In On the Six-Cornered Snowflake, Kepler reflects on the repeated appearance of hexagonal forms in nature, including snowflakes, honeycombs, and pomegranates. He wonders why these shapes appear so consistently and considers whether they arise from an inherent form, from the dictates of the material itself, or from an external cause, possibly divine. Throughout the text, Kepler connects mathematical structure with beauty, design, and order in the natural world.
My first stop in reading this text, also prompted by one of Vi Hart’s videos, was the realization that snowflakes always have six corners. I am not sure I knew this at some point in life, but I definitely did not remember it. Kepler’s questioning about whether nature follows an archetype of beauty or divine purpose immediately sent me to my chemistry background and to molecules with fixed symmetries, such as benzene, as well as to the special properties of water that make life possible here on Earth. Water molecules are polar and have a bent shape, which means they form hydrogen bonds at specific angles when they freeze. As these molecules lock together, they arrange themselves into a hexagonal crystal lattice, creating a structure with six directions of growth. I looked up what we now know about snowflake structure, and we now understand that the six arms of a snowflake come from this underlying molecular arrangement. The fine details and branching come from environmental conditions like temperature and humidity as the crystal grows. In this case, the six-ness of the snowflake seems clearly dictated by the material itself.
My second stop was the honeycomb. When I think about how well everything comes together in this structure, it is hard not to feel a sense of awe. The hexagonal cells allow bees to face outward for movement and safety, share walls so less material is needed, and maximize the volume available for storing honey. Without performing calculations, bees construct a design that appears incredibly efficient and elegant. I find myself wondering whether, even with our advances in mathematics, technology, and computational modeling, we would be able to design something that works better? I can see why Kepler pauses to consider the possibility of divine influence here. I also wonder whether all bee species around the world build similar honeycombs, or whether there are variations where this hexagonal form does not appear.
This Week’s Question
I find myself admiring Kepler’s ability to look closely at the natural world, notice mathematical structure, and hold onto a sense of curiosity and beauty long enough to investigate it. It feels like children do this naturally, noticing patterns and shapes in the world around them, but many of us seem to lose this way of seeing over time. I know I have.
My question is: Do you think maintaining, or reclaiming, this sense of wonder is an important part of our work in education, especially in how we invite students to experience mathematics? Do you feel that you still have this sense of wonder yourself and if not, how can we as adults get it back?
Hi Nichole,
ReplyDeleteIt is somewhat ironic that mathematics classrooms are so devoid of the recognition of patterns - this is one of the basis to this subject. I know that I ask my students to look for what happens, what did you notice that *I* did, and what commonalities might we see? The looks of horror or pure panic when I ask this instead of giving them answer is sort of terrifying.
I also know that some students hate (they have explicitly told me that it is my job as the teacher to tell them what to do - not impressed by the tone or the statement when I heard that one) when I ask them to try and think about a difficult problem. The idea of reclaiming the sense of wonder in mathematics is so important to me. How we slowly change this is not as easy. I really think it needs to start at an early age. As students advance in their mathematic journey, the same curiosity needs to be maintained but approached in a different way. Rather than physical discovery of patterns, I think the wonder needs to be shown and seen in the math itself. Each problem as a complex puzzle or solving a more relatable issue (ex. cost analysis, interest, debt, etc.). This is not to say that older students shouldn't explore the physical world around them, but I don't think is it necessarily prudent to utilize the same techniques across all ages. What a Gr .1 students find important to them is not the same as a Gr. 12 student. Priorities change and then sense of wonder follows it. I think matching these priorities with mathematics will support student curiosity.
Everette,
DeleteThis really connects with what I’m seeing too. Since moving from Grade 8/9 into K–6, I’ve been modeling routines like Which One Doesn’t Belong (WODB) and number talks for teachers. These help students get used to noticing and talking about patterns. I see the same panic you described, not just about noticing, but about the idea that there isn’t one right answer. In WODB and that you can make a case for any of them, you just have to back it up. That really challenges how many students think math is supposed to work.
I agree that curiosity needs to start early and then change as students get older. For older students, I like things like Would You Rather puzzles. Along with problems around money, interest, or real-life decisions, it feels like a good way to keep that sense of wonder.
Activities such as Would You Rather questions and mathematical puzzles are effective ways to keep curiosity and wonder alive. I am already noticing that my Grade 6 students are beginning to lose this sense of wonder, and I am intentionally trying to sustain it through rich mathematical experiences, such as puzzles like pentominoes and other open-ended problem-solving tasks.
ReplyDeleteReflecting on my own practice, I realize that I had also lost much of this sense of wonder early in my teaching career. At that time, I did not approach mathematics with curiosity or awe. However, through continued exploration, professional learning, and intentional reflection, I have slowly been able to reclaim it. This process has reminded me that wonder is not something that simply exists—it requires conscious effort. I need to regularly pause and truly look at mathematical ideas, patterns, and structures.
As educators, we must similarly encourage students to pause and observe, using a variety of modalities—whether tactile, visual, or otherwise—to explore patterns and deepen understanding. Creating space for this kind of intentional noticing may be key to sustaining curiosity and meaningful engagement with mathematics as students grow older.