Developing the “Character” of Mathematical Concepts

The Grasp the Meaning component of the GROWTH Framework™ focuses on the “character development” of mathematical concepts. Just as a storyteller delves into a character's personality, background, and motivations, teachers are called to deeply engage with a mathematical concept's essence. This goes beyond surface definitions to uncover its relationships, functions, and nuances. The foundational question at the heart of this process is: “What is this?”

This journey of understanding involves gathering realizations—the connections, definitions, and representations that form a rich, multidimensional view of the concept.

Understanding the "Rules of the World"

The Observe the Implications component encourages educators to explore the broader consequences, limitations, and applications of mathematical concepts. Just as a storyteller considers the potential outcomes of a character's actions, teachers ask, "What happens if…?" to uncover the rules and boundaries that govern how a concept behaves across different contexts.

This principle goes beyond surface understanding. By observing implications, educators gain insight into what can go wrong when a concept is misunderstood or misapplied. This exploration reveals the "rules of the world" in mathematics, highlighting how concepts interact with and influence one another.

By understanding these dynamics, teachers can adopt a more robust and reflective approach to instruction. They equip students to anticipate challenges, think critically, and approach mathematics with a blend of curiosity and caution.

Creating a Unified Narrative

The Weave Together Concepts component challenges educators to synthesize everything discovered so far by answering the question: "How does this all work in combination?" This integration process connects mathematical ideas while exploring the relevance, assumptions, and historical context that give each concept depth and purpose.

Weave Together Concepts emphasizes viewing mathematics as a cohesive and dynamic story. It transforms math from a collection of isolated facts into an interconnected system rich with meaning and purpose. By weaving these elements into a unified narrative, teachers illuminate the utility and beauty of each concept, inspiring students to see mathematics as a vibrant, evolving field.

Through this approach, educators not only deepen their own understanding but also empower students to think holistically, fostering curiosity and a profound appreciation for the interconnected nature of mathematics.

Collaboration as a Catalyst for Growth

The Teach with Peers component celebrates the transformative power of collaboration in teaching. Shared experiences, insights, and challenges not only deepen our understanding but also enrich our practice. Much like authors refine their stories by collaborating with editors or co-writers, teachers thrive when they engage with their peers to develop, enhance, and adapt their approaches.

At its core, Teach with Peers asks, “How can our shared knowledge enhance this teaching practice?” This collaborative mindset fosters a dynamic space where educators learn from one another, build on collective ideas, and co-create innovative solutions. By embracing teaching as a collective journey rather than an isolated endeavor, educators cultivate a supportive community that inspires growth, creativity, and resilience.

Amplifying Voices in the Learning Narrative

In the Hear Your Students component, effective teaching begins with active listening and careful observation. This goes beyond assessment—it’s about recognizing students’ voices as integral to the learning narrative. By genuinely hearing students, we capture their perspectives, struggles, and triumphs, allowing their individual and collective stories to guide and enrich our teaching approach.

Hear Your Students empowers learners to become co-authors in their mathematical journey. It creates space for their questions, ideas, and reflections, fostering a classroom culture where every voice matters. This collaborative dynamic deepens students’ engagement and strengthens their confidence as thinkers and problem-solvers, turning the act of learning mathematics into a shared, evolving story.

MathTrack’s GROWTH Framework™: Bridging Research Paradigms in Mathematics Education

The GROWTH Framework™ integrates two influential research paradigms in mathematics teacher training: Concept Analysis and Lesson Study, situated within the broader theoretical lens of Enactivism. This approach is rooted in the work of Varela, Thompson, and Rosch (1991) in The Embodied Mind, which introduced enactivism as a theory of learning. Since then, enactivism has gained prominence in mathematics education research (Ernest, 2006), positioning mathematics as a human construct shaped by experience, cognition, and societal concerns.

Enactivism emphasizes that cognition evolves in response to the world we inhabit, profoundly influencing how we understand and teach mathematics. Within this framework, knowledge is defined as adequate or viable action in the context of lived experience (Proulx, 2004). Perception, a core element of enactivism, is understood as an active process of categorization, shaped by prior interactions with the environment. As we encounter unfolding events, we apply perceptual and conceptual filters to generate insights, which are then tested and refined in the real world to produce experiential data (Boisot, 1998). This dynamic cycle constitutes “knowledge,” distinct from raw data, information, or understanding.

Enactivism underscores the inseparable nature of perception and action in lived cognition (Varela et al., 1991). Two critical components emerge: the individual cognizing agent (student/learner) and the environment with which the agent interacts. Through the GROWTH Framework™, educators learn to foster classroom environments that actively engage these interactions. Delivered in an online, job-embedded format, the framework equips educators to derive actionable insights tailored to their specific teaching environments.

GROWTH Framework™ as a Concept Study

The GROWTH Framework™ draws from the concept study methodology Dr. Brent Davis and colleagues introduced. Their research provides a pathway for exploring access, development, and study of collective Mathematics for Teaching (M4T) in collaborative environments (Davis, 2008a, 2008b; Davis, 2012; Davis et al., 2009; Davis & Renert, 2009; Davis & Simmt, 2006; Davis & Sumara, 2007, 2008; Simmt, 2011).

A concept study merges the collaborative approach of Lesson Study (Chokshi & Fernandez, 2004; Fernandez & Yoshida, 2004) with the disciplinary focus of Concept Analysis (Usiskin, Peressini, Marchisotto, & Stanley, 2003). This hybrid approach brings teachers together to collectively explore both pedagogical strategies and mathematical content, fostering deep insights into Mathematics for Teaching. By combining the strengths of collaboration and disciplinary analysis, the concept study creates a space for educators to refine their practice and enhance their understanding of mathematical ideas.

The Case for Teachers as Storytellers

Why should teachers embrace the role of storytellers? Storytelling, deeply rooted in human experience, fosters emotional connections and makes learning memorable. Research within the GROWTH Framework™—drawing from Enactivism, Concept Analysis, and Lesson Study—supports the idea that teaching is more than transmitting knowledge; it is about creating meaningful, context-rich experiences for students. Storytelling aligns seamlessly with these research paradigms by emphasizing relationships, narratives, and the active engagement of both teacher and learner.

Enactivism positions knowledge as a product of action and perception within lived experiences. When teachers adopt the role of storytellers, they weave mathematical concepts into narratives that resonate with students’ real-world experiences, helping them perceive mathematics as dynamic and meaningful. Similarly, Concept Analysis encourages deep engagement with the "character" of mathematical ideas, while Lesson Study highlights the power of collaboration to refine and enrich these narratives. Together, these paradigms underscore that teaching is an act of co-creation, where teachers guide students through stories that bring concepts to life.

This framing invites teachers to view teaching through this storytelling lens, exploring how to craft narratives that make learning meaningful, relevant, and accessible across disciplines. Mathematics, in particular, benefits from this approach. Far from being a static collection of rules, mathematics is a creative human endeavor—a story of discovery, resilience, and innovation. As modern storytellers, teachers have the unique ability to transform how students understand and experience this discipline. Through the GROWTH Framework™, educators learn how to use storytelling as a powerful pedagogical tool. 

The Purpose of Storytelling in Mathematics Teaching

Storytelling is more than an instructional strategy—it is a way to transform how we perceive and teach mathematics. The research underpinning the GROWTH Framework™ reveals that effective teaching requires more than content delivery. It involves crafting meaningful connections between concepts, experiences, and learners. Storytelling achieves this by situating mathematics within a narrative, highlighting its relevance, purpose, and interconnectedness.

Enactivism provides a theoretical foundation for this approach, emphasizing that learning arises through active engagement with the world. Storytelling embodies this principle by turning abstract mathematical ideas into relatable stories shaped by real-world contexts. Concept Study, as Brent Davis and colleagues described, builds on this by encouraging teachers to collaboratively analyze and refine their understanding of mathematical ideas, creating rich, nuanced narratives grounded in disciplinary knowledge.

By telling the story of mathematics, educators bridge the gap between standards-driven instruction and transformative learning. Standards provide a stable framework, but storytelling humanizes mathematics, turning it into a journey of inquiry and discovery. Teachers, as storytellers, play a crucial role in shaping how students connect with mathematics—helping them see it as not just a set of procedures but as an evolving, creative field that is deeply intertwined with their lives and communities.

In this era of optimism and innovation, storytelling empowers teachers to be more than content experts. They become custodians of a profound, culturally rich discipline, capable of inspiring students to see themselves as contributors to the broader story of mathematics. Through storytelling, teachers can foster a generation that views mathematics as a source of creativity, resilience, and empowerment.