This post will be updated weekly with quotes and references that I've found particularly impactful. This post may be more for me than for others, but I will be using this as a quick reference for any personal reflections to be done by the end of the course.
Week 1
Antonsen, R. (2016, December 13). Math is the hidden secret to understanding the world [Video]. YouTube. https://www.youtube.com/watch?v=ZQElzjCsl9o
"So my day-to-day definition of mathematics that I use every day is the following: First of all, it's all about finding patterns. And by "pattern", I mean a connection, a structure, some regularity, some rules that govern what we see. Second of all, I think it is about representing these patterns with a language. We make up language if we don't have it, and in mathematics, this is essential. It's also about making assumptions and playing around with these assumptions and just seeing what happens...And finally, it's about doing cool stuff."
Gerofsky, S. (2011). Seeing the graph vs. being the graph: Gesture, engagement and awareness in school mathematics. In G. Stam & M. Ishino (Eds.), Integrating gestures: The interdisciplinary nature of gesture (pp. 245–256). John Benjamins Publishing Company. https://doi.org/10.1075/gs.4.22ger.
"Contrary to earlier classroom norms, this study shows that the students who hold mathematics 'at arms length' and use the most restrained movements to gesture graphs are less capable of noticing mathematically-salient points than the students who internalize the mathematic and make large gestures. (Gerofsky, 2011, p. 254)
"Gestural work is not sufficient on its own, but when accompanied by focused teaching that helps make salient features of graphs visual, kinesthetic and audible, gesture can play an important role as both a mode of expression and an experiential learning resource." (Gerofsky, 2011, p. 254)
Nathan, M. J. (2021). Foundations of embodied learning: A paradigm for education (1st ed.). Routledge. https://doi.org/10.4324/9780429329098
"Education is basically about engineering learning experiences." (Nathan, 2021, p. 3).
"What makes mathematics difficult for many people is not their inability to understand the ideas, but to learn the meaning of the formal notation and how it describes these basic ideas and their variants." (Nathan, 2012, as cited in Nathan, 2021, p. 147).
"Linking metaphors allow people to offload the cognitive operations needed for a new domain, such as negative numbers, onto a previously grounded domain, such as counting numbers" (Nathan, 2021, p.148)
Week 2
"Observing the paths of the hands of both the blind subjects during the explorations of the tactile tools and the objects presented by them, we could see how they moved their hands intentionally to capture the particularities of the shapes in question and hence constructed for themselves an image of the objects involved in the activities. This intentionality shows the active quality of their touch and how images are made, not passively received" (Fernandes & Healy, 2013, p. 43).
"...as the students became more confident in articulating the mathematical properties of symmetrical shapes, in the multimodal images associated with the mathematical object that were becoming part of their memories, they were not wiping out any connections with the physical in favour of some kind of disembodied symbolic representation. Knowing about symmetry did not transcend feeling it" (Fernandes & Healy, 2013, p. 51).
"...our tendency to design learning scenarios for the blind relying exclusively on what we know about the learning trajectories of sighted might not offer them the best opportunities for mathematics learning" (Fernandes & Healy, 2013, p. 53).
"Vygotskii (1993) suggested that, in VI pupils, the substitution of their eyes with their hands may result in the emergence of perspectives that differ from those of sighted pupils, due to the difference in the sensory tool through which they access mathematics and construct mathematical meaning." (Stylianidou & Nardi, 2019, p. 344)
"Healey, Fernandes and Frant (2013) argue that the multimodal nature of mathematical representations, which meets the sensory needs of every pupil in the classroom, benefits not only the pupils with sensory impairments but also the pupils with no sensory impairments in that it allows them to develop arrange of ways to think mathematically" (Healy, Fernandes, & Frant, 2013 as cited in Stylianidou & Nardi, 2019, p. 344).
"We saw this invitation as potentially beneficial for both the VI and the sighted pupils: it would make the VI pupil feel that he is no more the only child in class who accesses mathematics differently from his peers" (Stylianidou & Nardi, 2019, p. 346).
Week 3
"Too often, the specific life, qualities, and character of a particular place become subordinated to the forcefully imposed "evenness" and uniformity of the grid geometry... We see how the grid is connected to notions of control and ownership" (Doolittle, 2018, p. 104).
"Euclidean geometry is often promoted for its practical value; the failures of the grid show that its practical value is limited to small, uniform regions of space time" (Doolittle, 2018, p. 108).
"Complexity theory and chaotic dynamics are some of the most flourishing areas of geometrical thought in our time, and they provide an extraordinary alternative to the old notion with which we started, the path of least resistance, through the new concept of chaotic control" (Doolittle, 2018, p. 116).
"[t]he non-trivial question is how to identify those critical moments, and in which direction to provide the nudge. The theory of chaotic dynamical systems provides a framework within which we can at least begin to approach such questions" (Doolittle, 2018, p. 117)
“It begins with noticing that teachers are in the squared-off boxes of classrooms for most (or all) of their teacher education experiences. That seems to follow logically and seamlessly from other, previous educational experiences, from preschool through to advanced university degrees, in which teachers and learners sit inside rectilinear rooms, at tables, chairs, and desks, and talk about things that are not present, within the straight-line, right-angled grid structures that are always present” (Gerofsky & Ostertag, 2018, p. 173).
“Can we absolutely reject the grid (in environmental education and in garden-based learning) when we are, at least in part, implicated and entangled in it — when it is an intimate part of ourselves and our way of being in the world?” (Gerofsky & Ostertag, 2018, p. 175).
“As new teachers, we desire in some way to command attention and control learners (for the purposes of learning, safety, order)— and at the same time, we recognize the illegitimacy of usurping the freedom of others. We are simultaneously within and beside ourselves and the persona of ‘teacher’ we are in the process of adopting. Rather than conforming to this persona, what other ways of being teachers might be possible? Can we dance or daydream teachers into being with a garden?” (Gerofsky & Ostertag, 2018, p. 180).
Williams, D. (2008). Sustainability education’s gift: Learning patterns and relationships. Journal of Education for Sustainable Development, 2(1), 41–49. https://doi.org/10.1177/097340820800200109
"I spent days staring at Escher's symmetry drawings such as his butterfly and lizard tessellations and from these, soon figured out how I could make a tessellation. I began experimenting with different shapes and after about 3 days I found a design that worked" (Thomas & Schattschneider, 2011, p. 202).
"It is interesting to note that Dylan's path to making his first tessellation parallels that of Escher. Escher, too, had no mathematical background for this task, and figured it out by studying several geometric tessellations by majolica tiles in the Alhambra. Later he studied a display of tilings in an article by mathematician George Polya, and from one of these, Escher produced his first tiling by lizards with 4-fold symmetry" (Thomas & Schattschneider, 2011, p. 203).
"Although I derive much of my inspiration from outside First Nations art, Mandala was my first truly cross-cultural piece of art. I enjoy doing cross-cultural art because art itself is one of the only practices that can be found in all cultures. Art is one of the things that makes us human, and bridging different cultural art forms helps me to feel the unity of humankind" (Thomas & Schattschneider, 2011, pp. 208-209)
Torrence, E. A. (2019). Bridges Stockholm 2018. Nexus Network Journal, 21(3), 705–713. https://doi.org/10.1007/s00004-019-00455-2
"Marjorie Rice, a woman with no mathematical training beyond high school, wrote to Martin Gardner in 1976 claiming she had found a new pentagonal tiling. Gardner sent Marjorie’s work to Doris for verification, and so began a mathematical friendship that lasted 30 years. Doris explained how Marjorie invented her own notation, and ultimately discovered many new pentagonal tilings" (Torrence, 2019).
Week 5
Dietiker, L. (2015). What mathematics education can learn from art: The importance of considering form and experience. Educational Studies in Mathematics, 89(1), 27–44. https://doi.org/10.1007/s10649-015-9592-3
“It is not an object’s attribute but the individual’s perception and interaction that is the locus of aesthetic.” (Dietiker, 2015, p. 2)
“Imagining mathematics curriculum as a story opens up the possibility of reimagining the mathematical activities by changing the setting.” (Dietiker, 2015, p. 6)
“Stories that seem to have no point…or are easily predictable are quickly abandoned.” (Dietiker, 2015, p. 9)
Kelton, M. L., & Ma, J. Y. (2018). Reconfiguring mathematical setting and activity through multi-party, whole-body collaboration. Educational Studies in Mathematics, 98(2), 177–196. https://doi.org/10.1007/s10649-018-9805-8
"Students' moving bodies also became meaningful aspects of the setting themselves and each other, beyond simply performing individual quantities moving and operation along the number line. Quantitative relationships were understood and talked about as spatial relations between students' home positions and bodies" (Kelton & Ma, 2018, p. 184).
"Thad suggested that if he held onto Morgan (two to his right) and Kian (two to his left) with either hand, he could just turn around and rotate them to their opposites. He then revised this to include the whole group... Maggie and Thad solve the problem from their respective physical and mathematical perspectives in the material arrangements of the space" (Kelton & Ma, 2018, p. 186).
"...Jeff, was presenting a whole interval to Ms. Collins that involved crossing his hands. Jeff explained that , in this case, half would need to 'go to the other side of the world' in order to complete the task. This solution imaginatively expanded beyond the walls of the classroom, wrapping and bending the whole interval in a great circle around the world while entailing an impossible journey for half." (Kelton & Ma, 2018, p. 191)
"Bringing this comparison into broader dialog with the field, we suggest that researchers and practitioners attend more closely to the ways in which different patterns of mobilities in mathematical activity might affect the negotiation and development of reconfigured mathematical practices in any instructional design" (Kelton & Ma, 2018, p. 193).
Week 6
Belcastro, S. M., & Schaffer, K., (2011) Dancing Mathematics and the Mathematics of Dance. Math Horizons. 18(3). pp. 16-20. DOI: 10.4169/194762111X12954578042939
"Bharatya Natyam the dancer’s lines end—they are cut off by abstract or representational mudras made by flexing the hands. This situates the dancer “in the world,” rather than extending beyond it into the “world of the gods.” In contrast, the ballet dancer’s lines extend toward infinity, symbolizing an endless extension over the natural world" (Belcastro & Schaffer, 2011, p. 16).
"Using just four symmetries - translation, mirror, reflection, 180-degree rotation, and glide reflection - we can create what is called the Klein four group, or Z2+Z2, by combining the symmetries pairwise" (Belcastro & Schaffer, 2011, p. 17).
Chase, S & Gerofsky, S. (2018, January 19). Sarah Chase Long Version [Video]. Vimeo. https://vimeo.com/251883173
“I feel like [math dance] can help with anxiety in a really interesting way because it takes so much depth of focus and concentration to have three different things going on at the same time and to be speaking in addition to that. So there’s no room left to feel worried and anxious about how you’re appearing to other people or even overly editing about your own choices and what you’re saying” (Chase & Gerofsky, 2018).
IOC. (2016, August 5). Rio 2016 Opening Ceremony – Parade of Nations [Video]. YouTube. https://youtu.be/N_qXm9HY9Ro
Vogelstein, L., Brady, C., & Hall, R. (2019). Reenacting mathematical concepts found in large-scale dance performance can provide both material and method for ensemble learning. ZDM: The International Journal on Mathematics Education, 51(2), 331–346.
Week 7
Futamuro, F. (2025). Writing a mathematical art manifesto. In Bridges 2025 Archive, Eindhoven, NL, 589-594.
“Mathematics creates art”; “Mathematics is art”; “Mathematics renders artistic images”; “Hidden mathematics can be discovered in art”; “Mathematics analyzes art”; “Mathematical ideas can be taught through art.” (Schattschneider, 2005, as cited in Futamuro, 2025).
"It is art that embraces the spirit, language and process of mathematics. Both maths and art are concerned with truth, but they differ in their ways of searching for it. Maths uses analysis and proof; art uses the senses and emotions. But maths can harness the spirit of creativity and art can be analytical. Together they form a great alliance for understanding the world around us." (John Sims, 2010, as cited in Futamuro, 2025).
Hart, G. W. (2024). What can we say about “math/art”? Notices of the American Mathematical Society, 71(4), 520–525.
“One cannot hope to approach the topic as in a text book with definitions and theorems already laid out. Instead, one must see it more like a challenge as a group problem-solving session, where one ponders examples and counter-examples and enjoys the communal process of beginner to sort through and make sense of an initially confusing cloud of ideas” (Hart, 2024, p. 521).
"Trained mathematicians are well positioned not only to appreciate the field and move it forward, but also to articulate what it is that makes mathematical art such a worthy human endeavor" (Hart, 2024, p. 521).
“Artists generally aim to communicate something to their viewers. In contemporary fine art, the message is often a social or political viewpoint, with the artist daring to push boundaries and speak truths not otherwise heard. Math/art is characteristically tamer” (Hart, 2024, p. 524)."Math/art is often used as a hook to engage students in new topics. After they are invested emotionally, mathematical conversations are a natural way for a teacher to gently introduce the technicalities" (Hart, 2024, p. 524).
“Those who have journeyed through mathematical lands have unique stories to tell of what they found and how they now see the world” (Hart, 2024, p. 525).
Martin, A. G. (2015). A basketmaker’s approach to structural morphology. In Proceedings of the International Association for Shell and Spatial Structures (IASS) Symposium 2015, Amsterdam: Future Visions.
"The geometry principle of introducing singularities in a mesh - each pentagon in a hexagonal mesh introduces positive convex curvature to a fullerene shell or a basket shape while a heptagon introduces concave curvature - applies equally in molecular structure and woven baskets. The shape of a basket can be defined in mathematical terms; a 3-D representation of hyperbolic functions (saddles) results from adding extra weavers, while subtracting from the numbers of weavers produces closed shapes." (Martin, 2015, p. 2)
"Porifera (sponges) are simple aquatic invertebrate whose shape seems to depend on nature’s ability to find the simplest algorithm which will generate the most extensive surface with minimal materials. As cells multiply their surface arrangement adds extra area to the organism. This is a matter of life and death for filter feeders like sponges and corals. Evolution is a great optimizer." (Martin, 2015, p. 6)
Week 8
"She would come into English class and she would say, ‘Guess what I watched on television last night?’ And then she would start to tell something she learned from a TV show. The big idea that I got from her is that you can learn from everything that you do. I treasure that so much." (Glaz, 2019, p. 248).
"My interest in these connections became aroused partly by the location of my office. For a stretch of time, the Math Department and the Art Department shared an area at my University. And so, I became very much interested in visual art, for which I had never had opportunities before. I had been a reader. Then I learned a lot about art and artists. I asked questions and went to art exhibits." (Glaz, 2019, p. 250).
"I was astounded by what Sharon Olds said. Not just saying that it has too many words, but assigning a number. Yet I have found that this happens to me when I read a poem aloud, that is when I get the feel of how many words may need to be eliminated." (Glaz, 2019, p. 251).
"[M]ost people who are not already oriented toward mathematics fail to notice that mathematics is a perfect model for what makes an activity human, as it involves the three main ingredients of what makes our species special: cognition, consciousness, and creativity" (Karaali, 2014, p. 38).
“Both poetry and mathematics may, in fact, be conceived of without or before language, but only with words will they become communicable and complete” (Karaali, 2014, p. 39).
“We encourage the timid to venture into the world of mathematics, as they too have experienced both mathematics and poetry, in very emotional ways, and now they have the chance to unleash these emotions and share with others” (Karaali, 2014, p. 44).
Week 9
"And the process itself offers insights: In creating an object anew, not following someone else’s pattern, there is deep understanding to be gained. To craft a physical instantiation of an abstraction, one must understand the abstraction’s structure well enough to decide which properties to highlight." (Belcastro, 2013)
"All knitting is the generation of global structure via choices made in local stitch creation" (Belcastro, 2013).
"The way an object is constructed, in any art or craft, highlights some of the object’s properties and obscures others. Modeling mathematical objects is no different: It requires that we make choices as to which mathematical aspects of the object are most important. When it’s possible to do so, I knit objects so that a particular set of properties is intrinsic to the construction" (Belcastro, 2013).
"Here is how I proceed. After deciding on an object to model, I articulate my mathematical goals (in practice, I often do this unconsciously). The chosen goals impose knitting constraints. This gives me a frame in which to create the overall knitting construction for the large-scale structure of the object. Then I must consider the object’s fine structure. Are there particular aspects of the mathematics that I can emphasize with color or surface design? Are particular textures needed? While solving the resulting discretization problem, I usually produce a pattern I can follow—my memory is terrible and I would otherwise lose the work" (Belcastro, 2013).
Belcastro, S. M., & Schaffer, K., (2011) Dancing Mathematics and the Mathematics of Dance. Math Horizons. 18(3). pp. 16-20. DOI: 10.4169/194762111X12954578042939
De Vries, G. (2016, August 18). Making mathematics with needle and thread: Quilts as mathematical objects [Video]. mathtube.org. https://www.mathtube.org/lecture/video/making-mathematics-needle-and-thread-quilts-mathematical-objects
"Using mathematical concepts and algorithms in the design of quilts can lead to endless variety. Recognizing mathematical concepts in quilts can surprise, inspire and delight" (De Vries, 2016).
Fisher, G. (2015). Highly unlikely triangles: Bead weaving. In G. W. Hart & R. Sarhangi (Eds.), Bridges 2015: Mathematics, art, music, architecture, education, culture (pp. 493–496). Bridges Organization.
Hawksley, A. J. (2015). Exploring ratios and sequences with mathematically layered beverages. In Proceedings of Bridges 2015: Mathematics, Music, Art, Architecture, Culture (pp. 519–524).
"Many recipes have fairly strict ratios of ingredients (for example, baked goods), while others, like those in beverages, allow for a great deal of flexibility" (Hawksley, 2015, p.519).
"[...], if the bottom layer has 5 teaspoons of simple syrup per unit of volume, and the top had 3 teaspoons of simple syrup per unit of volume, then the sweetness ratio is 3:5" (Hawksley, 2015, p.519).
"Many students struggle with understanding fractions even though they are a crucial skill that is generally taught fairly early in the mathematics curriculum as a precursor to later skills. Fractions can seem illogical and hard to conceptualize (Hawksley, 2015, p.523).
MATU. (2020, November 9). Miura Ori - Traditionelle Miura-Faltung [Video]. YouTube. https://www.youtube.com/watch?v=EEGmnKKKhrk
Nguyen, U. (2021, January 7). Origami Fashion with Uyen Nguyen Part 1 [Video]. YouTube. https://www.youtube.com/watch?v=i4AoN1DtH6I
Nguyen, U. (2021, January 7). Origami Fashion with Uyen Nguyen Part 2 [Video]. YouTube. https://www.youtube.com/watch?v=bD7vUhdyO34
Polster, B. (2020, June 20). What is the best way to lace your shoes? Dream proof [Video]. YouTube. https://www.youtube.com/watch?v=CSw3Wqoim5M
I like the idea of tracking quotes you found impactful. If we each did this, I wonder what we fragments of text would be in common.
ReplyDeleteI really like this idea, and I'm so interested in the quotes that stand out for you.
ReplyDelete