GeT: A Pencil

Month: June 2025

  • Rediscovering the value of Euclid’s Elements

    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.

  • Practices Empower our Students

    K-12 teachers shape the public view of mathematics.  They are the first representatives of the math community.  If we want the public and our future students to understand what math is really like, we need teachers who have acquired mathematical practices: habits, values, and characteristics that enable them to independently generate and access mathematical knowledge, reason and communicate mathematically, and create authentic mathematical experiences for their students. Thus, in teaching preservice teachers, we feel it imperative that we focus our teaching on helping them develop mathematical practices, some of which are found in the SLOs, such as: making sense of problems and persevering in solving them (e.g. SLO 2 & SLO 10), generating and utilizing mathematical representations (e.g. SLO 6 & SLO 8), reasoning mathematically and constructing viable arguments (e.g. SLO 1 & SLO 5), communicating mathematical ideas through the precise use of language and symbols (e.g. SLO 5), and understanding and critiquing the reasoning of others (e.g. SLO 1).

    Studying our students’ and our own math practices has empowered us to support our students in developing these practices. As we describe in Chapter 47 of the GeT book, we study practices through a cycle of inquiry that involves: identifying evidence of math practices in student, creating & implementing tasks to engage them in those practice, and assessing their practices. This allows us to identify micro-practices: fine-grained skills that contribute to the development of a practice. Awareness of micro-practices enables us to provide explanations, orchestrate discussions, create tasks and assessments, and provide feedback that helps our students acquire math practices.

    For example, by studying the practice of reasoning mathematically and constructing viable arguments (SLO 1) we identified a number of micro-practices that we can intentionally discuss, help students develop, and assess.  These include that mathematically proficient students:

    1. Justify claims and solutions through a convincing explanation as to why it is correct, rather than simply telling how they solved it. 
    2. Attend to the scope of a claim by providing examples and counterexamples, when they are sufficient, and broadly applicable arguments, when examples are insufficient. 
    3. Ensure their arguments are complete by identifying and attending to the burden of proof.
    4. Build their case by making valid and relevant claims. 
    5. Provide detail by supporting all of their claims. 
    6. Make valid deductions from appropriate axioms, definitions, and theorems to support their claims. 
    7. Logically structure their arguments in valid ways, appropriately using direct proof, proof by contradiction, proof by contraposition, or proof by induction techniques. 

    Knowing these, we help students develop them through tasks and assignments, discuss these outright, and use them in rubrics for grading and feedback.

    The inquiry cycle has become a part of our ongoing work as teachers.  It gives us a way of exploring other math practices and support students in developing them.  The benefit of identifying micro-practices goes beyond our geometry classes.  It allows us to support student development across the curriculum and provide early mathematical experiences that contribute to our students’ development of productive mathematical practices. 

    We encourage you to take a deeper look at how you implement mathematical practices in your own courses and begin your own study of your and your students’ math practices. For additional information on our cyclic approach and in-depth descriptions of aforementioned micro-practices, see our paper.

  • Member Highlight – Interview with Henry Escuadro

    • What is special about your GeT course? In 2-3 sentences, describe your GeT course.

    In the GeT course that I teach, we build Euclidean geometry from the ground up in the spirit of Euclid’s Elements. There is a strong emphasis on the axiomatic method which aims to give students a sense of the logical foundations of mathematics. The approach lends itself to a discussion of the nature of mathematics.

    • Who are your students?

    The GeT course I teach are taken mostly by mathematics secondary education majors and mathematics students. But it is open to any student who has taken an introduction to proofs course.

    • What are you most interested in learning/achieving through participating with the GeT: A Pencil community?

    Knowing what mathematics is and having the ability to do mathematics is important not just for individuals but for society. Everyone has the capacity to know and do mathematics. By participating in the GeT: A Pencil, I hope to contribute to building teacher and student confidence and belief in their ability to do mathematics.

    • What is your favorite book you have read in the last few years?

    I really like and enjoyed reading the book “Mathematics for Human Flourishing” by Francis Su and Christopher Jackson.