Kepler is a keen observer. This excerpt is not only about the geometry of snowflakes, pomegranates, and beehives, it is about pattern finding, exploration, wondering, and playing with mathematics; characteristics of mathematics for flourishing described by Su (2020). This playful investigation started with one question: why, if snowflakes are made of vapor (which is shapeless), always fall in the shape of a six-cornered star? Kepler starts his investigation by generalizing with another hexagonal shape found in nature: beehives. And to truly understand bees’ instinct, he detours with observing the patterns present in pomegranate seeds. With this exploration, Kepler does not come to the answer to his question (maybe it comes later in the book), but it allows him to deduce that bees purposely build beehives in the shapes to be as efficient as possible, navigating solidity and higher area/perimeter ratio. In this excerpt, I recognize many mathematical habits of mind describes by Cuoco and al. (2010) like finding, articulating, and explaining patterns, generalizing from examples, and creating, and using representations. This excerpt shows how nature can be a great mathematics teacher by developing habits of minds in a way that allows flourishing.
Stop 1: “But if you observe the bottom of each cell, you will notice that it slopes into an obtuse angle formed by three planes (Kepler, 1611/2010, p.43)”
Many times in this excerpt, I struggled to follow what Kepler was explaining. I still do not really understand how the shape he is describing looks like, even if there seems to be a representation of it. It is true that education is based on the sense of sight, which limits the different perspectives brought to investigate a concept (Gerofsky, personal communication, January 20, 2026). I wish I had this shape in my hand so I could touch it and feel it. Where is the obtuse angle? To make a connection with last week’s readings, it seems that I struggled to develop a metaphor/analogy of this shape by just using my sense of sight. I needed another perspective to understand it.
Stop 2: “The only figures that can fil up a plane without leaving empty spaces are the triangle, the square, and the hexagon. Of these, the hexagon is the most capacious, and bees avail themselves of this capacity to store their honey. (Kepler,1611/2010, p.61)
The way bees build their hive relies partly on the concept of equivalent figures. It would be interesting to use the example of the beehives to look into which shapes offer the smallest perimeter for its area. I like the connection to the beehive because my students could then see that humans are not the only animal using mathematical concepts with a goal to be more efficient. Also, I like the example because it does not only rely on the concept of equivalent solid. Technically, the circle would be the best shape to have less perimeter (use less wax), but since it is not as solid as straight lines, the hexagon is the most efficient shape in this situation. I think adding critical thinking to any mathematical concept is interesting. I believe it helps students see that when mathematics concepts are contextualized, it is important to think critically about them. Yes, a circle shape would require less wax for the same area, but is it the best choice for this situation? What are the other elements of the context we need to take into consideration? It offers a more holistic view of this concept, which is more aligned with my students’ ways of learning.
Stop 3:” This shows that the bee is endowed with this instinct by nature and as its particular property, and prefers to build in this shape than any other (Kepler, 1611/2010, p.61).”
It is fascinating that bees’ instinct creates such mathematical beauty. Like the hexaflexagons, the beehives’ structure is a hands-on mystery that makes people want to know more about and explore about. Bringing those activities in the classroom not only offers a different perspective to the concept, but it could allow students to re-connect with the playfulness of mathematics that is often left behind in more ‘traditional’ classroom.
Question:
What natural mathematical beauty are you fascinated with? Can you imagine yourself bringing it into the classroom?
References
Cuoco, A., Goldenberg, E. P., Mark, J., & Hirsch, C. (2010). Contemporary curriculum issues: Organizing a curriculum around mathematical habits of mind. The Mathematics Teacher, 103(9), 682-688. https://doi.org/10.5951/MT.103.9.0682
Kepler, J. (1611/2010). The six-cornered snowflake: A new year’s gift. Paul Dry Books.
Su, F. (2020). Mathematics for human flourishing. Yale University Press. https://doi.org/10.2307/j.ctvt1sgss
Noemi, I am really looking forward to reading these blog posts a few days from now (when others have had a chance to respond too). A small but important formatting suggestion: you may have written these two most recent posts in Word or some other format, and then pasted them into your blog posts (which is a great way to do things!) But there is one additional step at the end to make sure the text is readable and doesn't flow way off the edge of the screen:
ReplyDeleteOnce you have pasted the text in, highlight it all, and then go to the top right hand corner of the ruler bar on screen, click on the three dots, and choose the symbol of a T with a diagonal bar through it. That removes external formatting and fits the formatting to the blog. Let me know if that works for you, and if it makes the text stay within the margins of the blog! Thanks.
Thank you so much!
DeleteEvery since a previous course (I cannot remember which at this moment) I have been much more aware of fractals and the Fibonacci sequence. These were 2 things that I hadn't ever really paid attention to, but tend to see them more and more now. From taking pictures of trees and enjoying their fractal like structure from afar, to trimming my house plants and observing Fibonacci like sequences in the way the leaves grow, I have learned to make the connection between math and nature and to appreciate it much more. I also have noticed that my camera roll has much more nature pictures in it... lots of up close flower pictures to better see details. I love bringing nature into the classroom, I did so through a greenhouse/ garden project last year and would love to do something similar again this year, perhaps bringing these concept in more concretely.
ReplyDelete