This week I attended the last session of the Re-Imagining Mathematics Seminar put on by the E. Lando Virtual Learning Centre (Cynthia was organizing along with Jo-ann Archibald, Joanne Yovanovich and Leyton Schnellert). They discussed Kagann Jad, “The mouse Woman” who is the symbol of the Indigenous Math Network. Mouse woman is like math - helpful and ever present. She always finds a way to help us out. This connected for me with the activities this week. The activities – which were maybe not clearly connected to what we traditionally consider mathematics – had a way of sneaking math in. The math was helpful, if not front and centre.
I had an interesting conversation with a colleague this week regarding how to fit Indigenous “content” into mathematics. My colleague is struggling to see: how to ‘fit it in’, the value of taking time away from learning math skills to ‘play’ with cultural aspects of mathematics, how this all relates to student success – both in math classes at school but also in life after school that requires the pre-requisite math skills/courses etc. I tried (somewhat successfully I think) to bring a different perspective into the conversation. I pushed back on the idea that it is enough to teach Indigenous ‘content’, but suggested we need to rethink the way we teach mathematics, asking how we can bring Indigenous knowledge and pedagogy to the classroom, so that Indigenous (and in fact, all students, ethnomathematically) see themselves reflected in our mathematics classrooms. We also discussed a few ways of doing this that necessitate a shift, not a complete overhaul, of our programs. Saying all of this connected for me with Dr. Gerofsky’s statement in the introduction, “It takes a change of attitude to start noticing the mathematics inherent in all these skills, crafts, and practices, and to connect these physical ways of making with the concept and patterns of mathematical relationships.” Thinking differently about what counts and does not count as mathematics is part of this process of shifting attitudes. Sometimes, like Kagann Jad, the mathematics sneaks in!
The reading I am reflecting on this week is b) Uyen Nguyen (Bridges 2020) Folding fabric: Fashion from origami
The following two quotes from Nguyen, give some insight into the article.
“Fashion loves the juxtaposition of extremes.” (p. 93)
“I wanted to give my artwork functionality.” (p.93)
This paper explores Nguyen’s exploration with folding fabric, origami style, and using rigid paneling and heat settings to maintain the folds. The paper explains the techniques Nguyen uses and the types of designs.
The techniques used include the following:
- Adhesion of semirigid panels to all regions meant to remain planar and unfolded
- Panels constrained in a way that forces them into the desired configuration
- Adding snap buttons to force some vertices to connect
- Heat setting fabric while in folded configuration
- Sewing vertices together to counteract gravity
Where’s the math?
- Patterns, Geometry – understanding the variable stability of geometries, Fibonacci sequence (Nguyen created a series of Fibonacci skirts), measurement – fabric, weight, temperature, effects of gravity, impact of the distance between folds, dihedral angles, all of the mathematics involved in sewing (I’m sure there are more mathematical connections I have missed)
Here are a few images from the article.
I’ve used fashion and origami to represent various concepts in mathematics. While I use a lot of mathematics in my origami design process, having a more rigorous understanding of the mathematical constraints behind the construction of a fitted garment would enable me to improve my designs moving forwards, both with and without origami elements. I plan to design both clothing that can be fitted and practical as well as garments that are more conceptual and artistic. (Nguyen, 2020, p. 102)
Both this article and the viewing, “How Orbifolds Inform Shibori Dyeing” by Carolyn Yackel talked about the idea of constraints. Learning to work within constraints or using constraints to create something mathematically beautiful. I wonder how often we feel that constraints are negative, rather than embracing how constraints can add to the beauty?
I experienced working with constraints as I (continuing on with my theme this week of folding) chose to do the activity on Miura Ora Origami (the technique Nguyen uses in her fashion design).
Here is a bit of the process and the final result:
The first two rows were mind- bending and required resilience. But as I continued to repeat the steps to add more rows, I began to understand what I was doing. Eureka!!! It reminded me how exciting it is to successfully make your way through the struggle! The final two rows I was able to complete without following the steps. I got so much satisfaction in completing this folding activity. I showed everyone my final product. It was only by working within the constraints of the necessary folds that I was able to produce this beautiful product, within the constraints came the freedom of the design. (A binary statement?)
Although this type of origami would be outside the capabilities of young students, there are many simpler types of origami that young students are capable of creating.
What are the benefits of students learning to creatively work within constraints?
This week, I’m considering how to bridge the dichotomy between the physical world and the abstract, conceptual world. Folding paper, spatial reasoning, geometry… when we work with the physical, it strengthens our understanding of the abstract. I’d love to hear your thoughts on this. Many of the activities in this course are doing just that, bringing the physical world and the conceptual world of mathematics together – embodying mathematics! Through movement, drama, film, various kinds of art, poetry, novels the mathematics sneaks in, just like Kagann Jad, helping us out!
Morning Joy,
ReplyDeleteIf you and I were to have the conversation regarding incorporating "How to fit Indigenous content into math" before I started this UBC MAED Masters program, I would have a similar response to your colleague. Worried of the "time" taken away from the math, struggling to see how it "fits in". But because of this program, my attitude towards incorporating Indigenous content has change from the shared ideas, speakers, professors at UBC, who has helped us, gave us me ways and means to unlock my mind to reimagine what can happen and how it can happen. I love how in Cynthia's course last semester, just the story of Spirit Bear, showed how "data" can be used for "good. "Change in attitude" is not something that just happen overnight, it will take a lot of time, patience, conversations, modelling, support, on your end, which you totally have, to help your colleagues become confident in their own planning to incorporate Indigenous content in with the math.
The paper folding you shared in your pictures looks hypnotic. It is funny how I avoid buying pleated skit because I am too lazy to think of how to iron the folds to keep the skirt in pleated conditions. It is a true testament to you, when you reflected on your resilience when you worked on the mind bending paper folding origami. But once you pushed through the struggle you were able to persevere and see the beauty in your paper folding. Again, it relates to that idea of shifting in attitude. It is something that I am trying to help my secondary students change with math and other aspects of their life. Because when the going gets tough, what would my students choose to do? Will they persevere or will they give up. I always share my own stories, showing them how I struggled in my academics at SFU with failing grades and being on academic probation, but was able to make changes and persevere through. Students are always engaged, intrigued and fascinated with the stories and lived experiences that I share with them to help them realize the importance and value of a shift in mindset and a change in attitude.
Shifting attitudes is a long process, not one that will happen quickly for sure. What you said about sharing our own stories and lived experiences really resonated with me. This week, 2 of my boys (aged 18 & 20) have experienced some big life struggles. I realized that even though they are my whole world - they don't know all that much about my life before they were born. It's not often we tell stories of when we have struggled, or had to make big decisions that altered the course of our lives, for better or for worse. I found myself realizing this week that I needed to share some of those struggles I went through when I was their age, so they can feel hopeful that they too will make it through the struggle to a successful future. The importance of sharing our stories (and listening to others stories) cannot be overstated.
DeleteHello Joy,
ReplyDeleteI appreciate your questions and comments about struggle and resilience. Your example of how you conquer the challenges of creating the foldings in origami would be a great opportunity for students to understand that even math teachers struggle. As April commented, these conversations would support the growth of our students and help them not only to conquer mathematical concepts but also personal challenges.
Your Miura Ora Origami looks great. You did a great job.
Thanks Lida. You are right, if we are going to invite students into productive struggle, we need to inspire them with stories of our own challenges, struggles and victories. If they see that we allow ourselves to struggle, we can normalize that process for them. Hopefully, rather than being scared of the struggle, students will experience that in embracing the struggle, we learn.
DeleteHey Joy,
ReplyDeleteYour reference to Kaagan Jaad (Mouse Woman) would be a good one for probably every week in this course. Where does math sneak in and help us with dance, art, poetry, paper folding, fashion design? It’s interesting to note that the idea of math sneaking in to help out was voiced by Billy Yovanovich who is a Haida carver. When we were in our session on Monday with Cynthia, Jo-ann Archibald, and Joanne Yovanovich (Billy’s mom), I thought it was interesting that Joanne Yovanovich said there isn’t a Haida language (assuming none of the major dialects) for “mathematics” and it was discussed that it is just there, part of how Haida operate with the world, sneaking in to help them like the Supernatural Mouse Woman. I looked up Kaagan Jaad in Terri-Lynn Williams-Davidson’s book, Out of Concealment: Female Supernatural Beings of Haida Gwaii, and here is what she writes about Mouse Woman: She represents a kind heart, generosity, the gift of intuition, and the power to tap into that intuition.” (Williams-Davidson, 2017. p. 73). I like this idea of intuition and tapping into it in relation to mathematics, too. Have you ever had a time when you are working through a problem or task and you just feel that you should try a certain approach or method? And then you do it and it works?! Maybe Mouse Woman is sneaking in.
Good for you for having the conviction to see the conversation about shifting from the idea of fitting in Indigenous content to actually changing our pedagogies to incorporate a wider range of experiences for learning that accommodate Indigenous ways of knowing/FPPL to benefit the diversity of learners that we have in our schools. I’m sure being one of the relatively newer district teachers, it might be harder to offer these ideas that challenge the status quo. I commend you on seeing the conversation through!
I think some of your conversation with your colleague mirrors, to some degree, the struggle I voiced on my blog about how to present embodied ways of teaching and learning mathematics to colleagues who might have a very specific idea of what mathematics is and is not. I feel like I am being heavy-handed sometimes when I pull in the idea of needing to get through the curriculum as a colonial notion that does not support an anti-oppression pedagogy but it does make colleagues pause. On the other hand, if I just introduce colleagues to one activity that incorporates dance and math, for example, that is probably going to do little to shift practice and it will remain an isolated activity that is done once followed by a return to “regular” math. In this course, I am seeing that there are so many people working with the arts and the outdoors in connection to mathematics and these ideas of poetry and math, fashion arts, etc. are not that “out there.” How can I help teachers see how many resources there are to help them and that this is a well-researched area? How can the academics at the universities work to move these ideas out of research journals and into wider practice?
Your discussion on constraints and creativity made connections for me. I am a person who appreciates constraints and can work creatively from there. Without constraints, I can feel paralyzed by options and ideas that all bang around in my head. If last week we were told, “Design a mathematical poem structure and write poetry” I would have taken days to get at it. However, with the structure of the Fib, I had something to hold onto to write my first poems and then spring-board from to create my own structure. I feel like there are many students who benefit from constraints in the same way; constraints ground them so they can take some risks in creativity that they might not otherwise attempt.
I LOVE that you showed off your final paper folding to others! I sent my pictures to my dad (my mathematical collaborator - his response was “very cool”), posted to Facebook (one of my engineer friends also said “cool”), and tweeted the pictures! I’m probably going to take it to show my uncle when I drop his groceries off – he likes that kind of thing, too.
DeleteMy final thought on your post is to collegially challenge your idea of bridging the abstract and the concrete. When I think of a bridge, it connects two distinct entities and they stay generally separate and other things can flow back and forth between them but they don’t actually meet themselves. I continue to be intrigued by the “third space” idea that Karl Schaffer and Susan have discussed. I am wondering if there can be an abstract and a concrete tributary that feed into the same river – they can exist separately but also flow together where you don’t know which water came from which tributary. (I also had a flash-back to Heidegger and the bridge in the writing we read – what does the bridge represent? Has anyone figured it out yet?)
Wow Sandra, so many things to discuss! I’m going to start with the push back on the bridge idea – you are right. I think sometimes we need to move from one to the other (concrete to abstract), but it is not always in a linear way, or in the direction from concrete to abstract. Sometimes we circle back and sometimes I think the concrete and the abstract can happen at the same time, and sometimes we start with the abstract and move to the concrete… Thanks for pushing my thinking on this. It reminds me of the Königsberg bridges problem and Doolittle’s response to finding a solution in week 3 of this course. Throughout this course, I have been wrestling with the idea of binaries (week 2 I started making a list of the binaries I was coming across) and with the idea of perspective. Doolittles solution to Königsberg bridges is all about perspective. I’m realizing that part of our colonial way, is to only see one perspective or to consider one perspective more valuable than others. I am working to change this in my own thinking. Karl Schaffer mentioned it on Saturday as well, about the importance of perspective and how we learn when we change our perspective. That is what this course is doing for me, changing or perhaps widening my perspective. Helping me to dwell in a better, more authentic and complete way.
DeleteI agree with you that the idea that we need to ‘cover’ the curriculum is a colonial idea, strongly embedded in mathematics more than any other subject. I love that you call it out as a colonial notion that does not support anti-oppression pedagogy. If we are going to make systemic changes, we need to develop understanding about this.