Summary: Impossible figures are optical illusions. In 2D they create the perception of a possible 3D object, but the 3D form is impossible to form - that is, if we assume rods are straight and connections are at right angles. Fisher creates impossible shapes with beading, and because of the flexibility of the beading joints they become possible to make in 3D. She calls the beaded shapes highly unlikely shapes. You can see that the rods that make up the shapes have a twist. Here is an example of an impossible triangle and a beaded highly unlikely triangle.
As Fisher replicates more impossible shapes, and extends the process to make highly unlikely polyhedra she finds new patterns. For example, faces that twist in unexpected ways and beaded paths on the faces that converge or diverge in different ways.
Stop 1: Fisher’s article was published in 2015, but she mentions that impossible figures were discovered by Oscar Reutersvard in the 1930s and popularized in art and graphic design in the 1950s. This timeline struck me because a new setting (beading as opposed to drawings) provided a new perspective that led to new/deeper mathematical understandings decades later. If no one had tried a new medium, we would be lacking this new knowledge. Creativity and modality in math are meaningful.
Stop 2: This exploration fits the framework that Vogelstein et al. (2019) outlined in my week 6 readings. Fisher foraged for an already existing shape in art/math. Then analyzed and recreated it with beads. And, finally she created something new (the highly unlikely polyhedra.) It feels like a framework that extends off the shoulders of giants.
Question: I feel like beadwork of this complexity may be out of the scope of what I can do with students in the classroom, in terms of time and skill. What other settings (or media) might be doable with students to follow this framework and look at discovering new maths connections and meanings?
References:
Fisher, G. (2015). Highly unlikely triangles: Bead weaving. In G. W. Hart & R. Sarhangi (Eds.), Bridges 2015: Mathematics, art, music, architecture, education, culture (pp. 493–496). Bridges Organization.
Vogelstein, L., Brady, C., & Hall, R. (2019). Reenacting mathematical concepts found in large-scale dance performance can provide both material and method for ensemble learning. ZDM: The International Journal on Mathematics Education, 51(2), 331–346.
Nichola, your first stop is very interesting! The shift in medium from drawing to beading provided NEW perspectives that led to further ‘impossible’ designs. With that knowledge, that really should embolden us to explore the same topic in a multitude of ways with our classroom just to – at the very least – be able to compare the limitations of the form side-by-side (ie. Rational number drawings vs. Rational number dancing). This draws into my question from the week of which is the most useful – the students may actually be able to determine this as part of an exploration.
ReplyDeleteExpanding on this idea, it would be interesting to teach a concept, then as ‘homework’ the students think about which medium they feel would best express connections (ie. Dancing, drawing/painting, knitting/sewing, origami, poetry, sculpture – we would narrow it down by groups like “fibre arts, visual arts, performance arts, etc) then they would need to justify why they moved in that direction to demonstrate understanding of the concept. For those who are uncomfortable or have trouble deciding, there could be a ‘default’ selection that may be the most accessible (ie. Requires the least amount of tools/materials to complete, or is the most obvious/easiest to connect).
You can bet I will be trying this out in the future! Thanks for the thoughts!
After reading your week 6 connections from Vogelstein, I’ve really seen the value of foraging, dissecting and recreating in mathematics (I’ve been calling it ‘reverse engineering’ in science and textiles; this idea is analogous to exploring different math concepts in various medium).
I love the idea of showing up with strange laces to spark a mathematical conversation. To make this more accessible, I would print off (and laminate or not) 30 student copies of shoes with 8-10 eyelets, then use basic string the same length as a regular shoelace
https://cdn.myshoptet.com/usr/www.sneakergear.eu/user/documents/upload/mceclip0-30.jpg?1711713534 ) Then, try the exploration with them. Perhaps have them try to recreate the proof of the shortest shoelace lacing by measuring the lengths of the crossings and adding them up – maybe they could get to a point where they can generalize like Polster does and just end up counting slope lengths (+1, +2, +3, +4) instead to simplify.
Downgrading tools to everyday supplies I’m not afraid of experimenting with (aka, ‘destroying’ or ‘wasting’) helps me with time management and cost in prep, and allows my students to be part of the preparation process. I’m sure you’ve found this too!
Thanks for the thoughts Nichola!
That is a great idea Olly! Earlier in the course (with our previous reading groups) I mentioned that cost of materials for embodied activities could be challenge/limitation. Our school does not have a math budget, and perhaps I should propose for one so that we can access this learning more. (I know that as teachers we often supply materials from our own pockets, but that is only feasible for so much and so long.) I was also wondering if our PAC may be able to help with funding for "craft-type-supplies" specifically for math use. In the meantime (and just in general) simplifying the materials is a fabulous tactic!
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