By Laura Bell, Tom Dagit & Amanda Kadulski, CPM Teacher Researchers

We have long recognized the importance of helping students build meaningful connections between daily lessons, their individual learning, and the broader mathematical concepts they will encounter over time. While collaborative teamwork is a regular part of our instructional practice, we remain uncertain whether students consistently leave our classes equipped with the tools and understanding necessary to make sense of future mathematical content.

Our work suggests that the durability of learning is not a result of preserving student work, but of intentionally supporting students in consolidating their thinking.

Research into effective teaching strategies for fostering student understanding in mathematics has highlighted the importance of consolidation and student reflection as essential mechanisms for lasting learning. Aligning with the Building Thinking Classrooms (BTC) practices (Liljedahl, 2020), consolidation plays a key role in helping students make sense of their learning and connect new concepts to existing knowledge. A core tenet of BTC is the use of Vertical Nonpermanent Surfaces (VNPSs); however, the nonpermanent nature of students’ work challenges a common assumption that permanence is essential for learning. Over the last two years, our team has examined this assumption through CPM’s Teacher Research Community (TRC), operating from the claim that learning endures through intentional consolidation—not only through the permanence of written artifacts.

Do we really need a permanent record of student work to support understanding?

Our research on consolidation suggests that a permanent record alone does not guarantee sensemaking, retention, or transfer of student knowledge. While students may enjoy knowing their work is preserved and enjoyment or reassurance may contribute to mathematical understanding, they do not necessarily equate with it. Mathematical understanding develops when students engage meaningfully with mathematical ideas. While writing can support this kind of engagement, it is often insufficient on its own. Verbal sharing, listening to others, building on and critiquing ideas, and making connections between what has been shared are critical components in deepening understanding.

In the past, we had students record specific details of their work. Yet, this effort often went unused because it was not meaningful for them. Throughout our first year of research, we introduced and facilitated multiple consolidation methods (see Table 1; Bell et al., 2025). 

Table 1. Example Consolidation Methods Studied through the Teacher Research Community

Monitoring Chart	Having a chart with specific categories will help us sequence consolidation in a meaningful way.
Learning Logs/Self Reflections	Summarizing learning and reflecting on their thinking is one way for students to make connections and meaning during consolidations.
Lesson Defining Consolidation	Creating a working definition of “consolidation” with our students will allow them to understand the expectations.
Analyzing Work/Giving Feedback	Allowing students the opportunity to analyze peers’ work and provide feedback to them will require them to consider what they know about the concept and extend their understanding as they analyze other strategies and methods to record their thinking.
Four Quadrants	Providing students with some scaffolding and time to decide what is important for them to remember about a concept will strengthen their understanding and help students make their own meaning.
Mild/Medium/Spicy	Asking students to choose three problems from a list of varying degrees of difficulty and to identify why problems are categorized as mild, medium, or spicy will help them consolidate the information by comparing and contrasting problems.
Big Ideas Posters / Mind Mapping	Allowing time for students to collaborate on summarizing their learning from a unit, drawing on various ideas, and finding connections among concepts will help them see that the content is connected and concepts build upon one another.
Daily Brain Dump	Providing students with time to write down something they learned for the day and discussing as a class what they think the main learning goal was will create an opportunity for students to reflect on the concepts and make their own meaning of the goals.

After the first year of research, we were still unsure to what extent these consolidation strategies supported students’ learning and understanding. Over the course of our research, we have built consolidation strategies, aligning them to appropriate lessons. We focused on how the strategies support students’ learning and understanding.

This year, we have focused our research on how students are using the consolidation methods and which strategies are most helpful for them. We identified several themes related to students’ learning and understanding by gathering and analyzing various forms of evidence, including individual student exit tickets and teacher observations of teams’ discussions.

Through our analysis of students’ reflections on various consolidation strategies, 85% indicated that the act of consolidation, written or otherwise, was highly beneficial and deepened their understanding of what they learned. However, many students reported rarely revisiting their written work or reflections unless time was intentionally allocated in class or they were explicitly instructed to do so. Therefore, if teachers require students to write down their consolidations, they should allocate time for students to revisit and revise their ideas. For instance, one student explained that a Daily Brain Dump “allows me to put what I want to stick with me on paper.”  

Our findings suggest that learning becomes durable not because work is saved, but because students are supported in revisiting, refining, and connecting ideas during intentional consolidation. Without structured opportunities to synthesize thinking—through discussion, reflection, or comparison of strategies—the record remains static. As Kubasiak (2025) noted, students rarely return to photographs of their work, highlighting that permanence without purpose has limited instructional impact. Instead, students appear to benefit more from engaging in the consolidation process than from having access to a permanent record of their learning. For example, one student shared that Exhibit Visits help them consider multiple perspectives, stating, “I used to think _____, but then I saw other teams’ work and now I know _____. Seeing others’ work helped me to see what I was doing wrong.” Ultimately, it was the act of consolidation—thinking, revising, and making connections—that strengthened their learning, not the existence of a permanent record. These student voices underscore that durable learning is social, reflective, and student-centered, reminding teachers that authority in the classroom is shared: students become active sensemakers rather than passive recipients of knowledge.

How Does Visible Thinking Contribute to Lasting Understanding?

Our findings have clear implications for teachers’ day-to-day decisions. 

Teachers can stop assuming that students will revisit written records on their own—permanence alone does not ensure learning sticks. Instead, teachers should intentionally plan for consolidation opportunities that guide students to revisit, reflect on, and revise their thinking. This might include scheduling time for Daily Brain Dumps, incorporating Reflection Journals, check-ins, facilitating Exhibit Visits, or structuring class discussions using a monitoring sheet to sequence the discussion in a way that encourages students to compare strategies and make connections. The teacher acts as a strategic navigator during these discussions, carefully selecting and sequencing student work to create a logical path toward the lesson’s goal. 

By encouraging students to explain the ‘why’ behind their ideas and explicitly naming the connections between different strategies, the teacher transforms a collection of individual ideas into a coherent, shared understanding.  This process is vital for building student confidence; as students see their thinking evolve and align with the lesson’s objective, they gain internal certainty of ‘knowing that they know.’ They no longer rely on external validation, but find evidence of their accomplishment within their own thinking. By designing consolidation as an active and intentional process, teachers help students become more independent sensemakers and strengthen understanding that lasts well beyond the lesson.

VNPSs encourage students to lay a strong mathematical foundation through visible thinking and collaboration. Consolidation is the cement that binds those ideas together. When consolidation is embedded, for example, through Reflection Journals, Daily Brain Dumps, Exit Tickets, and class discussions, the thinking becomes internalized, making a permanent external record less critical. When students are given space to make sense of their learning in ways that are meaningful to them—rather than being told exactly what notes to record—the process becomes more useful in the long run. In this way, consolidation supports understanding that lasts beyond the lesson, even without a permanent artifact.

References

Bell, Laura, Tom Dagit, Amanda Kadulski, & Grahame Sorensen. (2025). Building Strong Consolidation. CPM Teacher Research Corps 11.0. [Unpublished manuscript]

Kubasiak, Robin. (2025, December). Rich problems on whiteboards: A powerful combination! CPM Educational Program News You Can Use. 

Liljedahl, Peter. (2020). Building thinking classrooms in mathematics: 14 teaching practices for enhancing learning. Corwin.

Schoenfeld, A. H. (2016). Mathematical thinking and problem solving (Studies in Mathematical Thinking and Learning Series). Mahwah, NJ: Lawrence Erlbaum Associates.