Zen and the Art of Knowing: Mastery in Mathematics Teaching

Introduction: The Zen of Teaching Mathematics

Teaching mathematics is more than just delivering content; it is an art form, a journey toward mastering the deep understanding required to truly engage and inspire students. A key word in that statement is elusive: the word mastery. What does it mean to master the deep understanding required to teach mathematics? Like the journey Robert M. Pirsig describes in Zen and the Art of Motorcycle Maintenance, where he explores the concept of quality through the dual lenses of the technical and the philosophical, teaching mathematics requires a similar balance. The true mastery of teaching lies in knowing mathematics and how to narrate its stories in a way that resonates with students. This is the craftsmanship of and responsibility for student maintenance, laboring to support student learners from “not yet” to thriving and joyful intellectual beings.

The Quality of Knowing: More Than Just Content

Classical and Romantic Understanding in Mathematics Teaching

In Zen and the Art of Motorcycle Maintenance, Pirsig introduces the idea that there are two fundamental ways of understanding the world: the classical and the romantic. The classical mode is rooted in logic, precision, and the technical mastery of details. On the other hand, the romantic mode is intuitive, creative, and focused on the beauty and experience of the whole. This duality defines the discipline of mathematics, and both modes are essential in teaching mathematics.

Classical Understanding

To teach mathematics effectively, teachers must have a technical and detailed understanding of the subject. In mathematics education research literature, this is referred to as Common Content Knowledge (CCK), Specialized Content Knowledge (SCK), and Pedagogical Content Knowledge (PCK)(Ball et al., 2008; Davis et al., 2005; Davis & Simmt, 2006; Scheiner et al., 2019; Schoenfeld, 2020). This type of knowledge, as outlined by Ball, Thames, and Phelps (2008), involves more than just understanding mathematical concepts—it requires knowing how to represent these concepts effectively in the classroom, anticipating student misconceptions, and using the appropriate pedagogical strategies to address them. Davis, Simmt, and Sumara's (2005) article further emphasizes that this knowledge is a unique form of applied mathematics, now broadly called "mathematics-for-teaching." This concept suggests that teaching mathematics requires a distinct form of knowledge that is actively constructed and adapted in response to the complexities of the classroom environment.

“We take a handful of sand from the endless landscape of awareness around us and call that handful of sand the world.”― Robert M. Pirsig

This classical approach to teaching knowledge is akin to the meticulous attention to detail that Pirsig advocates for in maintaining a motorcycle. Just as a mechanic must understand the machine's inner workings, a teacher must grasp mathematics's underlying structures and nuances to guide students effectively. The research highlights that this understanding is not static but dynamic, developed through continuous reflection and interaction with both the content and the learners. The Zen of being fully present, reacting to your environment, and achieving “satori” or “kensho” in the art of teaching.

Romantic Understanding

Pirsig describes the romantic mode of understanding as intuitive, creative, and focused on the experience of the whole rather than the details of its parts. You thought mathematics wasn’t creative? This approach is not about dissecting the world into its components but about perceiving it as an interconnected, living reality. In the context of mathematics teaching, romantic understanding is the ability to connect students to the beauty and relevance of mathematics, transforming abstract concepts into meaningful, lived experiences. This aligns with the idea of "mathematics-for-teaching," where knowledge is not merely transmitted from teacher to student but co-constructed in real-time through the interactions and experiences within the classroom. This perspective emphasizes that teaching is a creative, responsive act where the teacher must engage with the subject matter and the students as a living, evolving entity.

Just as a Zen master perceives knowing as an experiential, non-dual awareness, romantic understanding in teaching involves a similar kind of intuitive insight. It is about being fully present with the material and the students, sensing the underlying connections that might not be immediately visible through logical analysis alone. Schoenfeld (2020) captures this when he describes teaching as an inherently creative act requiring flexibility and the ability to adapt to the needs and curiosities of students. Scheiner et al. (2017) emphasized that effective teaching involves transforming content knowledge into pedagogical forms that are accessible and meaningful. This transformation is not merely a technical exercise but an artistic one—where the teacher draws on creativity and empathy and can narrate the discipline through storytelling to bridge the gap between abstract concepts and students' lived experiences. In the classroom, this means recognizing that mathematics is not just a set of procedures to be memorized but a way of seeing and interacting with the world. It is about making the abstract concrete, the complex simple, and the unfamiliar familiar.

"When we make the unfamiliar familiar, make the unknown known, make the uncomfortable comfortable, and believe the unbelievable, we can then expect the unexpected"~ Charles F. Glassman

In this way, a teacher with a strong romantic understanding sees mathematics not just as a subject to be taught but as a narrative to be shared—a story constantly being written in the minds and experiences of the students. This reflects the Zen principle of interconnectedness, where nothing exists in isolation, and every action, thought, or lesson has the potential to influence the whole. This understanding allows the teacher to be more than just an instructor; they become a guide, leading students through the landscape of mathematics with an intuitive and deeply connected awareness of the human experience.

The Journey to Mastery: The Teacher as a Craftsman

Pirsig’s journey across the country on a motorcycle is more than just a physical trip; it is a metaphor for the journey toward mastering quality. Similarly, the journey to becoming a master mathematics teacher is not a destination but a continuous process of growth, reflection, and improvement.

“Sometimes it's a little better to travel than to arrive” ~ Robert M. Pirsig

The teacher’s path is one of continuous learning, where each lesson and each student interaction is an opportunity to refine one’s craft and achieve a higher level of mastery. This is why MathTrack Institute builds all of its programming embedded within the work of teaching. Mastery is embodied in a teacher’s behavior; they teach who they are to students through modeling. Mathematical habits of mind or intellectual ways of being are modeled for students, not learned through rote memorization.

Mastery is not achieved overnight. It results from years of practice, reflection, and a willingness to embrace mistakes as opportunities for growth. Boaler and Dweck’s work (Boaler, 2015, 2024; Dweck, 2006) clearly shows the advantages of this mindset for mistake-making. Pirsig emphasizes that the journey is as important as the destination, especially in teaching. Each challenge faced in the classroom, each moment of uncertainty, is part of the teacher’s journey toward becoming a master. They should receive credit and credentialing competency for this journey, and at MathTack Institute, they do.

Teaching as Craftsmanship and Artistry

A teacher must approach each lesson with the same care and precision as a mechanic approaches a motorcycle. Every concept taught, and every question answered is part of a more extensive system that needs to be understood and maintained. A skilled teacher, like a skilled mechanic, knows when to be precise, when to be creative, when to follow the rules, and when to think outside the box.

If everyone is thinking alike, then somebody isn't thinking. ~George S. Patton

Pirsig argues that quality is not just about getting the job done but about doing it in a way that embodies excellence. In the classroom, this means creating informative and engaging lessons that challenge students while supporting them. Quality teaching ensures that every student learns and grows in their appreciation and understanding of mathematics. You know what this classroom feels like. You’ve felt it before.?

“The test of the machine is the satisfaction it gives you. There isn't any other test. If the machine produces tranquility, it's right. If it disturbs you, it's wrong until either the machine or your mind is changed.” ~ Robert M. Pirsig

When I speak about craftsmanship and artistry, I also refer to the human element of teaching. One of the critical messages in Zen and the Art of Motorcycle Maintenance is that quality is inherently tied to the human element. This means that the connection between the teacher and the student is crucial for meaningful learning. A teacher who has mastered the art of knowing in mathematics can create a lasting impact on their students. They don’t just teach the material; they inspire curiosity, confidence, and a love for learning. This is the ripple effect of quality teaching—it extends beyond the classroom, shaping students’ attitudes toward mathematics and their future academic pursuits.

Connecting to the GROWTH Framework: Grasping the Meaning (G)

The Zen of Grasping the Meaning in Mathematics Teaching

In the GROWTH framework developed by MathTrack Institute, the "G" stands for Grasping the Meaning. The word grasp has specific and Zen connotations in our work. It isn’t about seizing and holding tightly that mathematics is something that you can know everything about; it's about the idea that you actually cannot do this.

“The stuff we hold onto so dearly brings pain, only because we hold on to it in the first place. But, if we look closely, we’ll discover that nothing is gained from holding on.” ~Steve Hagen

This is the foundation upon which all other GROWTH framework components are built. For teachers, Grasping the Meaning involves not just seeking to know the content but deeply understanding it—knowing it in a way that allows them to let go of the rigid interpretations of the mathematical concepts. For example, to explain to a child why you can’t divide by zero, you could take a rigid approach that lacks an understanding of mathematics and the needs of novice students. The most likely one is, well it has always been that way. Not satisfying or correct. Zero was invented, and so was division, at different times. So there isn’t a scenario where either has always been that way. When you grasp the meaning of division, you can understand it as repeated subtraction. So division by zero is asking how many zeros you can subtract from a number like 12 till you run out of stuff. Well… you can do that forever and not get anywhere. Therefore, it's undefined.

Just as Pirsig’s concept of quality requires a deep understanding of motorcycle maintenance's technical and philosophical aspects, quality in mathematics teaching requires a deep understanding of the content, self, student, and pedagogy. Teachers must grasp the meaning of the material they teach, understand it from multiple angles, and know how to narrate it to students in a clear and engaging way. By focusing on Grasping the Meaning, the GROWTH framework ensures that teachers are not just content experts but true masters of their craft, capable of inspiring their students to achieve a deep understanding of mathematics. Like a Zen practitioner who seeks balance in life, a master teacher seeks balance in the classroom. The GROWTH framework helps teachers integrate their knowledge of mathematics with their teaching practice, seeking out that harmonious learning environment where students can thrive.

Conclusion: The Zen of Mastery in Mathematics Teaching

Mastering the art of knowing in mathematics teaching is a journey—one that requires dedication, reflection, and a deep love for both the subject and the art of teaching. Like Pirsig’s journey in Zen and the Art of Motorcycle Maintenance, it is a path toward achieving true quality, not just in the material taught but in the connections made with students. MathTrack Institute 's GROWTH framework provides the tools and support needed to embark on this journey, helping teachers to not just know mathematics but to truly grasp its concepts' meanings and in doing so, to change the lives of their students.