Reading 2: Excerpts from Johannes Kepler (1611/ 2010) On the Six-pointed Snowflake: A New Year's Gift.
In
The Six-Cornered Snowflake, Johannes Kepler takes a common observation-the
snowflakes falling on his coat-and elevates it into an acute inquiry about
pattern, form, and necessity in nature. He asks why snowflakes always turn out
to be six-cornered and pursues similar patterns in honeycombs and pomegranate
seeds. Instead of attributing such shapes to either chance or beauty alone,
Kepler argues that often form is really the result of material constraints,
efficiency, and spatial organization. The following article demonstrates how,
quite independent of any conscious design, mathematical order can arise
naturally from physical interaction, long before formal equations or symbols
are at stake.
Stop1:
I
paused when Kepler asserts that it is not possible for snowflakes to have six
sides by chance. “If six were not a necessary number, there would be snowflakes
with five sides and with seven sides—but there are not,” he argues. I did not
initially think much about this question, but Kepler’s insistence that
consistency demands explanation made me stop and reflect. What struck me here
is how Kepler treats observation as serious knowledge. He does not rush to
formulas or authority. Rather, his method involves observing, recognizing a
pattern, and seeking a motive for that pattern. This particular stop taught me
that great ideas in mathematics lie in recognizing something quite basic.
Stop
2:
The
second pause came when Kepler explains why bees build hexagonal honeycombs. I
realized that I had never fully noticed this before reading the article: hexagons
are the most efficient shape for filling a flat surface without leaving any
gaps, allowing bees to store the maximum amount of honey using the least amount
of wax. This realization altered the way I thought about the honeycomb pattern.
In the past, I thought the honeycomb pattern was either a natural, beautiful
concept, but it is, in fact, a useful and necessary concept. The honeycomb
pattern isn’t something bees intentionally make because they understand the
concept of mathematics. Instead, the honeycomb pattern simply becomes a natural
byproduct because it’s the most effective pattern in the physical realm. This
topic is vastly related to embodied mathematics. Topics like efficiency,
tessellation, and optimization are far more applicable in a hands-on setting,
where one can grasp how shapes fit into space. This stop caused me to realize
how many situations in mathematics become understandable when learned through
physical engagement.
Question:
Kepler
valued the process of questioning as much as the answer itself. In what ways
might our current approaches to math instruction unintentionally limit
students’ opportunities to engage in this kind of open-ended mathematical
wondering?
Our current approach to mathematics instruction often stifles students' opportunities for open-ended exploration due to an excessive focus on correct answers, grades, and speed. In many classrooms, educational success is narrowly defined by how quickly students can arrive at the “right” answer, which leaves little room for exploration, questioning, or alternative interpretations of mathematical concepts.
ReplyDeleteI remember one particular student who genuinely enjoyed my mathematics lessons and frequently shared her appreciation for my teaching style. Despite this enthusiasm, she often struggled to meet the traditional expectations of academic success as outlined by tests and timed activities. In contrast, she thrived in hands-on environments, particularly during cooking activities that emphasized measurement, sequencing, estimation, and problem-solving. Unfortunately, I did not yet possess the pedagogical tools to recognize these strengths as valuable mathematical resources.
Looking back, I realize that a greater awareness of embodied approaches to teaching and learning would have enabled me to better support students like her. By integrating tactile, movement-based, and real-life activities—or by establishing supportive spaces such as mathematics clinics where students could explore concepts at their own pace, I could have helped her connect her embodied strengths to mathematical understanding. Rather than labelling her as a “weak” or “dull” student, I could have reimagined her learning differences as alternative pathways to mathematical meaning.
This reflection underscores how our limited definitions of success can stifle mathematical curiosity and marginalize learners whose strengths do not conform to conventional academic standards. By embracing embodied and multimodal approaches, we can move toward a more inclusive educational landscape—one that values questioning, recognizes diverse forms of intelligence, and creates expanded opportunities for students to engage deeply in mathematics.
Hi Rosmy,
ReplyDeleteThanks for the interesting question! I think traditional approaches to mathematics instruction have emphasized the importance of a singular procedural method, leading to one correct, predetermined answer. While other disciplines may leave room for exploration through open-ended responses, math rarely does. Historically, math has been seen as "objective", with no room for interpretation, which can stifle students' intrinsic motivations or natural curiosity.
Additionally, classroom norms often lead to success in math being measured by speed and accuracy rather than depth of thinking, as students are frequently assigned repetitive worksheets with defined answers. These methods lead students to believe that math is strictly about following rules and procedures, rather than generating questions or constructing meaning.
Assessment practices also tend to reward final answers over the reasoning process. While we may tell students to "show their work", this is often used primarily as a way to identify where they went wrong if they do not obtain the correct answer. As a result, students may come to view questioning in mathematics secondary rather than a central component of meaningful learning. In this way, even well-intentioned instructional methods can constrain the kind of open-ended mathematical inquiry that Kepler viewed as fundamental to one's intellectual progress.
Rosmy, I love your insights in this writing! These get to the very heart of what mathematics, and observation, and regularity in the natural world, and curiosity really are. Great question to us as teachers as well, and thanks to Clementina and Sarah for your commentary. Sarah, as a science teacher, do you see things as similar or different in science pedagogy?
ReplyDelete