Conceptualizations of Effective Approximations of Practice 

By: Liza Bondurant, Ph.D., Associate Professor, Mississippi State University
Heather Howell, Ph.D., Research Scientist, Educational Testing Service

This is the first of two blogs centered on effective approximations of practice in mathematics teacher education. In a 2024 position statement on linking mathematics education research and practice, the National Council of Teachers of Mathematics (NCTM) states that “research should identify high-leverage, effective, equitable mathematics practices shared in useful, actionable ways” (NCTM, 2024). In this blog, we argue that approximations of practice offer a promising approach to addressing the NCTM’s call by highlighting frameworks that might form a useful starting point and drawing attention to the potential areas of tension that will need to be addressed. 

Background  

We began designing, implementing, and researching the structured use of approximations in mathematics teacher education in the late 2010s. Calls to transform teachers’ professional preparation through pedagogies of practice—including approximations—have existed for over a decade (Grossman et al., 2009). Practice-based approaches are not universally accepted and have been critiqued for privileging generic practices in ways that can be reductive (e.g. Philip et al., 2019). Despite this, practice-based approaches dominate the field and are well-represented in professional standards such as the Association of Mathematics Teacher Educators (AMTE) Standards for Preparing Teachers of Mathematics (AMTE, 2017). Although we will focus on approximations in this blog, we acknowledge that instructional interventions often involve a combination of pedagogies of practice. 

We have worked with a variety of scholars focused on approximations in mathematics teacher education through working groups hosted by the North American Chapter of the International Group for the Psychology of Mathematics Education (PME-NA) and an AMTE community circle. Early working group convenings sought to establish common ground around the definition and design of approximations (Howell et al., 2019). Over the next couple of years, we explored variations across the dimensions of approximations before, during, and after the pandemic (Bondurant et al., 2021; Bondurant et al., 2022a). 

In later years, our collective interest turned to considering critiques that had been leveraged around the use of approximations. The COVID-19 pandemic prompted increased use of digital simulations, which were broadly endorsed by professional organizations (AACTE, 2020). Because simulations offered a practice opportunity when many programs were unable to access schools, many programs overlooked skepticism and raced to adoption, setting aside any concerns that might have been raised about how realistically they depicted students and the likely implications in terms of their effectiveness as learning environments for teachers’ effectiveness in practice. Recent societal shifts have further underscored the need to examine the role of potentially stereotypical representations of students in teacher preparation (Lee et al., 2023). Many scholars in our group began to ask more pointed questions about approximations. Even though we ourselves were early adopters and enthusiasts, we worried that legitimate sources of concern such as whether representing students digitally can be done without misrepresentation (e.g. Baker-White, 2021), might not have gotten adequate attention during the rush to adoption. But we also wanted to lean into the potential for approximations to offer affordances as vehicles for teachers to encounter and work with students who are different from those they work with in real life, such as students with disabilities (e.g. Cohen et al., 2021). 

Designing Effective Approximations 

After several years of engaging with and learning from the diverse work being done, we recognized the need to collect and disseminate the innovative work occurring at the intersection of approximations and effective instructional strategies (Lee et al., 2025). We acknowledge that instructional quality means different things to different people. Gutiérrez (2012, 2018) advocates for a multidimensional approach that emphasizes the need to rehumanize mathematics by valuing each and every student’s identity and experience, questioning conventional viewpoints, and promoting a learning environment where all students can thrive as mathematical thinkers. Cochran-Smith and Keefe (2022) distinguish between different interpretations of effective teaching, addressing the structural and systemic policies that influence instruction. Alan Schoenfeld’s Teaching for Robust Understanding (TRU) framework (2023) views quality instruction as ensuring all students have access to high-quality learning experiences and opportunities to develop deep mathematical understanding, emphasizing inclusive classrooms, meaningful problem-solving, student voice, and the need to address systemic barriers and biases to foster an environment where every student can thrive (see Figure 1 below).  

Common themes across the conceptualizations of equity include prioritizing instructional effectiveness; valuing students’ variable backgrounds, identities, and experiences; and emphasizing the need for deep conceptual changes in teaching practices to ensure all students can engage meaningfully and thrive. All of these are areas that the field might choose to attend to more closely in the design and use of approximations. 

Figure 1. Teaching for Robust Understanding (TRU) Framework (Schoenfeld, 2023, p. 889) 

Well-designed approximations have the potential to prepare teachers to work with a broad range of students and to use effective practices in their planning, instruction, and assessment. Moreover, allowing teachers to practice in approximations instead of with children protects children from errors and inexperience. Approximations can provide mathematics teachers opportunities to prepare for recognizing instructional challenges and creating supportive learning environments. This can be addressed through the curriculum used within the approximation, such as providing opportunities to practice incorporating relevant tasks that draw on students’ funds of knowledge (e.g., Berryhill et al., 2024; Bondurant, 2022b), or situating learning within a real-life context that empowers students as change agents (e.g., Casey et al., 2023; Bondurant et al., 2022c). Instructional quality can also be prioritized through instructional strategies in approximations, such as providing opportunities to practice ensuring active participation, randomizing who is called on, using talk moves, amplifying student voices, and leveraging student work to inform instruction—all with the belief in, expectation of, and support for every student’s potential (e.g., Bondurant & Amidon, 2021). Finally, effective assessment strategies in simulations include providing opportunities to practice implementing blinded grading; allowing for revisions, retakes, or corrections; and offering constructive feedback that encourages learner growth (Jansen et al., 2024).

However, the relationship between simulations and instructional effectiveness is not simple. The core teaching movement has been critiqued as pushing aside critical instructional considerations (Philip et al., 2019; Daniels & Varghese, 2020). Digital simulations have been criticized for their representations of students (Baker-White, 2021; Bondurant & Reinholz, 2023), and many of the same arguments may apply just as strongly to in-person rehearsals in which pre-service teachers pretend to be students in ways that may reflect their own biases (Self & Stengel, 2020). It stands to reason that any built environment that attempts to represent human beings is at risk of misrepresenting them. Generative Artificial Intelligence may make this both better and worse. For example, as we point out in Howell et al. (2025), it may do a better job of representing English Language Learners (ELLs) who are fluent in other languages, essentially solving a point of significant technical challenge in traditional approximations that currently makes it difficult to practice teaching ELLs in simulated environments. But it may also introduce or sustain bias in the representation of students. In short, the potential of simulations to be transformative in teacher learning is cause for both enthusiasm and caution. The second blog by Dr. Jason Trumble will focus on how such approximations might be designed with intentional attention to the TRU framework dimensions. 

Acknowledgments 

If you are interested in reading more work in this space, please check out a newly published book we edited with our colleagues. Promoting Equity in Approximations of Practice for Mathematics Teachers addresses the persistent gap between theory and practice in mathematics teacher education by exploring how approximations of practice can be used as transformative tools to center equity in teaching, offering a comprehensive analysis of methodologies, frameworks, and future directions to inspire equitable and impactful practices in mathematics education. 

Thanks to Jason Trumble, University of Central Arkansas, for serving as an editor for the 2024 ARISE blog series and for working with Liza and Heather on their blog. Watch for Jason’s blog on the design of approximations.

Liza Bondurant, Ph.D., Associate Professor, Mississippi State University
lbondurant@colled.msstate.edu

Liza Bondurant is an Associate Professor at Mississippi State University. A first-generation college graduate, she began her career teaching secondary mathematics in upstate New York. Her research centers on bridging theory and practice to rehumanize mathematical experiences for marginalized students, focusing on embodied cognition, mathematics identity, and equity-focused pedagogies of practice. She serves as a From the Archives Department Editor for NCTM’s Mathematics Teacher: Learning and Teaching PK-12 and is on the Editorial Board of the Journal of Urban Mathematics Education.

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Heather Howell, Ph.D., Research Scientist, Educational Testing Service
heatherhowell@icloud.com

Heather Howell’s research includes the study of teacher content knowledge for teaching and teacher learning of educational practices such as discussion. Heather holds a MS in Mathematics, a MA in Mathematics Education and a PhD in Teaching and Learning with specialization in Mathematics Education from New York University. Previously, she taught mathematics in grades 9-12 and at the undergraduate level, mathematics teaching methods, and mentored mathematics teachers in grades 6-12 in New York City. She has led or co-led numerous NSF-funded grants, predominantly focused on the use of digital simulations to support teacher learning.