The Wonder, Joy, Beauty, and Power of Geometry
We see geometry in our world every day, whether in naturally occurring phenomena or something created by individuals. The study of geometry supports students in being able to broaden their understanding of mathematics through critiquing and understanding their world and exploring the wonder, joy, and beauty of mathematics; and it is crucial for students across grade levels. Students need multiple opportunities and experiences to dig deeply into geometric concepts that will help them succeed across other mathematical experiences and in their world. As stated in NCTM’s grades 3–5 Developing Essential Understanding of Geometry and Measurement, “Learners often struggle to understand aspects of geometric relationships, not because they are incapable of grasping them, but because they need frequent experiences and time to develop important thinking and visualization skills” (NCTM 2014, p. 1). Having multiple rich, rigorous experiences is key to students developing a robust understanding of geometry.
I recall getting so excited about student-teaching my senior year in college many years ago. I was finally going to be in a high school classroom teaching mathematics. My passion! But I did not want to be placed in a classroom where I would have to teach geometry. Even though I loved mathematics as a high school student, geometry was my least favorite mathematics course. I had basically memorized my way through it and never really worked to understand it. Well, you guessed it! I was assigned to a supervising teacher who taught only geometry. I could not believe it: geometry all day long. Her name was Ms. Edon, and she was amazing! She not only mentored me in effective practices in teaching mathematics and how to build positive relationships with students, but she fostered in me a deep love of geometry. My joy of teaching geometry grew so much that when I was hired for my first teaching position, I chose the schedule that had the most geometry!
Thus began my journey into teaching, and I have to say into learning geometry as well. As I taught geometry, my goal was for my students and me to deeply understand geometric concepts that spanned spatial relationships, proof, logic, axioms, theorems, constructions, transformations, and more. I studied every night! Through the years, this led to investigations into Euclidean and non-Euclidean geometries, coordinate geometry, explorations with fractals and the fourth dimension, examination of connections to algebra and the real world, and using literature such as Edwin Abbott’s (1992) Flatland and Norton Juster’s (2000) The Dot and the Line. My students began to explore their own research questions, create geometric designs, and delve into historical contexts. What and how I taught in those early years has evolved over time as I grew in my own understanding of geometry. Oh, to have those first students back again! But it is a journey, and I keep learning.
Maryam Mirzakhani was the first woman and first Iranian to be awarded the Fields Medal, one of the highest honors in mathematics. She loved art and stories as a child, but she didn’t see herself as a mathematician. Numbers were not her favorite thing, but then she “met” geometry! In the children’s book Maryam’s Magic: The Story of Mathematician Maryam Mirzakhani, Meghan Reid describes how Maryam felt when she first met geometry: “Geometry was different from any math she’d known before. Every number held a story. It made those numbers into shapes and those shapes into pictures” (Reid 2021).
Maryam explored her own questions, created her own number stories. It seemed she could relate her passion for art and stories through geometry. I found many of my students who had struggled with algebra concepts thrived in geometry. They often saw their world through geometry, whether it was the understanding of shapes or logic and justification. I also found that other students struggled with geometry. Through the years, as I studied more about how students learn geometry and learned from renowned researchers and teachers such as Pierre van Hiele, Dina van Hiele-Geldof, and Zal Usiskin, I began to see what might be contributing to students’ lack of understanding. The work of my instructors helped me understand some of the struggles my students had and how to support them.
The writers of the Catalyzing Change series remind us that experiences with geometry related to the development of areas such as spatial reasoning, shape classification, visualization, modeling, conjecturing and generalizing deserve greater attention across PK–12 mathematics programs and should be done “in a manner that is integrated and active, and capitalizes on the wonder, joy, and beauty of examining the world in which they live” (NCTM 2020, p. 90).
Engaging students in the history of geometry is also important. By doing this, we give them opportunities to gain a deeper understanding of the past that informs their present and helps them develop a richer experience in geometry that affects their future. I recall one of my students being intrigued when I shared about the three classic impossible constructions using a compass and straightedge that captivated mathematicians for centuries: squaring a circle, trisecting an angle, and doubling a cube. He decided that an angle could be trisected with just a compass and a straightedge, and he set out to explore. Although, of course, it cannot be done, the joy and power were in his exploration and all that he learned, and I learned as I met with him periodically to discuss his journey into this investigation.
Much of what is typically done in PK–12 geometry is attributed to Euclid, a Greek mathematician. We need to expand our understanding of the development of geometry beyond the Greeks to the cultures of others such as the Babylonians, Chinese, Indian, Islamic, African, Native American, Dutch, and many others. We also need to expand beyond men to women and non-binary people so that our students have a deeper understanding of the contributions of many cultures and genders that have contributed to the understanding of geometry. Studying the history of geometry provides an empowering opportunity for students to see themselves in geometry.
Let’s continue to dig deeply into understanding geometry through its wonder, joy, beauty, and power. We must do this for our students and for ourselves as learners.
Trena WilkersonNCTM President@TrenaWilkerson
Abbott, Edwin A. 1992. Flatland: A Romance of Many Dimensions. Mineola, NY: Dover Publications.
Juster, Norton. 2000. The Dot and the Line. San Francisco: Chronicle Books.
National Council of Teachers of Mathematics (NCTM). 2014. Developing Essential Understanding of Geometry and Measurement for Teaching Mathematics in Grades 3–5. Reston, VA: NCTM.
National Council of Teachers of Mathematics (NCTM). 2020. Catalyzing Change in Middle School Mathematics: Initiating Critical Conversations. Reston, VA: NCTM.
Reid, Megan. 2021. Maryam’s Magic: The Story of Mathematician Maryam Mirzakhani. New York: HarperCollins.