Reflection of the term

 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. 

Blog Post #17: Assignment #3 Submission

 

Photo of my artwork for Assignment #3 ('Horseman')

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.

 

Citations:

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/

Escher in Het Paleis. (2021, August 9). Horseman. Retrieved December 7, 2021, from https://www.escherinhetpaleis.nl/story-of-escher/horseman/?lang=en

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?

It might be difficult to introduce these ideas in a math class because the concept of personalities being associated with numbers would seem a little too trivial for the more complex concepts in a high school math class. One way that I could imagine this being brought up would be if the teacher were doing a demonstration of solving in class and they come across a single number as an answer, then they could briefly segue into a fun fact about the particular number (about how the Mayans had a particular personality associated with this number). Or if a teacher was asking a student to give a random number for an example, and the teacher knows the number personality off the top of their head, then they could briefly talk about that before going on with the solution. 

• 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.?

The number 30 has a celebratory personality to me because my mother, father and I all have birthdays on the 30th of different months. 

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/

Escher in Het Paleis. (2021, August 9). Horseman. Retrieved December 7, 2021, from https://www.escherinhetpaleis.nl/story-of-escher/horseman/?lang=en

Medium: 

paint, coffee, permanent marker on canvas

"Horseman" by M. C. Escher. 

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. 

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 

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. 

Blog Post #11: Euclid and Beauty

Why is Euclid and Euclidean geometry still studied to this day? Why do you think this book has been so important (and incredibly popular) over centuries?

Euclid and Euclidean geometry is still studied today because he wrote "The Elements". "The Elements" provides the basics of geometry. Within the 13 books, the books 1-6 are about plane geometry, books 7-9 deals with number theory and books 11-13 deal with third dimensional geometry. "The Elements" also begin with definitions and five postulates that set the ground rules for the following books. This book has been really important over centuries because it not only provides the basics of geometry, but the "standard of [proof] rigour was to become a goal for the inventors of the calculus centuries later." 

Is there beauty in the Euclidean postulates, common notions and principles for proofs? How can we define beauty if these are considered beautiful?

I am not sure if there is beauty in the Euclidean postulates, common notions and principles for proofs, but I can say that it is extremely useful for setting the ground rules for writing geometric proofs. For example, it specifies that you can draw a line between 2 points and assumes the existence of points, lines and circles. The second definition of beautiful in the dictionary is that something is "of a very high standard". In the supplementary article, it states that "Euclid's Elements is remarkable for the clarity with which the theorems are stated and proved. The standard of rigour was to become a goal for the inventors of the calculus centuries later." So, his proofs were of a high standard that became a role model for later mathematicians, which is beautiful.

Blog Post #10: Explication/ commentary on poem about Euclid

 Explication and Commentary on the poem:

"Euclid Alone Has Looked on Beauty Bare"

A reference from here: https://lifeorange.com/writing/writing2.htm

In the first stanza, immediately, I question: Who is Beauty? Knowing that Euclid is a mathematician that has written many proofs, Beauty is probably a reference to some mathematical truths that have yet to be discovered by other people. This is because the Beauty is described as "bare", like something that has been uncovered from the unknown. The following lines in the stanza seem like it is criticizing those who "prate" for foolishly speak of the "Beauty" without understanding what it is. In the second stanza, it seems like Euclid is the only one who is able to analyze the "Beauty", or mathematical truth, as he was able to "anatomize" its light. And because of this realization, he was able to make the "Beauty" real (proving her as a truth) as she can now set her massive sandal on stone. 

"The Euclidean Domain"

This poem seems to be ridiculing the original poem. In my opinion, the first stanza asks if Euclid was the only person who ever saw Beauty bare, or other variations of her. I think that this is basically acknowledging that there are other mathematicians out there that have probably seen multiple sides to the same mathematical truth and proven it one way or another. The second stanza asks how the poet knows that the Beauty exists and whether or not she can justify that by having seen the Beauty clothed. The third and fourth stanza states that the poet is calling herself one of the praters that have misunderstood the Beauty because she states that only Euclid has ever analyzed the Beauty correctly.

Parody:

Euclid alone has looked on Beauty bare. 

Let all who prate of Beauty speak their mind, 

And lay them beside Euclid upon Earth and continue 

To ponder themselves, while they stare 

At the Beauty, intricately drawn nowhere

In shapes of shifting lineage; let geese 

squawk and fly off into the distance.

O blinding hour, O holy day, terrible day,

When first the shaft into their vision shone

Of light anatomized! Together

have looked on Beauty bare. Fortunate they

Who, though once only and the but far away, 

Have heard her massive sandal set on stone.

Blog Post #9: Was Pythagoras Chinese? article reflection

Does it make a difference to our students' learning if we acknowledge (or don't acknowledge) non-European sources of mathematics? Why, or how?

I think that it makes a difference to our students' learning if we acknowledge non-European sources of mathematics if the information can provide support for students' understanding of why we use math in a relevant way. For instance, "The Jiu Zhang suahshu gave the reader the tools to solve problems and did not concern itself with proving the authenticity of the tools", whereas "The elements was nearly the opposite, as its chief concern was giving rigorous proof of ideas presented.". In this way, a students' learning is further supported because this gives the perspective that math is not only about proving what is true, but it is about applying it in problem solving.  Sometimes, to teach basic concepts to school students, a rigorous proof might be confusing and too complicated. Some perspectives from different cultures might give students and easier grasp on the concept than others. For example, when looking at the Greek vs. Chinese mathematicians ways of determining pi, we understand that the "Chinese proof requires little background knowledge due to its visual nature" using a method of circle division instead of squaring the circle as the Greeks did. 

What are your thoughts about naming of the Pythagorean Theorem, and other named mathematical theorems and concepts (for example, Pascal's Triangle... check out its history.)

I think that the naming of Pythagorean Theorem and other named mathematical theorems should be named based on which person brought a greater influence to its application across the world. In my opinion, I don't believe that it should be based on who necessarily discovered it first. I would argue this because the theorem was not only discovered and proven by a particular mathematician, but that mathematician recorded the theorem and made sure to spread influence so that people of all classes can benefit from this knowledge in their work and daily lives. It was discovered that Pythagoras had proven his theorem and became well known for it by introducing it to society in his cult; also 'European jesuit missionaries began entering China in early 17th century" to spread their scientific influence while being influenced by Chinese perspectives, in which the Europeans spread math knowledge around the world to implement science into improving the lives of many. Although I think that this is the way that theorems should be named, I believe that others who have contributed towards the proving of the theorem should be credited for their perspective as well because it is equally as valuable. 

Blog Post #8: Personal reflection on the process of Assignment 1

 I worked on Problem #10 which was the Ahmes' Loaves Sharing problem. I think that it's mind-boggling that the Egyptians used a method of false position to come up with numbers to this hypothetical problem of dividing up 100 loaves of bread between 5 people with 2 complex criteria: the loaf division has to have arithmetic progression and 1/7 of the 3 largest shares must be equal to the 2 smallest shares. 

The process of finding the solution to the problem was to reverse engineer the problem by multiplying each share by 100/60 and then adding all the entries together to ensure that they add up to 100. And then taking 1/7 of the largest 3 entry sum and ensuring that they do in fact equal the smallest 2 sum. 

As a reflection, this is a pretty interesting exercise that I would not be able to quickly accomplish by doing guess and check. I would definitely need to use algebra to solve the problem efficiently. 

Blog Post #7: Assignment 1: Solving ancient problems in ancient and modern ways

 

The Powerpoint Presentation from class is posted here: https://docs.google.com/presentation/d/1SFpx71zq2HDgFA1etofldv6vAquwBHJS/edit?usp=sharing&ouid=106697084074657608536&rtpof=true&sd=true

Blog Post #6: Egyptian fractions problem and reflection

Using the regular way of multiply the fraction to the totals before the death of a horse. This is the result: 

By using the Ancient Egyptian Unit Fractions visual. We are better able to visualize the distribution of horses among the three people, as noted here:

 

Blog Post #5: Egyptian division and reflection

 

As a reflection, Egyptian division uses multiples of 2 to get the answer, in which using this tactic to teach kids that struggle with division is useful (as discussed in class). The reason why is because you can see what goes into the number very easily as all the options you have are laid out in front of you. 

Blog Post #4: Response to readings on Babylonian word problems and 'algebra'

Reading on 'Babylonian Word Problems'

After finishing the reading, I found that practicality was considered through the convenience of the students' knowledge to solve the problems-- problems are not made too difficult for students to solve. In reference to the text, Babylonian mathematics seemed to be about word problems that had real life applications for trades training purposes. However, some of these real life applications did not really apply to the training at times, because an agricultural field worker did not need to make complex calculations with the work that they are usually assigned. In terms of generality, sometimes the situation is fit to practice the mathematical tool at hand and it doesn't really apply to the real life task. Pure mathematics seem like a conceptual form of mathematics that does not serve any hands on purpose to solve real world problems. While, applied mathematics is used to solve real world problems. I agree with the text that "Even when Babylonian mathematics is 'pure' in substance, it remains 'applied' in form." The reason why I agree with this is because Babylonians may have come up with really complex problems for agricultural workers that won't be applied on a daily basis; but if a problem were to arise that disrupted their day-to-day business, they would have the necessary tools and logic to solve them. Maybe the worker lost some seeds and they need to know based on the size of the field and the yield they calculated, how much they lost. These ideas relate to the history of mathematics because word problems play a large role in determining mathematics and its applications in the real world-- and how math has developed over time as "real life" situations have evolved. 

Reading on 'algebra;

From the reading on Babylonian 'algebra' from the Crest of the Peacock, I think that one could state a general mathematical principle in a time before algebraic notation using word problems to explain variables in the form of known objects. Mathematics is about generalization and abstractions because the mathematical equations/formulas can be used to solve real world problems. Other than the use of word problems portraying a situation giving real life examples to variables, I can't imagine a good way to show general or abstract relationships without algebra. 

Class Activity: Table 45

 

Blog Post #3: Response to reading of Crest of the Peacock introduction

Something that I really enjoy about math history is how it could provide some pretty useful context on why a certain theorem was looked into and then discovered. In my post, I will highlight three takeaways that I found super interesting from “The Crest of the Peacock”. The first interesting fact that I found was that the “Babylonians had developed a place-value number system, knew different methods of solving quadratic equations and understood the relationship between the sides of the right-angled triangle”. The reason why I find this interesting is because these two inventions are vital to the foundations of mathematics and geometry. In my own learning, I have used both of these concepts from grade school all the way up to my fourth year of university! The second interesting takeaway that I got from this text is that the “Arabic text... (which maybe loosely translated as ‘calculation by Restoration and Reduction')... refers to the two main operations in solving equations: ‘reunion’, the transfer of negative terms from one side of the equation to the other and ‘reduction’, the merging of like terms on the same side into a single term". I find this discovery to be quite intriguing because this tells me that there was so much thought that was put into something that is a strategy used when solving basic algebraic problems; where you gather the like terms onto one side of the equal sign and try to solve for the variable. Lastly, the Mayas "had accurate estimates of the duration of solar, lunar and planetary movements". Without the invention of telescopes and transportation into space, it seems nearly impossible to accurately predict something beyond the scope of observation. 

Blog Post #2: Why base 60 in Babylonian mathematics?

Why would 60 be a convenient, significant or especially useful number to use as the base for a number notational system, in comparison to the number 10? From the top of my mind, base 60 has more factors than base 10, such as 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60; while 10 only has 1, 2, 5, and 10. With 60 being divisible by more factors than 10, it may be easier to split it up into smaller subsections. Smaller subsections could account for more accuracy in measurement of time, etc.

How do we use 60s in our own daily lives in Canada and other cultures? All over the world, we use 60's in time by measuring seconds per minute and minutes per hour. And as discussed in class, the year consists of nearly 360 days which is almost divisible by 60. In our education system, when we learn about geography in school, where the geological coordinates that we use are also in base 60. Another example of base 60 being used in other cultures, would be the Chinese. 

According to Wikipedia, "in the Chinese calendar, a sexagenary cycle in commonly used, in which days or years are named by positions in a sequence of 10 stems and in another sequence of ten stems and in another sequence of 12 branches. The same stem and branch repeat every 60 steps through this cycle." It seems that the Sexagesimal is used frequently to measure time, angles and geographic coordinates because base 60 is divisible by 3, where as 10 is not. So when trying to express 1/3 in decimal form in base 10, you would end up having a repeating decimal. However, in a Sexagesimal system, 1/3 would be 20/60 which is basically 0.2 in the Sexagesimal system (https://blogs.scientificamerican.com/roots-of-unity/the-joy-of-sexagesimal-floating-point-arithmetic/). 

Blog Post #1: Response to Arcavi et al: Why teach math history?

In my opinion, math history should be incorporated into my own math teaching because it can give more context to why a proof or theorem exists by exploring what problem the mathematician was trying to solve given their background knowledge. It could be incorporated by using the mathematician’s background history as an anecdotal introduction to the lesson or main concept. It can also be incorporated in textbooks as a side blurb or fun fact to add more context to the information presented to better the understanding of the reader.

The first thing that I question is how under the affective predisposition towards mathematics section it states that “mathematics is an evolving and human subject rather than a system of rigid truths” because I believe that historical context of mathematicians could provide a better understanding as to why concepts were developed but, mathematics is primarily based on rigid truths to ensure that a proof or theorem is true in order for it to be useful in solving problems. The second thing that I agree with is that under the same section, “Not getting discouraged by failure, mistakes, uncertainties or misunderstandings…. Have been the building blocks of the work” because I can resonate with that in my own attempts to solve math problems; I am constantly faced with making errors and learning from them in order to deepen my understanding of the subject. The last thing that I disagree with is that “History may be tortuous and confusing rather than enlightening” because if the historical context provided is confusing and tortuous, then I believe that the information provided is ineffective or out of scope of the subject being taught.

My ideas have been further developed on the many ways that the history of mathematics can be taught in a classroom from when it is appropriate to assign a presentation or individual worksheet work. I have also developed a more humanized perspective on mathematics after reading this article, as it emphasizes that mathematics is developed through learning from errors of oneself and others.