Reference: Kepler, J. (2010). The six-cornered snowflake: A new year's gift (J. Bromberg, Trans.). Paul Dry Books. (Original work published 1611)
Summary: Johannes Kepler writes casually, as if to a friend or himself in a journal entry, about "why snowflakes, when they first fall... always come down with six corners with six radii tufted like feathers" (Kepler, 2010), p. 33). Kepler ponders its nature comparing and contrasting snowflakes with pomegranate seeds and a beehive, as it too is a six-sided shape; is it nature, possibly God, that has bestowed the bee with this capability of creating a hexagonal beehive? or is the hexagon formed due to the limitations of the material itself? possibly both? Kepler discusses the potential candidates of various space-taking shapes (ie. triangle, square and rhombus and their assembly into 3D shapes). Kepler concludes that the bee builds the hexagon by nature, and the founder of nature, God, has a purpose for its architecture. The purpose is broken down into three ideals that, Kepler himself confesses, are not strong individually: (1) the hexagon can fill up a plane without leaving gaps and is the most spacious amongst its competitors - the triangle and the square). (2) the hexagon shape is more cozy for the baby bee as the many obtuse angles more closely resemble a sphere. (3) the hexagon comb allows for efficiency for the bee's construction efforts as each cell shares a wall, the straight frames are stronger than round frames, and finally the gap-less design allows for increased insulation for the baby bees.
Stop 1: "Here, indeed, was a most desirable New Year's gift for the lover of Nothing, and one worthy as well of a mathematician (who has Nothing, and receives Nothing) since it descends from the sky and bears a likeness to the stars" (Kepler, 2010, p. 33).
This reference to Nothing is very interesting to me. He continues to reference nothing has he has brought his benefactor Nothing as a New Year's gift. I wonder if this is an inside joke between the two. I think his mathematical insights are the Nothing to which he keeps referring. The idea that Kepler just shows up with new thoughts and ideas brought out of the air and his natural observations to share as what seems like their agreement, and it all is immaterial but terribly important, allows for suspenseful irony.
I discussed this concept with my colleague that these thoughts all come from being keenly present in the world, capable of making observations that feel meaningful to them, then taking the time to reflect and put it into words. With a world of distraction/entertainment and very little mental-physical (mind-body) presence and even less personal reflection time... how could we expect to achieve the same level of breakthrough in the 21st century?
Stop 2: "...the first particles of snow adopt the shape of small, six-cornered stars, there must be a particular cause; for if it happened by chance, why would they always fall with six corners and not with five, or seven, as long as they are scattered and distinct, and before they are driven into a confused mass?" (Kepler, 2010, p.35)
I love how Kepler presents his idea, then immediately provides a counter-argument using probability. To expand on his idea, we should see the same variety (cornered-ness) of snowflakes as the variety found in an elementary school classroom full of different creative artists.
Perhaps I can recontextualize to our classroom environment: How can we provide a classroom experience based on a natural observation that is so engaging to the individual student that they would prioritize thinking about it to reach new insight?
Oliver,
ReplyDeleteWhat a thoughtful reflection and impressive piece of work with the playdough. I enjoyed following your curiosity and hands-on approach, as well as the persistence you showed throughout the process. You clearly demonstrated presence in both observation and reflection, and that in itself already speaks to the kind of breakthrough you are describing! At the same time, I agree with your concern about the challenges of the 21st century. With constant distractions from phones, social media, and even the convenience of AI, many people are gradually losing the habit of sustained thinking and reflection, let alone the ability to reach new insights. As teachers, one possible response is to design tasks that genuinely require originality and personal thinking, rather than ones that can be easily searched online. This connects well to our earlier course on mathematical thinking, where the emphasis was on reasoning, sense-making, and process rather than quick answers.
In terms of creating classroom experiences grounded in natural observation, I wonder whether spending more time learning outdoors could help foster this kind of presence. Designing tasks that connect to students’ lived experiences and interests may also make them more willing to slow down and engage deeply. Just as importantly, students need sufficient time in class to sit with ideas, struggle a little, and reflect without rushing to a solution. To model this kind of presence ourselves, I think we need to show students that careful observation, curiosity, and patience are valued. When teachers visibly take time to wonder, question, and reflect alongside students, it sends a strong message that thinking deeply is encouraged.
Hi Oliver, you brought up some really insightful questions! I agree that in our current time, the overuse of digital devices causes a lot of distractions and people can hardly focus on the little things around them. It is important for students to develop a tangible connection to the world and nature.
ReplyDeleteI like the point that you said about mental-physical (mind-body) presence. I think I can link this idea with The First Peoples Principles of Learning: “Learning ultimately supports the well-being of the self, the family, the community, the land, the spirits, and the ancestors” and “Learning is holistic, reflexive, reflective, experiential, and relational (focused on connectedness, on reciprocal relationships, and a sense of place)”. By developing the use of various senses, students will explore more meaning of lives, lands and spirits rather than only restricting themselves to the classroom and a piece of paper. Instead of telling students the formula and equations directly, letting them discover the questions from observation and research by themselves will be more beneficial. Especially in senior math class, most often the class structure focuses on solving the problems, but rarely offers students the chance to participate in math exploration. Thus, I’m wondering what kind of topics and activities we can design for senior students to engage their interests in the class.