From this semester, I learned about why math history should be incorporated into math education; how simple math and algebra was done in ancient methods; how to teach math in different ways, such as art, dance, physical interpretation of proofs; how mathematics and mathematicians were perceived at different time periods; and how to teach mathematical history to a class through our assignments. The insights I take away from these experiences is that there are many ways to teach mathematics and it isn't always completing worksheets and writing exams. I have also learned that mathematics is art and beauty, through the multiple art forms that math creates, from dances to creative tilings. I think that this course has been educational and insightful, so I don't really have any specific feedback other than that I enjoyed the course very much and learned a lot.
Sunday, 19 December 2021
Monday, 6 December 2021
Blog Post #17: Assignment #3 Submission
An explication for this assignment:
Biography of M.C. Escher:
Maurits Escher was not a traditional mathematician that made ground breaking discoveries in the field of mathematics; he was an artist who did not excel at math in primary nor secondary school but eventually would make discoveries in the field of symmetry groups and crystallography. He grew up in a family with a solid engineering background, which gave him a methodological approach to thinking, aside from his creativity. Switching from architecture to graphic arts by the recommendation of his professor in his post-secondary studies in Haarlem, he would be first introduced to woodcutting techniques, which will later have a huge impact on his work. He started from sketching patterns from his travels to imitate a regular division of the plane (which means that he would select specific objects and repeat them in a way where they would fit together like a puzzle across a 2-d plane and he would use a variety of objects such as fish, birds, and many more). Then, he would engrave those sketches into wood cuts to make prints by rolling paint onto the wood cuts and stamping them onto paper, either for experimentation or to make a living. Although he claimed that he wasn’t mathematically inclined, he still studied Polya’s paper on the 17 symmetry groups and created his own notations to better understand how he could incorporate these patterns into his tessellation studies. In 1941, he created his first notebook the “Regular Division of the plane with Asymmetric congruent Polygons”, which would lead to the development of his own categorization system that covers all the possible combinations of shape, colour and symmetrical properties. Unknowingly, he made a mark on crystallography research through his passion for art.
Horseman Tiling:
To give a definition of a valid tiling, each outlined shape must connect together at the edges with no gaps nor overlaps on a Euclidean plane—all of these tiles must also be countable (so you can denote the tiles as T1, T2,… all the way to Tn and group them in a set). And yes, although the reigns of the horse are technically overlapping with a helmet of another tile, this is written off as an artistic choice. The overall tiling is more important. This tiling image was originally a woodcut that presents a row of horsemen tilings going one way and the row above and below it going the opposite way as represented by the different colours. I found this tiling to be particularly interesting because multiple sources had differing perspectives on the way that this tiling was constructed. One source had said that this pattern is much like a mobius band that figure 8’s from going from one row to another and there is an intersection at a part of two rows in between (An example of a mobius band is taking a rectangular strip of paper, twisting it, and attaching the ends together so that if you trace your finger along the flat side you can trace over both sides of the paper an infinite amount of times); the intersection of the loop depicts an illusion of the brown horses in the foreground and white horses in the background and vice versa. Another way to interpret this pattern is that these horsemen have a glide reflection moving diagonally upwards. To see this, select a brown tiling on the bottom left corner, reflect it to the left over the y axis and move it upwards and continue the glide reflection in the same direction. This diagonal movement of the horsemen in either direction (you can reflect the other way as well) creates a patch that repeats itself throughout the tiling, meaning that you can map this specific group of tilings onto the next and the next. Within the modern day 17 wallpaper groups this would be group #4 denoted as “pg”. The “p” in the label is trivial and the “g” signifies a glide reflection.
Fun Facts about Escher through the lens of his Eldest son:
After discussing my project idea with Susan, she sent me very interesting interviews with Escher’s eldest son, George Escher, who is an engineer. In one interview she specified, George talked about his dad, at a CMESG (Canadian Math Education Study Group) conference in Halifax of 1996. He talked about how his mom and dad travelled to southern Spain in 1939, on the eve of the Spanish Civil War, to Alhambra and were entranced with the tessellations they saw there. As George told it, his parents hurriedly sketched as many of the tile patterns as they could, and then Escher spent several years in his studio playing with those sketches by abstracting and reworking them in the forms of birds, fish, elves, plants and other images that later became iconic of his mathematical work. So, the Islamic tilings of the Alhambra in Granada was a direct inspiration of Escher's work.
In a video interview: As a child, George said that his father would build mazes all over the house using furniture to create tunnels that zig-zag in the dark. He would also hear his father play Bach, in which this composer was a musical inspiration to Escher. And in one of Escher’s famous sketches of a hand holding a glass ball with the reflection of him and the background of his studio, George can point out specific parts of his father’s studio that he remembers as a child. He also noted that when his father was working on his art, the house would be very tense and he would demand silence, while working with his door locked. He also never wanted anyone to see what he was doing (as the kids were not even allowed to be in the garden in fear that they would peer through the windows). However, eventually, the tension would relax, and he would allow his family to look inside his studio and see what he was up to. His father would also work on wood cut works in his studio in a specific way– where he would sand the wood first, and then use a tool to make a light engraving for the outline before switching to harsher tools to define the print in the wood. And when he began to print the pieces onto a piece of paper, he would use an ivory spoon to press the piece of paper against the wood cut to ensure that the paint is fully pressed.
Landau, T. (2019, May 10). Classifications of Frieze Groups and an Introduction to Crystallographic Groups. https://www.whitman.edu/documents/Academics/Mathematics/2019/Landau-Balof.pdf. Retrieved December 7, 2021, from https://www.whitman.edu/documents/Academics/Mathematics/2019/Landau-Balof.pdf.
O’Connor, J. J. O. (2000, May 1). Maurits Escher. MacTutor. Retrieved December 7, 2021, from https://mathshistory.st-andrews.ac.uk/Biographies/Escher/
Schattschneider, D. S. (2010). The Mathematical Side of M. C. Escher. Notices of the AMS, 706–719. https://www.ams.org/notices/201006/rtx100600706p.pdf
Films on M.C. Escher. (2021). M.C. Escher - The Official Website. Retrieved December 7, 2021, from https://mcescher.com/about/video-on-m-c-escher/
Sunday, 5 December 2021
Blog Post #16: Response to Alice Major article: Numbers with Personality wrt Mayan math presentation
• Is this something that you might want to introduce to your secondary math students? Why or why not? If you would use these ideas in your math class, how might you do so?
• Do numbers have particular personalities for you? Why, how, or why not? What about letters of the alphabet, days of the week, months of the year, etc.?
Thursday, 18 November 2021
Blog Post #15: Assignment 3: Draft resource list and format
-Will give an introduction about tessellations/tilings in general -> Give some background on M.C. Escher
Reference list:
Landau, T. (2019, May 10). Classifications of Frieze Groups and an Introduction to Crystallographic Groups. https://www.whitman.edu/documents/Academics/Mathematics/2019/Landau-Balof.pdf. Retrieved December 7, 2021, from https://www.whitman.edu/documents/Academics/Mathematics/2019/Landau-Balof.pdf.
O’Connor, J. J. O. (2000, May 1). Maurits Escher. MacTutor. Retrieved December 7, 2021, from https://mathshistory.st-andrews.ac.uk/Biographies/Escher/
Schattschneider, D. S. (2010). The Mathematical Side of M. C. Escher. Notices of the AMS, 706–719. https://www.ams.org/notices/201006/rtx100600706p.pdf
Films on M.C. Escher. (2021). M.C. Escher - The Official Website. Retrieved December 7, 2021, from https://mcescher.com/about/video-on-m-c-escher/
Medium:
paint, coffee, permanent marker on canvas
"Horseman" by M. C. Escher.
Thursday, 11 November 2021
Blog Post #14: Assignment 2 takeaways from 4 presentations from each day (Nov. 10 & 15)
First day:
1) Nadine's Presentation on Permutations on Combinations: Church bells rung based on calculated permutations to produce different tunes.
2) Marlaina's Presentation on the History of Conic Sections: Historically recorded version of drawing circles by keeping one's elbow at the central point of the circle and rotating your hand in a circular motion.
3) Yiwei's Presentation on the History of Proportional Reasoning: The explanation about the translation of texts between Italians and Chinese on Rational Numbers to spread mathematical knowledge internationally.
4) Alan's Presentation on Measurements and Units: 3 pieces of corn can approximate an inch.
Second day:
1) Ivan's presentation on the History of Modern Measurement: The measurement of distance using only time by the Indigenous people, as we still use this method casually in modern day,
2) Austin's presentation on the History of Combinatorics: A particular combinatorics puzzle existed in different cultures and eras, from Ancient Egypt to an English nursery rhyme in the early 1700's.
3) Emilie's presentation on the History of Geometric Constructions: Origami Applications in Satellites in order for the device to fold up.
4) Victor's presentation on the History of Non-Euclidean Geometry: The fact that there is no proof or disproof for the parallel postulate.
Wednesday, 10 November 2021
Blog Post #13: Mathematics of medieval European universities
Three things that surprised me from the article:
1) "Numbers were identified with the various gods. He considered the odd numbers to be male and the even ones to be female. He made a strange distinction between the "divine number," a sort of general concept of number which existed only in the mind of the creator-god, and scientific numbers, which were the common numbers known to men on earth." - pg. 267 I find it surprising that numbers were once identified with gods, as if nobody could really know where they came from. I am also curious about how they determined odd numbers to be male and even numbers to be female.
2) "The arithmetic of the schools did not receive the wholehearted approbation of all the people." - pg. 271 Considering how mathematics is used as a basis in many fields (such as science, engineering, architecture, etc...), I find it mind-boggling that it was once received as something rather trivial and not of importance.
3) "Dialectic or logic became so important that it tended to obscure all the other arts." - pg. 270
Saturday, 23 October 2021
Blog Post #12: Dancing Euclidean proofs
Two things that surprised me or made me want to stop and think about the article:
1) I found it interesting that the "exploration of multiple representations and modes of cognition has been shown to enhance and deepen mathematical understanding building more holistic mathematical understanding through a knowledge of the equivalence of varied modes of representation." When I learned math, I mainly did practice problems to deepen my understanding and never really thought about physical activities as another mode of cognitive learning for this subject.
2) I also found it intriguing that there is a need to "spark an appreciation of the beauty of mathematics". I have always viewed math as a tool that was used to solve problems as I never interpreted it as an art form. To see that mathematics has beauty in it because it has an artistic interpretation is really eye-opening.
How might this kind of activity be helpful for math learning and understanding math history in a high school mathematics class? What kinds of constraints or obstacles might you encounter?
This kind of activity could be helpful for math learning and understanding math history in a high school mathematics class especially when learning about geometry units. This is because it can be a little bit difficult to visualize how angles are proportional to each other on paper and a physical interpretation could be easier to grasp. Constraints could be picking examples that are simple enough to embody and space to do physical interpretations with the whole class.