Hello everyone! Since we worked through UBC's reading week in February, we will not be meeting in
person over the next two Wednesdays that are the Spring Break for many schools in BC. But we will have one asynchronous virtual class during that time -- and here are the details of how it works!
First, there are two interesting and somewhat contrasting articles for EVERYONE in the class to read:
(a) Nemirovsky et al, Craft and the Origins of Geometry
(b) Harriss et al, Geometry in the Walnut Grove
These are articles that take on big ideas about the nature of mathematical understanding, embodied and computer-assisted arts, and the relationship between ideas and materials, in pretty accessible language with lots of examples. I hope they will spark interesting thoughts and conversation!
Here are guidelines for your participation in the virtual class:
1) Each person should join the conversation (in the "comments" to this posting) and should post at least 4 substantive posts over the course of the time of our virtual class, Mar. 19 - April 1. Please don't make each post too long -- from one to three paragraphs ought to be enough -- but use that space to offer lots of ideas.
2) Please don't summarize the articles! We will all have read it. Do respond, question, critique, expand on, and offer connections to what the authors are saying, and to what your classmates are saying .
3) Please don't stray far from the articles in your comments. Avoid going off on tangents. Do keep closely tied to what the authors are saying. As a rule of thumb, I recommend that you use a quote from the articles in three of your four postings.
4) One good strategy for a close reading of an article is to think about the "stops" -- the things that stop you in your reading for some reason -- and revisit these to explore why they make you stop and reread.
Have fun and enjoy the conversation! I will check in on how things are going and perhaps offer some comments as the virtual class is in progress.
I really liked this article, especially the way it challenges familiar assumptions about what it means to learn and understand mathematics. When I was reading this article, the very first stop I had was the passage: “Even for a straight line, one of the most basic mathematical concepts, there is a world of difference between the experience of just seeing it … and of making it” (p. 2). This quote immediately stood out to me because it offers a powerful new perspective on embodied learning. Instead of focusing solely on mathematical content or the final product produced through embodiment, the authors emphasize the process of making and the lived experience itself. The value lies not only in physical sensations, but also in what happens internally—our thoughts, attention, memories, and emotional responses. These mental and bodily experiences become another dimension of information, allowing us to think about and remember mathematical ideas in deeper, more meaningful ways than abstract representations alone.
ReplyDeleteThis idea strongly resonates with what we have discussed in class about embodied mathematics. Embodied learning is not simply a technique to help students grasp academic content more efficiently; rather, it is a form of experience-based learning that positions students as active participants in meaning-making. By engaging their bodies, students are invited to think critically, to reflect on their actions, and to produce creative forms of expression. The process described in the article, from the beginning of shaping an object to its ongoing transformation, mirrors how learning itself unfolds. Students are not merely arriving at an answer but are exploring, negotiating, and making sense of ideas as they go. In this way, embodied mathematics opens space for creativity, critical thinking, and agency, allowing learners to experience mathematics as something alive, personal, and deeply connected to how they interact with the world.
Hi Lee, I really like the ways you've connected this article to our class discussions, and specifically appreciate the way you've connected the way of "making" with the way students learn. It's not always straightforward, and the final product isn't always perfect, but the best kind of learning comes from those mistakes and imperfections!
DeleteMy final project explores learning geometry through crochet in which students "make" geometry instead of simply learning it as an abstract concept - which I think ties really well with your point on students exploring, negotiating, and making sense of ideas as they go, as that was really similar to my experience with learning crochet as well learning math.
In response to the second article, I find myself less compelled than I was with the first, and I’m starting to see how the two pieces meaningfully juxtapose one another as Susan hinted. The philosophy of perceptualism is new to me, and while I understand it as appreciating a place through the senses, free from self-conscious concerns, I find it difficult to reconcile with how we actually experience the world. In both mathematics and learning, we emphasize that understanding is never neutral—it is shaped by prior knowledge, context, and interpretation. This makes me question whether perception can ever truly be separated from meaning-making.
ReplyDeleteIt seems that the authors may be using perceptualism as a way to make mathematics more accessible, suggesting that people can engage with mathematics through experience or emotion rather than formal knowledge. While I see the intention, I’m unsure about this move—can mathematical understanding really emerge from perception or emotion alone, without interpretation or prior knowledge? There may be a sense of mathematical beauty or satisfaction that feels universal, but disciplinary knowledge is what deepens and enriches that experience. For example, a Goldie poem may read as just another poem if we do not recognize how it represents phi.
Critiques of perceptualism—particularly the idea that emotions are subject to rational evaluation—further complicate the claim of “pure” perception. At the same time, this framework risks overlooking how history, power, and racism shape what and how we perceive. The notion of encountering a place “unencumbered” feels like a privileged position that is not equally available to all.
This stood out to me in the description of the walnut grove in Fayetteville, where family history is centered without acknowledging the broader historical and racialized context of the space (America). I came across this article on the site’s development (https://talkbusiness.net/2026/03/1-billion-fayetteville-project-to-include-2400-homes-1-3-million-square-feet-of-medical-commercial-space/) and found myself questioning the framing of “legacy” and “authenticity.” It seems these ideas function as place-based branding, raising questions about whether history is being meaningfully honored or strategically mobilized.
I’m curious how you interpret this—do you think this development genuinely honors Noah Drake’s history, or does it reflect a more capitalistic approach that selectively preserves certain narratives?
That’s a really helpful connection, and it made me rethink how I was reading the article. I hadn’t considered the development context at first, but it does complicate the idea of experiencing the grove “unencumbered.” It seems like the article invites us to focus on sensory and aesthetic experience, but your point highlights how that kind of experience can also leave out important social and economic realities shaping the space.
DeleteThe author seems to be exploring how mathematics can emerge from perception and engagement with a place, rather than trying to give a full account of its history. But I agree that this choice matters, because what gets left out can shape how we understand the place. It makes me think that engaging with a site meaningfully might require both: experiencing it directly and critically questioning the larger contexts that are not immediately visible. In that sense, I think the development might be doing both—honouring certain aspects of Drake’s history while also selectively framing that history in a way that fits the goals of the project.
As I read “Geometry in the Walnut Grove,” I found myself stopping at the notion of mathematics as “an additional sense” by which one could experience and understand a particular place. The authors outline a process that begins with sensory experience, walking through the grove, observing patterns, and making sketches, and then moves into mathematical abstraction. The most intriguing part, however, is that it is used to discover and define patterns within the landscape, such as spatial grids, light, and perspective.
ReplyDeleteHowever, I stopped at this stage because it challenged my conventional perception of how mathematics is commonly presented in most learning settings. In the article, mathematics is not presented as a predefined concept to be applied, but rather as a concept that can be derived from careful observation of the environment. This challenged me to reflect on my own practice, in which, in most cases, I would introduce mathematical concepts before offering my learners opportunities to make observations and recognize mathematical patterns on their own. In the article, the authors emphasize the need for experience before concept, which aligns with contemporary views of learning.
I found myself stopping at Nemirovsky et al.’s assertion that there is a “world of difference between the experience of just seeing [a straight line] … and of making it”. In my math classes, we usually treat geometric concepts as static, finished products, but this article suggests that the "liveliness" of geometry is actually born from the “resistance of materials”. This completely reframes the act of learning, and it suggests that the struggle with a medium, what the authors call the “workmanship of risk”, is where mathematical understanding lives. Lee, I connected with your point about the "internal dimension" of making. If we only provide our students with "perfect" digital lines or pre-drawn diagrams, we might be skipping the very process where true cognition happens, as Nemirovsky notes that “we make shapes as they shape our bodies”.
ReplyDeleteMy second "stop" occurred in the Harriss article when they described mathematics as “an additional abstract ‘sense’” used to experience and understand a place. This idea that math can be a sensory tool for “landing and grounding” in a site like the Walnut Grove is a powerful shift away from seeing math as just a repository of objective data. Rosmy, this really speaks to your observation about the need for “experience before concept”. In my Business Ed and Math classes, I am often guilty of providing the "lens" of a formula or theorem before the students have even "sensed" the disorganized patterns of the environment they are trying to manage. By reversing this, we allow geometry to emerge from careful observation rather than being a predefined concept we force onto the learners.However, I also share the hesitation Anna expressed regarding the neutrality of this "pure" perception. Harriss et al. mention that a scene can be appreciated for its own sake when the observer is “unencumbered by self-conscious interest”. But as Anna pointed out with the $1 billion Fayetteville project, space and the math used to define it are rarely neutral, and they are often tied to power and commercial expansion. While the authors talk about a “direct engagement with the material world,” we must acknowledge that geometry can also be a tool used to reorganize or even sanitize the history of a landscape.
Kabula, what stood out to me in your reading of Nemirovsky et al. is the way you locate learning in friction, not fluency. That idea, that understanding emerges through resistance, risk, and bodily adjustment, feels especially important in classrooms where we often rush students toward polished answers. Your point about perfect digital lines is sharp because it asks whether we sometimes remove the very instability that makes geometric thinking possible.
DeleteI also really liked your connection to Harriss et al. and the phrase of math as an “abstract sense.” It suggests that mathematics is not just something we apply to the world after the fact, but something that can deepen perception itself. At the same time, your final caution is crucial: perception is never fully innocent. Space is already political, and mathematical ways of seeing can illuminate a landscape, but they can also participate in erasing its histories. That tension makes both articles feel much more complex and useful.
One idea that really stopped me while reading Geometry in the Walnut Grove was the claim that mathematics can function as “an additional abstract ‘sense’” for experiencing a place. I find this idea compelling, but also a bit unsettling. On one hand, I appreciate how this reframes mathematics as something that emerges from and deepens perception, rather than something imposed afterward. It resonates with what Rosmy and Kabula mentioned about experience preceding concept. But at the same time, I wonder whether calling mathematics a “sense” risks naturalizing it too much—as if it were neutral or universally accessible in the same way as sight or touch. If mathematical perception depends on prior knowledge, training, and cultural background, then it may not function as a “sense” in the same way the authors suggest. In that sense, I find myself aligning with Anna’s critique: perception is never fully unencumbered, and what we are able to “see” mathematically is shaped by what we already know.
ReplyDeleteThis tension becomes even more interesting when I read it alongside Nemirovsky et al.’s idea that “we make shapes as they shape our bodies.” Here, geometry is not something we perceive first, but something that emerges through interaction and making. Compared to perceptualism, this feels less about accessing a pure experience and more about acknowledging that knowledge is always formed through entanglements—between body, material, and context. It makes me wonder whether these two articles are actually offering slightly different starting points: one begins with perception and moves toward abstraction, while the other begins with action and shaping. I’m curious how we might reconcile these—does meaningful mathematical understanding begin with sensing, with making, or with both at once?
Amy, I also stopped at the "abstract sense." The difference in both articles for me is that Nemirovsky et al. felt immediately imaginable to me. When they write about the difference between seeing and making a line, I could picture it vividly, because this is how children often learn shape: by drawing, folding, stretching, pressing, and adjusting. Their argument feels bodily and intuitive.
DeleteHarriss et al., by contrast, felt more distant to me. When they describe mathematics as an “additional abstract ‘sense,’” I understood the idea intellectually, but it also felt a bit exclusive, almost like they were describing a sixth mathematical sense that not everyone naturally has. That made the article interesting, but also less accessible. Where Nemirovsky grounds geometry in ordinary acts of making, Harriss risks making mathematical perception feel rarefied or elite.
In addition to the previous discussion, two moments in the first article that also particularly stopped me are the authors’ claim that “we make shapes as they shape our bodies” (p. 10) and the account of the dodecahedron created through “two hours of workmanship of risk and workmanship of problem-solving” (p. 13). Together, these passages challenge conventional understandings of application and problem solving in mathematics education. In many curricula, problem solving and application are treated as key indicators of mathematical achievement, yet in practice they are often reduced to calculating numerical solutions to word problems. Such tasks implicitly frame mathematics as abstract, detached, and oriented toward correct answers rather than meaningful engagement.
ReplyDeleteThis reduction also shapes how mathematics is perceived in everyday life. When people think about the application of mathematics, they often immediately associate it with quantitative data, measurement, or computation. The article disrupts this narrow view by positioning making, shaping, and bodily engagement as mathematical acts. Through craftwork, mathematics becomes an activity that unfolds through interaction with materials, risk-taking, and improvisation, rather than merely the execution of predefined procedures.
Artwork, in this sense, offers a powerful bridge between mathematics and application beyond numbers. It connects the classical knowledge students encounter in classrooms with the community knowledge and lived experiences they bring from outside school. By creating mathematical objects—such as a clay dodecahedron—students are able to represent mathematical ideas in ways that reflect their personal understanding and cultural ways of knowing. This form of expression allows learners to demonstrate understanding not only through written answers, but through processes, decisions, and embodied experiences.
Moreover, this approach reframes experience-based learning in the modern mathematics classroom. Creating and constructing objects has a more immediate and lasting impact on learning than memorization alone. Engaging the body, materials, and imagination lays a stronger conceptual foundation and invites students to see mathematics as alive, relational, and deeply connected to human experience.
I also noticed this same idea in this article, especially the idea that “we make shapes as they shape our bodies.” I especially loved this quote because it takes mathematics from being only about paper and pencil to being about our own bodies. I think this quote made me realize how narrow our traditional ways of teaching mathematics can be. I also found the example of clay dodecahedron to be compelling. The description of ‘workmanship of risk’ and ‘problem-solving’ reveals that actual mathematical thinking is not about achieving rapid, correct results, but about engaging with uncertainty, adjusting, and learning through the process. I found this highly challenging to my previous conception of problem-solving as structured and predictable. What I think is particularly interesting is how this relates to the student’s experience. As you mentioned, arts and crafts allow students to bring their own cultural knowledge into math. That’s much more meaningful and relevant, as opposed to the idea of some abstract concept of math.
DeleteOverall, your response has helped me think more deeply about how we can move beyond the traditional problem-solving approach and create an experience where the math is alive, embodied, and relevant to the world.
I am truly captivated by how mathematics is applied in this context. The author’s idea of using "the power of knowing with numbers, shapes, and forms" to create a "personal experience of design" on page 153 really struck me. We often see math as a rigid, universal truth, but the authors encourage us to view it as a personal tool that empowers us to translate our unique visions into tangible creations.
ReplyDeleteThis prompts me to think about my role as an educator. When teaching students about shape and form, are we just handing them formulas, or are we giving them the means to express their own view of the world? If "beauty is in the eye of the beholder," as the adage goes, then could math be the "accessory" that allows each student to convey their version of beauty?
It makes me wonder if we sometimes focus too much on the "numbers" and overlook the "personal experience" behind them. The authors didn't just find a grid but explored what it represents. This approach suggests that mathematical understanding is a dynamic way of "shaking hands" with the world, as the metaphor signified on page 154. It’s thrilling to see math as a gateway to deeper engagement and personal expression.
Just reading through everyone’s initial thoughts, it seems that many of us paused at similar moments—particularly the idea from the Harris article that mathematics can function as an additional sense. I think Amy points to a key distinction between the two articles: one suggests that mathematics can be naturally experienced as a kind of sensory perception, while the other emphasizes that mathematical understanding is constructed through interaction and meaning-making.
ReplyDeleteReflecting on Susan’s discussion about avoiding dichotomous thinking and resisting the urge to force ideas into binaries, I find myself wondering whether these perspectives actually need to be separated at all. Is it possible that mathematics is both something we can experience intuitively and something we actively construct through social and contextual engagement? Perhaps these views are not opposing, but complementary—where an embodied, sensory awareness of mathematics provides a foundation that is then shaped, refined, and expanded through interaction with others and with the world.
This raises further questions about what it might look like in practice to honour both perspectives in mathematics education. How might we design learning experiences that tap into students’ intuitive, sensory ways of knowing, while also supporting the collaborative processes through which mathematical meaning is developed?
What I found most interesting in the Nemirovsky, Bunn, and Silverton article was its attention to language. The authors treat words like shape, form, and especially shaping as more than descriptive terms. They use them to rethink geometry as a lived, material, and processual practice rather than a fixed abstract system. I especially appreciated this because the move from noun to verb subtly shifts the whole argument: geometry is no longer about static objects but about acts of making, transforming, and sensing. In that sense, the paper is not just about craft and geometry, but about how language organizes what counts as knowledge. At the same time, one limitation is that this conceptual richness sometimes feels more persuasive than historically conclusive. The language opens exciting possibilities, but the archaeological evidence does not always fully support the broad claim about the origins of geometry.
ReplyDeleteLooking back at the thread, I am struck by the tension between Sushi’s observation that the "shaping" of a line feels immediately imaginable and Amy’s concern that calling math a "sense" might naturalize it as something neutral or universal. In classrooms, I see this play out constantly - students can often imagine a shape when they hold it, but they struggle to activate that "additional abstract sense" required to see geometric patterns within a complex landscape like the Walnut Grove. We should see intuitive sensory awareness and active construction as complementary. For my students, this mathematical sense isn't a gift they naturally have or lack, but a "flexibility of reaction" developed through what Nemirovsky calls the "workmanship of risk".
ReplyDeleteClementina, your question about whether we are just handing out formulas or giving students a means to express their own beauty really resonates with the example of the clay dodecahedron. When a student spends hours on a task where the quality of the result is not predetermined, they are engaging in a "process of growth" where the design emerges from the "initiative of active materials". As Sushi noted, learning is often located in friction and instability, not just fluency. In an ICT context, if I simply fix a student's broken code, I am removing the very "resistance" that makes deep logical thinking possible. We need to protect that messy space where a learner is "shaking hands" with a problem, ensuring that math remains an activity of interaction and improvisation rather than just the execution of a predefined procedure.
ReplyDeleteAnother stop I made while reading Nemirovsky is at the end of “Crafting Shapes,” as this section provides a very important conceptual basis for the article. Here, the authors introduce a distinction between “shape” and “shaping,” where “shaping” is a continuous, process-oriented activity, in contrast to “shape,” which is static. This concept of distinction challenges traditional views of geometry as the study of static, abstract shapes by highlighting the role of materials, society, and the body in “shaping” shapes. At the same time, it is emphasized that shapes cannot be understood without reference to the context of “shaping,” thereby highlighting geometry within the wider context of human activity.
Pausing here allows for further engagement with this conceptual shift before moving into the more complex philosophical arguments that follow. It provides a space to think critically about how the traditional approach to geometry tends to privilege the final product over the process by which it is achieved. In this way, the reading emphasizes the process of shaping, allowing for a more dynamic understanding of mathematical knowledge, one that considers the interplay between the material and the conceptual. This is a significant pause in the discussion, as it raises the question in the reader's mind about how this could be done in the context of teaching geometry. I wonder how adopting a process-oriented view of “shaping” might challenge dominant epistemological assumptions about geometry and influence pedagogical approaches in mathematics education?
This is a fascinating discussion — thanks for all your ideas so far! I want to introduce another aspect of these two articles into the conversation:
ReplyDeleteWhat are your thoughts on the very different materials and ‘making’ processes involved in the Harriss et al and the Nemirovsky et al articles? The contrasting approaches to these things seems very interesting to me…
I really appreciate how this conversation is moving toward seeing these ideas as complementary rather than opposing. Anna’s point about avoiding binaries helped me rethink my earlier hesitation about mathematics as a sense. I’m starting to see that what the authors call an “additional abstract sense” might not be something purely natural or given, but something that develops through experience and interaction—which connects closely to what Kabula described as a “flexibility of reaction” built through workmanship of risk. In that sense, maybe mathematical “seeing” is not immediate perception, but something that is gradually shaped through engagement with materials, problems, and contexts.
ReplyDeleteRosmy’s distinction between “shape” and “shaping” also helped me think about this differently. If we take “shaping” seriously as an ongoing process, then perception and action might not be separate starting points at all—they could be happening together. As we make something, we begin to notice new details, and those observations then shape how we continue making. This makes me wonder whether what we call “intuition” in mathematics is actually something that emerges through these cycles of interaction, rather than something we bring beforehand. In practice, this seems to suggest that designing learning experiences is less about choosing between sensing or making, and more about creating conditions where students can move back and forth between the two.
Thinking about Susan's question, in Nemirovsky et al., the materials seem to push back against the maker. The idea of “workmanship of risk” highlights how shaping involves resistance, uncertainty, and ongoing adjustment. The material (like clay) is not passive—it actively participates in the process, and mathematical understanding seems to emerge through that negotiation between body and material.
ReplyDeleteIn contrast, in Harriss et al., the “material” feels quite different. The landscape is something the designers observe, interpret, and abstract from, rather than something they physically manipulate in the same way. Even when they engage through drawing or geometry, the process seems more about seeing patterns and representing them than working through resistance. This makes me wonder whether the type of material matters for how mathematical thinking develops. Does working with resistant, unpredictable materials (like clay) lead to a different kind of understanding than working with visual or spatial patterns in a landscape? And how might we bring both kinds of experiences into the classroom—so that students not only see mathematics in the world, but also feel it through making and resistance?
A key difference between the two articles is how they conceptualize both materials as well as the act of making. In Nemirovsky et al, the materials like clay in pottery are active and unpredictable participants in the process. Like Amy has pointed out, the materials seem to "push back", and shaping becomes an embodied, improvisational interaction between the maker, the materials, the tools, and the environment. Making is also inseparable from culture, social, and historical contexts, and it unfolds through the "workmanship of risk" and continuous problem-solving.
ReplyDeleteIn contrast, Hariss et al. works with materials that are more controlled, such as digital models, geometric concepts, and CNC fabrication. While the process still begins with embodied, perceptual engagement (sketching, sensing the site), the "making" shifts into a space where materials are shaped through mathematical frameworks and parametric design, allowing for precision and replication.
The contrast between these two papers highlights varying relationships between thinking and making. Nemirovsky et al. emphasizes making as emergent and co-constructive, where knowledge arises through direct engagement with material resistance and variability. Hariss et al. presents a process where making moves between intuition and abstraction, with mathematics acting as a bridge translating sensory experience into design and then back to a physical form. Personally, I find myself more drawn to Nemirovsky et al.'s view of making as emergent and co-constructed with the materials. The idea that materials push back and that meaning develops through interaction feels more authentic to how learning and creativity actually happen, especially in hands-on practices like craft. In this sense, knowledge is not just applied, but generated through the act of making itself.
My response to Susan’s question is similar to Sarah’s. In many ways, Nemirovsky et al. resonate with themes we’ve encountered in other readings, particularly around the co-construction of knowledge and our shared commitments to collaborative meaning-making. By contrast, Harris et al. feels newer in its framing, positioning mathematics as a kind of bridge—connecting sensory experience, design, and creative practice, as Sarah suggests.
ReplyDeleteI wonder if our responses are shaped by our relative unfamiliarity with design processes, subtly orienting the conversation in one direction over another. It would be interesting to hear how artists themselves understand and engage with mathematics in their creative practices, and whether their perspectives might expand or complicate how we interpret these ideas. I also find myself thinking about the hesitancy that often arises when people feel unfamiliar with either math or art. Rather than positioning mathematics as something to be brought into artistic spaces, there may be value in collaborating with artists to surface and articulate the mathematics already present in their work.
I really appreciate your point about our responses potentially being shaped by our familiarity with different practices. I hadn’t thought about it that way before, but it makes a lot of sense, especially since many of us seem to gravitate more easily toward Nemirovsky et al.’s approach! It does feel more immediately relatable, whereas Harriss et al. requires us to step into a design process that might be less familiar.
DeleteYour idea about collaborating with artists rather than positioning mathematics as something that needs to be brought into artistic spaces also really stood out to me. It shifts the perspective from seeing mathematics as something external that needs to be added, to something that is already embedded in practice but maybe not always explicitly named. That makes me wonder if part of the challenge is not just helping students learn math differently, but also helping us as educators learn to notice the mathematics that is already present in different forms of making.
I really appreciate both Sarah’s and Anna’s points, especially the idea that our responses may be shaped by our familiarity with different practices. I also found myself initially drawn more to Nemirovsky et al., perhaps because the emphasis on material interaction and “resistance” feels closer to how we often understand learning. However, I am starting to wonder whether Harriss et al. might be working with a different kind of resistance—one that is less physical, but still present in the process of translating perception into abstraction and design. Even if the materials are more controlled, there seems to be a kind of negotiation involved in deciding what to abstract, what to emphasize, and how to represent experience mathematically.
ReplyDeleteBuilding on the earlier thread, I see a direct parallel between the "material pushback" in Nemirovsky et al. and the more "abstract resistance" Amy just mentioned. In Nemirovsky et al., the material, like clay, is an active participant in an “improvisational intra-action” where the maker must respond to the “initiative of clay”. This resonates with Sarah’s point that learning feels most authentic when it is co-constructed with the material's unpredictability. When my students are physically building a geometric model, they are not just applying a formula - they are in a “workmanship of risk” where the quality of the result depends on their ongoing judgment and care.However, the "making" in Harriss et al. feels different because it shifts into a space of “mathematical frameworks and parametric design”. As Sarah pointed out, this process uses math as a bridge to translate sensory experience into a physical form through tools like CNC fabrication. Even if the physical resistance of a digital model feels less visceral than wet clay, I agree with Amy that there is still a negotiation involved in deciding how to “abstract and re-apply” a landscape. Perhaps the goal isn't to choose one type of material over another, but to provide a “gateway to deeper engagement” as Clementina suggested. Whether we are “shaking hands” with a physical grove or a digital simulation, we are trying to move students away from math as a “rigid, universal truth” and toward math as a personal, generative tool. By engaging with both the messy resistance of clay and the precision of parametric design, students can see that mathematical understanding is a dynamic process of “shaping” rather than just a static study of shapes.
ReplyDeleteAs I read the article on Nemirovsky Crafts and the Origins of Geometry, it resonated deeply with what I have read from Isaiah 64:8 and Jeremiah 18:1-6 in the bible, on the potter and the clay, while growing up. One quote from the article that struck me profoundly was on page 10, which says, “We add that design not only results from the initiative of active materials but also from travelling with the wheel, bartering vessels, dwelling in traditions...” This beautifully illustrates how design isn't just a solitary endeavour, but it's profoundly influenced by community, culture, and environment. This insight prompts a necessary reflection on education, especially in the realm of mathematics.
ReplyDeleteAs educators, we often rush to label students as failures in math without pausing to consider the rich tapestry of influences that shape their learning journeys. Every student carries a unique blend of experiences, from family dynamics to cultural backgrounds. Recognizing these factors encourages us to shift our perspective. Rather than viewing setbacks in math as personal shortcomings, we can cultivate a more empathetic and supportive classroom environment. This means integrating math with students' interests and connecting it to their world may help them rediscover themselves.
Reading through this thread, I’m starting to think about the idea of “resistance” in a different way between the two articles. In Nemirovsky et al., resistance is tangible: the clay pushes back, the form collapses, and the maker has to respond in the moment. This makes the “workmanship of risk” visible, and it’s easy to see how mathematical understanding might emerge through that physical struggle. But in Harriss et al., I’m wondering if the resistance is less visible rather than absent. It seems to show up in the difficulty of deciding how to abstract a lived experience: what to include, what to ignore, and how to represent something like space or perspective in a mathematical way.
ReplyDeleteThis makes me think that the two articles might not just differ in materials, but in where the difficulty of making is found. In one, it’s in the hands and the material, while in the other, it’s in perception and representation. I’m curious what this means for teaching... do students need more opportunities to encounter both kinds of resistance? It seems like physical resistance might help ground understanding, while conceptual resistance might help refine and extend it.
Hi Sarah, I think you really highlight the tensions around the concept of restriction, which is also mentioned in Kabula’s response below. It seems that, as a group, we’re uncovering more differences between the two articles by paying close attention to the language they use—like Sushi’s point about shifting geometry from a noun to a verb, and Li’s mention of mathematics as metaphor.
DeleteI really appreciate how you, Amy, and Kabula discuss resistance across the two articles; that perspective has helped me understand them more deeply. I especially like your openness to the possibility that multiple forms of resistance can coexist.
It makes me wonder what that might look like in a mathematics classroom. Perhaps it could involve creating space for students to work within mathematical structures while also questioning and reinterpreting them—through discussion, multiple solution paths, or connecting math to personal and cultural experiences. At the same time, there could be room for physical or embodied forms of resistance, where students engage with mathematical ideas through movement, gesture, or interaction with materials. In this way, resistance—both conceptual and physical—wouldn’t be something to eliminate, but something productive that supports deeper engagement and meaning-making in mathematics.
One part of the article Geometry in the Walnut Grove that strongly resonated with me is the authors’ description of mathematics as a metaphor. In particular, the statement that “this process of moving from a complex world into a simpler abstraction, studying what happens, and then returning that understanding to the world is at the heart of mathematical applications and science” (p.156) reframes mathematics in a deeply human and creative way. Rather than viewing math as a set of rigid rules, the authors emphasize abstraction as a purposeful act—one that allows us to pause, simplify, and see patterns that would otherwise be hidden in complexity. I appreciate how this idea positions mathematics not as an end, but as a movement between the real world and abstraction.
ReplyDeleteThis connects to my reflection on how mathematics is often taught and learned. Sometimes we focus too much on fully understanding a single concept as a complete and polished whole in order to appreciate its beauty. In doing so, we risk overlooking the potential that emerges when mathematics is investigated as a project constructed from small pieces. When we allow space for partial understanding, exploration, and iteration, mathematics becomes more dynamic, creative, and meaningful. The article’s emphasis on process highlights this idea beautifully.
Another moment that stood out to me is the discussion of balance, especially when the authors state that “reason balances and enriches the artistic impulse” (p.166). This notion of balance is critically important when applying mathematics in real-world or creative work. Mathematics is not used simply to display technical skill or sophistication, but to analyze situations and reveal different perspectives. In this sense, mathematics becomes a tool for thinking rather than a performance of correctness. This balanced approach reminds us that meaningful mathematical application lies in dialogue—with art, with place, and with human experience—rather than domination.
Hi Lee, I really appreciate your idea of mathematics as a movement between the real world and abstraction. It made me think that this process also involves a kind of resistance—not from materials, but from the difficulty of deciding what to include and how to represent it.
DeleteYour point about allowing space for partial understanding, exploration, and iteration also really stood out to me. I think this is powerful, but also challenging for teachers, since we are often used to thinking that teaching means helping students reach full understanding. Creating space for partial understanding can feel uncomfortable, but perhaps these incomplete and evolving ideas are exactly what allow students to engage more deeply with mathematics.
I’ve been thinking about how two articles look at geometry in different but connected ways. Nemirovsky et al. say geometry comes from our hands-on experience with materials like clay and natural fibres. They suggest that when we work with these materials, we really connect with them, which creates a kind of blending of our minds and the physical stuff we’re working with. So, geometry becomes something we truly feel and experience, not just a bunch of formulas.
ReplyDeleteOn the other hand, Harriss et al. see geometry as a tool that helps artists express themselves better. They use things like CNC-milled wood and large-scale land art to turn materials into abstract ideas. For them, geometry is more like an outside perspective—a math framework that helps us understand the world around us.
What really fascinates me is how these different viewpoints show the various ways we create. One focuses on the hands-on experience and physical interaction, while the other uses math to find deeper patterns. This contrast really opens our understanding of both art and geometry, showing us the many ways we connect with the world, and it amazes me.
I have been thinking a lot about how "conceptual resistance" in my math class actually mirrors the physical "pushback" of clay. In Nemirovsky et al., the material is an active participant in an “improvisational intra-action” where the maker must respond to the “initiative of clay”. When my students hit a wall in a math problem, they are often navigating that same difficult movement from a “complex world into a simpler abstraction” before trying to return that understanding to the world. I stopped at the point in Harriss et al. that math is not just for “ready reckoning” but serves as a “gateway to deeper engagement”. This suggests that the "resistance" in an abstract space is the actual intellectual struggle of deciding what to prioritize and what to ignore when we build a model.This ties directly into the “workmanship of risk” that Nemirovsky et al. discuss, where the quality of the result is not predetermined but depends on the judgment and care of the maker. Lee, your point about math often being reduced to “calculating numerical solutions” is so true in our current secondary system. When we remove the risk of being "imperfect," we remove the very thing that makes math a “lived, material, and processual practice”. In my own practice, I want to move toward what the authors call “hylonoesis” - an approach that acknowledges how every student’s “shaping” of a concept is unique to their own history, tools, and even the "weather" of the classroom that day.
ReplyDeleteTo answer Susan's question, what strikes me most about the contrast between Harriss et al. and Nemirovsky et al. is how differently they position materials and making in relation to mathematics. Nemirovsky et al. foreground making as an open, lived process in which mathematics emerges through bodily engagement, sensation, and responsiveness to materials. Crafting clay, drawing lines, or shaping vessels is not aimed at a predefined mathematical object; rather, mathematics is felt into being through ongoing interaction. The process is flexible, contingent, and deeply personal. What matters is not the final form, but how the maker experiences geometry as it unfolds through touch, resistance, adjustment, and improvisation. Mathematics here is inseparable from embodiment and time, and it can “go anywhere” depending on the individual and their relationship with materials.
ReplyDeleteIn contrast, Harriss et al. begin with a clearer intention. Their project has a defined artistic and pedagogical goal: to create a land‑based artwork that uses mathematical ideas—particularly geometry—as a structuring resource. Making is still exploratory and perceptual, but mathematics is explicitly mobilized to support and enrich the final expression. Digital modeling, parametric design, and careful planning play an important role in shaping an outcome that is meant to communicate something powerful to others. In this sense, mathematics functions as a connective and generative tool that strengthens the artwork and makes it legible from multiple perspectives.
Together, these two approaches reveal complementary ways mathematics can live in making. Nemirovsky et al. emphasize learning mathematics from within the process, through feeling and forming, while Harriss et al. show how mathematical thinking can be used in the service of an expressive goal. One privileges emergence and personal sense‑making; the other balances intuition with structure to invite shared meaning. In practice, we should see mathematics not as a single way of knowing, but as something that shifts depending on whether the journey itself, or the communicative power of the destination, is placed at the center.
In response to Susan’s question, I would say that the two articles offer very different understandings of what “materials” and “making” mean, which also influences what is meant by “geometry.”
ReplyDeleteIn the Nemirovsky et al. article, “making” is very material and embodied. The emphasis is on craft activities such as pottery, where the shape is developed through continuous interaction with the body, tools, and materials. The “process of shaping” includes preparation, use, and manipulation, and is also characterized by uncertainty. The authors also highlight that “making” is not just about creating an object but also about continuous interaction with the material, which plays an important role. This also raises issues of “workmanship of risk” and “problem-solving” in the process of “making.”Geometry, in this understanding, is not abstract but is developed based on “lived,” “cultural,” and “material” experiences.
In contrast, Harriss et al. take a more conceptual and design-led approach to making. While their project begins with sensory engagement with the location, it rapidly becomes abstract through the application of mathematics, geometry, drawing, and computer modelling. In this case, the materials are not only physical but also conceptual, as mathematics and computers serve as guides in the making process. Geometry is used as a perspective through which the location is understood and transformed, ultimately leading to the creation of the physical form. Overall, my understanding is that the main difference is that, in Nemirovsky et al., the emergence of geometry is through material engagement, while, in Harriss et al., geometry is used to shape the making process. The two studies, therefore, illustrate how mathematics is lived and constructed in very different ways.
While exploring how mathematics and craft pass a message, in the two articles, I’m struck by a beautiful tension: is the message meant for the maker or the world?
ReplyDeleteFor Nemirovsky et al., the message is an internal dialogue. Mathematics isn't a set of rules passed from teacher to student; it is a discovery emerging from the body’s response to material. They describe "shaping" as a processual genesis where we experience geometry through the "active touch" and the "initiative of clay." Here, mathematics is a lived, sensory experience—we feel shapes into being as they, in turn, shape our physical movements.
In contrast, Harriss et al. view mathematics as an outward expressive tool. It acts as a "connective and generative" resource used to structure a message for an audience. They describe geometry as a "natural toolkit" that helps clarify and communicate intuitive impressions of a landscape. While a project might begin with a private feeling, mathematics provides the rigorous structure needed to make that feeling "legible" and shared through a final artwork.
Ultimately, these perspectives reveal that mathematics can be both a silent, bodily realization and a powerful language for public expression. Whether we are listening to the resistance of materials or using a "mathematical sense" to design for a community, we are navigating a spectrum between personal sense-making and shared meaning.