gather evidence while students work. Selecting and sequencing allow us to use that evidence intentionally, and connecting helps the class make sense of the mathematical relationships among the responses.

Use Math Instructional Strategies to Respond to Evidence

The final step is the one that makes checks for understanding genuinely formative: deciding how to respond. The same check should not automatically lead to the same instructional move.

If most students demonstrate strong understanding, we might invite them to compare strategies or generalize the relationship they noticed. If responses are split, we might select contrasting representations for discussion. If many students share the same misconception, we might pause for a new example, return to a representation, or revise the task so the important idea is easier to see.

Sometimes the best response is not a whole-class explanation. We may ask a follow-up question, pair students strategically, confer with a small group, or hold onto the evidence when planning the next lesson. What matters is that the information changes something about what we do.

This is also why an exit ticket, while valuable, cannot be our only check for understanding. Evidence collected at the end of the lesson can shape tomorrow’s instruction, but evidence gathered during the lesson gives us an opportunity to respond today.

Use Math Instructional Strategies to Move Learning Forward

Checks for understanding help us make instruction more responsive because they keep student thinking at the center of our decisions. A whiteboard, quick poll, turn and talk, strategically designed multiple-choice question, or exit ticket is only as useful as the mathematical idea it reveals and the instructional response it supports.

The goal is not to stop instruction every few minutes or sort responses quickly into correct and incorrect. The goal is to create purposeful moments when every student can show enough of their thinking for us to make a better decision about what should happen next.

Strengthen Mathematics Discussions with the 5 Practices

The connection between checks for understanding and productive discussion is especially important. We cannot purposefully select, sequence, and connect student ideas unless we know what students are thinking. Thoughtful monitoring allows us to move beyond calling on volunteers and build a discussion around evidence gathered across the room.

Coaching That Counts is kicking off the 2026–2027 open workshop season with Orchestrating Productive Mathematics Discussions Using the 5 Practices on October 13 in Cherry Hill, New Jersey. During this full-day workshop, we will explore how to anticipate likely approaches, monitor with purpose, and use student work to build discussions that advance important mathematical ideas.

Ready to strengthen the way you notice, interpret, and connect student thinking? Use the 2026–2027 one-day workshop registration form to register for the 5 Practices workshop or explore the full workshop schedule.