As I was preparing to hand in assignment 1, the annotated bibliography, I came across a study from Thailand by Firmansyah, Sunardi, Susanto, and Ambarwati (2018). Reading their introduction made a few things click for me across our MAE3 program in terms of the rigidity in student processing of particular mathematical procedures.
To sum up briefly in bullets:
Student memorize procedure, and imitate how the teacher has solved problems without understanding.
When presented with different questions, students don't know where to start.
I've experienced this many many times especially in Workplace Math 10.
“The ability to think critically and creatively plays an important role in modern society since it is capable to make people become more mentally flexible, open and easily adapt to various situations and problems.”(Firmansyah et al., 2019, p. 1)
To stretch the students' understanding and allow them to gradually move from imitation to understanding underlying procedures, modifying the problem slightly and allowing students to apply what they have previously been able to challenge themselves with forces them to see baked-in patterns: What is similar? What is different?
This is the foundation of Variation Theory, that we've previously read about from a paper by Askew (2011) in Week 3 of EDCP 550 (our second course).
In addition, elementary and middle school students do this well through worksheets called Derived Facts like "If I know... then I know..." to establish connections between similar concepts/ideas. Here's a screenshot from the NCETM Five Big Ideas video that we discussed in Week 7 of EDCP 552 (course 4).
This also relates to Thinking Mathematically by Mason et al. (1982) in terms of the students addressing problems with an Entry-Attack-Review sequence and the idea of problem-posing explained by English et al. (2005) in which students generating a similar but perhaps simplified problem on their own but with a twist to observe similarities and differences in the underlying concepts. This key skill is quite important for everyday life! (Both of these references were studied in EDCP 552 as well)
References
Askew, M. (2011). Variation theory. In Transforming primary mathematics (pp. 75–88). Routledge.
https://doi.org/10.4324/9781315667256
English, L. D., Fox, J. L., & Watters, J. J. (2005). Problem Posing and Solving with Mathematical Modeling. Teaching Children Mathematics, 12(3), 156–163.
Firmansyah, F. F., Sunardi, E. Y., & Ambarwati, R. (2019). The
uniqueness of visual levels in resolving geometry of shape and space content
based on van hieles’s theory. Journal of Physics: Conference Series, 1211(1).
https://doi.org/DOI:10.1088/1742-6596/1211/1/012076
This week we discuss the merging, bridging and connections of the two worlds (the arts and the sciences). As an artist, musician, craftsman, athlete, mathematician and scientist I try to merge all I know and have experienced to develop understandings of each of these 'worlds'. I imagine my knowledge in these facets like tools to be used when the time is right regardless of its initial designated discipline much like using a fork to comb ones hair (shoutout to Ariel). Each skill is transferable and needs to be acquired with this mindset in place - one of infinite applications - rather than for a very specific context.
This week Vi Hart's video on using a mobius strip to invert a musical tune was beautiful and intriguing. If only it were this easy to do while playing music on a piano, I would love to hear what would happen to a few classics by Chopin (Nocturne in E Flat Major Op.9 No.2, or Nocturne in C Sharp Minor No. 20, for example). I imagine that over time, the surprising effect would wane as I would better able to simulate in my brain what the song would sound like before the tune would become inverted - but to get to that point of familiarity with song inversion, it would take some serious effort!!
If I were to translate this effect into mathematical concepts, based close inspection of the special music box paper, I think it would be like taking C5 as the y=0 line (...sorry to impose a grid upon an already grid-like structure of the musical staff...we've talked about liberation from this cage last week) and creating a reflection about the x-axis, essentially assigning all musical values a positive and negative equivalent, then multiply the entirety by negative one.
How could I trick my brain into doing this automatically? Perhaps using technology... Start with the original (this arrangement was by Rowena Young and contains no key signature unlike the original in E Flat Major):
Grab only the melody line, and exclude the clef so as not to trick oneself:
Then invert with a drag of the mouse:
Aside from the note direction, signage of sharps and flats and numbers... this is pretty much ready to play (along with the left hand inverted similarly)!!!
Try it yourself before you hear my sight-reading performance!
Here's the results:
Vi Hart represents sounds using various pen strokes, then transposes the visual and audio to different harmonic keys overlaying them to develop a unique performance art project. Their art is quite fascinating!
For my assigned Bridges activity, I got excited about representing integers using different base systems. Inspired by Melissa Schumacher's Celtic knot artwork, I wanted to combine the two ideas myself using a dot and two types of lines system, but expand to a based four system to see how it would flow. However, what I realized through experimentation is that the base 3 system is directly related to the three line types, but not the number of digits in the row.
So, while playing around with different base numbers, I created a new number system instead using base 3.
I wonder what a base four drawing would look like given the intricate style of drawing of Celtic Knots.
A few wonders I have before I start...
Could there be a different stroke for Celtic Knot systems that would allow me to draw one beyond base 3? (0=cross, 1=vertical impediment, 2=horizontal impediment, 3=both horizontal + vertical impediment?)
Does even number base systems work with Celtic knot patterns?
Would the numbers in which (4^4)x1 also resemble a loose strand of knot similar to Schumacher's?
To fulfill my art project around one of the Bridges 2023 Activities, I was lucky enough to find one developed my our very own professor, Dr. Gerofsky! A Wurzelschnecke is a spiral created by using the hypotenuse as the leg of the next triangle. Start with a triangle of legs 1 unit x1 unit, repeat this process about 15 more times and you get a spiral that very nearly touches itself but would overlap at the next iteration.
Dr. Gerofsky's exploration with the Wurzelschnecke shapes and different iterations for toys, jewellery, furniture and the like got me curious about what it would sound like to create a musical instrument with the same interesting ratios of the Wurzelschnecke's hypotenuses.
I went about trying to build a frame for the musical instrument, but realized that I wouldn't know how to tune it with proper tension even if I did create the frame. Instead, I turned to developing an artifact with Claude.ai to simulate the sound of the ratios between all the triangles!
I was curious about this expanding relationship between using the side lengths to build onto the next shapes, but I didn't want to re-use the triangle... so I needed a shape that would have a face that would expand depending on the other two faces.
I decided a fitting candidate would be the trapezoid with sides at 22.5 degrees from the top (a = short parallel side). Both sides and top are the same length, and these lengths determine the expanding base (b = long parallel side) length each time.
A few questions I had about relationships between the expanded shapes:
1. As we generate an expanding trapezoid, what number does the ratio of a/b approach?
2. At what distance do the sequential pieces exactly cover the previous piece? Is this distance the same for each sequential pair?
3. By what factor does expansion happen for each trapezoid?
Reference: Dylan Thomas & Doris Schattschneider (2011) Dylan Thomas: Coast Salish
artist, Journal of Mathematics and the Arts, 5:4, 199-211, DOI: 10.1080/17513472.2011.625346
Summary: This article resembles an autobiography of the Coast Salish artist Dylan Thomas. In it, he acknowledges his teachers and influences that helped him develop his unique style of combining Coast Salish traditional shapes, mathematics and modern elements. Thomas was inspired by Susan Point's spindle whorl piece (a tool used to spin wool for weaving used by the Coast Salish people) which exhibited rotational symmetry. Another major inspiration was M.C. Escher and his tessellations. Studying these artists, Thomas developed:
Sacred Cycle - a cyclical piece depicting three salmon exhibiting three-fold rotational symmetry.
Salmon Spirits - an extended tiling piece with salmon rotating in four-fold symmetry.
Ravens Housepost - a flowing vertical pattern of raven heads rotated 180 degrees repeating.
Eagles Housepost - a repeating vertical pattern of eagle heads rotated 180 degrees from a central face.
Horizon - a piece depicting a salmon mirrored in the vertical and horizontal plane whose tails overlap such that they create a perfect circle in the centre.
Mandala - a cyclical piece with concentric circles in which squares have been inscribed with the Coast Salish circles, crescent and trigon motifs, inspired by Hindu and Buddhist traditions, exhibiting four-fold rotational symmetry.
Infinity - a group of salmon flowing clockwise in an ever shrinking tessellation with each salmon taking up a wavy triangular space, exhibiting four-fold symmetry with each quadrant exhibiting dilation symmetry inspired by M.C. Escher's print Smaller and Smaller.
Stop 1: "I spent days staring at Escher's symmetry drawings such as his butterfly and lizard tessellations and from these, soon figured out how I could make a tessellation. I began experimenting with different shapes and after about 3 days I found a design that worked" (Thomas & Schattschneider, 2011, p. 202).
This quote from the author demonstrates focus, determination, and perseverance. From my interpretation, Thomas was strongly motivated through his inspiration. We have innate abilities that allow us to figure things out regardless of formal teaching and terminology - this might be a really good example of why teaching outside of student means could work as long as they are hooked! Visual art might be a strong contender for this possibility of over-reaching to inspire because students would have something study, all the steps can be figured out in a two dimensional plane without needing to imagine any hidden critical pieces.
I would love to see what his work looked like as he progressed through the days sketching. The process of creation is something I'm fascinated with.
Stop 2: "It is interesting to note that Dylan's path to making his first tessellation parallels that of Escher. Escher, too, had no mathematical background for this task, and figured it out by studying several geometric tessellations by majolica tiles in the Alhambra. Later he studied a display of tilings in an article by mathematician George Polya, and from one of these, Escher produced his first tiling by lizards with 4-fold symmetry" (Thomas & Schattschneider, 2011, p. 203).
Similarly to stop 1, it's interesting to hear that M.C. Escher too had a similar experience of not being introduced to tessellations and tilings through mathematics, but just figured it out by personal intrigue and fortitude. Do all students have this kind of experience of perseverance in their lives?
Stop 3: "Although I derive much of my inspiration from outside First Nations art, Mandala was my first truly cross-cultural piece of art. I enjoy doing cross-cultural art because art itself is one of the only practices that can be found in all cultures. Art is one of the things that makes us human, and bridging different cultural art forms helps me to feel the unity of humankind" (Thomas & Schattschneider, 2011, pp. 208-209)
This idea of art being a practice found in all cultures is something else I'm interested in as it is not part of Bishop's (1990) six mathematical activities found in all cultures (below), but I argue that it should be! There are so many mathematical decisions that go into creating artwork that perhaps it might actually be a culmination of several of these individual math activities rolled into one.
Wonders:
Does art work for every type of mathematical concept?
Can I isolate the concept of fractions, for example, and develop art from it?
Could I isolate the concept of logarithms and develop art from it?
Reference: Bishop, A. J. (1990). Western mathematics: The secret weapon of cultural imperialism. Race & Class, 32(2), 51–65.
https://doi.org/10.1177/030639689003200204
