Nick Sayers Interview Reflection

Stops

1) At 06:42, Nick discusses how he learned to write basic computer programs and described this as a sort of “math by stealth”. Programming requires logic, structure, and mathematical thinking, but not numbers or arithmetic, which were the aspects of math he previously disliked. This challenged me to think about the values of mathematics we transmit onto our students. Like Nick says, in school, math is often represented primarily as numbers and arithmetic, especially in early grades. Students who struggle will often internalize the message that they are “bad at math”, similar to how students who struggle to draw may consider themselves “bad at art” (I’m certainly guilty of that one!). However, there are many different forms of mathematics including logic and patterns that students may connect with more deeply, which do not involve numbers and arithmetic the same way. Many students never reach advanced math courses like calculus, not necessarily because they lack mathematical thinking, but because they were filtered out early through narrow definitions of what is considered a strong mathematical ability. Nick’s experience highlights how important it is to broaden students’ exposure to different types of math before they label themselves as “not a math person” based on the institutional privileging of certain knowledge over others. 

2) At 34:43, Nick explains how bicycle gears create spirograph patterns as you cycle. I found this fascinating, linking math and art but also adding in engineering qualities as well. Geometry can emerge naturally through motion, looking at how rotational speeds, radius, and mechanical ratios can generate intricate designs. This example is powerful because many students want to know “how things work”, but struggle to focus on abstract definitions, equations, or numbers. I think this is a really effective way to see how geometry can emerge from motion and how math is not just numbers and arithmetic, but can also generate beauty. 

3) Around 1:25:00, Nick describes a workshop in which students built their own cameras, though the images appeared upside down. He was able to explain why that inversion happened in a way that was simple and easily accessible even to children, and then used their questions of how to get it right side up to further expand the project using the upside-down house, eventually leading to creating a bifocal lens. I thought this was really effective and powerful, revealing the importance of investigations and finding ways to answer students’ questions in meaningful ways rather than brushing off non-curricular questions or only focusing on the complex mathematics.

What does this artist's work offer you in terms of understanding math-art connections, and what does it offer you as a math or science teacher?

Nick Sayers work allows me to see the value and beauty in math-art connections, revealing an accessible means to learning mathematical properties. I’ve run into many students who say they “aren’t a math/science person” but are very interested in art. I think by introducing students to mathematical art, they can see that they may actually be mathematically inclined through means that have not been emphasized in traditional classroom practices.

Nick’s work invites students into a world and perspective that may not have been previously available to them. As a science teacher, his work challenges how I think about the mathematical processes I teach. Complex processes can be explained and observed through simple, everyday objects if looked at with a bit of creativity and connected to meaningful experiences in students’ personal contexts.

Question for Nick Sayers:

What do you think schools and teachers can do differently in early grades to prevent students from internalizing an identity in which they are “bad at math” or “bad at art”?

Reading Response Week 7 - Writing a mathematical art manifesto

Reading: Fumiko Futamura. (2025). Writing a mathematical art manifesto. In Bridges 2025 Archive, Eindhoven, NL, 589-594.

Summary:

In this paper, Futamura describes a proposed workshop to lay the foundations for a mathematical art manifesto. The main goal of the workshop is to develop a shared understanding on what mathematical art is as a form of human expression, separate from its usefulness as demonstration, illustration, or pedagogy. Shaped by the historical traditions of artist manifestos of the 20th century, Futamura establishes how manifestos identify an established paradigm, criticize it, and introduce a new one. Participants in the proposed workshop collectively define mathematical art, evaluate selected artworks, consider which paradigms may be rejected, and draft the elements of a manifesto.

The paper also provides historical context for mathematical art and acknowledges the argument that all artists utilize mathematics, referencing artists who engage with mathematical ideas through metaphor, structure, or process. Futamura also questions the argument that mathematical art is a subfield of mathematics, instead proposing it as an art form of human expression. Overall, this paper uses the proposed workshop to provoke reflection on the characteristics and values of mathematical art as a potential artistic movement.

Stops:

1) I really appreciate the idea raised in this paper that all artists use math, knowingly or not. While not everyone has the vocabulary, formal education, or intrinsic interest in mathematics, everyone uses it in some capacities. I think being able to think of how our students use math in their lives through their interests and experiences help us to develop lessons in which it becomes more accessible and valuable to them.

2) I also appreciate how the workshop is designed to allow participants to discuss what counts as mathematical art. In the classroom, a teacher may be questioned for implementing arts-based activities because schools often have a set understanding of what “counts” as math, and this is something that has been passed on to students as well. Reframing what counts as math challenges the traditional definitions and constraints and may help students see mathematics as creative and personally meaningful.

Question:

If you asked students to write a “math manifesto” describing what math is and is not, what do you think they would say? Would that change at different grade levels? What does that tell us about how they experience mathematics?

 Final Project Draft Outline and Annotated Bibliography - Sarah Johnstone

Reading Response Week 6: Mathematics & dance, movement, drama and film

Reading: sarah-marie belcastro & Karl Shaffer (2011). Dancing mathematics and the mathematics of dance.

Summary:

Belcastro and Shaffer describe the ways in which mathematical ideas are intrinsic to dance; music is divided into counts, and counts are used to mark timings in which particular movements are done. They discuss that complex patterns arise when dancers play with rhythm, and that various dance styles and traditions have their own way of using mathematical concepts.

The paper describes the use of symmetry in dance, and the various types of symmetry such as translation or mirror reflection. While dancers and choreographers may not recognize the exact mathematical terminology, they have a deep understanding of performing these mathematical ideas and therefore likely have a better understanding of them than some with formal education on the topic.

Belcastro and Shaffer also describe how explicit mathematics shows up in the history of modern dance, as the work of Rudolf Laban describes the direction, types, and qualities of movements within the kinesphere, noting coordinates on the axes in relation to certain dance steps.

Dance is typically intertwined with music and therefore sound, with repeating rhythmic patterns, leaving endless mathematical opportunities. Importantly, the authors note that dance is not solely mathematical and must take on a life of its own, revealing the humanness of both in order to not limit the possibilities within either mathematics or art forms such as dance.

Stops:

1) “Dancers and choreographers might not use this terminology, but they do have to be adept at performing symmetries” (p. 2)

This quote really highlights a consistent barrier within education and even more so within academia. In order to actually engage with mathematics, we often are required to understand numerous terminologies, which may not be consistent between regions, cultures, or languages. While some may argue knowing extensive mathematical vocabulary is a basic, fundamental “skill”, it sometimes restricts access to mathematics instead of allowing everyone equal opportunities within it. This can result in success within mathematics to become exclusive only to those with consistent and high-quality mathematics education within one culture and language. Understanding mathematics and recognizing mathematical terminology are very different things, and being unfamiliar with the technical words does not necessarily hinder one’s ability to engage meaningfully with mathematical ideas.

2) “Dance is many things, sometimes all at the same time: artistic expression, ceremony, social interaction, political protest, expression of sexuality, a form of athletic competition, physical exercise, theater, psychological catharsis, community event. And sometimes it’s the interplay of these elements with mathematics that engages us as artists – and, we hope, engages the audience as well” (p. 5)

I appreciate how this quote touches on the humanness of dance and mathematics, and I think it can be related to schooling and education as well. Our lessons are not solely one thing – math class is rarely solely about math. Schooling, and our lessons, involve real people with real lives, and the way we can find and understand math within our own contexts outside of the classroom is the key to finding meaning within them. Sports, dance, music, and so many other interests that students have are heavily intertwined with math, and by bringing those contexts into our lessons and into our classrooms, we can guide our students to recognizing the importance and creating their own mathematical meanings.

Question:

How can we introduce mathematical terminology while also ensuring we are not restricting meaningful engagement to only to those familiar with the vocabulary and symbols involved in mathematics instruction in our classroom contexts?

Reading Response Week 5: Developing Mathematics Pedagogies that Integrate Embodied, Multisensory, Outdoors and Arts-Based Modalities

Reading: Dietiker - What Mathematics Education Can Learn from Art: The Assumptions, Values, and Vision of Mathematics Education

Summary

In this reading, Dietiker proposes that mathematics curriculum is truly a form of art as mathematical content can be presented in narrative terms, allowing the mathematical experience of students to be rewritten and artfully crafted into a mathematical story. Dietiker proposes this reimagination of mathematics education after analyzing years of curricular reform, which has revised content but done little to improve the mathematical experience of students.

Dietiker asserts that more importance should be placed on exploration than on discovery, that more value be placed on surprise and anticipation rather than control and predictability, and emphasizes aesthetic ways of knowing and learning. These aesthetics describe an individual’s response to an experience, the act of making something of the experience, and the motivating influence that can advance an individual through challenges and setbacks, allowing one to sense the value and truth – part of what it means to think and understand mathematically. Claiming current mathematical practices to be “too tidy”, this paper challenges assumptions about mathematics and the values of mathematics education with respect to the current realities of teaching. Pointing out that artists develop and achieve their “ends through their means”, Dietiker highlights the importance of the process of mathematical understanding opposed to rote memorization or linear procedures, thus revealing the importance of “untidying” the mathematical experience of students.

Mathematical stories are interpretations of how the mathematical content emerges and changes through a series of events, reflecting the ongoing changes in understanding of the ideas under discussion. Introducing “characters” or mathematical properties can allow for the development of prior knowledge into more sophisticated explanations and allows for the integration of plot twists in which anticipation builds and shifts away from the predictability of traditional practices. Dietiker describes that students’ experiences and anticipation for what was to come were equally important as her learning objectives, and that stories can integrate both logic and aesthetics by focusing on both the intelligibility of what is being presented but also whether the story guides students to continue their explorations to satisfy their anticipations. 

Overall, Dietiker’s paper proposes that educational challenges can be met by applying an artful lens, describing her own experiences with developing mathematical narratives as a way of improving the mathematical experience to inspire wonder and grab attention through surprise.

Stops

1) “However, as attractive as this goal may be, part of the problem with the mathematics classes described earlier may be that they are too tidy. Coming to understand a mathematical idea is generally not a tidy affair, as evidenced by written accounts of mathematicians” (p. 2).

As someone who thrives on structure, routine, and predictability, this quotation really forced me to reflect on the way content is presented to students. However, I agree that coming to understand mathematical ideas, or any challenging unfamiliar content, is anything but “tidy” for students. They often struggle, make mistakes, and have various setbacks before grasping a meaningful understanding, though our practices often involve linear presentations of material, rely on memorization, place little value on prior conceptions, and frame knowledge development as “tidy”, simple, or straightforward, when in fact it is the opposite. Instead of trying to simplify content delivery to fast-track students to coming to the previously determined correct answers, more emphasis needs to be placed on the process to give students the time and space to work through their understandings or misconceptions to make the learning process more meaningful. 

2) “Stories integrate both logic (e.g., Does the story make sense?) and aesthetic (e.g., Does the story move me to continue reading?)” (p. 4)

This reminds me of previous readings I have done on conceptual change in science education – the idea that we must replace one’s prior conception which may be incorrect or incomplete, with a more scientifically accurate one. Literature in conceptual change claims that for this shift to take place, the new conception must be intelligible, plausible, and fruitful – that is, students must understand the presented idea, they must find it to be a possible explanation, and they have to see the value in adopting that scientific conception. Stories would allow for these shifts to happen more organically but adopting a logical stance in which it makes sense to students but also allows it to be fruitful as it motivates students to continue to develop their understandings to better comprehend the big picture.

Questions

1) What are some other ways we can we “untidy” traditional classroom practices?

2) How do we, as educators, balance the desire for structure and predictability to guide our students in their mathematical experiences while also ensuring we provide the space for their own explorations? How can this be achieved under the constraints of the school system?

Reading Response Week 4: Mathematics and the Arts Introduction

Reading: Berezovski, Cheng & Damiano (2016). Spinning Arms in Motion: Exploring Mathematics within the Art of Figure Skating.


Summary:

This reading emphasizes mathematical modeling as a key goal in mathematics education, in which students apply their knowledge to real-world scenarios and phenomena. The authors highlight the importance of incorporating contexts that may be meaningful and engaging for students, such as art and sport; in this paper they focus specifically on singles figure skating spins. Two workshops are introduced: one designed for elementary and middle school pre-service teachers, and another for high school pre-service teachers. In both workshops, students analyze a diagram of a figure skater during a spin that includes relevant angles and lengths. 

The authors note that this represents a simplified version of a complex real-world system, as the model focuses primarily on one arm motion rather than full-body movement, demonstrating best practices in mathematical modeling instruction. Using both the diagram and dynamic geometry software that depicts the changing positions of the upper and lower arms during the spin, students are asked to respond to a series of mathematical questions. This approach allows students to engage with a more accurate, dynamic representation of motion, rather than relying solely on a static two-dimensional model that fails to capture movement over time. Overall, the paper illustrates how teachers can design meaningful models and contexts that draw on students’ interests and experiences, while integrating mathematics with artistic forms such as figure skating.

Stops

1) “Mathematical modeling is recognized as an important content area to be learned in school mathematics. Students are expected to apply their knowledge of mathematics to different real-life situations and solve real-life phenomena” (p. 625).

This quotation really stood out to me in this paper (even though it’s the first sentence!!) because I strongly believe it captures the foundation of meaningful mathematical and scientific learning. Modelling helps students visualize processes while also recognizing the importance behind what they are learning, shifting schooling away from rote memorization and toward understanding real-world phenomena. Through the incorporation of modelling, embodied or multimodal approaches can also be integrated into classroom practices as well, allowing instruction to become more accessible for students may not thrive under traditional methods. By using modelling regularly, we can move away from viewing it as a form of enrichment and instead recognize it as a necessary component of authentic learning.

2) “Usiskin [4] has advocated simplifying complex real-life problems for the purposes of mathematical modeling in the school curriculum. Consistent with this suggestion, we focus on the motion of one particular body part, the arms. Also for simplicity, we ignore the additional distance traveled by the rotation of the skater’s body” (p. 625).

This quote stood out to me because it directly addresses one of the reasons teachers may be hesitant to incorporate modelling into their practice. I have heard other educators suggest that using real-world contexts would be “too complicated”, and I recognize that I have likely been guilty of this thinking myself. By giving ourselves permission to simplify complex scenarios or focus on only certain components of them, we are still able to integrate students’ contexts and experiences with meaningful mathematical representations, standards, and competencies. This reframing helps make modelling more manageable and realistic within typical classroom constraints. 

Questions:

1) What are some potential limitations or hesitations teachers may have to incorporating modelling into classroom practices?

2) How might teachers design modelling activities in ways that are accessible and intelligible to all students with different personal contexts and lived experiences?