Week 9: Mathematics & traditional and contemporary practices of making and doing

Reading Reflection 

Kallis, S. (2014). Building for Change from the Ground Up. In Common threads: Weaving community through collaborative eco-art. essay, New Society Publishers.

The excerpt of this book begins with a brief introduction of what community garden initiatives in Vancouver and asks the questions" "Beyond food, what if we began to make some of the things that we need for ourselves again as a cultural and social norm?" (p. 19.) Newer towns are being built around the idea of humans as consumers but could also be geared towards supporting production and consumption. Kallis suggests using our power as purchasers to ask questions about what we are buying and dictate the type of consumerism we want. Currently, our needs for food, clothing and shelter are being met by others who are doing the work, likely in factories. Through striving to meet our own needs we can learn these skills that have been lost. 

There is a quote box on the last page of this excerpt from a community participant that really stood out to me (see below). I have seen students on their brand new iPhones, online shop in the middle of class and use their parents credit card to purchase whatever they want. This was crazy to me as I would never have been allowed to do that when I was growing up. When children and youth have these abilities to buy whatever they want when they want it, it definitely becomes an issue and an ongoing cycle of purchasing the newest trends even when the item they previously bought is fully functional. Even for me now as an adult I need to think about purchasing something for quite a while before I do, unless it is something I need. 

Quote seen in Kallis (2014, p. 22)

Thinking about making our own food or clothing, I was thinking about how I know school gardens are in some of the schools around me but since I was in high school they have taken out the textiles program. I grew up sewing with my mom and loved taking textiles in high school, in grade 8 and 9 we had a combined 'home -economics' course with both textiles and foods and in senior classes we could take each course fully. In these classes we learned a lot of life skills and learned how to make our own clothing - I still have some of the things I made. I think this may just be due to lack of teachers for a textiles course where I live but I wonder how else we can teach students to make things of their own. 

Questions 

• How can we teach students to ask the questions about where their items are coming from when they have the power to purchase whatever they want at the click of a button? 
• What are some other ways aside from school gardens that we can teach and encourage students to relearn these lost skills and make things for personal use? 

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Activity: Multi strand braid 

It is very snowy here today so I couldn't forage for any items, I decided to try the multi strand braid. I know how to braid my hair with 3 strands but have never tried more than that. 

I started with a 7 strand braid as seen the video. This was challenging for me, I used different colours to see all the seven strands but struggled to hold all strands in my hands and found that if I let them go the braid would be quite loose. I think this would have been easier if I had used a thicker string like macrame yarn as seen in the video. 

setting up the braid 

Beginnings of a seven strand braid 

seven strand braid

I then tried to make a four strand braid, this one was easier to keep ahold of and I was able to feel the weaving and patterns in this. I liked how naturally I was able to complete this braid as it required less concentration and frustration. 

four strand braid 

Reflection

Knowing that this 'activity' has been done for tens of thousands of years is empowering, doing the craft made me "really appreciate how lovely and skilled these technologies are" (Jerofsky, 2024). I think it is fun that we are able to continue on these crafting techniques and teach younger generations about them. Learning these skills allowed me to see the skill and concentration that many generations have gone through to learn. These crafts must have true integrity to be able to withstand and be passed on for so many years, I wonder how long they will continue to be learned and passed on. 

As I was trying the braiding I was able to see the patterns and counting involved in the movements. For the four stranded braid there was a clear over under, under over pattern. With the seven strand braid, while this one was less natural for me, I could count the under one over two pattern on each side. 

I currently work in classrooms of all ages in four different schools so don't have my own group of students. I have seen some quite complex mathematical ideas associated with braiding  (like this or this). I was thinking about how braiding could be used in an elementary classroom aside from looking at patterns and I thought about skip counting. Students could braid and count each time they put the left strand over. 

Connecting to a larger project idea for upper elementary, students could braid old fabrics together and make rugs or sleeping mats. These could connect to a social justice project as well and students could give away the sleeping pads to those experiencing homelessness. These could also be made into dog beds for animal shelters or students could even make braided sit spots for their classrooms outside explorations. In each of these activities the students would need to calculate how much materials they would need and how large they need their mat or rug to be. 

While I didn't try the upcycled activity this week, it reminded me of when I was in grade 5/6 and we participated in a project where we were to take recycled or items that couldn't be used again and made them into something new. Like in the introduction we were tasked "to make useful and delightful things from what would otherwise have been considered waste." This is something that I still remember today and I still have the project that I made, it is a great way to connect to Earth day and allow students to see the ways that they can use objects that are often considered garbage or useless. 

Final Project Draft: Weaving in Mathematics

 Please click the link here or image to view final project draft. Completed with Kaitlin Burns. 

Week 8: Math and Fibre Arts, fashion and culinary arts

 Reading Response 

Sarah-Marie Belcastro describes how she has converted mathematical shapes into knitted objects. She has been knitting since a child but as adult began knitting mathematical objects, starting with a Klein bottle. The article outlines her process to creating the Klein bottle including materials chosen and shortcomings. At the end of the article she explains that she is still working on designs for the Klein bottle and continues to be challenged with this mathematical knitting. The article showcases photos of Belcastro's knitting projects as well as where her inspiration came from. She explains that these objects are good teaching aids as they can be physically manipulated. She also explains that knitting connects to geometry as there are stitches, increases and decreases, rows and columns. The article discusses other mathematical objects that Belcastro has knitted and the process/struggled involved with them. She explains the design process for knitting mathematical objects: choose an object, articulate mathematical goals, consider objects fine structure, produce a pattern. 

In explaining the process of creating Klein bottles, Belcastro explains that her shortcomings were aesthetic or mathematical. The knitting was ugly due to the materials and tools she had chosen to use and there was difficulty in creating the mathematical shapes. As someone who knits and crochets often I related to the feeling that final products did not turn out because they don't look the way I wanted them to. I also have had issues with my knitting because I dropped a stitch early on and didn't realize it or somehow I added a stitch that messed up the line. While I haven't tried to knit mathematical object necessarily, I have experienced issues in my knitting due mathematical errors (counting stitches). 

 As I have knitted since I was a child I have always wanted to try and put knitting into my classroom. In EDCP 550 I completed my final project on knitting blankets and sleeping pads for the homeless out of old clothing. I am still thinking of more ways that knitting can be brought into the classroom. After last week reading about the comic being written on a Mobius band, I am thinking about how students could knit or sew a mobius band and then add a story to it so that the reader can physically move the story along as they read. I also think just making the Mobius band would be a positive way for students to be able to understand it better (although more of a time commitment than making one out of paper). 

I love the idea of knitting mathematical objects and am inspired to try it, I just am unsure what I would do with the object once I was finished it however I will continue to explore the math involved in knitting. 

Questions 

• Do you knit, crochet or something similar? Have you tried to use this with students before? 
• If you don't knit yourself, how would you go about incorporating knitting into a math class? 
• How can we encourage everyone to try these crafts especially when they see the binary between what crafts for 'girls' are versus activities for 'boys'? 

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Activity

Connecting to the introduction for this week, my mother taught me to knit when I was young. I engaged in a lot of different craft skills and even had a little weaving set where I made coasters. I have continued to develop some of these crafts as I am older. Since starting this program I have been thinking more consciously about how these crafts connect to mathematics. These crafts continue to hold this binary between what women can do and what men can do and I have seen this in classrooms even with young boys who don't want to knit or make bracelets because it is a girls activity. 

Kaitlin and I are doing weaving for our final project for this class. I thought this was the perfect opportunity to try out a weaving activity on my own! I also really liked that the document had a variety of applications for different grade levels. 

I started by making my cardboard loom and adding the warp threads. This part was simple and quick. 

Cardboard loom and warp threads

Next, I added my first and second colours. I started with two colours to copy the design that was on the document however it was a challenge to keep track of all the yarn. 

First few rows alternating colours 

I then decided to try with only one yarn as I was using a multi-colour yarn anyways. This was much easier to work with and keep track of and still looked pretty okay with the many colours of the yarn. 

Halfway through using only one yarn now 

My warp threads were getting squished 

I realized that my design was getting quite tight and in the future I would need to try and be a little looser with my tension. 

I like the activity of trying to weave this way, it ended up being a little more challenging that I thought it would be however I do know how I could adapt the weaving in a few different ways to make it easier for those that may struggle. I enjoyed the chance to try out some weaving as this isn't something I have really done since I was a child. 

week 7: Math and Poetry

Reading Reflection

Saex de Adana, F. (2018). Surfing the Möbius Band: An Example of the Union of Art and Mathematics. Bridges 2018 Conference Proceedings, 423-426. 

The reading begins with an introduction to the Mobius band, explaining different examples for understanding that the band is a continuous loop without a distinct inside and outside loop. It also explains that the Mobius band is a non-orientable surface, meaning that when the beginning of the band is reached it will be in an inverted rotation. The paper explores the story of Silver Surfer, where the Mobius band is used in the plot of the story as well as for the layout of the comic story. The story takes place on a Mobius band and the reader must follow the band to read the comic in order. Silver Surfer is immersed in a loop that causes him to repeat the same events over and over again. The story shows the reader what a single-sided geometric figure like the Mobius band is. This paper brings forth the excitement that has come from this mathematical figure and has been used multiple times. 

The paper briefly outlines different storylines that use the Mobius loop as a part of the plot, these include Star Trek episodes, films and stories. After reading this I got looking online at different fictional uses of the Mobius band and couldn't find a ton easily. I thought it would be cool to teach students about the Mobius band, have them create one with paper and then look at the story, show or video that has a fictional aspect to it to allow students to understand what the band is a little better. It may also be a fun activity to have students create their own Mobius band comics where the characters are caught in a loop or sorts. As much as I love the idea of these activities and can see how they could easily be used in an elementary or middle school math class, I continue to struggle with how we can implement them into more secondary classes. 

I liked how the comic used the Mobius band as both as an idea for the plot as well as for the layout of the story. I got thinking about other mathematical shapes that could be used in this way to help students understand them better and as a fun way to connect art with math. I was thinking about different 2D and 3D shapes, or perhaps something similar to what was explored in the week 2 Kepler reading with rhombic figures of pomegranate seeds. I'm still pondering about what other objects or shapes could be used in a similar way aside from standard geometric shapes. 

Questions

• What are some other mathematical 'objects' that could be used to shape the plot or layout of a story? 
• How can we bring these sorts of stories, poems, and mathematical art activities into high school classes when there are time constraints for learning curricular concepts? 

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Activity 

For the activity this week I chose to write a Fib poem. I always struggle to come up with a topic to write about when it comes to these sorts of activities. I decided to write one about my dog because she was sitting with me as I did this activity and one about dancing. I had spent my day in a PE class learning hip hop dance moves, this brought a lot of joy to me especially since all of the high school students were engaged and participating without any complaints, it was a great feeling so I wrote about how learning this new activity felt. 

PAISLEY

My 

Dog 

Big ears  

Lots of love

Full of energy 

Always excited to see me 

 DANCING  

 Joy 

 as 

I move

my body, 

Dancing to the beat 

Learning new things is a fun time 

I then thought that first poem writing was fun so tried one of the second method as well, this was inspired by my bluebird day skiing on the weekend. This method of writing a poem was quite a bit harder for me, and I found that as I wrote the same words over and over again they seemed to look weird and incorrectly spelled to me. I don't think my whole poem makes sense with the word combinations but here it is! 

Bluebird Day 

Blue skies over snow 

Skies blue, snow over 

Skies snow, blue over 

Snow skies over blue 

Snow over, skies blue

Over snow, blue skies 

Over blue, skies snow 

Blue skies over snow 

 

Week 6: Mathematics and Dance

In the introduction for the week, Gerofsky explains that we cannot make abstract shapes with our bodies but we do have imaginations, and imagination is where these concepts come from. This made me think about how much math really connects to our imaginations and how using our imaginations at all ages (not just early elementary) can add another level of engagement and excitement to math classes. 

In her Ted Talk, Malke Rosenfled asks herself the question "I wonder if there's math in what I do?" pondering if math exists in her world outside of the math classroom as well. This is definitely a question that I have seen come up over the course of this class and am still asking myself that question as I live my daily life. I really found myself thinking of this question as I read this week. 

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 Reading Reflection

Belcastro, S.-M., & Schaffer, K. (2012). Dancing mathematics and the mathematics of dance. The Best Writing on Mathematics 2012, 79–92. https://doi.org/10.1515/9781400844678-010

This reading explores the connections between artistic dance and mathematics. Sarah-Marie Belcastro and Karl Schaffer are both mathematicians and dancers, they suggest that math is intrinsic to dance as each dance form uses math concepts in their own way. Examples of these connections are explained with aspects for the readers to try on their own and gain a deeper understanding. One focus is on how symmetry can be seen in dance including how symmetries interact with one another and three dimensional symmetries. Some other mathematical concepts explored are geometry, spatial paths of dancers and conceptual problems like game theory. Rhythm is explained as how the math sounds coming from music that repeats its rhythmic pattern. 

While I grew up figure skating, I found that much of my reading this week was connecting mathematics to my experience skating. While I skated in a variety of different disciplines including free skate, another aspect is ice dance. In this we learned different foxtrots, waltzes, tangos, etc. These all had music that accompanied them and involved a lot of counting to stay on the correct timing as well as the beautiful patterns that the dance leaves on the ice. I don't do this kind of skating anymore but I will always remember the counting and discipline involved in ice dance to ensure that everything was smooth. 

Towards the end of the reading, Belcastro and Schaffer state that "even when a dance has a strong mathematical element, we let the dance take on a life of its own" (p. 20). This stood out to me as dance is not fully viewed through a mathematical lens however the math can be seen in the art. There isn't a need to always look at dance in a mathematical way as it is its own art and beauty. I think this applies to many different art forms or other mathematical connections - while we can see them in a mathematical way it is equally as empowering to acknowledge the beauty of the art on its own and allow the art to take its own shape regardless of the mathematical intent. 

In the PE class I am teaching right now, we have a Jamaican dance company coming to dance with the students next week. I am excited to see how they use math in their dance, if it is presented intentionally or more subtly.  

Questions 

• Have you used dance in your teaching? (either with math or without) 
• What math have you noticed appears in the activities you do in your daily life? 

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Activity 

The lesson plan that I chose to look at was Math in your Feet which is an exploration of math where students work together to figure things out themselves. 

I think this would be a good introduction activity for all ages to incorporating dance and math together as it gets students moving and dancing in an activity that doesn't require a lot of math work and allows students to participate in small groups.

I don't have my own classroom or group of students and my nephew who I often have tried activities with is quite sick this week so I tried the activity on my own. While this lesson requires groups of people to complete this activity I thought about how it could be done with only one person. I found that while it is possible to do the activity of moving in the square on your own it definitely is not as fun, I tried to put a few of the cards together in longer sequences and see if I could memorize and finish the sequence without messing up - this was quite fun and I could see it being fun with a group of people too. 

Some more extension ideas could be to create different shapes on the floor and have students try to create patterns that could exist within those shapes. For a more high school math class students could explore the different combinations and patterns of using a few movements in the squares and four movements. 

In their Ted Talk, Stern and Schaffer explain that physical activity in the classroom can be an opportunity for all ages and all disciplines. This is something that I think is often forgotten when students enter higher and more academic grades. Using an activity such as this one can be incorporated into math lessons but also can be used to encourage students to get moving in the classroom even if for a short amount of time. These allow students to use their imaginations as well as see math beyond the sheet of paper or textbook work that they may be used to. 

Week 5: Developing Mathematics Pedagogies that integrate embodied, multisensory, outdoors and arts-based modalities

This week I really connected with the reading but found myself puzzling over the activity. I chose to do an activity that put me outside of my comfort zone with the 'dancing' aspect and found that it took me a lot longer to complete this time. I've mentioned this in previous weeks but I wouldn't consider myself a really artsy person and find that any sort of body moment where others are watching and I am without explicit instruction is a stretch for me. I grew up figure skating and dancing but was not overly dramatic or comfortable with the 'interpretive' freestyle when there were others around. I chose the movement activity this week in hopes that I would be pushed outside this comfort zone even if my housemate thinks I'm crazy as I do random body movements in the kitchen.  

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Reading Reflection

Dietiker, L. (2015). What mathematics education can learn from art: The assumptions, values, and vision of Mathematics Education. Journal of Education, 195(1), 1–10. https://doi.org/10.1177/002205741519500102

While I was reading this week I found that I had a lot of stops where I was putting stars on my notes either because what I was reading stood out to me or connected with me. 

The reading begins with an introduction of how math has been referred to as "dry as dust" "monotonous" or "flat-lined" and how the arts can help to re-inform math experiences. There is an explanation of aesthetic ways of knowing and learning. To challenge current assumptions of math this paper explores how math can be artful and aesthetic. Aesthetic is described as "an individual's response to a mathematical experience" (p. 2) which is said to help mathematicians sense if a result is correct or not. This describes aesthetic to be essential to mathematics whether it is a positive or negative response. Through understanding mathematical aesthetic, the paper discusses using different art to structure aesthetic math. Dietiker uses stories as an example, outlining the various parts of a mathematical story and how it can make math a verbal art. Mathematical stories include characters, setting and plot, much like a regular story. Using stories in math classes can help to 'shake up' the familiarity of traditional math classes. 

The idea of using aesthetic in math was interesting to me, when I first think of aesthetic I think of the way I want things to look and if they look nice then I would consider them aesthetically pleasing. In explaining her experience with math Dietiker (2015) explains a fond interest for math and puzzles growing up which led her to studying math in college and developing a desire to share this with students. This resonated with me as this is largely similar to my experience with math. I used to love doing puzzles with my dad and now as I have spent the last month helping my mom clean out some of his things I found a lot of the puzzles and mind-teasers that we used to do together and am reminded of my love for the puzzle aspect of math. None of my siblings enjoyed these but it was something that my dad and I did together all the time. This was a happy memory that stayed with me as I read and did the activity this week. 

Dietiker (2015) states that math classrooms where students and teachers work is tame and harnessed are something other than mathematical as math is generally untidy; she says that "creating aesthetically-rich mathematical classrooms requires untidying the mathematical experiences of students" (p. 3). 

This stood out to me, mainly because I had not thought of math as tidy or untidy before. When I think of my general math work - in high school or university - I think that my work was generally tidy and "aesthetically pleasing" because I like the neatness of lining up my equal signs and erasing mistakes. When I think of my teaching experience though I love engaging students in math classes where they are able to move around, use different materials and try to come up with a solution to the problem on their own (as we have learned the benefits of this throughout our courses). To me this means embracing mistakes and using these as learning opportunities.

Finally my last connection was to using stories in math. When I was completing my undergrad I was involved in an applied study where I had the opportunity to go into schools and tell students stories. I ended up doing a research project connected to this and explored storytelling and math. I explored different oral stories as well as picture books that tell stories about math concepts. This sparked my desire for including stories in math classes and have used them a few times but find that I often get caught up in needed to get things done and forget about this research and work that I did and how impactful it was. 

Questions

• What do you think it means to untidy the math classroom in your own classroom? 
• Do you use stories in math lessons or is it something you are willing to try? 

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Activity 

I chose Sarah Chase's illustration of moving 3 against 2 as the basis for my mini lesson idea. I found myself trying many different combinations of this and becoming confused as I tried to count and move both arms at the same time. While I was doing the movements I was also spending time trying to draw them out to see if my movements were correct with where the arms should line up. I then thought of how students could do this together, having one person do the first number and the other doing the second. I found myself trying to think of how these types of movements could connect to multiplication or division however am still questioning that one. 

Here are some of the ones I tried to draw out, it got a little confusing when I tried to include larger numbers so didn't draw those ones. 

I found that some of them worked with multiplication

 but not all of the movement combinations. 

After moving my arms over and over again I connected this movement to help understanding angles or the unit circle - not directly with what was shown in the video but with the way the arms were used. 

I really struggled with this activity and wanted to give up many times but here is an idea I came up with. 

Unit Circle inspiration

Ideas 

Students will learn about the unit circle on paper and then have a chance to explore it with their arm movements, this will allow them to add a multisensory aspect to the math concept and potentially help them understand or remember it better. 

Guiding Questions 

• How can we understand the unit circle with our bodies? 
• What connection does the unit circle have to angles? 

The Story 

• I am still brainstorming ideas for this part... I can think of stories to go with elementary math concepts but am struggling to come up with one for this. 

Integrating Embodied learning and other learning 

• Students learn about the unit circle and triangles, how to construct the unit circle on paper 
• Imagining that their arms are equal to the radius 1, they can create a larger scale unit circle 
• This can help them to 'memorize' the unit circle with their body being the y-axis and arms stretched out to the sides being the y-axis 
• Using this new tool for understanding the circle students can work together to try and solve math problems that require the unit circle 

Possible extensions

• Ask students to think of what other math concepts they can understand with their body 

Idea for unit circle with body

Final Project Proposal

Hello! 

Here is the proposal for my final project, completed with Kaitlin Burns. 

 EDCP 554 Final Project Outline