Week 9 - Critical Reflection - Mathematics & traditional and contemporary practices of making and doing

    Over the past several days I found I was not only reflecting and making connections between the introduction, viewings, activities, and readings for this week but also thinking back over the experiences we have had over the whole course. Themes that emerged from my stops/pauses could be connected backward through the course.

Intergenerational Teaching and Learning

    The idea of intergenerational teaching first came up to me this week in the reading “’The spirit of the medicine will lead us back’: How Avis O’Brien is guiding Elders to weave their first cedar hats”, an article by APTN, and has been a theme over the course for me. So many of the activities we have undertaken have given me the option to engage with crafts that I have previously explored and have been taught to me by mom or grandmother (or both), specifically knitting and quilting. Even in the first week, I went to my little garden which connects me to my mom and my dad as gardening and growing food was (and is) something we do as a family. I explored art through quilting, learned about some of the mathematics of knitting, learned about mathematical ways of seeing and describing patterns in quilts, and realized that I had previously knit a mathematical object! When we were braiding this week, I thought of how my dad gave my sister and I string and a pile of harvested onions and asked us to figure out how to braid the onion tops together with string for support so the onions could be hung for storage.

Throw-back to Week 4 -
bringing in quilting to mathematical art endeavors

    Often we think of intergenerational learning going in one direction, from older generations to younger generations. One of the spinners in the film “The art and geometry of rope making and yarn plying” noted that when she spins, she feels connected to all the spinners that came before her. However, in the APTN article, it is noted that weaving (with cedar, specifically) has been a cultural practice that “went to sleep” due to colonial laws imposed on Indigenous peoples and the people who are now Elders did not have experience learning and practicing weaving. Now it is people in younger generations, such as Avis O’Brien, who have learned to weave and are bringing the cultural practice to the Elders.  For me, this connected to being able to virtually meet and learn from Evelyn Vanderhoop, a Haida weaver and artist, in my elective course HGSE 371 last summer. Even though she comes from a line of weavers, having learned from her grandmother and mother, she also travels to museums where woven pieces that were taken from the Haida are located to study the designs and patterns that may not have been seen in Haida Gwaii in recent times and brings the designs back to her woven robes, reintroducing them to the Haida of all generations.  (Evelyn has a really wonderful Youtube channel where she posts videos of her simultaneously weaving and telling stories or teaching history from a Haida perspective – check it out here).

So, how do I relate this to the teaching of mathematics and bringing “ancient and ancestral technologies” into mathematics classes? By giving our students the opportunity to play with and explore traditional ways of making things while exploring the mathematics, we are connecting them to generations before and an aspect of being human and providing for ourselves. Susan notes in this week’s introduction, by making we can “really appreciate how lovely and skilled these technologies are, and how important they have been for human thriving throughout the ages.” By making sure we explore ancient techonlogies that are common across cultures but also from individual cultures (especially those represented in our students), perhaps we might give students an avenue to re-awaken some cultural practices and knowledge in their own families. I imagine students going home and talking about their learning and families perhaps taking time to explore connections to family history of making. The fact that we explore mathematics in it might be a side benefit to connecting to culture.

Question: What do you find are the benefits of exploring ancient technologies through mathematics classes? What about modern technologies?

(Aside: I have been struggling throughout the course with the privilege that is embodied in some of the activities and this came up again this week. It almost seems contradictory to speak of privilege in relation to making culture, especially when it is based in ancient technologies. Usually I am connecting the concept of privilege to time. This week it came up for me in the video Vested Interest. Following Sharon Kallis through the process of making her vest, right from visiting the farm to collect fleece to adding pockets and the final stitches, showed what a time-consuming process it is. Kallis has this time because she is being paid as an artist-in-residence. Maybe this vest project is an extreme example of creating without being consumers first, but I think of single parents who are working multiple jobs to feed and cloth multiple children and maybe also be taking care of aging parents or other family members. It’s nice to say we need to connect to the land in order to take care of the land, but in a system that oppresses large groups of the population in many different ways, who are the people that can embrace the way of being of foraging and upcycling and creating and making or even writing mathematical dances, art, or poetry? I tied this back to the classroom because it is here where we can provide that time to connect with the land and connect with traditional practices of making and technology and give students those experiences as a way to decolonize education and a small part of society.)

Mathematics as Inherent to and Inseparable from Artistic Endeavours

The idea of mathematics being inherent to and inseparable from various human endeavours really came to the forefront to me in the week about dance and I really connected with Karl Schaffer’s idea of the third space where math and dance combine in a way that each supports the other and where it is hard to distinguish between the two ways of knowing. I can see this in visual art, paper folding and sculpture, music, knitting, quilting, drawing, and everything with which we experimented over the course. Last week Joy and I had an exchange of how the Indigenous Math Network has Kaagan Jaad (Haida Supernatural Mouse Woman) as a symbol for the network and the connections Billy Yovanovich, a Haida carver and artist, makes the connection of how mathematics is like Mouse Woman, sneaking in to help us out. I see how mathematics is inside all of the activities we have done, helping us out but also how the craft/art/activity helps, in turn, in supporting our mathematical understanding. They work together in an interwoven way that squishes both together so they can’t be separated.

Kaagan Jaad as photographed for
Out of Concealment: Female Supernatural Beings of Haida Gwaii
 by Terri-Lynn Williams-Davidson
(Image from: https://thelasource.com/en/2019/10/21/out-of-concealment-the-interconnectedness-of-femininity-the-supernatural-and-the-environment/)

I, along with Joy, was in a meeting with the Lower Mainland Math Contacts yesterday and we got into an interesting discussion about Indigenous ways of knowing and Indigenous cultural practices and mathematics. One of the members shared a beading project she did with students alongside and Indigenous Education teacher. Even though there were specific mathematical outcomes to the activity related to patterns, she noticed so much more mathematics coming to life in how students were creating their projects – counting, measuring, comparing, and much more. Her take-away from the project was that we can start with an activity or project and then look for the mathematics, rather than starting with a math concept and developing projects from there.

Question: In planning activities with your students, are you finding that you start with an activity and find the mathematics or start with the mathematics and develop/find an activity that supports the math or a bit of both? What do you prefer?

I connected with that idea of starting with the activity and letting the mathematics support it because of how I have experienced mathematics as an inherent part of the activities we have done. I could feel the mathematics. I noticed this in the video about the rope making in Norway and how the ropemaker seemed to feel the tension in the materials as she was pulling the rope. I know when I played with the raw fleece to ply it, I was feeling (sometimes more successfully than others) for how fast and hard I could tug on it to pull the fibres without them disconnecting from one another and then getting a feel for how tightly I needed to twist and ply to get a stable piece of plied yarn. I know I feel the patterns when I dance, too, and hear and feel patterns of notes when I play music. I have discovered and learned that feelings of having mathematics inside my body, right down in my core, when I engage in different human activities is what is meant by embodied mathematics.

Week 9 - Reading Summary

 APTN article, April 11, 2021: ‘The spirit of the medicine will lead us back’: How Avis O’Brien is guiding Elders to weave their first cedar hats

This article form APTN, 'The spirit of the medicine will lead us back': How Avis O'Brien is guiding Elders to weave their first cedar hats, centres around Avis O'Brien, who is Haida and Kwakwa̱ka̱ʼwakw. After a time of being disconnected from her Indigenous culture, even wanting little to do with her culture, in her words "I still embodied the impacts of our history", she learned to weave cedar from her sister, Meghann O'Brien. She speaks of how the experience of learning to weave began to remove the shame she had previously felt about her culture and she "became open to her cultural practices."

At the time the article was written, O'Brien had returned to her home territory of the Ligwildawx peoples and was working as a land-based cultural empowerment facilitator. She has been teaching We Wai Kai and We Wai Kum Elders to weave with cedar, specifically to weave cedar hats. She notes that this helps "'give back something that was taken from them throughout history.'" She also speaks of cedar having healing powers and that it is an essential part of life, from birth to death, for practical, artistic, and spiritual purposes.

Some of the Elders had never made something out of cedar before participating in O'Brien's workshop. One Elder was going to gift her first cedar hat to her grand-daughter. O'Brien has received gifts, such as smoked fish, from an Elder for teaching her how to weave. The author comments on the power of the exchanges between generations. O'Brien comments on how it is the medicine of cedar that is healing and helping do the intergenerational work.

Image: https://www.aptnnews.ca/national-news/the-spirit-of-the-medicine-will-lead-us-back-how-avis-obrien-is-guiding-elders-to-weave-their-first-cedar-hats/

Week 9 - Activity continued! Twining fleece

 Thursday night rolled around and I was watching the viewings for this week.  In the video, Vested Interest, there were scenes of a sheep farm and bags of fleece were being inspected.  I had a desired to touch and work with fleece, especially since I had also seen felting in person at my worksite in an after school session on Wednesday.

So, I went and dug around in my storage locker for a bag of raw (though washed and carded - but undyed), unspun fleece. So random, right? Or at least that is what I can imagine my former 10-year old students saying if I told them about having a bag of fleece in my storage locker!

Well, not so random.  My parents live in Armstrong, BC. Their next door neighbour is a sheep-sheerer (originally from Australia). When he sheers some sheep or alpaca, if the owner of the animals doesn't have use for the fleece he sometimes calls my mom and asks if she wants some of it. The first time she took some, she and my sister tried washing it themselves in the bath tub but it was a dirty job (see the Vested Interest video). They found that there is a place in Calgary that will wash and card the fleece and even spin it if you wish. So that is what they do with any fleece they get now.  Sometimes a sheep-raising neighbour takes hers to Calgary and takes anything my mom has with her. (Looking for a niche market business? - maybe set up washing, carding, and spinning fleece in British Columbia!)

I had visions of doing some natural mason jar dyeing with my students and then felting rocks for paperweights - think of all the math!  So, I brought a bag (one of many my mom has) back to Vancouver. I even tried some of my own natural dyeing in the summer of 2020 when I was in Armstrong to experience what it was like, even though I used a vat and hot plate to do the dyeing, rather than mason jars like I envision doing with students. Here are the results on spun yarn using pink yarrow and white yarrow flowers and stalks:

I haven't decided what to make with these yarns just yet.

OK - so back to Wednesday night and interrupting my video watching to find fleece.

Once I had a handful of fleece, I came back to my work area and decided to try to play with twisting it/twining it as shown in the Art and Geometry of Ropemaking video.

I twisted it until it had a will to twist upon itself and then started twisting the two strands away from me while twisting them over each other towards me.  The first result was very fine. I tried again and didn't make the two strands as thin as the first time, hoping for a bit more obvious definition.

What I remembered from the video was about getting "zero twist" or finding the maximum twists per length so the twine doesn't unwind and is stable.  After a few twists, I got a 'feel' for how each had to snug up against the one before. Sure enough, the resulting "twine" did not unwind!

Remember I mentioned seeing the felting process in person at my worksite on Wednesday? The Literacy and Assessment Teacher brought in Janice Novakowski for a workshop on creating felted story maps for story workshop. She's coming back in April so the teachers can make felted figures to go with the story maps.

Check out Janice's tweet about the session!

Week 9 - Activity - Braiding with no earworms

 So, what people (other than April) might not know about me, is that my guilty pleasure is watching The Bachelor on TV. When people find this out, they often act surprised. I think it is a form of escapism - I hardly remember any of the names once the TV goes off, so I'm not super invested but still quite dedicated. It all started with a teacher friend and then 2 teacher friends (we all lived within 4 blocks of one another) - Monday night ritual: leave school at a decent hour, get some exercise (together or alone), and gather at one of our places for tea or wine, snacks, and The Bachelor/Bachelorette. If we had mindless school tasks to do (cutting things out, etc.) we might do that at the same time. We don't all live so close anymore and have different schedules than we used to for various reasons so it gradually fell off.

So, what does this have to do with this week's activity?

Me in my pigtails selling lemonade.

Well, there were TWO episodes of The Bachelor on this week. But there is also homework and projects to do.  Thanks to April's brilliant mind, I found myself doing homework and watching The Bachelor at the same time! OK...the activity took more visual attention than I thought so I really have to say I listened to the Bachelor while I did the activity!

I chose to do the braiding. I taught myself to braid my hair when I was in grade 1 (I am sure I had watched my mom braid my sister's hair to get a 'feel' for it). I wore pigtails and would play with them during carpet time at school. I remember being so proud that I figured it out and announced it when I got home from school. My mom's reaction was more about "why weren't you listening to the teacher?" than excitement that I had learned to braid, but I still felt very accomplished! It is a very distinct memory for me.

I watched the braiding videos that were recommended to us (all except the Seven Strands of Alphabetically Braided Crows - sorry, no more earworms for me! (see poetry week)). I already know how to 3-strand braid and the 5-strand and 4-strand seemed pretty straightforward. I was most intrigued by the 7-strand double braid and although the steps in the pattern (over the adjacent, under the next two) made sense I really wanted to see how it came together.

I dug out some 1mm beading cord I had and gave the 7-strand double braid a go and this is what I got:

7-strand double braid - 1st attempt

My total braid is about 15cm long and has some mistakes in it. It took me the whole second hour of The Bachelor to get through this!  The cord is quite slippery but also a little bit rigid and the braid wanted to flip around on me (I had it tied to a key ring which was safety-pinned to a pillow). The process and brain work was satisfying and I like looking at parts of it but I knew I'd be coming back and trying again.

Braiding material!

At work the next day, I noticed a pile of approximately 1m lengths of cotton string on my desk that was collected at the end of a circle investigation last week in a school (testing out an activity from the project I am doing with Cassie). Braiding material!  Into my bag it went.

The Bachelor second episode, here I come!

Tuesday night, I tried the 7-strand double-braid again, with string this time. There is a mistake it in (if I lost the "rhythm", it was hard to sort the strands out again) but on the whole, the cotton string was more satisfying to work with.  Here is the result:

7-strand double braid - 2nd attempt

Then I worked through braids with 3 strands (so I'd have a picture), 4 strands, and 5 strands:

3-strand braid

4-strand braid

5-strand braid

I found the 5-strand braid most satisfying because of the process and the result.  I could "choke" up on the braid with how I held my fingers and get a tight, even "layering" of the strands. The final result is quite firm but also flexible. It feels really strong.

The 4-strand was interesting just to try the process of the pattern. I found the little twist in the middle (2 over 3) interesting and hard to hold onto and keep twisted while making the next two moves. I really had to keep talking myself through the moves, out loud to start (with The Bachelor on pause) and then in my head. Once I got a few centimeters away from the starting point, it was easier to maneuver the strands rhythmically.

3, 4, 5, and 7-strands done!  I could move on...

...onto more braiding.

Wednesday night with no silly TV to watch, I got curious about what the 5 strand braid might look like in different colours. I pulled out the beading cord again and cut a length of blue and one of magenta (which I called 'red' in my head) each of which I folded in half, and then I cut a gray.  I tied them together so they were from left to right, blue, red, gray, red, blue. I started braiding. This was interesting because the colours prompted a little chant in my head that didn't stop even when I had a rhythmic flow going - blue, blue, red, red, gray, blue, blue, red, red, gray. It was interesting to see that the pattern started on the left the first time, the right the second, and back to the right and so on.  Here is the result:

5-strand in coloured beading cord

I started to write this post after braiding the 5-strand coloured braid. However, I remembered that one of the videos mentioned Celtic braiding and I had to investigate that.  The result was a quick exploration of a Celtic Bar and here it is:

Celtic Bar

I'm interested in trying this again with some material that will show the definition better. The pattern to this was a neat combination of folding loops over and behind and crossing them over each other alternating with doing the same with single strands.

I'd like to think I am done so I get on with other things but I have a sneaky feeling I might be trying out some other braiding/knotting crafts. Check back for edits to this blog post!

Week 8 - Critical Reflection - Mathematics & fibre arts, fashion arts and culinary arts

 

A section of a quilt my mom made for me.
(I bought the fabric in Hawaii for her,
she re-gifted it back to me like this.)
I have more terminology to describe
what I see in quilts after watching
the presentation by de Vries.

   I got most involved with the activity this week and found the reading and viewing a little harder to engage with, which I am disappointed in because this week connects to artistic endeavours I regularly undertake. I can explain some of it with mental exhaustion right out of the gate on Monday with several during the day and after school meetings and it didn’t let up all week. I think that is why the paper folding got done – there was mental energy involved but I could convince myself to engage with it. “At least go try to fold some paper before bed, then you have done something!”  The exhaustion seems to have interfered with “stops” and rather, I found myself engaging then processing and adding layers to the processing as I went through more materials. In the end, I have, rather than stops, some overall thoughts, a few of which I will put down here and hope the others come out in responses to others.

Concrete-abstract support system

A section of a quilt my mom made for me

    In this week’s introduction, Susan quickly reminded us of our discussions about the binary of concrete and abstract before moving onto a discussion about binary based on gender and social class. However, the concrete-abstract binary came to mind in my reflection on the activity, reading, and viewing. Or rather, I noticed how this binary was really challenged in the artistic crafts we looked at. In the reading I did by sarah-marie belcastro, she mentioned how she designs a pattern for her mathematical objects before starting her knitting because she does not want to “lose the work” by relying on a not-so-reliable memory. I thought the blurring of the binary was apparent in this case; she needs a concrete pattern to record her thinking in order to create a concrete object. Any abstract manipulation in her head (which I mused earlier in the course can be argued to be concrete) is tested and confirmed through the concrete. I watched the Gerda de Vries Quilts as Mathematical Objects presentation (after falling dead-asleep sitting up during a viewing of the one on shibori dyeing!) and noticed a concrete-abstract support system in play. De Vries talked about her own quilts and I felt like there was a lot of imagination through a “what would happen if…” kind of thinking and then that got tested with the concrete material of fabric pieces. I have only a little experience with quilting but do enjoy playing around with it when I have time (more often when I visit my mom who is a quilter). I know I play with blocks in my head sometimes before I move them around to see if what I imagine will actually work. I could also feel the abstract-concrete playing together when I folded paper; sometimes I would stop and picture how the rest of a row would have to be pushed and pulled and prodded and what it would look like before I made the paper move. I can tie this back to Susan’s introduction and our discussions in the dance week with the idea of the third space where abstract and concrete work in support of each other in a way that it is hard to extricate one from the other.

Quilt in progress
These 'stars' were not my original design but
when I did some imagining (abstract)
and playing (concrete) these stars appeared.

I also imagine and play with how the units can
come together. I'm trying to decide if I separate
each unit with a strip of fabric or keep
them together like in the first picture.

How do you see a concrete-abstract support system at play in your classroom or your life?

Gender binary and the mathematics in handicrafts and culinary arts

I found myself considering the gender and social class hierarchies that are thought of in binary terms and what shifts may or may not have happened to date. I am still working this out and welcome engagement and debate. The cynic in me wonders if the shift to valuing the inherent mathematical activity in handicrafts is because of men taking them up and men examining them for mathematics. I can already challenge that idea by noting that I witnesses through reading and viewing, two women, belcastro and de Vries, examining the mathematics in knitting and quilting. However, I can find evidence of the work of men in handicrafts traditionally done by women being held up higher or given more accolades. Sometimes it is very subtle. For example, de Vries showed a quilt in her presentation and referred to the quilter as “he.” In her next sentence, she specifically said, “this quilter is male” after using “he” again and there was just enough inflection in her voice that held the work up. The accolades are not just in handicrafts. I think of the food industry, too, where awards seem to favour male chefs and restauranteurs (and the industry as a whole is dominated by men). According to an article in the Jakarata Post, only 2.7% of Michelin starred restaurants are run by women. When I went looking for artists working with paper after engaging with the paper folding, I looked at male artists that had been suggested by Peter Beck, 4 out of the 5 artists he referenced being male. I realize I am veering away from the mathematics connections here, but can bring us back to my earlier point in wondering if the mathematics in traditionally female-endeavours is valued because of male participation.

From the Jakarta Post 

How might we engage in conversations about social justice related to gender binaries through engagement in fibre arts, culinary arts, and fashion arts in our mathematics classrooms?

Bringing fibre arts, fashion arts, and culinary arts to our classrooms

While I very much value the place of these arts in a mathematics classroom, and have done so in bringing cardboard loom weaving to a patterns exploration (I’d show you a picture but I didn’t realize Showbie dumps classes after 5 years so I have lost some student work pictures), I know the struggle to make students, families, and colleagues understand how it is mathematics. Susan talks about a “shift in attitude” and I get that – but it is sometimes hard to get that shift in others. I have a number of teachers piloting MathUP, an online teacher resource by Marian Small, in their classrooms. One teacher told me how she was released for remedy the other day and had a TTOC in her classroom. She left a MathUP lesson for the TTOC, who is a trained secondary math teacher, and got to chat with her after school. The teacher reported to me that just the idea of playing with mathematics through manipulatives stretched this secondary trained TTOC. Not all colleagues have the flexible thinking that members of this cohort do. I could definitely offer workshops where teachers experience these arts and making mathematical connections first hand, but who is going to sign up for the workshops? The people who like to push their thinking and explore different places where math lives. Then there are the parents and students, too.  When I did the weaving, I first started the students off on it and then it became something they needed to schedule themselves (I had the students making their own afternoon schedules – things they must do and other things they could do and it had to be a mix to get everything accomplished over time). There was a group of, for lack of better term, good students that dismissed the weaving product as unimportant. Only when I really pushed them to explain how they were going to demonstrate their understanding of increasing and decreasing patterns without having completed the project, did they start the buy-in (I actually would have welcomed an idea for an alternate project but they didn’t offer one). In no way am I using this challenge of buy-in or inability to shift thinking as a way out of incorporating these arts into mathematics classrooms, but only recognizing the effort it can take to convince/educate others into the value of and place for these arts.

Do you think I am being overly cynical about colleagues, parents, and students? Is there more potential for shift than I am giving these stakeholders credit for?

My knitting of a mathematical object

At first I almost dismissed belcastro’s knitting of mathematical objects as something I would never do. Then I realized, I have knit a mathematical object.  Here is a toy cat I made for a little girl who calls me “Auntie” when she turned 5:

Kitlyn - I see mathematical objects in her body parts

The cat’s name is Kitlyn. I reflected that in knitting the head, it was making a spherical object (I won’t quite call it a sphere). It was challenging but I remember that it really appealed to my mathematical brain. I think making the arms from an almost flat piece to one that decreases at the end (hands) to force a cylindrical shape was a kind of mathematical object creation, too.

Week 8 - Reading Summary

belcastro, s-m. (2013). Adventures in mathematical knitting. American Scientist, https://www.americanscientist.org/article/adventures-in-mathematical-knitting 

 All images are from the article.

 This article in American Scientist is written by sarah-marie belcastro with whom I had the opportunity to interact on Zoom two weeks ago when Susan invited us to the Dance, Movement, and Mathematics group meeting. She has co-published a couple of books with Carolyn Yackel (who was the presenter in “How orbifolds inform shibori dyeing” video that was an option in this week’s viewing): Making Mathematics with Needlework (2007) and Crafting by Concepts (2011).

In this piece, belcastro writes about knitting mathematical models, noting that these knitted models make good teaching aids because they are flexible and are easily manipulated. Additionally, she comments on how a knitter must understand the mathematical concept well in order to craft a model but also how the knitting process itself provides mathematical insight.

Knitting as Geometry

Knitting geometry -
rectangular mesh/grid

belcastro describes the geometry of knitting and likens it to a grid or mesh within a coordinate system. She indicates that shaped knitting is a manifold, meaning that the coordinate system is only consistent in a local section of a piece, but not over the whole piece. Knitted objects can “stand in for 2D objects” just as paper is often used, and therefore can act like the skin of 3D objects.  In this way, belcastro is able to make knitted Klein bottles, a mathematical object that in which the “inside is contiguous with the outside.” She has been working on improving knitted Klein bottles for over 20 years.

Left: image of a Klein bottle; Right: Knitted Klein bottle

Getting the Math Right

belcastro insists that the mathematics in knitted models be “mathematically faithful” to the mathematical object. The model may end up highlighting some mathematical features over others and belcastro takes this into consideration when she plans the construction; she makes her patterns in order to feature certain properties. When knitting surfaces, which is the bulk of her mathematical knitting work, she has criteria for the projects in order to maintain the mathematics: there must be no edges or bound stitches, any transition between yarns must be invisible, and textures must be consistent to the mathematical object. For this last criterion, this means that if a surface transitions from outside to the inside, as in a Klein bottle, then the texture on the inside and outside of a knitted model must be the same. Conversely, if more than one texture is needed, the knitted model must reflect this.

Some of belcastro’s models have one mathematical design knitted into another. The example she gives of this is having a Mobius band knitted into a knitted Klein bottle. This leads to a set of challenges such as the difficulty of creating smooth lines out of rectangular stitches, the inherent difficulty of the rectangular mesh created by knitted stiches (as opposed to square mesh, that might be more ideal to work with), knitting a line into a bumpy surface, and knitting multiple line segments into a curved surface. Some of these challenges are illustrated through discussion and photos of models of Torus and Torus knots.

Knitted torus with curved lines embedded in the design

 Design Process

The article concludes with a description of the design process that belcastro uses. First she decides on the model and what the goals are for the mathematics (what mathematics should be featured or highlighted). Next, she plans both the large structure and the fine structure. Finally, she produces a pattern as the last step before knitting.

Week 8 - Activity - Paper folding: Miura Origami and more

I may have a new hobby!

Well, on Wednesday night, I sure didn't feel this way.  I used the first video Susan linked for us for the Miura Origami but got quite confused when the demonstrator's hands went off the screen.  I made some mis-folds which made things more confusing and the paper was becoming soft and I was ready to give up.  Here is the result of the first attempt:

First attempt at the Miura Origami type folding

I walked away to do dishes and other chores. I couldn't quite let go of the paper folding conundrum so before I went to bed, I decided to watch and work with the second video that Susan posted. Ah...much better.  Fewer vertical folds and bigger "pieces" to push in and out from a similar sized paper. Apparently I needed this scaffolding and the result was crisp and the experience satisfying - I could go to bed feeling accomplished!  Here is what had me feeling this way:

Fewer starting folds was more successful

Well, I didn't actually go to bed - I should have gone to bed.  However, YouTube was suggesting videos to watch and I came across an art teacher (Marquette University High School) who has lessons for his students posted online (starting spring 2020). I went to his YouTube channel and looked at his video library. There was a series of videos called Cut-Fold that caught my eye:

Peter Becks video library on YouTube

I noticed that the Cut/Fold Vol. 1 video image was a design that Cynthia showed me that was her "room" for the Virtual Family Math Fair this year. I browsed through a few videos and discovered that Mr. Beck references some paper artists at the end of Vol. 5.  I had to check a couple of them out online.  I'll talk more about that later.

After staying up too late Wednesday night, I really thought I would be in bed early on Thursday night, especially since I had to be online at 7am Friday morning for a full-day workshop with Katie White. Walking by my work area, my paper folding caught my eye and I decided I needed to try another fold. I first started with the Cut/Fold Vol. 1 and got this:

And I couldn't leave it at that.  I returned back to the original task, this time using the number of initial folds on my first try but employing a larger piece of paper. This experience was so satisfying! I started understanding what I was doing and what would be coming next. I had another video playing in the background and I heard that demonstrator say something along the lines of "The paper will want to do what you want it to, it just needs some prodding." That's how this attempt felt.  At least down to the last "row" which was the most difficult because there was nothing to stabilize it below. That video I was listening to also taught me that paper stretches, especially on the fold lines (but origami paper less so - it is designed to maintain more of its original structure) and that what these folds result in is "tessellating surfaces" - cool term, right? Here is the result of staying awake too late on Thursday night:

When the tessellated surfaces are all folded up, it reminds me of the start of a hexaflexagon!

When I was in my online workshop on Friday, my folded paper designs were next to me. Sometimes, I want to keep my hands a bit busy when I am listening and not note-taking (hexaflexagons have been good for this!) and I took a small piece of paper left from cutting down other paper and started folding. This wasn't successful - my fingers are big and the folds were small and the paper was getting soft as it did on my first attempt. I think it could be really cool to fold a small version but I think I need a "bone" folder to help crease the paper tighter and "punch" the folds in the right direction...or maybe stronger paper?

Tiny paper folding - got too soft to work with

OK, so back to those artists I researched.  Below are two that caught my particular attention.

The first is Li Hongbo who makes sculptures out of folded paper. Some of them look solid but they can be stretched out in all sorts of amusing ways.

from https://www.instagram.com/lihongboart/?hl=en

from https://www.instagram.com/lihongboart/?hl=en

Watch this one-and-a-half minute video if you have the inclination:

The second artist is Matt Shlian. He is a sculptor, too, who works with paper and other materials.  Here are a couple of his paper designs from his website Matt Shlian :

Aren't they incredible?!

And my favourite find - Matt Shlian on Sesame Street making paper sculpture with kids (it's just one-and-a-half minutes - take a look):

I have looked online at DeSerres art supplies - they have cutting mats, knives, and folding tools in stock in Vancouver.  This might be my reward for finishing my reflection post later!

Week 7 - Critical Reflection - Mathematics & poetry and novels

 Before reading further, if you haven’t read my reading summary, please read at least the summary poem “Poems are made by erasing.”

    This may have been my favourite week of the course so far; at least that is what the hours I have spent playing with and writing poetry this week tell me. It might be tied with the week were we re-created art and I got sucked into flow with tessellating trapezoids. My brain (and body) is a bit exhausted from late nights not noticing the clock but still needing to get up in the morning. Please forgive me if my writing is discombobulated in this post.

    When I reflect on my classroom practice with poetry, I see that I connected math at least implicitly, if not explicitly. I admit I was somewhat neglectful of having kids engage with reading poetry (but when I had teaching partners, they were good at that), but loved exploring the writing of poetry with them. I had a book from Scholastic that I used a bunch that taught different forms (haiku, cinquain, limerick, acrostic, ballad, and others) through examining a few examples and then scaffolding the writing (then I had students develop a book of their own poetry with at least one example of each of these and a bunch more of their choice). We were mostly counting and noticing patterns – of lines, syllables, stress, rhythm. So, I can see the mathematics connected to poetry through these experiences but this week opened up my understanding to more ways that math and poetry live together in this world.

9x9square poem by JoAnne Growney
cited in Glaz, 2019

    I appreciated how Susan, in the introduction this week, laid out three ways that “mathematics inspires poetry and novels” (p. 1): they might be about mathematics or mathematicians, they might connect mathematical ideas/concepts and human experiences (analogies, metaphor), or mathematics might be used to create/shape the writing or to analyze it. Without this guidance, I may have needed more time to figure out what mathematical poetry (my focus, I didn’t examine other literature this week) is or is not. I am not sure I have it completely figured out but Susan’s ideas helped with listening to and reading (and creating) poetry this week. I found some examples of these in my reading/viewing/creating this week. The understanding that the subject matter is what can make a poem mathematical is demonstrated in Sarah Glaz’s poems on the Bridges Math and Art 2021 Virtual Poetry Reading website. The two poems Death of Euclid and Archimedes are about the title mathematicians. (I suspect there is something mathematical about the structure of these poems as well, but I didn’t examine them that way.) Amy Uyematsu’s poem Countdown brings together counting and numbers to ‘deal with more serious political issues’, specifically shootings in the U.S. To me this is connecting mathematics and human experiences. I think the final idea, that of mathematics informing the structure of poems, is what I enjoyed exploring most through my own poetry but also finding it in others (when it was obvious). Daniel May used the Fibonacci sequence to structure his poem A Fibonacci Poem for the After where the poem grew in syllables and then contracted again based on that mathematical pattern. In the interview by Sarah Glaz of JoAnne Growney, there are a few square poems where the number of syllables on each line is matched by the total number of lines (e.g., 5x5square or 9x9square). I enjoyed creating Fibs and played with having a pattern of Pascal’s triangle structure a poem of my own creating. When creating found poems as my reading summary, I tired a 9x9square poem for the Romania section (as the poem in that section was of the same type) and it was really difficult (I had the added constraint of using someone else’s words) and time constraints had me abandon it. I am sure if I had dedicated more time to examine other poems, I would have uncovered intriguing ways mathematics structure them.

My poem structured on Pascal's Triangle

    It seems like we can define mathematical poetry pretty broadly (though some people want more constraints than others – I felt JoAnne Growney and Sarah Glaz differ on what is and isn’t mathematical poetry). So, to that end, my question is:

What is NOT mathematical poetry?

Notes for the 'stop' at 
"Poems are made by erasing" in Glaz 2019

    One of my biggest stops, big enough to involve a whole page of stops and scribbles, was when I read the section in Glaz’s interview with Growney titled “Poems are made by erasing.”  Before I had read more than the title I had scribbled “I erased so much or scratched out --> connection to math – draft solutions, back up, try something different – then make elegant.” The poem this section is framed around is called How Did It Come to This? and references Picasso’s art as “notions of a bull…in the end a few lines” (Glaz, 2019, p. 251). In the interview, Growney talks of a writing prof who shared her own draft in class and said, “’That poem has one-third too many words’” (p. 251). Here I scribbled, “Like Mozart – Eugene Oschady.” Eugene was a cellist in the VSO and a member of a local string quartet. At a quartet performance, he talked to the audience about how Mozart’s music was criticized for too many notes but Eugene argued that Mozart’s genius was knowing when he had the exact right number of notes. He played a motif from Mozart’s Symphony No. 40 and added just one extra note in the middle and end of the motif and it made the audience chuckle because sounded silly. Ironically, when I composed the found poem for this section of Glaz’s interview, I had way too many words and was going to add more. I stopped, thought about it, and scratched things down to the minimum that would get the message across.

Editing down "Poems are made by erasing" found poem

    The connection to math education here is the idea of math drafts.  We allow students first, second, third, or more drafts in writing. When do we do that in math class? I was introduced to the idea of math drafts in an NCTM webinar that I only caught the end of and haven’t gone back to watch the whole thing. However, the part I did see gave me a good idea about the concept. The idea is that students take a break from a problem and circulate the room looking at other students’ incomplete work, offering feedback and gathering inspiration for their own work. They may go back to their own work and scratch things out, start again, or have a new idea that breaks them out of being stuck.

    We can also give feedback to students on being more elegant or efficient. I don’t think elegance or efficiency should be the first goal when students are learning a concept, but as they are ready, they can develop this. For an example, I had a student who understood division using the partial quotient strategy but would only divide by 2, 5, or 10 (as I remember it – anyway, the numbers were small) even if dividing by 100 or 200 or 500 might have been more efficient. She didn’t mind working through long columns to get to her answer. I could tell she understood the concept and was able to slowly nudge her to increasing her initial numbers to be more efficient. I think this is like erasing words in writing poetry – getting rid of the extraneous to get at the essence. (If I had the inclination, I’d edit this blog post back by 1/3!)

Questions to ponder

What is NOT mathematical poetry?

What similarities do you notice between the creation of poetry or other writing and working with mathematical problems and solutions?

I’m curious to know what possibilities primary teachers see in writing mathematical poetry with students who have, generally, a smaller lexicon to draw from and relatively limited experience with math patterns and concepts compared to older students.

Week 7 - Reading Summary

Glaz, S. (2019). Artist interview: JoAnne Growney. Journal of Mathematics and the Arts, 13(3), 243-260. DOI: 10.1080/17513472.2018.1532869

 

This reading is an interview conducted by Sarah Glaz with JoAnne Growney at the 2017 Bridges conference and through follow-up conversations. It is framed around 10 of Growney’s poems in 9 sections.

Growney talks about exercise and a good night sleep both being ways she clears her mind when writing poetry.  When I was part way through reading the article, I needed to get out for a walk, despite the quite chilly evening; my body was tired, my brain was tired, and I needed steps for our Meandering Math Teachers Walk for Joy challenge team!  The walk worked.  Maybe too well. Not only did I finish the reading, I decided to do my summary. And not just a quick summary, I had to follow a creative impulse to summarize each of the 9 sections in a found poem using text from the article (and now it is 12:45am and I am just typing this up from my hand-writing). I tried to use Growney’s words as much as possible but sometimes Glaz’s slipped in.  The introduction is made from purely Glaz’s words.

I consider the poems drafts. Some were re-worked but remain drafts. Others are what developed on the first try. You’ll understand why I point this out when you read my reflection.

 

Introduction

Mathematics and poetry:
complimentary, supporting,
conflicting.
C. P. Snow described and lamented
the division.
Many of us seem to have crossed
the great divide -
the creation of a community
of both poets and mathematicians.

 

On the farm

One hundred forty acres,
a white frame house,
a good woman worth three good women.
There isn’t really any history
of advanced math in my family
or of poetry.
A career to fall back on –
I had gotten too much education.

Four teachers

Flamboyant, silver, fearless woman who ahd gone to Columbia.
The geometer had three daughters – kind hearted man.
The one who loves Old English and late-night TV – each thing is our teacher.
A stressful teacher made errors, demanded we write perfect mathematics in perfect English sentences.

 

 Everything connects

Everything connects is the way my mind works
The history of the vacuum cleaner, Roe v. Wade, and Billie Jean King
I put them all together.

Nervous energy, I need to clear my head
Exercise, a good night sleep
The clutter gets cleared.

 

Poems are made by erasing

 Early drafts have far too many words.

I read a poem aloud,
I feel how many words
need to be eliminated.

  

Romania

A tiny organization
'Teachers for Tomorrow’
for teachers to help students
learn English
in Indian and Romania.

I mostly presented poetry.
We read,
Puzzled over,
Discussed
English translations
Of Romanian poetry.

Found a poem
Related to mathematics
By Nichita Stanescu, a particular favourite.
Delighted!

  

Math and poetry – the perfect blend

Play with both meanings,
mathematical and non-mathematical.
Six is a perfect number –
the sum of all its proper divisors.
Six is a perfect number
of husbands.

  

Counting the women

Mathematics -
a man’s world.
Learn how to live in it
in terms acceptable to men.

Look around the room.
If you can’t count how many women
in one glance,
smile.

Intersections – poetry with mathematics

Collecting math-related poems
Tried the blog
It just kept on going
Makes mathematical poetry
More welcome
Visible
Threads of a mathematical idea
Pattern
Structure
Counting
Recursion
Permutations
Sometimes silly
Sometimes evocative
A back-and-forth
Between my poems
And the poems
Of others
A meeting place

Week 7 - My Mathematical Poetry (activity)

 I started my mathematical poetry exploration with Fibs.  I am playing around with PH4 and a couple other ideas; playing around enough that dishes are unwashed and tomorrow's lunch is yet to be made.

I'll start with two of my Fibs and edit the post should other poems come fully to life.

*Edit - added another Fib later the same night so now there's three!

Can you figure out the inspiration for this Fib?

*Edit - Last night after posting the Fibs, I played around with structuring a poem on Pascal's triangle:

I decided to pattern the syllables of a poem on the first 5 rows of Pascal's triangle.  It was a very interesting exercise, not in the least for having to find a 6-syllable word that worked with the subject!  The inspiration was the weather I saw when I looked up from playing with mathematical poetry and peered out my window!  I finished the poem tonight after giving my brain some incubation time to play with words and to evacuate that earworm that was invading all my thoughts!

Here is my Pascal Poem:

(I'll finish my reading sometime - just not tonight!)