In the article Adventures in Mathematical Knitting, Sarah‑Marie Belcastro explores the relationship between mathematics and knitting, or what she calls mathematical knitting. She walks readers through her process of creating a knitted Klein bottle, a mathematical non-orientable surface that has no clear inside or outside. Belcastro discusses both the design process and the challenges that arose from the materials used as well as the mathematical constraints of translating smooth mathematical surfaces into knitted fabric. Because knitting is made of discrete stitches arranged in a grid, mathematical shapes must be approximated through careful adjustments in stitch placement and structure.
Belcastro also describes characteristics important for topological mathematical knitting, such as surface texture and the placement of curvature, which are controlled by increasing or decreasing stitches. Through this process, knitting becomes a way to physically model complex topological surfaces and better visualize abstract mathematical ideas. She also situates her work within a broader history of mathematical fiber arts, noting earlier examples of knitted mathematical surfaces and the growing interest in representing mathematical concepts through crafts such as knitting and crochet.
Stop #1: " One reason is that the finished objects make good teaching aids; a knitted object is flexible and can be physically manipulated, unlike beautiful and mathematically perfect computer graphics."Physical manipulatives and crafted mathematical objects have the potential to help students internalize abstract mathematical concepts in ways that purely symbolic or digital representations cannot. When students physically construct objects such as Klein bottles, hyperboloids, or cones, they engage in spatial reasoning and embodied learning, allowing them to experience properties such as curvature, orientation, and continuity through touch and construction. For example, knitted or crocheted models have been used to represent complex surfaces such as Möbius strips and hyperbolic planes, making ideas that are often difficult to visualize more accessible and intuitive. Daina Taimiņa famously developed crocheted models of hyperbolic planes so that students could explore hyperbolic geometry tactilely, and students reported that interacting with these models helped them better understand the concepts being taught.
Belcastro writes this quote after explaining the difficult design and creation process of the Klein bottle. I stopped here because it reminded me that many mathematical formulas and concepts we take for granted today took years—sometimes centuries—for mathematicians to discover, refine, and validate. In contrast, expectations of student progress in mathematics often feel contradictory to this reality. I am reminded of the idea of slow pedagogy, an approach that emphasizes deep, attentive learning and reflection. Realistically, however, teachers are often pushed forward by prescribed curricula and limited time. How, then, can teachers navigate these timelines while still engaging in slow pedagogy? How slow is slow? Would allowing students even a few minutes to pause, wonder, and explore truly cause detrimental harm to their curricular progress? Perhaps the more important question is how we might move beyond this binary of slow versus fast learning altogether.
Hi Anna,
ReplyDeleteThank you for these deep reflections. You’ve highlighted a major tension in teaching: the pressure to cover a massive curriculum within a very limited timeframe. It’s a struggle we all face.
I think your point about slow pedagogy is vital, but as you noted, we have to balance that with the fact that students' learning rates differ. I’m currently mentoring a student who is practicing two-digit subtraction at school while mastering fractions at home with me. It made me wonder if the 'one-size-fits-all' pace of the classroom is shortchanging her.
To address this, perhaps we can combine differentiated learning with the physical manipulatives Belcastro mentions. If we had a 'mathematics clinic' or dedicated lab time, students could use these tactile tools—like the knitted models—to explore at their own pace. This would allow those who are 'lagging' to build that physical memory of the math, while others move toward more abstract concepts.
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ReplyDeleteHi Anna, I love your focus on "slow pedagogy" and the idea that deep mathematical breakthroughs, like Belcastro’s Klein bottle, are often the product of years of lingering thought. In our secondary classrooms, we usually treat math as a race toward a final answer, yet your reflection reminds us that the true "aha" moments happen when we allow students the tactile space to fail and iterate. This connects so well to the Radakovic et al. (2018) piece we read, where they argue that math isn't just a static body of knowledge but a dynamic "process of knowing". When we let students physically manipulate a knitted surface, we are validating their sensory intuition as a legitimate form of mathematical authorship.
ReplyDeleteYour point about teachers being pushed by prescribed timelines is a reality I struggle with daily, but perhaps the "multiplicity" mentioned by Radakovic is our way out. If we view math as a "humanistic mirror," then those few minutes spent wondering over a hyperbolic plane aren't a distraction from the curriculum; they are the foundation of what makes the math personally authentic.