Week 7 Reading Reflection

 Article Summary:

Futamura's article Writing a Mathematical Art Manifesto, starts by highlighting the first art manifesto by Filippo Marinetti, the Manifesto of Futurism. Emphasizing the inspiration from Karl Marx and the use of strong imagery and fiery language. Aiming to introduce a new aesthetic, his manifesto was naming and rejecting an old paradigm just as much as it was about defining Futurism. Manifestos became a way for artists to distinguish themselves, during the modernist age. Becoming an art form in itself, manifestos were used to announce radical ideas and new directions to push the status quo. 

In preparation for workshop, they discuss the beginnings of the foundations for a mathematical art manifesto, centering the perspective of creative human expression apposed to its usefulness as demonstration, illustration or pedagogy. Following the process of naming and rejecting an old paradigm to establish a new one, they needed to ensure not only an understanding of mathematical art but how it additionally rejects an artistic belief or norm. 

The workshop structure:

- Facilitator introduces manifestos and goals 

- In small groups discuss: What is mathematical art, or math/art, as a physical manifestation of creative expression? 

-Groups share ideas

-Facilitator hands out images of artwork, in small groups discuss: Are the following works considered mathematical art according to your understanding of mathematical art as a physical manifestation of creative expression?

-Groups share ideas

-Facilitator goes more in depth of structure of the manifesto, presenting examples of rejected artistic paradigms. 

- In small groups discuss: What established paradigm or tradition in art might we be rejecting through declaring that mathematical art or math/art is art? Specifically, what criticisms do we have about the established paradigm? 

-Groups share ideas

- Facilitator leads reflection and creation of an outline for a mathematical manifesto

They acknowledge that some may argue that this contrasts the welcoming nature and inclusive spirit of many mathematical artists as seen at the Bridges Conference. Stating that most if not all artists use mathematics either consciously or unconsciously. 

The article covers a recent history of art that uses mathematics, recognizing that mathematical art can be found far back in the history of many cultures before framing the focus of this article on visual artwork produced in the west in the 20th and 21st century. Many mathematical developments inspired artists in the 20th century, in these works the mathematics was not the focus but used as metaphors with transcendent experiences. Mathematical models have captivated and inspired artists who have created guided by intuition and pattern not necessarily a strict following of a mathematical equation. Artists re-discover mathematical ideas like the Mobius Strip and Riemann Surfaces, through artistic experimentation. 

The introduction of computers fostered fascination with visualizations and mathematical curiosities, some images were just visualizations, but some mathematicians and computer sciences started to use computers as its own artistic medium. Carlos Sequin presented "Art, Math and Computers: New Ways of Creating Pleasing Shapes" at the first Bridges Conference, explaining his process of using a computer to virtually design complex geometric shapes similar to minimal surfaces that could be sculpted out of wood. 

Art with underlying mathematical themes and processes became popular in the mathematical community in the 21st century, with support of established organizations like Bridges. Mathematical Art is still performed among people who primarily identify as artists, but increasingly there have been mathematicians who have found artistic expression through mathematical art. 

In discussing what is mathematical art the article shares that mathematical art recently has be characterized as a subfield of mathematics. In Farris' 2021 article Mathematical Art as a Discipline he claims that mathematical art has become a discipline in its own right. George Hart places math/art within mathematics, and a liminal space in between. Mathematics connects to art in a variety of ways. They ask what if the focus on mathematical art as a form of creative expression over its practical purpose? 

The article shifts to share the characteristics of a manifesto, sourcing from Hart's 2024 article What Can We Say About Math/Art"? he posed the question of if a mathematical art manifesto should be the goal, sharing that he believes that there is not enough consensus on the core values of mathematical art. Futamura shares that while the first Futurist Manifesto was written in 1909 over 50 more were written over the next decade, concluding that a group with common views can come together and write separate manifestos. While there are many different styles and content that can come together in a manifesto, from poetic forms to short numbered declarations, there is a general three step formula developed by Marinetti that many follow. 

Steps:

1. Identify and define an established Paradigm in art.

2. Criticize the paradigm ( with strong passion)

3. Introduce a new paradigm to counter the old (again with strong passion).

Marinetti declares that a good manifesto "must have violence and precision, a specific accusation and a well-defined insult, wit and bombast, and a sense of style that mimics the art itself." (Futamura, 2025, p. 592)

The article shares specific artworks that may or may not be considered mathematical art, these are works that are not by known mathematical artists, followed by quotes from mathematical artists. 

One of the works shared in the list was John Sims, Seeing Pi, felt fitting with Pi day coming up and was the only fiber arts based piece on the list. 

Math artist John Sims is living a life of pi. QUARTZ. (2022, July 20). https://qz.com/quartzy/1571241/pi-day-math-artist-john-sims-creates-a-life-of-pi

"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, Artist

(Futamura, 2025, p. 593)

Futamura concludes the article sharing examples of rejected paradigms from art manifestos, and that many more can be found in 100 Artists' Manifestos edited by Alex Danchev.

 

Stop 1: "if an artist decides that their art is mathematical art, then it is." (Futamura, 2025, p. 590)

I wonder if in their intention of this statement they were saying, mathematics just is and the line of weather art is mathematical art is in the artists acknowledgement of such, because it is present, it exists, so is it the naming, acknowledging, and seeing of it by the artist that is the determining factor. 

Stop 2: "'Mathematics creates art'; 'Mathematics is art'; 'Mathematics renders artistic images'; 'Hidden mathematics an be discovered in art'; 'Mathematics analyzes art'; 'Mathematical ideas can be taught through art'" (Futamura, 2025, p. 591)

I really enjoyed this quote, the lens for art and math are so rooted in each other and I appreciated the way this quote summarized that. 

I was intrigued by the works by Felix Klein and Alfred Clebsch that the article mentioned and looked up some of their works and saw similarities of these curved images and structures to the work Nick Sayers shared in their interview. Multiple points in the article there was mention of violence in writing  manifestos, I found this to be a jarring word, after my first read through I wondered if its use was more intended to be provocative and invoke a reaction or feeling, if that's what their intent with the manifesto and with their use of words like violence in their description of it, when I started to reframe it with that as the intent I additionally considered the nude art pieces that Nick mentioned in the interview and how that could also be considered provocative to invoke a reaction or feeling.

Question:

What do you think defines mathematical art for you? Does it just exist? Are there specific defining features that are needed?

Futamura, F. (2025). Writing a Mathematical Art Manifesto. Bridges Conference, 589–594.

Math artist John Sims is living a life of pi. QUARTZ. (2022, July 20). https://qz.com/quartzy/1571241/pi-day-math-artist-john-sims-creates-a-life-of-pi

Week 6 - Reading Reflection

Article Summary:

Campbell and von Renesse's article Learning to love math through the exploration of Maypole patterns they demonstrate how seeing Maypole dancing through a mathematical lens can excite Liberal Arts students to explore mathematics more deeply. Starting with sharing their foundational perspectives on their math class. Later delving into the mathematics discovered by the Fall 2016 class and Julianna Campbells independent study. Exploring how many non-equivalent ribbon patterns are there, given the number of dancers and numbers of colours?

Julianna Campbell was a student in Prof. Von Renesse's 'mathematical explorations' class, at a small public university, this course was a general core class for non-math majors, the course description is based around meta-goals and doesn't require specific content goals, 

The meta -goals for the course include that students will:

• appreciate mathematics as a human endeavour which is one of our most fundamental intellectual pursuits
• strengthen their reasoning skills and become better problem solvers.
• strengthen their skills in reading, writing, argumentation and speaking.
• will become more self-monitoring, reflective learners and take greater personal responsibility for their learning.
• approach mathematics more positively and gain a balanced perspective of mathematics.
• improve their mathematical confidence.
• develop awareness of the negative impact of broadly held societal views.
• be capable of and interested in considering mathematics outside of the confines of the classroom, understanding the value of lifelong learning in mathematics

(Campbell and von Renesse, 2019, p.132)

The classroom pedagogy centered around inquiry, von Renesse shares a definition of inquire from a study by Laursen et al., defining inquiry-based learning courses to be "characterized by:

• Learning goals focused on problem-solving and communication
• A curriculum driven by a carefully constructed sequence of problems or proofs
• Driving toward a small number of big ideas
• Course pace set by students’ progress through this sequence
• Class time used for a mix of active and collaborative problem-solving tasks
• Instructors who guide student work instead of delivering information."

(Campbell and von Renesse, 2019, pp. 132-133)

Campbell reflects on the course as the first time she was encouraged to ask questions and search for answers without the fear of seeming less intelligent, motivated by the desire to understand and the motivation to question. Sharing that every time they thought they had reached the end of questioning Prof. von Renesse would ask ""But how do we know for sure? Will this always happen?'"

Prof. Von Renesse alongside colleagues profs. Fleron, Hotchkiss, and Ecke developed content and pedagogy materials for using inquiry in teaching mathematics for liberal arts courses, which can be found at www.artofmathematics.org. 

This mathematics class was stable as a fully integrated learning community with an English composition honors class. Working alongside colleagues Jennifer DiGrazia, they were able to model how to be students in each others subject area, to take risks, be curious, and make mistakes. The topic chosen for this course was from the arts, this gave students who felt negatively about mathematics a chance to start new and experience mathematics differently. 

Julianna Campbell who co-authors this article as previously mentioned was a student in this course. Campbell had experienced math as a source of struggle and insecurity, until experiencing Westfield State university's math and English learning community. Feeling failed by the route memorization and formality of her K-12 math education, it wasn't until she was encouraged by Prof. Von Renesse to follow her natural instincts of curiosity to question and struggle with mathematics that she found joy in mathematics. Campbell shares the insecurity she felt first walking into that class, feeling confused by conversations about factors, questioning her idea of factors, to the encouragement she was met with and the progress she has made to have completed an inquiry-based mathematics independent study, presented at a math conference and co-authored this paper with her professor. 

The class was spent in small groups initially searching for conjectures and taking steps to prove them, with students volunteering to share work without fear as there was and understanding that each mistake led them closer to an answer, being encouraged to develop an argument around their answers and thought process. 

Campbell describes herself as the other breed of mathematician, saying others can quickly solve a problem and logically find a solution, but she is strong in areas of inquiry, curiosity, and perseverance with mathematics. She says mathematicians like herself have her to be encouraged to pursue mathematics in K-13 education, and it wasn't until collage that she found her place in mathematics, stating students are experiencing a misinformed perception of math, it can be used to enrich our lives, to answer questions your students truly care about, creatively and as a means of bringing students together as a team.

The article goes on to describe a mathematics classroom that is active, engaged, with arguments, discussions and developments of proofs, where math assignments are pages long of handwritten theorems, filled with inquires, points of confusion, and proof of hours of hard work; a math class of non math majors. Working in small groups on particular problems, before gathering as a whole class to discuss ideas, Prof. von Renesse would listen to ideas, ask questions, but let students lead with their findings. Students collaborated to understand and figure out problems that initially seemed impossible, doing things that now felt new and exciting.

Campbell recalls the day that Prof. von Renesse brought a tall PVC pipe and a bag full of colour stripes of ribbon, and the class danced the Maypole dance, launching a 6-week undertaking for the class and a semester long independent study for Campbell. Creating space in the class they assembled the pole and practiced dancing, creating an over-under pattern. Once the pattern had been danced and the ribbon was wrapping the pole, the students observed and were asked "what do you notice? what are you wondering about?" Without fear students could share their observations and conjectures, no idea wrong, just a step closer to a bigger lead. They explored what happened if leaders held all one colour and followers all another, if some leaders held some of each colour and followers some of each as well. considering how these alterations and manipulations changed the picture. As they continued questioning they determined they needed to represent a Maypole dance without actually doing the dance. This was considered to be their biggest challenge, there was no known answer, it was new, students worked together to explore different ways to represent this 3D representation into a 2D image, they considered different viewpoints of above and to the sides but it didn't look right. A student named Austin figured out a way, he 'cut' the pole vertically and unwrapped the picture so that it could be viewed as a rectangle, this was later termed by Prof. von Renesse and Campbell as the tree representation. Austin showed the class how to draw the dance and later another student, Olivia helped refine the technique by using graph paper to help make the picture neater and more systematic, supporting students quest to search for patterns.

Originating from a traditional folk dance, the maypole dance involves a tall wooden pole with various coloured ribbons attached at the top, from the bottom of the pole people hold the ribbon and dance in patterns. "Figures 4 and 5 show an example of a ribbon pattern with 3 couples. Assume that there are 3 black ribbons (called 1, 2, and 3) and 3 red ribbons (called A, B, and C). Each black ribbon is paired with a red ribbon. All of the people holding black ribbons move to the right (i.e. mathematically positive when looking at the pole from above) and go first over the red ribbons. At the same time, the people holding red ribbons move left (i.e. mathematically negative from above) and go first under/inside the black ribbons. After this first step, the black ribbons would go under while the red ribbons go over. This pattern keeps alternating. While the dancers move around the pole, the ribbons will cross over and under to generate different patterns depending on the colour and order of the ribbons." (Campbell and von Renesse, 2019, p. 136)

With the tree diagram determined by the students, von Renesse introduced an efficient way to label the different dances with letter representation. With the expanding scope of patterns hand-drawing these representations became more challenging, leading to independent student and professor exploration of excel sheets to attempt to automize drawing the ribbon patterns.

The following definitions were determined throughout their process.

"Definition 5.1 (Tree Representation): We call the representation of a ribbon pattern using diagonal ribbons as in Figure 4 the tree representation. The ribbon pattern on the maypole is ‘cut vertically’ so that the cylinder can be unwrapped to see the full pattern. Leaders are represented by numbers and followers are represented by letters. Leaders are defined as ribbons which first ‘go over’ in the dance. Accordingly, followers go under first in the dance. In the image, 1, 2, 3 are leaders and A,B,C are followers; we can also see that the 1-ribbon first passes over the A-ribbon, the 2-ribbon passes over the B-ribbon, etc.

It was a big step for the class to notice that the tree representation doesn’t need to be ‘coloured in’. Although it is tempting to colour these diagrams, and often very helpful when looking at specific examples, this diagram is more useful for proving results in its general form.

Definition 5.2 (Letter Representation): We can also represent a ribbon pattern by the shorter letter representation 1A2B3C. We call this in short the letter representation. Here the letters and numbers are used as described in Definition 5.1. 1A2B3C is the basic unit of all dances for 3 couples. If we want to talk about a specific pattern, we can, for instance, write BWBWBW for the patterns resulting from 3 black ribbons (leaders) and 3 white ribbons (followers).

To avoid having to draw too many ribbon patterns by hand, we explored a Microsoft Excel sheet to find the coloured patterns more quickly. You can find the excel sheets at https://www.artofmathematics.org/node/705. This led to a new definition.

Definition 5.3 (Screen Representation): We used Excel to automize the pattern drawing and received a pattern in which the ribbons were now horizontal and vertical instead of diagonal. We call this the screen representation of a ribbon pattern. Figure 6 shows how the tree representation relates to the screen representation. The leader ribbons (numbers) are now horizontal while the follower ribbons (letters) are vertical. 

In the screen representation on Excel, we can easily extend the pattern horizontally to see the larger pattern.

Definition 5.4 (Fundamental Domain): A fundamental domain of the ribbon pattern is defined as a smallest region that repeats itself inside the whole pattern. See Figure 7 to see a fundamental domain of 1A2B3C in screen representation. Notice that a fundamental domain has to exist since the dance repeats itself, but that it is not unique." (Campbell and von Renesse, 2019, pp. 137-139)

Initially the classes guiding questions sought to understand how the maypole dance worked. Hoping to predict the ribbon pattern for any given dance without having to dance, to determine this they narrowed the dance to four and then six dancers. With a growing understanding of the dance, students wondered how many different ribbon patterns there are given the number of ribbons. The class manipulated the letter representation to determine if new geometric patterns would emerge and if there would be commonalities with the original pattern. Identifying four ways, the letter representation could be manipulated with the ribbon patterns being essentially the same. 

The article goes on to define what equivalent ribbon patterns are and discuss proofs of four corresponding theorems. Later seeking to determine how many different ribbon patterns are produced by specific numbers of ribbons and colours, they use powers with a as the base representing the number of colours and b is the exponent representing the number of ribbons. They use this alongside their previously established definitions and proofs to identify the different ribbon patterns, concluding the following.

Number of ribbons- Number of colours: Number of different ribbon pattern

6-1:1 ribbon pattern

6-2:5 ribbon pattern

6-3:7 ribbon pattern

6-4:5 ribbon pattern

6-5: open problem, pictures do not have much visible structure, not as aesthetically pleasing

6-6:1 ribbon pattern since all ribbon patterns, are colour changes of each other the pattern is a checkerboard.

(Campbell and von Renesse, 2019, p. 150)

Stop 1:

"Students are experiencing a misinformed perception of math." (Campbell and von Renesse, 2019, p. 134)

This was more of a sad stop, I agree that many students experience a misinformed perception of math. I sometimes as a secondary teacher feel I am fighting against the math ideas and identities students have already started establishing for themselves. Throughout this program I realize just how misinformed my perception of math was, I am still wrapping my head around the various connections that are and are still to be made with mathematics. With the videos this week, specifically thinking of the TedX talk with Karl Schaffer and Mr. Stern I felt myself trying to see the math before they fully revealed it, I thought of common multiples but hadn't fully made the connection to fractions and fraction subtraction before Schaffer revealed it. I find myself feeling the furthest distance from embodied mathematics with music and performance and am hoping to continue to bridge more of that distance and start to find ways of utilizing these components in my teaching practice.

Stop 2:

"In our class we sat in small groups, working together on a particular problem, and then gathered as a whole class to discuss our ideas." (Campbell and von Renesse, 2019, p. 135)

This sentence was a stop for me as the description reminded me of a very specific time in my math education. I had left an engineering program, spent a year taking a variety of classes before switching to a different university and going into a math degree and my first course was a proofs course. In that course I experienced a style similar to this, where we were regularly working in small groups all contributing to working through a proof together on one of the chalkboards around the room, before coming back to the class as a whole and having each small group present their proof to the rest of the class. I remember in that class not really knowing my classmates as I was joining in a small university into second and third year courses, but all of my courses with this prof felt like a little community, her style of teaching did just as this article said, it welcomed mistakes as a part of the journey of learning and through that built connection among students. While we didn't experience embodiment in the way that this article described, many other key features discussed aligned strongly with my experience in courses taught by this Professor. 

I apologize for the length of this summary/post and the amount of directly quoted information, I struggled to summarize their definitions and proofs without feeling that the details needed to follow the work were getting lost, and found it very interesting to wrap my head around these proofs and wanted to include those details for folks who were also curious. If interested in the more detailed information of the proofs and accompanying visuals check out page 139 onward.

Question: How do you think we can work against students experiencing a misinformed perception of math? 

Campbell, J., & von Renesse, C. (2019). Learning to love math through the exploration of Maypole Patterns. Journal of Mathematics and the Arts, 13(1–2), 131–151. https://doi.org/10.1080/17513472.2018.1513231 

Week 5- Activity Reflection

Trying Out Extending the Activity:

I chose to primarily explore Sarah Chases illustration of moving 3 against 2. Initially needing a few attempts to start to get the hang of it, I appreciated her tip of assigning the different arms different non-numbers to make it a bit easier to follow the cyclic progression.

After feeling I got the hang of that I tried to think of what else could be represented through this type of movement, and wondered about using it for fraction multiplication (proper fractions multiplying by proper fractions). Where using your two arms can represent the multiplication of the two numerators and then your two legs could represent the two denominators. This idea felt more clear to think than do, I picked small numbers for both numerators and denominators 1/3 by 2/5 I also picked something that wouldn't simplify because I wanted to focus on using all my limbs first before considering what a simplifying fraction through this illustrated movement might look or feel like. Ultimately I found it challengeing using all four limbs to keep track of where my arms were in the count and my legs but manged to still try out movements connecting to proper fraction multiplication.

After more time with the movements I started to wonder about using numbers that were not prime, not to determine their product but to consider divisibility, so looking at something like 2 and 4, then 12 and 3, I found myself needing to use my fingers to track what cycle of the divsor I was on in my movement so I could focus on executing the correct movement for the count. I found it worked where if after one cycle of the divided your dividend and divisor were both going to start back at position 1 at the same time then the number was divisible. I wondered how this may look if divided among two people, one as the divisor and one as the dividend to make it a group activity, leading further to the thought of doing it with a class and having multiple students be different divisors and perhaps a student or the teacher be the dividend, and as a bigger group go through the movements and when the dividend is done and returning to the first position again to stop and see what divisor students are also at the first position. Challenges I would anticipate from that would be coordinating the pacing of the movements but feel a slow beat could help keep everyone at the same pace. 

I took some time to consider Ali and Colins work as well, as I loved learning more about the Bridges works last week and greatly enjoyed their video. I was thinking back to a few years ago when I had some advanced students who I was trying to create extension projects for, these students were also very engaged with artistic and alternative expression, and I was trying to find a way to create a project that they could make that would express the rules of simplifying polynomials through artistic expression. After multiple attempts to create an example, I sat with them and left it open for them to create anything that would visually communicate the simplification of polynomials without giving an example, partly because I wanted to see what they would do and partly because I was struggling to create an example that I felt represented it well. One of those students made a book, and all the key components of like terms, the degree of the terms, etc. were all communicated through such detail it took me multiple times going through it to notice the level of detail they attained and communicated. 

A video from 2024 when I received this book from the polynomial unit extension.

Edit: The video wasn't working when I looked back at this post, try this VIDEO LINK if you are interested.

The Start of the Curriculum Idea:

Sarah Chase's mathematical movements, reminded me of my first week reading by Gerofsky, Seeing the graph vs. being the graph. Currently my grade nine students are working through our Linear Relations Unit, there is an art integration project I have used before first with only extension students, and then the following year with the whole class. This is a unit that is current for myself in the moment and one I am more familiar with having taught. I used this as the focus for my curriculum idea.

I wrote my ideas out on paper for more of a flow (if parts are hard to read in the image let me know and I will type out a digital version).

From 2024, the first year I tried having extension students draw, determine equations, and digitally graph their drawings

From 2025, a student both completed the digital graph of their drawing and created restrictions for all of their equations.

Week 5 Reading Reflection

 Article Summary:

Riley et al. worked in the article Movement-based Mathematics: Enjoyment and Engagement without Compromising Learning through the EASY Minds Program, to determine both teacher and student perceptions of the Encouraging Activity to Stimulate Young Minds (EASY Minds) program, which is designed to increase physical activity and enhance learning and engagement using movement-based learning activities. This study was a 6 week intervention

The article established the problem of low mathematical engagement and its resulting declining achievement scores, highlighting the prominent factors tied to student engagement to include teacher influence and the pedagogies employed in mathematics. This work focused on utilizing the integration of physical activity into math lessons, emphasizing that this program has an additional aim of increasing physical activity to support the benefits it has on children's physical, mental and cognitive health. Acknowledging both the importance of teachers in the delivery of interventions such as this and the challenges with the skills and knowledge to integrate it effectively. 

The study was a 6 week intervention after selection from the EASY Minds cluster randomized controlled trial. The teachers involved were trained over a single day professional learning for the interventions delivery, promoting two types of mathematical lessons: activities that used physical activity for development of procedural fluency of fundamental number operations such as students recall of multiplication tables while skipping, and activities focused on looking at mathematics in the world around the school such as estimating and measuring distances. Grade 5/6 classes were selected from eight public schools in New South Wales (NSW), Australia and randomly assigned intervention or control groups. During the intervention groups professional learning, they received a resource pack of equipment to help promote physical activity, and a small example for lesson ideas from each strand from the NSW syllabus for mathematics and were directed to  embed movement-based learning in their mathematics program at least three lessons per week for all six weeks. Continued intervention support was provided through weekly emails, members of the research team made three lesson observations followed up by a discussion about a 3 scale self-evaluation/ activity log: 1) mathematical concepts reinforced throughout the movement-based activity, 2) activity levels and transition management, and 3) engagement by students with the activities. The control group maintained their regular mathematics program. At the completion of the intervention the teacher and their selected two students each from higher, middle, and lower achievement, participated in a focus group and discussion approximately two weeks after the completion of the intervention program. Where semi-structured discussion frameworks were used to facilitate discussion around perception of the program, and later verbatim transcripts were analyzed by an independent researcher not previously involved. 

The results were separated into enjoyment and engagement of mathematics lessons and the quality of learning experiences. The perceptions of the program by students and teachers were positive, with increased enjoyment and engagement in the mathematics lessons. Highlighted by many students was the increased time outside, in the fresh air, and having more fun. Sharing that it helped them concentrate, focus better, and reduced talking, off-task time, and other distractions. Students from all levels reported finding EASY Minds beneficial, additionally students perceived their teachers to have enjoyed the program for trying new things and not having to deal with as many discipline problems. All teachers interviewed perceived the program to be enjoyable and engaging, they were all planning to continue with the EASY Minds approach. Teachers additionally commented on wishing to see the program extended to other subject areas as well, or even whole-school level. Teachers acknowledged the benefit of the resource equipment and the running around to organize equipment that they no longer had to do, and one suggested a pooling of lesson plans to share and make more accessible for everyone. 

 The study acknowledged limitations as the need for the professional learning day to prepare and the physical resources teacher acquired through the program, all teachers involved were teachers prepared to embrace the EASY Minds movement-based learning approach. The professional learning was given by researchers who specialize in physical activity or mathematics, making future replication or application of this project to be financially challenging. 

The shift from worksheets based activities and teaching that students perceived as dull, repetitive, and uninteresting, making it easy to be distracted to a 14% increase in on-task behaviour during active math lessons, and students finding their teacher more innovative with different interesting activities for their learning, shows the the EASY Minds program can support changing students attitudes towards math and increase students physical activity.

Stop 1: 

"Teachers were only given a small sample of lesson ideas to encourage creativity, autonomy and ownership of lesson content" (Riley et al., 2017, p. 1657)

I appreciate that they gave the teachers a set of lesson plans to have a jumping point but I would be curious to know in the professional learning how much time there was for the opportunity for teachers to discuss ideas an collaborate together to create ideas and activities. When planning more physical or "alternative" math activities I find the amount of time it takes to plan, prep, and execute to sometimes not be feasible, where the upfront work can be multiple hours for just a single hour lesson, where as if I am able to have the time to collaborate or bounce ideas around with someone it goes much faster and feels significantly easier. I appreciated as well that later in the article there was acknowledgement of the "run around" that often is required with incorporating activities like this if you do not have the equipment provided, as well loved their suggestion to share lesson plans to make it more accessible for everyone. The application of the grade levels from the article are lower than the grade levels I teach and I am curious how teachers of higher grades (ten and up) feel about the ability to implement components of this? 

Stop 2: 

"Teacher attitude towards mathematics is a key predictor of students attitudes towards mathematics."(Riley et al., 2017, p. 1668)

I am curious of this in the context of if the EASY Mind program were to implemented by a teacher who did not enjoy components of the movement-based activities. Would their students still see some improvement in engagement and perceptions, or would the teachers attitudes towards math in that context prevent that shift? 

Question(s): 

Have you used movement-based activities in your mathematics lessons? If so what grade level and what have you done? If you haven't how could you incorporate it for the future (and what grade level)? What challenges have you found in what you've done or anticipate in what you would like to try?

Including images of some outdoor movement-based activities I have tried in the past:

Creating shapes with specific areas and calculating pre-drawn ones dimensions and area with chalk. (Grade 7)

Recording Basketball shots to do students sports stats. (Grade 8)

Throwing paper airplanes and measuring their distance to gather data to determine mean, median, and mode of their flights. (Grade 7)

Riley, N., Lubans, D., Holmes, K., Hansen, V., Gore, J., & Morgan, P. (2017). Movement-based mathematics: Enjoyment and engagement without compromising learning through the EASY minds program. Eurasia Journal of Mathematics, Science and Technology Education, 13(6), 1653-1673. https://doi.org/10.12973/eurasia.2017.00690a 

Project Outline: Roots, Ratios, and Repairs

Fast Fashion has created a disconnect between people and their clothing. This divide has altered our relationship with clothing, and shifted people away from a focus on lasting quality and proper maintenance, for a relationship of convenience and replacement over repair. There is empowerment in caring for your clothing and maintaining items to have a long closet life. This project will look to utilize mathematical concepts to support and empower students to care for their clothing through exploration of mending, and upcycling materials they already have to give clothing new life.


Search Strategy:

Throughout my search for information and literature I primarily utilized Google Scholar and the UBC Library Collections. Initially Google was used to find blogs and webpages discussing mending and textiles to search for inspiration to narrow ideas and begin the search for academic literature. Starting with general literature on mending and searching for already established articles linking textile mending to mathematical learning, there were no results that supported the directions I was hoping to explore. Shifting towards searching for works that looked at learning that incorporated textiles and mending, textiles and math, as well as works that would provide more insights into processes involved in textile works.


Background:

In Nova Scotia there is an hour a week in the schedule for grade seven to nine students for something called integrated learning time, this is a curriculum-less hour that each school may choose their approach for implementation to create cross curricular and interdisciplinary projects and engagement for students. My school spends the last two months running a program developed by one of the teachers which we call “Skills Rodeo”, where teachers pick an activity, project or skill they would like to run for students, students then give a list of what they would like to sign up for and get sorted into groups, these groups are not separated by grade levels but by interests based on what students listed as their preference. This is an approximately six week program broken down into three two week sessions. Previous sessions I have run for this program were, Google Sheets: Coding, Formulas, Games, and Fiber Arts: Crochet and Knitting.


Topic and Plan:

In this project I would like to focus on textiles, upcycling, and mending. With the set up of the Skills Rodeo program my school runs I plan to break this into three varying projects that differ in focus but build on each other, offering students the chance to take any one of the three possible two week sessions or to take all three and participate in a 6 week cumulative textile project. As the skills rodeo happens at the end of the year I will try out the textile projects with a smaller group of students in my school's GSA. My continued research will focus on structuring this project, and creating support for guiding students while still giving them flexibility and choice across the various stages as to how they will engage and implement the skills and activities.

Nova Scotia Curriculum Connections:

As this format would work across multiple grade levels the related mathematical curricular components it would relate to also span across grade levels.

Grade 7: Measurement, students will be expected to develop and apply a formula for determining the area of triangles, parallelograms, and circles.

Grade 8: Number sense, students will be expected to solve problems that involve rates, ratios, and proportional reasoning.

Grade 9: Geometry, students will be expected to determine the surface area of composite 3-D objects to solve problems.



Annotated Bibliography

Bairy, S., & Inamdar, N. (2026). Enhancing middle school mathematics through interdisciplinary integration: A 21st-century approach. Discover Education, 5(1), 22. doi:https://doi.org/10.1007/s44217-025-00877-w

Bairy and Inamdar advocate for a 21st century approach to middle school mathematics through interdisciplinary integration, within their article they discuss a micro-project and a variety of activities focusing on hands-on real-world challenges, enhancing conceptual understanding, and student-engagement. Within their methods they highlight the key elements for the design and planning of a micro-project, implementation, assessment methods, as well as 10 activities highlighting the various connected disciplines involved in each activity.

The structure and procedure of this article to guide the creation of these project structures provide strong factors for consideration in the development and implementation of mathematical projects. Additionally, it is beneficial to have examples of activities and interesting to notice the consistent presence of art alongside all of the mathematical activities.


Fisher, G. (2025). A stitch in line: Mathematics and one-stitch sashiko. Taylor & Francis Ltd.
https://doi.org/10/1080/17513472.2025.2469529

In Fisher’s review of ‘A Stitch in Line: Mathematics and One-Stitch Sashiko’ she highlights Seaton’s various project examples noting that she maintained emphasis on mathematics, Fisher shares that throughout the book there are mathematical connections across a variety of levels starting at a middle school level and increasing in difficulty as the chapters progress, providing a brief note of the core math concept of each of the eleven chapters and their corresponding embroidery projects.

The variety of mathematical concepts starting at middle school levels and progressing with direct project connections suggest this book to be a valuable resource with more direct and accessible integration into student learning as well as an opportunity for interdisciplinary work with art and textile courses.


Geetha, B. & Judia Harriet Sumathy, V. (2013). Extraction of natural dyes from plants. International Journal of Chemistry and Pharmaceutical Sciences, 1 (8), 502-509.

Geetha and Judia Harriet Sumathy demonstrate the use of a variety of plants and moderates that can be used in the extraction of plant based dyes for textile application. They consider the environmental impact of synthetic dyes and the push with global concerns to re-explore eco-friendly alternatives for this process as a driving motivator for their research and acknowledge the use of natural dyeing throughout art, history and indigenous traditional knowledge.

Including a natural dyeing of textiles component to this project prompts connections for participants with the environment around them and provides a tactile experience. In exploration of saturation or shades of colours and for replication of the colours production will rely on detailed recording the ratios and process.


Manoj, T. & Prabir, J., (2023). Apparel manufacturing measures and calculations. In R. Chattopadhyay, S. K. Sinha, & M. L. Regar (Eds), Textile Calculations (pp. 275-299) Elsevier ScienceDirect eBook - Engineering 2023, & Elsevier All Access Books.

Manoj and Prabir connect the planning, production, and postproduction to measures and calculations, from plant set-up and facility design, to manufacturing operations providing multiple direct calculations and formulas centred around textiles in the context of manufacturing plants. Primarily focusing on the optimization of time and resources, providing multiple ways to determine fabric utilization percent giving considerations of the calculation by weight or by length.

The calculations focused on material waste and optimization can be applied by students to conceptualize what it takes to make textiles and to encourage thoughtful planning of material usage. Other calculations provided insight into the development of the commercialized processes more commonly seen in today's textile productions.



Willett, J., Saunders, C., Hackney, F. & Hill, K. The affective economy and fast fashion: Materiality, embodied learning and developing a sensibility for sustainable clothing. Journal of Material Culture. 2022, 27(3), 219–237. https://doi.org/10.1177/13591835221088524

Willet et al. explores shifts in peoples relationships towards clothing and its impact on their consumer behaviours as they are immersed in a workshop on the making, mending and modifying of clothing, from a perspective that considered clothing to be low-cost, to considerations of their complexities, viewing them as precious. They highlight a value-action gap where the ethical beliefs and attitudes held by people do not often equate to consistent actions. In addressing this, the impact of peer group support in maintaining change is determined as invaluable, for people have an inherent desire to conform to group norms, within the workshop of the study it was not the direct teaching but the immersion into the workshop and the space for participants to create their own knowledge, and cultural meaning from their experiences participating in the workshop.

The collective experience and influence of peer group support, highlights the value of collaboration and the impact of shared experiences on the participants. Shifting of attitudes and affect requires more than direct instruction or information, but connection, and direct experience.

Week 4 - Activity Reflection

This week I selected Infinite Loop, Thread on Mobius Frame by Skylar Cheung from University of Toronto from the 2020 Bridges Joint Mathematics Meetings . I was drawn to this as I have found myself with mobius strips on the mind recently and with this weeks video of  Vi Harts, Mobius Music Box. I was drawn to the colours and the continuous flow from one to the next, there is no start, end, or break which complimented the mobius frames continuous side in a way I found very visually and mathematically satisfying to look at.

When it came to my recreation it was not as seamless. I wanted to ensure I kept the visual flow with the colours so I spent time selecting embroidery thread to use. However, all of my colours were seperate threads, I tried to glue them to the frame I was using but had little success and ultimately had to tie all the different coloured thread together to create a long thread that would have the colourful progression I wanted. 

When I finished wrapping my mobius frame with the thread, I felt something was off. I kept going back to the original piece by Cheung, and wondered if it was the ends of the threads tied together poking out that gave me that off feeling. It wasn't until the next day that I realized... it was not a mobius strip. I had twisted the frame too much and so there were two sides. Tried to brainstorm what the best way to fix this was, ultimately I unraveled my thread and adjusted the cardstock I used as a frame to become a TRUE mobius strip. I felt the ends of the threads were poking out even more after rewrapping it, tried to use some liquid glue to keep them down as much as possible, but am happy with the colour progression and properly structured frame.

In learning about the Bridges conferences and how transdisciplinary they are I was very intrigued to look at the art, I wondered from my experience as a knitter about if you could create a mobius strip when knitting in the round. When learning to knit, I recalled always being told to make sure your stitches were not twisted before joining in the round. I wondered if there was a twist to your stitches and you join in the round if that would create a mobius strip (this was after my first attempt at the mobius strip activity). I got some scrap yarn and cast on enough stitches to join my work and knit a few rows to see what would happen. 

Unfortunately I made the same orientable strip I had when I first tried to make the activity for this week. I am still curious if there is a way to set up knitting or crochet to actively make a mobius strip, but that is for future exploration. 

Bridges work of inviting in a diverse group and creating community to produce and engage with transdisciplinary works felt healing and tied to sustainability education, it remind me of the following quote from last weeks reading. 

"The whole is more than the sum of its parts: The essential properties of a

living system are properties of the whole, which none of the parts have; these

properties arise from the interactions and relationships among the parts.

Therefore, properties of the parts can be understood when the whole is

understood." (Williams, 2008, p. 42)

A bit of a random tangent:

My go to hobbies have generally engaged with fiber arts, I have crocheted for multiple years and got into knitting last year, found myself quickly hooked as it felt very logical and mathematical to me and throughout the year I enjoyed progressing with learning increases, decreases, lace detailing, double knitting, and most recently cabling. One of my friends husbands and I call each other "Math friends" as most the time we are together we get on some math topic and talk about whatever cool math thing we have seen, heard, or done recently. I started talking to him about knitting and how much it felt math-y to me and showed him the patterns I had been working on and discussed the movement and construction of the garments, he was intrigued and has since started knitting as well! Now our chats cover a lot of math, knitting, and knitting math. The coming together of math and art that Bridges highlights has also been present in my life and friendships and I enjoyed the connections.

Some pictures of my knitting projects that made my math brain excited.

(I call the scarf the inverse operations scarf)

Williams, D. (2008). Sustainability education’s gift: Learning patterns and relationships. Journal of Education for Sustainable Development, 2(1), 41–49. https://doi.org/10.1177/097340820800200110 

Cheung, S. (n.d.). Skylar Cheung. The Bridges Organization. https://gallery.bridgesmathart.org/exhibitions/2020-joint-mathematics-meetings/cherylskylar

Week 4 - Reading Reflection

Column Summary:

Fenyvesi's Bridges: A World Community for Mathematical Art is a feature of the mathematical communities article, and it is not the first time this column has featured the Bridges organization. The column discusses the previous feature from the 2005 conference that took place in Banff, Canada. Diving into the intertwined artistic and mathematical ties that pull this conference together to be the interdisciplinary hub it is known to be, through theatre, formal lectures, an international mathematical art exhibit, a mathematical music night, and a math art workshop series. The work started in that Banff conference was the beginning of the journal of mathematics and the arts with the support of the Bridges community. The Bridges community has always been a transdisciplinary and intercultural platform, never having to shift to include the arts in its approach, as it was always integral to these projects.

​

The beginning Bridges conferences were hosted by South Western College. The work of Reza Sarhangi, an immigrant from Iran with connections in both science and culture, brought attention to the complex cultural roots of mathematics, looking at the joint efforts of mathematics and artistic communities from ancient times. Specific research into Persian history highlighting the work of Abul Wafa al-Buzjani, who educated craftspeople in geometry, the decorative motifs of Persian art of this period required constant training and consultation with mathematicians, as the complex geometric patterns needed advanced knowledge in the area. Sarhangi was more than a teacher before immigrating; he was a graphic artist, drama teacher, playwright, theatre director, and props designer, giving him firsthand experience of the complex and collective artistic processes. Using his knowledge and experience to introduce creative forms of study to change how mathematicians were being educated. Creating a course on connections between mathematics and the arts. He attended art and mathematics conferences alongside artists, architects, and other experts in applying mathematics creatively. These conferences led the way to many interdisciplinary papers uniting perspectives on topics.

Before there was the Bridges conference, there was the international society of the arts, mathematics and architecture. Three of the first four directors of Bridges had previously been involved in ISAMA. Many groups connected to form the background for science and art communities involved in Bridges across multiple continents. Creating more conferences, festivals, societies, and associations that are closely tied to the Bridges organization.

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The column goes on to reflect on the review by Solomon Marcus of the first bridges conference. Marcus, an established pioneer in interdisciplinary and transdisciplinary areas with mathematics, advocated for broader and deeper artistic analysis within the discourse of the Bridges conference. The Bridges conference has followed this, increasingly including new mediums for mathematical art displays and lectures. Marcus speaks to the perspectives both artists and scientists gain from seeing the achievements of the other and its ability to broaden each other's horizons. Bridges' goal was to bring research alive. Bringing together artists, mathematicians, computer scientists, and educators, Bridges wanted the attendees to get more than just the content of the papers but to get an experience that integrates art, dance, and other performances.

​

Bridges has gone barely sixty participants to annually 250-300 global participants and thousands of audience members. The conference has shifted to be held in tourist destinations, adding in day trip opportunities, having been held in multiple different countries. The Bridge's expansion has created a form of math-art tourism, and with international media, has strengthened many local math-art communities.

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The diversity of the topics featured at Bridges conferences entices a large, diverse audience. It is a platform for exchanging experiences, and it encourages teachers to engage in a more math-art approach. The Bridges collection has become the largest exhibit of mathematical art in the world. This transdisciplinary program pushed standards, but faced challenges balancing the conference's openness with academic legitimacy. Bridges had to develop their own way to uphold this, ensuring that submissions are judged by the standards of their own field, using different juries to review works and a transdisciplinary program committee for reviewing conference papers. Bridges works to bring in new ideas and voices every year.

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The column closes, highlighting the value of research, learning, and creativity that can strengthen awareness towards interlocking systems in preserving a sense of exploration and inquiry essential for rational and creative activities.

Stop 1: 

"I use community in its most complete sense - including adults, children, artists, university professors, art lovers, and local people- for the wealth of conference activities could only ne accomplished though each and every individual present." (Fenyvesi, p. 35)

Throughout this column there was a consistent and repeated emphasis on community and the coming together of diverse groups. I found this repeated emphasis to be one of my favourite things in this reading. The transdisciplinary focus and willingness/enthusiasm of Bridges conference to push and explore was exciting, finding places for so many experts in their fields to share and connect. 

Stop 2:

"For medieval craftspeople, creating the decorative motifs common to the Persian art of this era demanded not only constant training, but also regular consultation with mathematicians. Indeed, decorating the inner as well as the outer, spherical surface of a cupola with ties featuring highly regular, yet still extremely complex geometric patterns would have required advanced knowledge of geometry." (Fenyvesi, p. 37)

 

Sarhangi

After reading this in the article I was quickly searching for images of these tiles, I found multiple in work by Sarhangi, and added two from one of his articles (linked below the image if you are interested in looking at more). I am curious as to what consultations would have looked like between the craftspeople and the mathematicians, and the processes that the craftspeople worked through in the creation of these tiles. I wonder what role the mathematicians played within the planning of these designs, were they primarily involved in the execution of the designs, in their composition or actively engaged in both? 

Through this weeks videos I realized just how little I had connected math to music beyond patterns, and fractions/rhythm. I had some experience in band in school and some hobby experience with playing guitar, but have limited knowledge on more theory, and composition. I appreciated in the videos that there was more compositional links through the sound braid (which was fascinating but felt a bit above my head at times) but also lighter connections with the mobius music box which is easier to see moving around (and so fun to crank). I wonder how much work some of these things would be to integrate into non-math classrooms or if they are already doing it. I try in my class to highlight how topics connects outside of "just math" and realize I have not had this conversation with colleagues about if they seek to highlight the links to things such as math in their courses. Seeing the musical connections of the videos has peaked my curiosity for what Bridges theatrical performances looked like as admittedly that is not somewhere I had previously considered the mathematical links before. 

Question: What ways have you linked art and math in the classroom? Have you made musical links to math in your classroom, if so in what ways?

Fenyvesi, K. (2016). Bridges: A world community for mathematical art. The Mathematical Intelligencer, 38(2), 35–45. https://doi.org/10.1007/s00283-016-9630-9

Sarhangi, R., Jablan, S., & Sazdanovic, R. (2005). Modularity in medieval Persian mosaics: Textual, empirical, analytical, and theoretical considerations. Visual Mathematics, (25). https://symmetry-us.com/Journals/sarhangi/