This is the impossible triangle made possible by twisting the beads.
The exploration of the impossible triangle led the artist to try different impossible shapes. The artist successfully created an impossible square, an impossible hexagon, and other impossible figures, even one that was never drawn before (impossible polyhedron)!
“An impossible triangle and other similar impossible figures are only impossible to construct in 3D if we assume the edges are straight and the connections are right angles (Fischer, 2015, p.100).”
This quote reminded me of Doolittle’s (2018) argument for a diversity of geometry, ones that allow us to get ‘off the grid’. In this article, because the artist was able to think outside of the grid (not assuming the edges and angles are straight), the artist was able to create an impossible triangle. This is a very good example of how the grid can fail to represent realities sometimes, and that other geometry could help us open our mind and be a more ‘truthful’ representation of a reality. Because the artist decided to follow unconventional geometry, this creative exercise led to the exploration of mathematics not many thought about before (we know it by the given name unlikely figures!). I wonder if we could reach this level of creativity by solely solving problems. It feels like bringing explorative arts activity in the classroom is more approachable than problem-solving to encourage students’ creativity. Since students are not enough reminded of the importance of creativity in mathematics, they often get really anxious when I ask them to be creative. Then, if we start with those types of activities, we can change the classroom’s mindset to have students who embrace creativity. When they do recognize the importance of creativity, we might see more and more creative approach to problem solving. I believe that creativity is one of the most important qualities a mathematician can have because we know that many mathematical concepts/theorems that we use today came from mathematician who dared thinking outside of the box.
“The right photo in Figure 3 shows my second attempt, the first successful highly unlikely triangle (Fischer, 2015, p. 101).”
This quote resonated with me because it reminded me of the concept of productive struggle I have been looking into for my school this week. Productive struggle, define as overcoming a challenge with the purpose of learning or progressing, is important for students to truly learn and understand the concept, rather than just learning procedures (Sangiovanni et al., 2020). Although, many students do not see the importance of struggling to learn. In my classrooms, this looks like students who are giving right away just because they do not know how to get to the answer right away or do not understand a concept right away. Because I teach grade 9-10-11, it is really difficult for me to break this habit since my students are so used to it and never had to struggle that much to get good grades before.
Thus, I think that using mathematics activities based on arts and craft could help me re-story my students’ idea of learning. In the article, the artist was not successful in the first attempt, but learned from it and was successful in the second attempt. I am assuming that Carolyn Yackel, from the video How Orbifolds Inform Shibori Dyeing (G4G Celebration, 2021), and Uyen Nguyen, from the video Origami fashion (YOUmediaChicago, 2021), also struggle a lot to be able to have such in-depth knowledge about their different patterns (origami clothing or dyeing). All three artists had something else in common: their passion kept pushing them forward. Artistic creation might not develop this type of passion in all of our students, but if it does reach some of them to some degree, it should be accessible to them. If passion makes my student learn to struggle and understand its importance, it is something certainly worth using in my classroom.
Questions
What mathematics do you see in beading and creating 3D shapes with beading?
How do you foster creativity in your classrooms?
References
Doolittle, E. (2018). Off the grid. In Gerofsky, S. (Ed.), Geometries of liberation. Palgrave. https://doi.org/10.1007/978-3-319-72523-9_7
Fisher, G. (2015). Highly unlikely triangles and other impossible figures in bead weaving. Proceedings of Bridges. (pp.99-106)
G4G Celebration. (2021, January 27th). Carolyn Yackel - How Orbifolds Inform Shibori Dyeing - CoM Oct 2020. [Video]. Youtube. https://www.youtube.com/watch?v=hjtc9LJ5ItI
Sangiovanni, J. J., Katt, S. & Dykema, K. J. (2020). Productive struggle: A 6-point action plan for fostering perseverance. Corwin.
YOUmediaChicago. (2021, January 07th). Origami Fashion with Uyen Nguyen Part 1. [Video]. Youtube. https://www.youtube.com/watch?v=i4AoN1DtH6I
“I believe that creativity is one of the most important qualities a mathematician can have because we know that many mathematical concepts/theorems that we use today came from mathematician who dared thinking outside of the box.”
ReplyDeleteYour first stop really resonated with me, Noemi. I also feel strongly that creativity and out of the box (or off the grid!) thinking is an important quality of mathematicians. It can be a struggle to foster this quality in a traditional classroom setting with a focus on textbook problems. Arts-based approaches provide an excellent opportunity to flex that muscle, but as you mentioned, students can be hesitant since it is not familiar to do this in mathematics. I agree with you that the more we add in these types of activities and open-ended problems, the better our students will be at thinking creatively to solve problems. This is again an overall benefit to mathematics learning, regardless of the connected curricular outcomes associated with the activity itself.
I’ve also been making my way through Productive Math Struggle by Sangiovanni et al. this year and have found it to be a helpful tool in finding ways to encourage that mindset for students who might be resistant to challenges.
I’ve never tried beading 3D shapes. I am not sure where I’d start. Did you try any this week?
I try to foster creativity in my mathematics classroom by offering open-ended, flexible problems, encouraging exploration and discovery through introductory activities, encouraging sharing of various strategies, and providing project choices that allow them to apply their learning and mathematical concepts in unique and creative ways. I think I have room to grow in this area and have appreciated the many ideas that have been sparked through this course to provide more tactile and kinesthetic creative opportunities for my students.