At the end of March, I had an amazing opportunity to host a full-day math workshop with a group of students in Grades 7 and 8. Given that I had complete freedom over choosing the content and delivery, I wanted to design an event that would provide students with hands-on exposure to some of the captivating mathematics not usually seen in elementary and secondary classrooms, but which is still connected to the curriculum. I decided to centre the entire day around the beauty and power of math. Exploring everything from mathematical proofs to introductory topology, the workshop was aimed at giving students a sense of how beauty appears in mathematics and how its power extends far beyond utilitarian purposes. It was an incredibly enjoyable day and I hope to do it again soon! Below is an outline of how the session unfolded, along with links to several of the BHNmath digital tools we used.

We started the session off with a thinking classroom activity with students collaborating in random groups on a problem solving task at whiteboards. It was immediately evident that this group was ready for action and that an engaging day was ahead of us! Following the activity, we gathered to chat about how it was currently a very exciting time for mathematics… Within the previous two weeks, the first aperiodic monotile was discovered by Kaplan, Smith, Goodman-Strauss and Myers, and two high school students, Calcea Johnson and Ne’Kiya Jackson, proposed a valid proof of the Pythagoren Theorem that uses trigonometry! 

From there we dove into a deep discussion on beauty in general. When considering where we often see beauty, students referenced nature, people, relationships, and art. Using musical examples such as Beethoven’s Sonata Pathétique, we analyzed how tension and resolution evoke emotion. Visual art examples such as Loretta Gould’s Prayer for Our Future and Henry Ossawa Tanner’s The Banjo Lesson  provided powerful evidence of Colonel Chris Hadfield’s claim that “when we really want to communicate, we use art.”

The discussion soon turned to the question of how (or if) the qualities of beauty that we noted (evoking of emotions, internal interactions, tension and resolution, purity, truth, etc.) appear in mathematics. Back to the boards! Through a series of investigations in number theory emerged an appreciation and, in some cases, obsession with the concept of mathematical proof. Not only did students feel firsthand the satisfaction of expressing undeniable mathematical truths through the use of drawings and manipulatives, they also experienced the power of algebra in generalizing their results. The finale of this introduction to number theory was a proof using mathematical induction (yes, with Grade 7/8 students) that the sum of the first n odd numbers is n².

Our chat about square numbers led us to the world of 2-dimensional geometry, where we reviewed how the so-called Pythagorean Theorem is based on the areas of squares. This idea wasn’t new to anyone in the room, but instantly became more intriguing when we asked if it could be extended to three dimensions. Using a digital tool to explore, it was evident that the 3D version of theorem holds if rectangular prisms of equal depth are built off of the right triangle’s sides. Pondering if it worked for cubes led us directly to Fermat’s Last Theorem. Enter Andrew Wiles and his emotional journey to prove that age-old enigma.

Continuing our chat about squares took us to the Fibonacci spiral and its relation to the golden ratio. Afterall, how could we skip that when talking about beauty in math! We used a digital tool to explore the golden ratio/rectangle and another digital tool to construct and compare the Fibonacci spiral and the golden spiral. Of course, we also took a look at where such spirals appear in art and nature!

Our journey into topology began when we saw how the golden rectangle appears in regular dodecahedrons and icosahedrons. Students proved that there are only five Platonic solids before exploring Euler’s famous polyhedron formula. Finally, we extended this idea to nonrigid objects and concluded with a cheesy joke about how we spent a whole day showing why coffee and doughnuts go so well together (they have the same Euler number)!