A Practical Approximation for Math Teacher Education

By: Jason Trumble, Ph.D., Associate Professor of Education & Interim Associate Dean, College of Education, University of Central Arkansas
Taylor McFelson, Teacher Candidate, University of Central Arkansas
De’Jon (DJ) James, Teacher Candidate, University of Central Arkansas

Learning to kayak on calm waters--an analogy for approximating practice (Grossman et al., 2009)

The promise of approximations to enhance teacher learning is exciting, yet their design and implementation require careful consideration. This blog builds on the previous post from Bondurant and Howell (2025), where they defined and examined approximations in alignment with the Teaching for Robust Understanding (TRU) framework (Schoenfeld, 2023). An approximation is a structured, scaled-down version of a teaching practice that allows teacher candidates to try and practice specific skills in a low-risk, supportive environment before applying them in real classroom settings (Schutz et al., 2018).

In this post, I collaborate with two students who participated in an approximation I created for a classroom assessment class aimed at middle and secondary teachers. First, I will describe the elements of effective approximations and introduce the TRU framework. Next, I will provide a brief overview of the approximation itself; then, my coauthors will share their experiences engaging in the approximation and describe how their experience aligns with the TRU framework. 

Elements of Effective Approximations  

Grossman et al. (2009) use the analogy of “learning to kayak on calm waters” (p 2076). This sentiment is appropriate as it encourages teacher educators to set the context, minimize pitfalls, and focus on practice fundamentals as they design approximations. Just as a beginning kayaker strengthens their paddling muscles and learns how to navigate calm waters without the wind and waves battering their boat, structured approximations allow preservice teachers (PSTs) to develop their teaching muscles in safe and controlled learning contexts. As I began thinking about implementing this approximation with my PSTs, I considered each of these elements.  

• Task Authenticity: Approximations must represent authentic tasks that are relevant to the daily work of a teacher. (Grossman et al., 2009). An approximation task design should veer away from the tediousness of learning theory and focus on the actions of teaching with a focus on honing teacher candidate practice. Decomposition of the task is also an element of authenticity one must consider when leading approximations. Teaching is complex, so decomposing the tasks to build individual skills is appropriate as long as one supports building the skills for holistic teaching excellence (Dack & Tomlinson, 2025).  

• Preparation for the Approximation: To prepare for approximations, teacher educators must ensure that candidates have an understanding of the task and are able to conceptualize their role in the learning space. They should also be equipped with the tools and support for engaging in the approximation before it starts. This includes prepping and coaching candidates and the peer audience on appropriate behavior during the approximation. 

• Context of the Approximation: Approximations can happen in various contexts, including teacher education classrooms, K-12 classrooms, and interactive digital simulation environments. It is important to remember that depending on the candidates’ experience and level of expertise, the context must be the least restrictive and intimidating environment. Grossman et al. (2009) explain that approximations that happen in real K-12 classroom settings under the watchful eye of mentors are the most authentic. However, it is appropriate to utilize approximations in varied environments to build candidate experience and confidence.  

• Scaffolding within the Approximation: Scaffolding is a vital component of any learning experience, and that is no different from approximations. Scaffolding can include controlling factors that might negatively impact the experience. Like when a participant role-playing as a student hyperbolizes their role to frustrate or derail the candidate. Having the facilitator or mentor there to mitigate situations in the moment is vital.  

• Immediate Feedback and Reflection: The final, and possibly most important, element of designing approximations is the feedback and reflection that occurs after the experience. Feedback that is immediate and focuses on decisions, actions, and teacher moves helps the candidate recognize and assimilate best practices. Feedback can take multiple forms. One-on-one consultation with the mentor or supervising teacher is a valuable tool. Discussion with role-play participants can also add to the nuance of the experience of the candidate. Meaningful reflection is essential in this process as the candidate can consider the experience and communicate ways they might change or improve.  

Elements of the TRU Framework 

“TRU is a framework for characterizing powerful learning environments in crisp and actionable ways. It provides a straightforward and accessible language for discussing what happens (and should happen) in classrooms, in professional preparation and Professional Development (PD).” (truframework.org)   The TRU framework includes five dimensions of classroom activity as shown in Figure 1 below.  

Figure 1. Teaching for Robust Understanding (TRU) framework (Schoenfeld, 2023) 

Approximation for Informal Assessment 

In my classroom assessment course, I help middle-level teacher candidates explore informal formative assessment through observations, listening, and conversations. While they can recognize these strategies in experienced teachers, they struggle to use informal data to support decision-making on future instruction. So, I created an approximation where my preservice math teachers designed an informal assessment connected to a mini-lesson on a single objective aligned with a state standard. In this approximation, the focus was to develop an activity for learners that allowed the PST to observe, listen, and talk with students while formatively assessing and collecting data using a self-designed checklist. Peers role-played as students, simulating a sixth-grade mathematics classroom setting. 

My students first analyzed various checklist structures before designing their own. They could use a physical paper checklist or a digital checklist designed for an iPad. Each checklist included the PST’s lesson objective and a key for efficiency in recording data. During my class, each PST introduced their topic and gave instructions for the task they asked the students to do, sharing grade level and objectives for authenticity. Afterward, we discussed observations, experiences, and recorded data. Role players shared feedback, and the “teacher” reflected in writing, refining their approach to informal assessment 

After engaging in the approximation, I asked two of my PSTs to share their experiences. Both are focused on middle-level mathematics and are seeking to graduate in December 2025. They each start with an overview of their experience, then discuss how they viewed their experiences in alignment with the elements of the TRU framework.  

Taylor’s Experience 

To prepare for the approximation, I created an informal assessment to evaluate students’ abilities to solve for unit rates in real-world contexts. My goal was to use this data to determine whether students were ready to move forward into the next unit on proportionality or if they needed additional support reviewing unit rates. I developed a checklist to document each student’s ability to solve unit rate problems and to track how actively they participated in the activity. The checklist provided a snapshot of their mathematical thinking and engagement. During the mini-lesson, all of the students demonstrated a solid ability to solve unit rate problems and showed high levels of participation throughout the experience. 

My experience supported the Content Domain by grounding the assessment in foundational mathematical skills—specifically, solving and applying unit rates. Unit rates are a key building block in understanding ratios and proportional relationships, so it was important to assess whether students had a strong grasp of these concepts before moving on. The task required students to engage with essential equations and real-world conversions, keeping the focus on meaningful mathematical content. 

The approximation aligned with Cognitive Demand because it included a range of problems that varied in difficulty. Some problems were more straightforward and familiar, while others required more critical thinking and careful application of strategies. This balance allowed students to experience productive struggle—the questions were clear enough that they knew what process to use but complex enough that the answer wasn’t immediately obvious. This helped deepen their understanding and gave me valuable insight into how they approached problem-solving. 

To ensure Equitable Access, the approximation was done in pairs. This setup allowed students to share responsibility for the work and created a space where everyone could contribute. I was also able to observe how students collaborated—who was leading, who was supporting, and how they communicated their thinking. The paired structure made it easier to see who was engaging with the content and where additional support might be needed. 

I intentionally stepped back from direct instruction to give students more Agency, Ownership, and Identity in their learning. They were responsible for solving the problems on their own, without my guidance, during the task. This allowed my participants to rely on their own reasoning. By encouraging independent thinking, they were able to express mathematical identities more confidently. 

This approximation was a strong example of Formative Assessment in action; in fact, it was focused on informal formative assessment. Because I was practicing and the experience itself was ungraded and low stakes, the participants could focus on the math without the pressure of earning a grade. I used the checklist to observe their thinking and gather evidence of their understanding. It gave me a real-time look at who was ready to move on and who might need more support. This type of informal data collection can help guide my instructional decisions. 

This experience showed me how powerful informal formative assessment can be when it’s used to guide instruction and support students in meaningful ways. I was able to observe not just what students knew but how they approached the task, and I gained insight into their confidence and participation as well. Below is the checklist (Figure 2) that I completed during the approximation. I am glad I had the chance to practice collecting data before implementing this in a class with real students.   

Figure 2. Taylor’s checklist

DJ’s Experience 

To prepare for the approximation, I created a checklist to track student progress during the mini-lesson. The checklist included three categories to represent student understanding: mastered, approaching mastery, and not mastered, along with a comment section to note any students who might need additional support. This tool helped me gather formative data and also gave me a rough idea of how much time would be needed for the lesson—something I think is especially helpful for early-career teachers learning to manage time and pacing effectively. 

The purpose of this task was to help us begin using informal assessment intentionally—to observe, listen, and make sense of student thinking during instruction. Informal formative assessment isn’t about assigning grades. Instead, it’s about tracking progress and noticing patterns that can inform how and when we adjust our teaching. I used quick notations—“Y” for mastered, “A” for approaching, and “X” for not yet mastered—to keep it simple and efficient while still giving me usable data. 

This approximation really helped me focus on content by encouraging students to think deeply about proportional relationships, specifically aligned to the Arkansas Mathematics Standard 7.PR.5: “Compare two different proportional relationships represented in different forms.” The lesson involved matching proportional relationships shown as fractions, decimals, ratios, and word problems. This activity helped students connect prior knowledge with new concepts, encouraging them to analyze relationships between different forms and think flexibly. 

The group work also allowed students to engage in mathematical conversations—discussing why certain representations matched and exploring the reasoning behind each decision. These discussions helped students build disciplinary habits like making connections, communicating mathematical reasoning, and developing a deeper understanding of the topic. 

The activity challenged students to interpret and compare proportional relationships across formats. It wasn’t just about finding the right answers—it required students to slow down, read carefully, and explain their reasoning. I saw how students had to wrestle with the idea that a ratio like 3:4 could be connected to a fraction like ¾ or a decimal like 0.75, and how all of these might appear in a word problem. 

By designing a task that wasn’t too easy or too hard, the approximation hit that sweet spot of productive struggle. Students stayed engaged, and the peer collaboration provided just enough support to keep the momentum going. 

I intentionally designed the activity so that every student could engage, regardless of which form they felt most confident with. Some students might be more comfortable with ratios, while others do better with decimals or visual models. By using a variety of forms, I gave everyone an entry point. 

Working in small groups also helped. Students could take on different roles and explain their thinking to one another, which not only helped reinforce understanding but created space for peer learning. This collaborative environment supported multiple ways of accessing and processing the math. 

After the matching activity, we opened the floor for discussion. Students shared how they made decisions and justified their thinking. It was powerful to hear different approaches and to watch students take ownership of their learning. I think it also gave them confidence—realizing that there isn’t just one way to think about proportional relationships and that their voice matters in the classroom. 

The checklist helped me keep track of how each student was engaging with the material in real time. I didn’t just mark answers—I made quick notes about where students hesitated or showed strong reasoning. This information gave me insight into how I might adjust my instruction next time—whether to review certain forms, pair specific students together, or slow the pace for deeper exploration. 

The post-lesson reflection was also valuable. I could look back at the checklist data alongside the class discussions and begin thinking more critically about how formative assessment informs the next steps in teaching. This experience helped me better understand what it means to use informal assessment as a real-time tool—not just for collecting data but for making responsive teaching decisions. It also showed me how much can be gained from designing activities that are thoughtful, equitable, and rooted in real classroom practice. 

Conclusions and Learning 

Although they both write about their experiences, each student learned valuable lessons during the approximation. In her reflection, Taylor discussed how she would modify her checklist to be more efficient. “If I were to use this checklist again, I would do it on paper. I found doing the checklist through my iPad more challenging than I thought it would be. I would also have organized my checklist by table or groups instead of alphabetically.” DJ also commented, “I would revise it to include a section indicating whether students participated in the formative assessment. This addition would help determine if they should receive full credit for the assessment.” 

This approximation offered a valuable, low-stakes environment for PSTs to experiment with informal formative assessment while strengthening their ability to teach mathematical content and make instructional decisions. Through thoughtful design and alignment with the TRU framework, teacher candidates were able to observe, reflect, and refine their practice in meaningful ways. This experience underscores how carefully constructed approximations can serve as powerful tools in preparing educators for the complexities of real-world teaching. 

Thank you to my co-authors for their contributions and willingness to share their thoughts and experiences.  

Acknowledgement

Thanks to Jason for serving as an editor for the 2024 ARISE blog series and for working with Liza Bondurant and Heather Howell on their blog, Conceptualizations of Effective Approximations of Practice.

Jason Trumble, Ph.D., Associate Professor of Education & Interim Associate Dean, College of Education, University of Central Arkansas
jtrumble@uca.edu

Dr. Trumble is an Associate Professor of Education and Interim Associate Dean. Before becoming a teacher educator, he served as an elementary teacher in Texas and California. His research examines the connection between teaching and technology, specifically focused on STEM development, teacher education, assessment practices, and emerging technologies. Dr. Trumble was recently elected as president-elect for the Society for Information Technology in Teacher Education (SITE). A recent publication is Theoretical and Practical Teaching Strategies for K-12 Science Education in the Digital Age.

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Taylor McFelson, Teacher Candidate, University of Central Arkansas

Taylor McFelson is a senior at the University of Central Arkansas. She is in the process of completing her Bachelor’s of Science in Education in Middle-Level Education with concentrations in mathematics and science. She hopes to become a middle school science and mathematics teacher in the public school system. 

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De'Jon (DJ) James, Teacher Candidate, University of Central Arkansas

De’Jon (DJ) James is a senior at the University of Central Arkansas, majoring in Middle-Level Education with a focus on math and social studies. Passionate about shaping the future, DJ is dedicated to making a lasting impact on students and equipping them with the skills and knowledge they need to succeed in their future endeavors.