I learned a lot in this course. I especially liked the classes when we learned about the ancient ways of doing math that are so alien to us. With writing systems so different, I would think that their thinking process would have had to of been different as well. I remember learning that the Mesopotamian's did not have the concept of angle, and had to think of different ways to solve problems. Who is to know if we are missing some glaringly obvious construct and are therefore limited in our understanding of our world?
I liked when Susan brought up the point that no one knows what do do when dividing by 0. It broke my brain, and now I wonder if this is the symptom of an imperfect math system. I also think of Goidel's incompleteness theorem, and about how there is a hole in maths. I wonder if this hole's existence is also a symptom of the same thing? Perhaps the students in 442 two hundred years from now will learn about us, and how we were so glaringly missing something that it seems incomprehensible to not understand.
Wednesday, December 13, 2023
Course Overview
Math Art Presentation Reflection
This project was really cool in that it made me recognize a skill I have that should help me in my career as a math teacher. I was a self taught computer coder, and have decent knowledge to make computer programs. It was difficult for me to think of ways that I can incorporate coding into my teaching career, and I expected to not code much in my future. But now I realize how useful this medium could be for me to convey ideas to my students via art. I never though of computer coding as an art, but I realize that I can use my computer coding to create art!
My favourite piece to come out of my project, as I mentioned in my presentation, was Jupiter's perspective of the solar system, and this is the art piece that I've chosen to officially submit.
Wednesday, December 6, 2023
Orbit Executable
Please download this program and play around with it (doesn't work on Mac though)
https://drive.google.com/file/d/1aeCC08e7us883Vd6z_PnMjnvSWpa9k3M/view?usp=drive_link
Thursday, November 30, 2023
Art History of Math - Orbital Mechanics, Heliocentrism, and Vector Addition
Our art project reflects the history of astronomy and actualizes historic beliefs using the mathematics of vector addition. There was a time in human history that people believed that all the heavenly bodies orbited around the Earth, an idea called geocentricism.
We have been taught that the earth and the heavenly bodies
orbit the sun, but why do we think this? Occam's razor states that the simpler
answer is the solution. If we were to monitor the paths heavenly bodies take as
they pass through the sky, we would expect these paths to be simple.
Tracking the orbits of heavenly bodies from Earth, their
paths are very wonky. It is only when we assume that the sun is the center of
the orbit do the paths of the planets become nice ellipses.
By using the math of vector addition, we were able to
simulate the paths heavenly bodies take while moving through the sky from the
perspective of different planets. The assumption of Geocentricism creates
beautifully complex patterns in the paths of the planets. We extended this to
allow for the perspective of each planet being at the center of the solar
system.
The paths of the planets are beautifully complex, and it is
only when the Sun is the center of the solar system that the complexity
dissolves to simplicity.
References:
https://journals.sagepub.com/doi/epdf/10.1177/002182860203300301
https://link.springer.com/referenceworkentry/10.1007/978-3-319-02848-4_926-1
https://link.springer.com/referenceworkentry/10.1007/978-3-319-02848-4_67-1
https://link.springer.com/chapter/10.1007/978-94-011-4179-6_10
Wednesday, November 22, 2023
Medival Islam Mathematics
This article was really interesting! the first thing that stood out to me was the abundance of visual proofs. The fact that these proofs were originally done in a different language, yet I could easily understand them just from the visuals shows how math is this beautiful art that transcends language and cultural differences.
I particularly enjoyed looking at the mathematical 2D art. Seeing how such beautiful, intricate designs could be created from the simplest of rules blows my mind. Just by inscribing a square and rotating it over and over again to infinity, a beautiful pattern emerges that resembles traversing down an inter-dimensional cooridor.
Finally, I enjoyed learning about the Bagdad House of Wisdom both in class and in this article. Personally, I prioritize the synthesis of knowledge over the discovery of facts. I enjoy combining ideas into bigger and better things. The Greek, Hindu, Syriac-Persian, and Hebrew texts were the building blocks upon which synthesis was performed.
Friday, November 17, 2023
Numbers with Personalities
When I think of the Ramanujan story, I think that when he says each of the integers were his personal friends he meant that he knew an intimate facts about each number. He knew that 1729 was interesting because it was the smallest number that could be written as 2 different sums of cubes. He knows interesting facts about the numbers as a person in a small town knows interesting facts about all the people they walk past. I understand that Major's paper implies that he saw personalities with each number and befriended those personalities, but I personally do not believe that to be the case.
The only aspect of this article that I would implement in my classroom is the idea that the words we use to describe the numbers are not the same thing as the numbers themselves. It is important to connect the numbers to the words at a young age while the concept of quantities develops. However, once the idea has been cemented, it is important to break down that idea and understand the quantities in their true abstract form. By seeing quantities as abstract, we have accomplished the greatest mathematical breakthroughs. 0, negative numbers, imaginary numbers, etc.
Unfortunately I do not relate to this article in any way. I understand that I must be open to these ideas for I will have students enter my class who relate strongly to these ideas. In my class, I will act similar to an atheist who is accepting of those who have faith. While I do not agree, I would never dampen their belief system.
Thursday, November 9, 2023
Trivium & Quadrivium
Three things that stood out to me in this reading were:
1) The distinguishing between liberal education and technological training. I feel like in today's world, we are much more focused on the technological training so that students can "contribute to society." I do not describe myself as anti-capitalist, but I do recognize how much the arts is pushed aside in order to make profits. At the University of Alberta, the engineering building got a brand new floor with interesting patterns engraved in stone. This flooring cost more than the yearly budget of the entire performing arts department. I feel that this idea of knowledge as a liberal education has been lost in favor of progress. When student's ask "when will this be useful?" I realize that I don't need a practical answer, the concept of being intrigued by its beauty should be enough.
2) The importance of Easter in keeping math alive through the European middle ages. The amount of times this article referenced determining Easter was intriguing to me. Members of the clergy had to have math knowledge solely to solve this singular problem. I realize how different things are important at different times. In our time, it is the math of binary operations and logic that fuel our society, it would be interesting to read a similar article in the future that looks back upon our years and recognizes the fascination with a certain aspect of math.
3) The intertwining of numerology and mathematics. It is well known how much religion influenced the European middle ages, but I did not realize how much religion influenced the math at the time. This article paints even Pythagoras in a different light, for I now see how strange math becomes when it becomes deified. To think of numbers not as interesting in themselves but the key to unlocking their god. Maybe I shouldn't be so critical, because one could argue that the current scientific method is doing a similar thing. All humans want to access the secrets of the universe, some just think they will take a different form than others.
Thursday, October 19, 2023
Dancing Euclidian Proofs
I found it interesting how it took time to choreograph this dance, and how there was a trial and error aspect to it. At first the proofs did not seem to lend themselves to dance, but after changing some base assumptions of how the dance should look things fell into place seamlessly. This embodies the process that all mathematicians go through, where roadblocks keep being hit until we have an epiphany moment that changes everything. This makes me think that not only were they able to dance the proofs, but they unlocked a new way of doing the math. Similar to the invention of symbolic algebra over classic geometric. They felt that things were "off" when dancing improperly, perhaps by dance we could solve some of the stickier problems in math.
I enjoyed the discussion of metaphors. The critique of their bodies not exactly representing the mathematical constructs was something that crossed my mind while watching them. But they make excellent points how our visual representations of math objects is imperfect as well. When one comes to term with the facts that no matter what tools they use, they will be imperfect representations, we can include so many different types of representation. And with these new modes, we may gain new insights.
And I think this idea of imperfect representation is one that is the most applicable to the classroom. Educators are trying to convey knowledge to students, and use a myriad of methods to do so. The words in a lecture, the drawings on a board, the feel of 3D printed shapes; all of these are imperfect yet useful. It is ignorant of us to decide that some modes are more imperfect than others. If I want to say that I tried my best to convey an idea to my students, then I must use all of the tools at my disposal.
Why Euclid is Beautiful
The beauty in Euclid's work does not come from the math that he discovered. In fact, he discovered very little of the math. Instead, the beauty comes from the connections that he made between all known math at the time. One could call this the "meta-math." He was able to sort through postulates that were seemingly inconsequential to one another and see the structure beneath.
Euclid has been so important for millennia because his structure showed mathematicians how to rigorously prove their ideas. The result of one's theory is only as good as the assumptions one makes at the beginning. "Garbage in, Garbage out." But with Euclid's structure, math became safeguarded against garbage entering the system. This garbage-less system is the source of beauty in his works.
Another source of beauty comes from knowing how much can be done with so little. By making only five assumptions (and a few "common knowledge postulates") all of human mathematical knowledge was laid out on the table. This beauty is like the beauty of a tree's branches spreading out from its trunk.
The final source of beauty came far after his death, when the fifth postulate was removed. For after its removal, an entire new field of knowledge sprung into existence. Or perhaps it was already there waiting to be discovered. This miraculous event is in my mind equivalent to marveling at a tree, digging through the soil underneath, and popping out into another realm with trees of its own.
Beauty in a system is simplicity. Euclid became the first person to show how complex ideas are just made from simple statements, and how those statements have the power to change entire fields of math.
Wednesday, October 18, 2023
Babylonian Algebra
The algebraic notations we use today are new inventions that have enabled us to discover many things. However, one could argue that by implementing these symbols, we have lost sight of the true mathematics. Many students leave high school math thinking that math only exists in the symbolic form, and that anything that is not explicitly stated via the symbols is not math. The invention of these symbols has helped, but they has also become a sort of false idol.
The invention of i as a solution to a quadratic that has no real solutions is a perfect example of how symbolic algebra has allowed us to leave the constraints of our reality to understand math at a deeper level. Symbols allow us to see past our plane of existence to what logically does exist. Our senses no longer dictate our understanding of math.
But we have swung too far on this pendulum. Students now do not value their senses to perform maths. Math exists in front of our eyes yet we do not look for it. Most math in history was done geometrically with physical objects. And even the modern theories we have today are often best described using visuals. Even imaginary numbers themselves are easily conveyed as 2D numbers. Just because a proof involves our physical senses, does not make it any less worthy.
It is up to us as educators to bring the perception of math back in harmony between our senses and our symbols.
Was Pythagoreans Chinese
By framing a knowledge as your own, it creates a perceived superiority over others. During the cold war, the ability for the USSR to put a satellite into orbit did not rouse cheers from the Americans, but despair. Cultures often compete with each other over which is more advanced, and this is often measured by which has the greater academic knowledge. We are now living in a global culture, which means that there should be collaboration, not competition, between nations. If we continue the false narrative that knowledge was only developed by "our" culture, then we lose the ability to work together. In competition there is no sharing of resources between sides. If our global culture is to overcome the challenges faced in the upcoming years, we must work together and recognize that all cultures have knowledge to bring to the table.
I am torn about the naming conventions of certain theorems. On one hand, I recognize that calling it "Pythagorean Theorem" emphasizes a the competitive approach to knowledge I mentioned above. However, I do believe that if it is taught with the emphasis that it was not only discovered in western society we can acknowledge that a name is only a name. Many cultures have discovered the concept of a chair, and thus many cultures have different words/names for the concept. It would be problematic if western students were taught that only their culture were smart enough to invent chairs and that is why they call it chairs. Hence, calling it Pythagorean is not a problem as long as the concept is introduced through an international lens.
Thursday, October 12, 2023
Two visual Proofs of Pythagoras
Proving the Pythagorean Theorem with a CIRCLE - YouTube
The second is a physical demonstration using water:
Pythagoras Theorem Practical Proof - YouTube
Friday, October 6, 2023
Assignment 1 Reflection:
This presentation went incredibly well. Not only was I able to understand the fundamentals of my question, but I was able to use my pre-existing knowledge to devise an activity that showed why my historical concept was so important.
What pleased me the most about this assignment was that I could play to my strengths. Most of my peers are mathematicians, while I am only an engineer. At first I thought that this was going to be a major detriment to become a successful teacher. But I learned that I could use this knowledge to my advantage and play to my strengths.
Engineering is applied math, and this assignment has raised my confidence. I will be able to bring a new perspective to my student's education. Though my engineering lens I will be able to show them relevance that traditional word problems could never attain.
Tuesday, September 26, 2023
Differences Between Egyptian and Babylonian Numerals
The two number systems are that they both use place based notation to represent larger numbers. However, place is much more important in the Babylonian system because the symbols alternate between two simple shapes. In the Egyptian system, while they did seem to order their symbols in terms of place, the did not necessarily need to. Their symbols were all different, so if one was to shift the order around it would still be the same number. For example, shifting the Egyptian number system would only rephrase the problem to 50 + 100 + 1 + 1 + 10 000. This would not work in Babylonia, the number behind the symbols would be lost.
In Roman numerals, place only kind of matters. They overcomplicated their symbols by having the adjacent symbols impact one another. These additional rules caused basic addition and subtraction to be much more difficult than a place based system.
Surveying in Ancient Egypt
In my engineering degree, I have learnt much about surveying. I have had jobs surveying, and have completed many courses on the subject. What surprised me was how little about this history I was taught. As I am seeing more often through the lens of this course, the majority of my STEM knowledge has been warped by western-centrism. I was only taught about how the Europeans used surveying, and the history of it was only taught back to the middle ages of British Kings. This article is amazing, and I feel robbed of this knowledge. I will ensure that my students will not suffer the same fate.
We have so much evidence of these different measurements, and the article mentions what they Egyptians knew about angles. I wonder how little they actually knew. Not every brilliant idea is recorded, so I can't help but think that there is something that the evidence is not telling us. We see that they have seqets for vertical angles, but that does not explain how they determined lateral angles. Perhaps there was a rule of thumb that surveyors of the time used, which was only passed on by word of mouth between generations.
The text mentions surveying techniques being used by priests and cultists. I wonder how much spiritual emphasis was placed on these mathematical concepts. We have heard the story of Pythagoras' math religion, but I know nothing about how math impacted Egyptian religion. Again, this is due to the western lens I have been taught through, but I wish to delve deeper to uncover the answer.
The Amazing Solution Behind the Russian Peasant Method
The Russian Peasant Method and the Egyptian Method are identical, except that the Russian Method uses a math trick to determine which factors of 2 must be used.
As a recap, to solve 9x13 using the Egyptian Method we find 1x9=9, 2x9=18, 4x9=36, 8x9=72. We then determine which combination of these equations gives us 13 9's. Finally, we sum up the right hand side of all of these equations. The end result is 13x9 = 1x9 + 4x9 + 8x9 = 9+36+72 = 117.
The issue in the method is that we must determine which combinations of these equations gives us 13 9's.
When we first look at the Russian Method's algorithm, there are two steps that seem chaotic: Only paying attention to the odds on the left, and ignoring all the decimal remainders. BUT, these two steps actually compensate each other perfectly. The remainders are actually accumulated in the odd rows while we simplify the expression.
Let's take advantage of the fact that x*y = 1*x*y = 2/2 * x*y=x/2 * 2*y.
In equation 1, we have expanded 13 x 9 = 6 x 18 + 1 x 9. Note that we have converted this multiplication problem to a multiplication and addition problem, where we added a remainder. The most important note is that the remainder is EQUAL TO THE RIGHT NUMBER IN THE ROW ABOVE.
Equation 2 does not introduce a remainder, only halves and doubles the two numbers in the multiplication.
Equation 3 converts the multiplication aspect into two additions, one being a remainder and the final being the largest doubling we reach.
In the Russian Method, we add the Right numbers in the rows where the Left numbers are odd. But what we are actually doing is adding half of the Right numbers in rows that are not integers. It just happens that these two statements are equivalent.
Even rows will not create a remainder. Even rows with a remainder are just represented using the row above, and this row above must be odd because it created a remainder.
Therefore, the Russian Method is the same as the Egyptian Method but uses this neat remainder trick to determine which combinations of equations are needed!
Solving Division Using Egyptian Methods
I decided to solve y=27/6 using the Egyptian method. Starting with doubling:
1x6=6
2x6=12
4x6=24
8x6=48
We stop here because 48 is larger than 27. The only numbers that we can combine and not exceed 27 is 24 alone. Even 24+6=30. Therefor, the whole number part of y is 4.
27-24=3, so we are currently missing 3. Now onto the halving:
(1/2)x6=3
After only one iteration, we have found the missing 3, and we get it by adding (1/2) to y. Therefore,
y=4 +1/2 = 4.5
Tuesday, September 19, 2023
Babylonian Word Problems
Word problems have the potential to deepen understanding in students, but it also has the ability to further the divide between a student and the subject matter. By needlessly adding a layer of complexity to a problem, students may disengage. These problems will only maximize their potential for learning when the problems reflect the real world authentically, not just needlessly shoehorning a random scenario that is not likely in the real world.
Done properly, world
problems can show a student the practicality of the knowledge they are
learning. A student can see that this knowledge has a use in the real world. By
implementing multiple word problems into a single unit, it can also show
students just how versatile and general math concepts can be through a process
of un-abstraction. Again, if this is done un-authentically, a student could see
this as “See! This topic is so useless that my teacher could only come up with
this random scenario.”
Educators have been making
up inauthentic word problems that only test learner’s knowledge in a practicality
vacuum for thousands of years. Even in Babylonian times, these problems focused
more on exemplifying a pure math concept instead of focusing on how one would
actually apply math in a real world problem. But just because this problem has
existed for so long, doesn’t mean that it is unsolvable. I believe that if the
proper time and energy is put into formulating these problems authentically,
students will truly benefit from them.
Wednesday, September 13, 2023
Base 60 Examples
These are the five examples I found of 2 base 60 numbers that multiply to equal 45:
45 = 6 * 7,30
45 = 30 * 1,30
45 = 2,24 * 18,45
45 = 8 * 5,37,30
45 = 40 * 1,7,30
Tuesday, September 12, 2023
Crest of the Peacock: Reflection
All my experience with historical bias is about devaluing cultures. I have not spent much time evaluating the bias in devaluing the impact of other cultures on an abstract institution such as math.
The most impactful part of this reading was seeing the traditional Eurocentric view of math’s trajectory vs a more global perspective on the subject. It was only after seeing the two diagrams that I realized how non-sensical the idea of a “dark age” was. It was almost like reading propaganda, “European maths are so fundamental that the entire global production of knowledge shut down.”
I am excited that this course takes down my pre-conceived notions that were instilled during my formative years. I am also excited to stop these biases from being transmitted to the next generation of learners during my career.
Monday, September 11, 2023
Why Base 60: Response
At first glance 60 is superior to 10 because of the large number of factors it has. The factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60; that is 12 factors! The number 10 only has 1, 2, 5, 10. Also notable is that the factors of 60 contain the low more commonly used numbers. Very rarely do I have more than 6 of a thing on my person. Just right now I have 4 pens, 1 helmet, 5 notebooks, and 2 waterproof clothes in my bag.
Having many factors could be important because it allows for simpler fractions. 60 divided by all its factors will produce nice non-decimal numbers. I could imagine an ancient tribe seeing a decimal number as their math “breaking.” So, through a trial-and-error method, a base 60 system caused their math to break the least.
We still use base 60s for time keeping. After some research, we also use it for measuring angles and China uses base 60 for many of their astronomical systems. The closest I can think of is that there are 12 inches in a foot, but that is not exactly base 60. There was a push for a metric time-keeping system, but it seemed that humans were much less willing to accept it. It seems like time is so intrinsic to human perception of our world that we are unwilling to change it. This reminds me of how there was a pushback to the ideas that time was a dimension of spacetime.
My research has shown that this idea was more or less the acceptable idea of the justification for the base 60 system. I have enjoyed learning about these Babylonian systems via my own private research in the past, and parts of this knowledge has luckily stuck in my brain. I am very happy that I am in this class, officially learning things that I have only dabbled in.
Saturday, September 9, 2023
Response to Constantino's Integrating history of mathematics in the classroom: an analytic survey
As I’ve been teaching math to students, I have always tried to sprinkle the history of math into my lessons. Yet, my current stance on the matter does not line up perfectly with Constantinos. Constantinos gives the impression that history of math should be its own class. That it should take up as much class time as the traditional math itself. This lends itself to criticism 07, that there is not enough time to teach both sides of math. History of math is a spice that enhances the traditional curriculum; a little goes a long way. When introducing the topic’s problem, framing it in a historical lens will foster further by-in from the students. After the introduction of a topic, I do not normally continue pushing the historical lens.
The number of examples
and resources provided by Constantinos is amazing. As I was reading, I created
a file with these alternative subjects. I am unsure if I will be able to fit
them into the pre-existing BC curriculum. If I find myself with pedagogical
freedom in my future, I will use these examples to further foster my student’s
growth.