The Alchemy of Learning: Mathematics Peripheral Vision for Teaching

Introduction: The Alchemy of Teaching

Teaching mathematics is more than just delivering content; it is an art form, a journey of transformation for both the teacher and the student. Thus, teachers become guides, leading students through the landscape of mathematics. Shifting from the perception of a teacher as a guide versus an instructor changes everything. To help us understand the implications of this shift, let’s look through the lens of three key metaphors: alchemy, horizon, and journey.

The first is alchemy, the guiding metaphor of The Alchemist by Paulo Coelho, which I’ll use to weave my points together. I’m defining alchemy as a seemingly magical process of transformation and creation rather than the precursor to modern chemistry (also known as turning lead into gold). Horizon will be defined as the limit of a person's mental perception, experience, or awareness of surroundings. Finally, a journey is a transition from one place to another, often a time of uncertainty and discovery. Holding to these definitions throughout, we’ll explore together how Alchemy, or a magical transformation, relates directly to the art of teaching mathematics.

The Alchemist’s Journey for Teaching

One of my favorite books about an epic journey is The Alchemist by Paulo Coelho. The main character, Santiago, goes on a quest to discover his personal legend. Who wouldn’t want to find your legend?

"Everyone, when they are young, knows what their Personal Legend is. At that point in their lives, everything is clear and everything is possible". ~Melchizedek in the Alchemist

The personal legend represents an individual's unique path in life—the ultimate goal they are meant to achieve. This resonated with me for teaching, which feels more like a calling than a career for many. The value of a journey is only fully realized if you are present while it happens along the way. Sometimes, you must look around and be thankful you are on a journey. The “looking around” and understanding the value of the journey is vital for the journey of learning.

The ability to look up or understand the value of the journey of learning a discipline like mathematics is a teacher skill. Researchers call this "horizon knowledge" (Ball, 1993; Ball & Bass, 2009) or the peripheral vision needed in teaching mathematics. It's one thing to teach an isolated concept about subtraction. It takes a different skill set to understand that subtraction is part of a journey across grade bands and levels of sophistication, fraught with difficulty. According to research, this type of knowledge is essential and completely lacking in traditional teacher preparation. The good news is that this can be developed with teachers while they are teaching.

Four Main Skills for Enhancing a Teacher’s Peripheral Vision

The Alchemist’s Journey provides a meaningful reference point that highlights this type of teacher skill. These themes will be the personal legend, interconnectedness as knowledge, overcoming obstacles, and finally coming back to the journey as the destination.

Personal Legend as the Horizon: Santiago’s quest to discover his Personal Legend symbolizes the ultimate goal that drives us forward. Santiago’s personal legend changed, similar to how we, as educators, evolve our awareness of the ultimate goal of mathematics teaching. As I’ve written previously, mathematics is about gaining wisdom and inhabiting an intellectual way of being. This mirrors Santiago’s evolution from initially pursuing material wealth through gold to enlightenment, wisdom, and the importance of following one’s heart. For math educators, there are broader goals for learning concepts. But, if you are focused exclusively on the procedure of long division with your students, you’ll miss the ability to connect long division to the journey of division throughout the curriculum. As teachers gain expertise from their journey in teaching, their mathematics peripheral vision can evolve in complexity if they are aware that it is important. Mathematical ideas have connections, and the journey gives freedom and purpose to every lesson and a clue that leads to a deeper, more comprehensive understanding for both teachers and students.

"Education is the key to unlock the golden door of freedom." - George Washington Carver

Interconnectedness of Knowledge: A central theme in The Alchemist is the "Soul of the World," a force that connects all living things and reflects the unity of the universe. Some call this energy, some call it quantum entanglement, and others call it the holy spirit. A peripheral vision in teaching is being aware of the interconnectedness of mathematical knowledge, where each concept is linked to others, forming a vast, coherent system. When teachers have this peripheral vision or acceptance of interconnectedness, they can better engage students to see these connections and how each experience with that concept contributes to the overall journey. When instruction lacks this, you will hear from students: “When will I ever use this?” Topics, in isolation, lack the type of coherent utility that would logically prompt this type of question from students. They are right to ask it. For example, understanding multiplication as repeated addition is actually tied to humans' longitudinal journey in developing more sophisticated ways of counting quantities. Multiplication also applies when counting quantities that we do not know, which is Algebra. Like all mathematical concepts, the vision can illustrate how foundational concepts are embedded and will grow into more complex, coherent mathematical ideas.

Overcoming Obstacles: Learning is hard. Santiago’s journey is fraught with challenges, from being robbed in Tangier to facing the life-threatening dangers in the desert.

“There is no learning without some difficulty and fumbling. If you want to keep on learning, you must keep on risking failure—all your life.” ~John W. Gardner

These obstacles test his resolve and ultimately lead to his personal growth. This is often called grit in education (Duckworth et al., 2007). Grit is defined as perseverance and passion for long-term goals. Similarly, in the classroom, students often face difficulties that obscure their view of the broader mathematical landscape. Recent research (Hwang & Nam, 2021) suggests that enhancing grit is possible, primarily through the growth mindset research of Dweck and Boaler (Boaler, 2015, 2024; Dweck, 2006). If your students fear making mistakes or let the natural trials and tribulations of the learning journey impact their self-identity as a learner, their ability to learn and grow is significantly impacted (Boaler, 2024). Teachers' crucial role is to help students build this long view of resilience. A master educator, as a guide, can show students that these obstacles are not insurmountable barriers but opportunities for growth and deeper understanding. The Alchemist relates this well,

"There is only one thing that makes a dream impossible to achieve: the fear of failure.”

The Journey as the Destination: One of the profound lessons Santiago learns is that the journey itself is the true treasure. While this may sound similar to the personal legend connection, I see it as nuanced. The personal journey is about the purpose of mathematics learning; the peripheral vision of an educator allows that understanding to come across in the learning process. The journey as a destination connection is that the process of learning—the exploration, the mistakes, the discoveries—is as important as the final understanding of a concept. However, to have the most long-term impact on the learner, it must be experienced, not just read about. For example, we could read all about the nuances of electricity, how it works, and how it travels. But until an exposed wire has zapped you, you don’t understand its true power or know how to handle it.

"One learns from books and examples only that certain things can be done. Actual learning requires that you do those things.” — Frank Herbert

Master mathematics educators encourage students to embrace this journey, fostering a sense of curiosity and a love for learning that transcends the immediate goal of solving a problem or passing a test. In one way, the public school model in the US has provided us with the most cost-effective way of giving students an abundance of experiences all day long. But, without embracing the journey and allowing students to get their hands-on experiences, it becomes a day of reading about everything and experiencing almost nothing. Mathematical habits of mind and intellectual ways of being are modeled by adults for children, and these become intellectual ways of operating throughout life.

The Alchemy of Teaching: Transforming Knowledge

What is Alchemy again? For our purposes, alchemy is a seemingly magical process of transformation or creation. This metaphor resonated deeply with me as I have often heard mathematics referred to as if it is some mystical magic that makes sense of chaos– mathemagic. Teaching mathematics is not merely about conveying information but about the intellectual ways of being that transform knowledge into action.

The least virtuous point of Alchemy was to turn lead (which was abundant) into gold, which was valuable. The pursuit of wealth and power is different from a scientific and virtuous pursuit. Alchemy is used as a metaphor for transforming the self in the book. The lead in learning is the abundance of opportunities to learn; the gold is turning basic understanding into profound and valuable knowledge. Or, even more precisely, transforming the “not yet” of a student into a hopeful, joyful, and lifelong learner. A master teacher can help students take the foundational concepts they learn and see how they can be transformed into powerful ways of being to understand the world. This transformation requires more than just technical knowledge—it requires insight, creativity, and the development of peripheral vision for those concepts.

Connecting to the GROWTH Framework: Reveal the Horizons (R)

The GROWTH framework developed by MathTrack Institute is a methodology for supporting teachers’ journey to becoming master educators. The “R” stands for Revealing the Horizons, a series of experiences that expand an educator's peripheral vision for teaching mathematical concepts. As teachers’ capacity expands, they can guide and even narrate the interconnectedness of mathematical concepts. Only then can they convey the importance of the journey of learning as a vital part of their students’ intellectual and personal growth.

“Research shows that you begin learning in the womb and go right on learning until the moment you pass on. Your brain has a capacity for learning that is virtually limitless, which makes every human a potential genius.” — Michael J. Gelb

Understanding and studying the scope and sequence of topics and how those topics change in sophistication over time is valuable for this journey. This is best done while teaching and with other peer educators. The landscape of mathematics is beautiful, and only master educators will appreciate its elegance and share its interconnectedness with others.

Conclusion: The Alchemy of Learning

I believe Santiago’s journey teaches us that the true treasure lies not in the gold he seeks but in the wisdom he gains along the way. In the classroom, mathematics peripheral vision helps students discover the broader connections and deeper meanings surrounding the concepts they learn. It’s about transforming the way they see mathematics, not as a series of isolated problems with the goal of getting the correct answer but as a journey toward understanding the world around them.