Group Response
Students: Jimena, Juma, Shannon
Artwork, Artist:
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| 12 Golden E-Chains Spiral, Philippe Leblanc |
Jimena took the main responsibility for assembling the artwork. Prior to this, we met to discuss whether this would be a viable task, what materials would be needed, and to lay out the production process of the piece itself. We also looked into how to extend this for the classroom. We tried to understand the way the artist calculated the lengths and positions of each chain. There were also practical considerations about the available box sizes and how to bead sizes might relate to this.
The original artwork was recreated with purchases from Michael’s and Amazon. So, some aspects of our piece were chosen out of practicality (for example, Leblanc’s work used a square-based rectangular prism whose lateral faces were in the proportions of the golden ratio). We began with spreadsheets to calculate bead colour amounts. As we were determining the lengths of each chain and the position of each chain, we found that we were trying to honour the golden ratio that appeared in Leblanc’s work as well as the golden spiral that determined each chain’s place. In reality, we were able to mimic the bead count in the five longest chains; however, the remaining chain lengths were more arbitrary. We worked within the constraint that we wanted 700 beads in total and projected possible lengths that seemed plausible for the shorter chain lengths. To determine the placement of the chains, a printout from Leblanc’s information on the Bridges Math site was used. Another alteration of the piece is that we chose our own colour scheme of 10 colours.
We decided to follow Leblanc’s inspiration of irrational numbers. The first thought was to build a project around pi. However, this didn’t play out in a way that we anticipated. So, we pivoted and focused on square roots. Using either straight edge and compass or paper folding, we could create lengths that were irrational. We were also guided by the Grade 11 curriculum, which requested that we order irrational numbers. From this, we decided to ask the students to fold a Theodorus spiral from a single, long strip of paper. The plan was that the spiral served as the base for our class-focused extension to the art project: a square root mobile. The hope was that this would mimic the beads of Leblanc’s chains. (In a real class, we think that we would have also asked students to create them using different colours for each digit of the decimal expansion of each root, provided it didn’t get too confusing visually.) We have them hung from a ring so that the magnitude of each chain is reflected in relation to the bead chain that represented root one. (We felt this provided a visual to the ordering aspect that is required by the Grade 11 Pre-Calculus syllabus.)
Individual Reflection
From participating in creating this art project, I learned a number of things.
Mathematically, I explored the catenary more than I had previously. I used technology to enable this exploration and tried to build on this in relation to chain length. I didn't find that I followed through much more than that. Consequently, I didn't find this project as mathematically involved as some others had the potential to be. The other (debatable) mathematics involved was in the actual spreadsheet skills that enabled determining the correct amounts of beads were not trivial (and had the potential to be transferred to statistics). This, for me, added to the depth of the recreation of an activity like this for students.
I do feel that I gained more from this project as a student and as a teacher, and I cannot really separate the two experiences as cleanly as the question prompt suggests.
As a student in the 342 class, the group nature of the project is where I feel I had the most to learn. Yes, there was a cool little art project involved, but the logistics involved in organizing a group provided some challenges. Dividing up work so that each person made a valuable and fair contribution was a consideration. Also, there needed to be the overall 'now it's done' evaluation, and it can be hard to come to a consensus about that.
As a teacher candidate, I feel I learned the most. We recognized straight away that getting people to put beads on strings is not really a valuable math learning experience. So, the question of what to do for an activity weighed heavily. First of all, I wanted to make a history of pi activity out of it. Pi is one of the poster children of irrational numbers, and one that influenced Leblanc as well. However, some research revealed that, even with very directed scaffolding, the activity might not be as fruitful as I'd hoped. This prompted the square root mobile. This one had potential; however, the challenge with this is that the links to the original artwork may not be immediately evident. So, there was work on the teaching side of it to maintain those links. The actual amount of preparation for a five- to seven-minute activity was also quite heavy. So much work goes into planning and preparing an activity. However, the production of a model ties the ideas together visually.
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| The first fold of the spiral of Theodorus, the planned class activity. |
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| Continuing the Theodorus spiral |
The satisfaction of this project is the mobile that would have been a product of each group in the class. Mathematics is about relationships, and each string of beads of our class mobile means so much more when it is in relation to the other strands. The context gives it meaning. The biggest difficulty for me was that we did not have the time to put it together in class. (The teacher-sided effort that was invested did not pay off as I'd hoped.)
I do hope to bring these ideas into a classroom as a way to fill out the Rainbow of Why (Dave Stuart Jr). I intentionally chose materials that were reasonable accessible in a classroom. But I could see a potential downfall in the time that might need to be invested in a project like this. However, if the planning is there, then it has the potential to be managed. Math gives people a way to engage with the world. And not only in practical ways. These relationships find their ways into entertainment and art and participating in an interdisciplinary activity like this helps to foster mathematics as a lens through which to view the world.
