Written over 2000 years ago, The Elements is an achievement of historical significance, possessing a clarity, rigor and superior organization that set the standard for the axiomatic development of every field of study. No other mathematics book has been published as many times or read by as many people, spanning multiple millennia, continents and languages. In our chapter of the forthcoming GeT volume we describe how we use Euclid’s masterpiece as the scaffolding for a course designed for future educators that dovetails nicely with the GeT SLOs.

Composed of thirteen books of axioms, definitions, propositions and their proofs, Euclid’s words are familiar, yet unexplored, and the axiomatic structure offers a well-defined entry-level experience. Euclid’s proofs are verbose and lack much of the symbolic language and notation we have come to expect, providing a rare opportunity in the curriculum to recognize the underappreciated brevity and clarity of algebraic notation as a ‘technological invention.’ As Bach’s experience of transcribing the string pieces of Vivaldi for the organ was transformative for the young composer’s skills, we flip the classroom and give students hands-on practice as they present and explain the proofs to their peers. In doing so, they transcribe Euclid’s wordy proofs into a more economical mixture of words, symbols and notation to develop the skills necessary to evaluate geometric arguments and understand the relationship between definitions, axioms and theorems. Students present most of the Book I propositions in class, but some constructions and justifications are completed with GeoGebra using our book of lab projects as their guide.

With a solid foundation of Neutral geometry established by the halfway point of Book I, our path takes a detour to reconsider our preconceived notions of line, circle and straightness. Here we use hands-on workshops to explore Euclidean axioms and propositions within the context of spherical and taxicab geometries. These side trips reveal flaws in Euclid’s reasoning and open the door to a discussion of models as well as desirable properties of axiomatic systems. After dipping our toes in non-Euclidean waters, we return to the non-Neutral second half of Book I to work through the propositions on parallels and comparative area that ultimately culminate in the Pythagorean Theorem and its converse. A careful study of the masterful development of Book I is an ideal way to develop deep roots in this geometry, and furthermore, provides an opportunity to shine a light on the chasm that existed between algebra and geometry for nearly two millennia.

After touring through Euclid’s geometry of the plane with selections from Books II, III and IV, we turn to the dramatic resolution of centuries of attempts to prove the Parallel Postulate. With a firm footing in Euclidean geometry, students have the tools needed to tackle hyperbolic, finite and transformational geometries as well as the four impossible constructions from antiquity. Along our curated path, the transitions to and away from The Elements are designed to push the narrative forward while consistently reinforcing the fundamentals and seamlessly interlacing the stories of the mathematicians who give life to the rich history of geometry. We find that this path offers a compelling and coherent narrative for our students to follow while providing essential resources for our future educators to utilize as they enter their own classrooms.