Rethinking Formative Assessment Math Practices

Formative Assessment Math Practices

When researchers Paul Black and Dylan Wiliam published Inside the Black Box (1998), they made a compelling argument: assessment should be used to improve learning, not just to measure it. They introduced the concept of formative assessment, describing it as the process of gathering evidence about student understanding and using that information to adjust instruction in real time (Black & Wiliam, 1998). Their research demonstrated that when teachers regularly engage in responsive formative assessment practices, student achievement, especially for lower-performing students, significantly improves.

This signaled a fundamental shift from summative assessment, which focuses on evaluating student learning after instruction is complete, often in the form of tests, standardized exams, or final projects. Summative assessments play an important role in measuring overall achievement, but they do not inform instruction in the moment. Formative assessment, by contrast, is an ongoing process, allowing teachers to identify misconceptions, adjust lessons, and provide targeted support before learning moves forward (Wiliam, 2011).

Since the time of Inside the Black Box, formative assessment has become a core instructional practice across content areas for many teachers, often taking the form of exit tickets, polls, quick writes, and informal checks for understanding, such as traffic light cards. These tools provide valuable, real-time snapshots of student understanding, offering teachers quick indicators of progress and areas that may need review. 

However, while they are useful for gauging surface-level comprehension, they do not always reveal the reasoning behind student responses. Many formative assessment math practices, for example, focus on measuring outcomes rather than uncovering how students are thinking, making it difficult to determine whether students truly grasp a concept (Heritage, 2010).

As an example, misconceptions can go unnoticed when students arrive at the right answer through memorization or procedural shortcuts rather than conceptual understanding (Burns, 2021). This means that teachers may unknowingly move forward with instruction before students have built a strong foundation of understanding.

To fully realize formative assessment’s original promise, educators need tools that go beyond surface-level responses and uncover the reasoning behind them. Think-alouds and student interviews provide just that. They offer structured opportunities for students to verbalize their thought processes, making their understanding visible in ways that many other formative assessment practices cannot.

Integrating these strategies provides a way for teachers to identify student misconceptions in the moment allowing them to adjust their instruction more effectively. This shift represents a broader transformation in assessment, moving from assessment as measurement to assessment as a tool for learning.

Think-Alouds as a Formative Assessment Math Practice

A think-aloud is an instructional strategy where an individual verbalizes their thought processes while engaging in a task like reading or problem-solving. Traditionally, think-alouds have been used by teachers as a modeling strategy, where they demonstrate how proficient readers or problem-solvers navigate complex material (Leighton, 2017). For example, a teacher might read a passage aloud and detail their thoughts on predicting content, deciphering vocabulary, or making inferences, giving students an example of metacognitive thinking.

However, think-alouds do not need to be limited to teacher modeling. When students engage in think-alouds while problem-solving in math, they externalize their reasoning, allowing teachers to assess not just whether they arrive at a correct answer but how they arrived at that answer. Leighton (2017) describes think-alouds as a valuable formative assessment math practice, enabling educators to identify misconceptions, gaps in understanding, or alternative problem-solving approaches that may not be obvious in written responses alone. 

Despite being very effective, one-on-one think-alouds can be difficult to implement in typical classroom settings because of the time they take to implement. While the insights into a student’s cognitive processes are rich, scaling think-alouds for a full classroom is often impractical. To address this limitation of one-on-one think-alouds, researchers have explored strategies that allow multiple students to verbalize their thinking at the same time. 

One such approach is collaborative verbalization, where students work in pairs or small groups to discuss their strategies and thinking while solving problems. This method is rooted in cognitive apprenticeship theory, which emphasizes the importance of articulation and reflection in learning (Collins, Brown, & Newman, 1989). When students are engaged in these types of structured peer-to-peer discussions, they are more likely to refine their reasoning, clarify misconceptions, and develop a deeper understanding of mathematical and conceptual ideas.

Whole-Class Think-Alouds (WCTAs) build on these principles by structuring student-led verbal reasoning within a full-class setting. Teachers facilitate WCTAs by having students pair up and verbalize their reasoning while solving problems. While students discuss their thought processes, the teacher circulates, listening for key insights, noting misconceptions, and gathering formative assessment math data in real time. This method allows teachers to assess multiple students at once while still gaining rich qualitative data about student understanding (Hicks & Bostic, 2021).

Incorporating student-led think-alouds and collaborative verbalization strategies into formative assessment math practices is a way for educators to move beyond traditional assessment methods that often only capture correct or incorrect answers. These formative assessment math practices provide a deeper understanding of student learning by making student thinking visible, allowing for targeted and responsive instruction (Leighton, 2017; Hicks & Bostic, 2021).

Student Interviews as a Formative Assessment Math Practice

Student interviews are one-on-one or small-group conversations in which teachers ask students to explain their thinking about a concept, problem, or skill. While student interviews are widely used in educational research to understand student cognition (Leighton, 2017), they are less commonly applied as a classroom formative assessment math practice. However, teachers who conduct student interviews gain insight that is difficult to capture through written work alone (Burns, 2010).

Unlike written formative assessment math practices, student interviews allow teachers to listen to students’ reasoning in real time. These conversations can range from a brief, informal check-in lasting a couple of minutes to a more structured discussion with targeted questions designed to uncover students’ understanding. Interviews also give students a chance to clarify their own thinking. Sharing their ideas out loud helps students identify gaps in their understanding and refine their reasoning. This aligns with metacognitive research, which suggests that explaining one’s thought process strengthens conceptual learning (Leighton, 2017).

Despite their benefits, conducting one-on-one interviews for every student in a class can feel overwhelming, particularly in time-limited instructional settings. To address time constraints, teachers can conduct small-group interviews instead of individual ones. In this approach, students work in groups of three or four, discussing their reasoning while the teacher listens and asks guiding questions. This structure allows teachers to gather insights from multiple students at once while fostering peer-to-peer learning.

Another option is peer-assisted verbalization, an approach grounded in cognitive apprenticeship theory (Collins, Brown, & Newman, 1989). In an example of peer-assisted verbalization called Think-Pair-Share, students contemplate a question individually, pair up and take turns explaining their reasoning to a partner, and then share their insights in a whole-class discussion. This method helps students organize their thoughts, build confidence in articulating ideas, and prepare for teacher-led discussions in a low-stakes environment. While students are paired up, the teacher can be circulating and listening, identifying patterns in student thinking and misconceptions that can inform future instruction.

The Benefits of Think-Alouds and Student Interviews

Formative Assessment Math Practices Provide Instructional Insights

One of the most significant advantages of think-alouds and student interviews is their ability to make student thinking visible as it develops. Heritage (2010) describes formative assessment as a process that unfolds within instruction, rather than something that happens afterward. These strategies allow teachers to recognize gaps in understanding immediately rather than waiting for written work to reveal them. 

Creating More Equitable Formative Assessment Math Opportunities

Traditional assessments often favor students who express their thinking well in writing, making it difficult to gauge the understanding of students who process information differently. Think-alouds and interviews provide an alternative means of demonstrating learning, offering all students (not just those who excel at written expression), a way to share their reasoning.

Hicks and Bostic (2021) emphasize that verbal reasoning assessments create a more inclusive classroom, particularly for multilingual learners and students with learning differences who may have strong conceptual knowledge but struggle with written responses. These strategies provide a more accurate and equitable reflection of learning by capturing students’ thought processes verbally, rather than relying solely on written work.

Supporting Metacognition and Student Engagement

Think-alouds and student  interviews do more than help teachers assess understanding. They also help students become more reflective learners. When students articulate their reasoning, they engage in metacognitive processes that deepen comprehension and strengthen problem-solving skills (Leighton, 2017). Verbalizing their thoughts requires them to slow down and consider whether their approach makes sense, helping them become more aware of their own learning strategies. This process encourages students to monitor their thinking, recognize inconsistencies, and refine their approaches. As a result, they not only develop stronger reasoning skills but also gain confidence in expressing their ideas.

Building a Classroom Culture of Discourse and Reasoning

Integrating think-alouds and student interviews into classroom practice fosters an environment where reasoning and discussion are central to learning. When students are regularly asked to explain their thinking, they become more comfortable engaging in academic discourse, listening to different perspectives, and refining their ideas based on new insights.

Hicks and Bostic (2021) highlight that classrooms where students are expected to discuss their reasoning see increased engagement and more productive mathematical conversations. These interactions help students develop the ability to analyze different problem-solving approaches, challenge assumptions, and construct more robust explanations, which are important skills that extend beyond the classroom.

Formative Assessment Math Practice Implementation

Implementing Think-Alouds as a Formative Assessment Math Practice

Think-alouds are most effective when they are intentionally structured, and used consistently. Before introducing them into instruction, teachers must consider the learning goal. Think-alouds work particularly well for multi-step problems that require students to make decisions, such as choosing an efficient strategy for multi-digit multiplication or determining how to compare fractions with different denominators. 

To introduce think-alouds, teachers should begin by modeling the process themselves. For example, when solving a problem on the board, the teacher can verbalize their thought process. This explicit modeling not only shows students how to approach a problem but also helps normalize the practice of verbalizing mathematical reasoning. After repeated exposure, students will see this as a natural part of doing math, and will become more comfortable thinking aloud and explaining their own strategies.

Once students are familiar with teacher-led think-alouds, they can begin engaging in structured think-alouds themselves. Younger students, particularly in K-1, may need visual supports, gestures, and oral sentence stems to help express their thinking, while older students in grades 2-3 may transition to more complex verbal or written explanations.

For early elementary students, verbal and nonverbal prompts might include:

For students in grades 2-3, who are developing verbal reasoning and early writing skills, prompts might include:

Over time, students should be encouraged to rely less on sentence stems and develop their own ways of explaining their thought processes. 

Embedding think-alouds into daily routines and familiar classroom structures provides natural opportunities for students to verbalize their thinking without taking up large portions of instructional time.

For example, quick “turn-and-talk” think-alouds during math talks, where students explain their reasoning to a partner in 30–60 seconds, provide frequent, low-pressure opportunities for students to practice verbalizing their thinking. In partner problem-solving, students can reinforce their understanding by discussing their strategies with a peer.

Math journals can extend this process, allowing students to build on their verbal reasoning by transitioning to written explanations or drawings.

Implementing Student Interviews as a Formative Assessment Math Practice

While think-alouds occur naturally within a lesson, student interviews require more structured planning. These one-on-one or small-group discussions give teachers the chance to listen closely to student reasoning in a way that written work often does not capture. The goal of an interview is not to assess whether a student can calculate an answer, but rather to understand how they arrived at it.

Defining the purpose of the interview is essential for making it meaningful. Teachers should consider what they are hoping to learn: Are they diagnosing a common misconception? Checking for flexibility in problem-solving strategies? Assessing whether students understand relationships between numbers? Clarifying the purpose helps shape the structure and focus of the conversation.

Interviews can be conducted individually or in small groups, depending on classroom time constraints. While individual interviews provide the most detailed insight, small-group interviews allow multiple students to discuss their thinking while the teacher listens and asks follow-up questions. To maximize efficiency, teachers can:

• Rotate interviews so each student is seen at least once every one to two weeks.
• Use math stations or independent tasks so the rest of the class remains engaged while interviews take place.
• Conduct exit interviews, pulling a few students aside after a lesson to discuss their reasoning briefly.

As with think-alouds, student interviews should focus on open-ended, thought-provoking questions rather than procedural ones. Instead of asking “What answer did you get?”, teachers should prompt students to explain their reasoning:

• “Can you show me two ways to solve this problem?”
• “What do you think will happen if we change this number?”
• “Can you explain what this problem is asking?”

When first introducing student interviews, teachers should explicitly model what a strong mathematical explanation sounds like. Recording a sample interview, either with a student or through a scripted demonstration, can help students see the difference between a vague response and a detailed explanation. With practice, students will become more comfortable articulating their thinking.

How to Use Think-Aloud and Interview as a Formative Assessment Math Data to Adjust Instruction

Think-alouds and student interviews are only useful if the information they generate is used to shape instruction. After gathering insights, teachers should analyze student responses to identify common misconceptions (e.g., students believing multiplication always makes numbers bigger), student strategy flexibility (e.g., Are they trying different approaches or sticking to one method?), and confidence levels (e.g., Are students hesitating or struggling to explain their thinking?).

Once patterns emerge, instruction can be differentiated accordingly. For students struggling with foundational concepts, for example, additional small-group instruction using hands-on materials (e.g., place value blocks or fraction strips) may be necessary. If students are using inefficient strategies, teachers can facilitate strategy comparisons, prompting students to explore multiple methods to determine the most efficient one. For students demonstrating mastery, teachers can provide extension tasks that challenge them to apply their knowledge in new ways.

Measuring the success of these strategies requires looking beyond correct answers. Teachers should observe whether students are becoming more engaged in mathematical discussions, demonstrating greater flexibility in their problem-solving strategies, and responding more accurately to teacher interventions. A shift from rote procedures to thoughtful reasoning signals that think-alouds and student interviews are making a meaningful impact on student learning.

From Answers to Understanding

Formative assessment math practices should serve as more than just a tool for collecting data. They should actively inform instruction, uncover student thinking, and shape meaningful learning experiences. Traditional formative assessment math practices, while valuable, often fail to capture the depth of student understanding, leaving teachers to infer misconceptions rather than directly addressing them.

Integrating think-alouds and student interviews as formative assessment math practice allows teachers to gain direct access to student reasoning, leading to immediate, informed instructional adjustments. These strategies promote deeper mathematical discussions, help students develop metacognitive awareness, and create a more equitable assessment process that values diverse ways of demonstrating understanding.

Although implementation of these alternative formative assessment math practices may require some adjustments to start, the long-term benefits make these strategies worth the effort. A good starting point is to introduce one structured think-aloud per lesson, and to plan for periodic student interviews, giving students time to get used to the process. As they become more comfortable verbalizing their thinking, these formative assessment math practices will start to feel like a natural part of the classroom routine, helping students articulate their ideas more clearly while giving teachers better insight into how they learn.

Shifting formative assessment math practices from a product-focused approach to a process-driven one ensures that assessment actively contributes to learning rather than simply measuring it. Classrooms that center on reasoning, discussion, and reflection provide students with opportunities to express their thinking, refine their understanding, and engage in meaningful mathematical discourse. When formative assessment math practices capture what students know and how they think, instructional decisions become more precise, and student learning becomes more deeply rooted.