This Pi Day, Let’s Look at Geometry from All Angles
Every year, on March 14, students, teachers and schools develop a sudden, short-lived interest in geometry. Some classes prepare by following along with the Pi Day countdown. Actual pies are made. Schools host evenings showing off their math students’ projects and prowess.
For one moment in time, we care deeply about geometry, and concepts like pi, the unending, irrational number that represents the ratio of the circumference of a circle to its diameter.
Pi, written as the Greek letter π, is a constant that shows up in calculus, trigonometry and geometry, and has prompted global contests to see who can remember the most of its unending digits. The current record holder is India’s Rajveer Meena, who recited 70,000 digits over ten hours.
Appreciation for Pi Day, and for the subject geometry in general, could go much deeper than sampling pie flavors and memorizing numbers. In an article for the New Yorker, mathematician Steven Strogatz writes that the beauty of pi is that it “puts infinity within reach.”
Yet, for the other 364 days of the year, students don’t seem to carry forward this appreciation for geometry. Why not, and what can be done to boost appreciation for this element of mathematics, which has so many real-world applications?
In Elementary School, Arithmetic Eats Geometry’s Lunch
In human history, geometry was one of the earliest branches of mathematics. Humans have long been interested in understanding shapes and spatial relationships. Young children develop some understanding of space and shape on their own, but early childhood educators can do a great deal to promote children’s spatial and geometrical thinking.
Unfortunately, in elementary school, especially in the primary grades, teaching children basic arithmetic dominates math time to a point where it often crowds out geometry. Recent research in The Mathematics Educator also points to decades of reduced emphasis on geometry in schools. When elementary teachers do tackle the topic, they often confine themselves to teaching terms, like names of shapes, without going deeper.
The researchers also noted that drawing can be a natural tool for children to learn more about shapes and spatial relations, but it appears to be underutilized. In their study, children in first through sixth grades made drawings to show what they knew about geometry.
Most frequently, the students showed pictures of geometric forms, either explaining how they were made or illustrating them as part of something in the environment, like a robot or Planet Earth. Other aspects of geometry were rarely brought up, especially creating patterns with forms (like tessellating), estimating measures or converting from one set of measurement units to another.
Drawings in the study also suggested students had not made the connection between the coordinate plane and geometry, which is very important for understanding how geometry and algebra are connected. These gaps in children’s understanding lead to a big disconnect later–and may be the root cause of why teachers rarely explore societal issues like geometry’s role in the unethical practice of gerrymandering voting districts.
School geometry looks different from real-world geometry
“Geometry is disconnected from more than just real-life applications, it's disconnected from some very basic elements of our humanity, like our vision and intuition and ability to spot patterns,” says Dan Meyer, former high school math teacher, Ted Talk speaker and director of research for the curriculum company Amplify.
Meyer points to an overemphasis on formal logic in traditional geometry, which comes at the expense of students getting to put their own spatial reasoning skills to work. ”People have tons of spatial reasoning but we say, that’s not helpful, you need to calculate,” he says. “People have a basic need to understand patterns and explain the unexpected, but we say, we have no use for that, you need to prove things we tell you are already true using this kind of weird structure called a two-column proof using language and notation that is alien.”
State tests’ narrow vision of geometry doesn’t help, notes Steve Phelps, a veteran math educator and instructional coach in the Cincinnati metro area. “Teachers are going to teach what gets tested, and they’re going to teach it and assess it in a way that is how they test it,” he says. “That’s one of the things that keeps teachers from engaging their kids in dynamic constructions, is how things are assessed at the state level.”
Phelps says teachers can teach the subject in a more dynamic way without a drop in scores. “It makes for a richer experience in geometry.”
For example, when he was both a math teacher and a track coach, he had his students determine how to mark the field for the discus throw, using their understanding of angles and a 250-foot-long tape measure.
Observing geometry in art and architecture also can bring ideas home to students, he says. In his time as a math coach, he saw “very little” 3D geometry, which helps students with spatial awareness. “Kids are situated in three dimensions, but we never ask them to do things in three dimensions,” Phelps says.
How to Spark Student Interest Beyond Pi Day
Despite the challenges, Phelps and Meyer agree there are steps teachers can take to make geometry more hands-on and real-world oriented, beyond baking pie.
“Really just start at more basic versions of the problems in our textbooks and build from there. Show a bunch of shapes with no numbers and ask what’s the same or different about them,” says Meyer. “Give them one fact about the shapes and ask what else must be true. Play a skeptic and ask them to convince you. Tap into the kinds of human abilities we've developed for a long time like vision and intuition and build off that foundation towards the more abstract stuff.”
Phelps likes the following tools:
- TinkerCad: Building 3D objects and coding them with scripts, made like puzzle pieces.
- GeoGebra: a math program/graphing tool he uses to do “dynamic instructions, animations, make conjectures, solve problems” and more in geometry. Here he taught teachers how to make 3D models of rooms, so students can try it.
- A mid-1900s curriculum from a German math educator, Tatjana Ehrenfest-Afanassjewa, that’s still a curricular guide today. Phelps explains it “taps into students developing spatial awareness” with projects like “how many desks will fit along that wall.”
- University of Cambridge’s curriculum resources
- Would you rather — the math version which asks students to justify their math decisions with reasoning.
Finally, Phelps adds that some students simply aren’t at the educational/developmental level to understand complex geometry tools yet.
“If you try to teach a kid about proofs before they’re ready, that’s not going to be helpful to a kid,” he says. To determine a child’s level, he refers to an invaluable tool from the Ohio Department of Education with Van Hiele’s model with five levels of geometric thinking.
On this Pi Day, and the rest of the year, we can all consider a more holistic view of the importance of geometry. As Meyer says, “I’d want to help students understand that being good at geometry doesn’t mean getting rid of the stuff that makes them human like vision and intuition and thinking more like a robot. Being good at geometry requires the stuff that makes them human.”