Course and Blog Reflection

 Since reading through all of my previous blog posts it is clear that my idea of math history has changed over the past few months. At the beginning of the term, I thought of the history of math very linearly, but now it is clear that is not true. The history of math is very intertwined, with similar discoveries being made by different civilizations at different times. The true origins of these discoveries also heavily depends on who you ask, which I find very interesting. The one thing that has stayed constant, however, is my stance on the role of math history in the math classroom. I still believe that there is already way too much content in highschool math and that learning math history would over-complicate already complicated concepts. That doesn't mean that I don't think math history is important, because I do, I simply do not think that the high school math class is the proper arena for the subject. 

I thoroughly enjoyed this class, it was a very pleasant change of pace from my typical Sauder courses, but all classes can be improved. My biggest critique would be the blog-format, I enjoyed creating blogs but it was hard to keep up-to-date on assignments when all of my class due dates and reminders are on canvas and one class had it on blogger. I think it would be a lot easier for non-education students to navigate the course if assignments and due dates were posted on canvas. 

I would like to thank Susan and all the students in the class for making this class so much fun and so informative! 

Assignment 3 Reflection

 Assignment Three was extremely fun for me. It gave me the opportunity to learn about the history of Coding together with two of my best friends. Coding and computer science have become increasingly important in recent years, even for myself, in finance, computer science will play a large role in my career. Not only will I need to understand computer science to effectively evaluate tech companies, but a lot of my day-to-day tasks will become automated with the help of computer science as well. My main takeaways from this assignment were:

1) The history of coding is incredibly intertwined

    i) The main languages of coding today have built upon the previous coding languages

2) There are only a few main players in the history of coding

    i) You see a lot of familiar faces in the history of coding, Bill Gates, Oracle, Nokia, IBM, etc.

3) The history of coding is incredibly young

    i) Unlike most other mathematics subjects, coding only really started to gain traction in the 1950s.

4) The history of coding is a lot more privatized than other math histories

    i) The history of most math subjects is largely dominated by scholars and academics, not coding though.         The history of coding is dominated by private companies that are worth billions of dollars.  

 

The History of Coding

The topic that our group decided to research is the history of coding and computer algorithms. What we noticed right away is that the history of programming languages has a couple key trends that we wanted to highlight through our format. Those trends are that, as with every other topic we have learned about throughout the semester, each succeeding programming language is influenced by or improves on a previous language in some way and that the core concept of coding is to make tasks increasingly simple and efficient. Furthermore, unlike many of the typical mathematics concepts, the development of coding has been rapid and is continuing to accelerate.  We also realize that computer programming is a quite recent development with the first programming language being in 1854, but the more significant advancements beginning in 1956 – less than 100 years ago. Given this, in addition to informing through the introduction of how and when each programming language was developed, we also wanted to highlight how quick the evolution has been and show how each language has been a building-block to its successors. Examples of this would be the development of Python, which is a successor to the ABC programming language or C++, which is an extension of the previously introduced C.

We found that these trends become obvious as we explore the progression of programming languages through time and that each advancement is quickly overshadowed or replaced by another. For this reason, we decided to keep the scope of our project very broad and display our findings through a roadmap that walks us through the history of coding. This way, we get to learn how coding has developed and the aforementioned trends will come to light as we move through time - the progression of coding has been swift, languages take ideas from previous ones and each succeeding language strives to become more simple.  We hope that this way, we will be able to cover the high-level important events in the history of coding while also highlighting how each event ties into each other. 

https://prezi.com/elxw8xbxtwxd/?utm_campaign=share&utm_medium=copy

Assignment #3 Outline

Topic: History of Coding and Computer Algorithms

Format: Visual Road Map

Bibliography: 

Bureau, T. G. (2020, November 5). Ada Lovelace Day: Commemorating the world's first computer programmer. TechGig. https://content.techgig.com/ada-lovelace-day-commemorating-the-worlds-first-computer-programmer/articleshow/78642521.cms

Encyclopædia Britannica, inc. IBM develops FORTRAN. Encyclopædia Britannica. https://www.britannica.com/technology/computer/IBM-develops-FORTRAN

Encyclopædia Britannica, inc. LISP. Encyclopædia Britannica. https://www.britannica.com/technology/LISP-computer-language

Hayward, D. (2020, October 7). A Brief History of Coding - BDM Tech Guides. BDM Publications. https://bdmpublications.com/brief-history-coding/

The History of Visual Basic. http://www.johnsmiley.com/visualbasic/vbhistory.htm

History - Open Dylan. https://opendylan.org/history/index.html

How Alan Turing Cracked The Enigma Code. Imperial War Museums. https://www.iwm.org.uk/history/how-alan-turing-cracked-the-enigma-code

McCracken, H. (2014, April 29). Fifty Years of BASIC, the Language That Made Computers Personal. Time. https://time.com/69316/basic/

McFadden, C. (2020, September 5). The Origin of Algorithms We Use Every Single Day. Interesting Engineering. https://interestingengineering.com/origin-algorithms-use-every-day

Mkhitaryan, A. (2017, October 13). Why Is C# Among The Most Popular Programming Languages in The World? Medium. https://medium.com/sololearn/why-is-c-among-the-most-popular-programming-languages-in-the-world-ccf26824ffcb

Python History - javatpoint. www.javatpoint.com. https://www.javatpoint.com/python-history.

The rise of C++. Nokia Bell Labs. http://www.bell-labs.com/about/history/innovation-stories/rise-c-plus-plus/

Nasar, Audrey A. (2016) "The history of Algorithmic complexity," The Mathematics Enthusiast: Vol. 13: No. 3 , Article 4. https://scholarworks.umt.edu/tme/vol13/iss3/4

The Golden Age of Medieval Islam

 One thing that I found very interesting from this reading was the inception of the word "algorithm". I had never put much thought into where the word came from, so it was interesting to find out. Apparently al-Khwarizmi's book "The Book of Addition and Subtraction According to the Hindu Calculation" was so important that it was the first Arabic math book to be translated into Latin. Today the word "algorithm" is used to describe the act of calculating something, but it is derived from the Latin interpretation of al-Khwarizmi to "algorismi". 

It was also very interesting to learn about al-Khayyami's achievements with ratios. Specifically, I had no idea that his view of the ratio of the diagonal of a square to the side (sqrt 2) or the ratio of the circumference of a circle to its diameter (pi) eventually lead to the introduction of positive real numbers. This is an achievement that has surely stood the test of time because we are still using positive real numbers today. 

Finally, it would be impossible not to mention al-Kashi's correct calculation of 2pi to 16 decimal places. Calculating the perimeters of inscribed and circumscribed polygons in a given circle having 805,306,368 sides. He even went as far as to describe how close he wanted his approximation to be and was correct. The time and effort that he put into this calculation show the true dedication that he had to his craft. 

Trivium & Quadtrivium

"Numbers were identified with the various gods. He considered odd numbers to be male and the even ones to be female."

I found this to be very interesting and it made me wonder what the thought process was here. This is obviously me stereotyping, but generally, speaking girls are better behaved in grade school. Once both genders are grown up I think it evens out a bit but during school, it is almost always the boys that give teachers a hard time. This is why this analogy makes sense to me, with boys being slightly more "odd" and less put together and females being more "even" and polished. 

"To qualify for a degree, he was required to participate in public disputations, either defending a proposition or opposing one defended by another student"

There was a couple of things that I found interesting. Firstly the use of the word "he" because back then university was only provided to men and not women, that is certainly one aspect that the modern university has changed for the better. Secondly, I find it interesting that the university used to require students to argue their opinions. I feel that this is something lacking from the modern university, where at times it feels like students are just being graded on their ability to memorize facts and figures. 

"education as the sole occupation of the first thirty-five years of a man's life. He would have the first twenty years spent on gymnastics, music, and grammar, the next ten on arithmetic, geometry, astronomy, and harmony, and the next five on philosophy. "

This was an opinion of Plato's. The first thing that sticks out to me is the subjects that he chose, particularly gymnastics and music. It is clear that Plato believed that a man should focus the majority of his education on the arts, which I found interesting. Additionally, this is a very long time! I have been in school for sixteen years and that is already a long time. On paper, it would appear that Plato's theory would result in a well-educated, more successful society, but I am not sure that is the case. My opinion is that the economy would suffer because people aren't entering the workforce until a later age and because there is no diversity of knowledge. Yes, a university education is important, but we also need trades workers and I don't think philosophy and astronomy are very important for those roles. 

False Position

My word problem: Two of a number and its square root are equaled to 36.




2x + √x = 36

Try 4: 2(4) + √4 =

8 + 1 = 9

36/9 = 4, therefore multiple 4 by 4

2(16) + √16 =

32 + 4 = 36




It worked!

Was Pythagoras Chinese?

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

I think that it is important to study the history of math in a fully holistic manner. It doesn't make any sense to me why someone would study Greek math history but not Chinese? Are their discoveries any less impressive or important? I would think not. The arguments behind teaching math history as part of the curriculum; understanding the trial-and-error aspect of math, understanding that math is always evolving, would surely also apply to Chinese math history. I guess it is true what they say, history is written by those that create it and in our western world we are more attuned to the western history of math, I think that should change though. 

What are your thoughts on the naming of the Pythagorean Theorem, and other named mathematical theorems and concepts (for example, Pascal's triangle)

The naming of proofs and math concepts is a complicated subject. On one hand, the Babylonians and Egyptians understood the relationship between sides of a triangle prior to Pythagoras, so why should he go down in history as the master of the triangle? Is it because his way of thinking was very western? Prior generations had possibly placed less importance on individual fame and recognition, placed less importance on a formal proof process and Pythagoras has reaped the rewards of such. On the other hand, where do you draw the line? If not the Pythagoras Triangle then what? The Egyptian triangle? But then the Babylonians would be up-in-arms. I think that there is no solution to this problem and if Pythagoras was the one that made the theory famous, I'm fine with that.       

Why Teach Math History?

Pre-reading ideas about whether, why, and how math history should or could be incorporated into math teaching. 

As one of the few non-education majors, I am not sure I can say how I would implement it into "my own" math teaching. I can, however, put myself into the shoes of a high school student and think about what I would want to learn about. That being said, I see math history playing a minimal role in high school math education. I think that the history of math can be helpful if it provides context to complicated techniques, showing how a certain type of math was discovered could create real-world applications for math in the minds of students. Other than that though, I think it would just confuse kids and overload them with information that they don't necessarily need. 

Things that made me "stop and wonder"

"History can be torturous and confusing, rather than enlightening", this is exactly what I was trying to say, this has been phrased a lot more eloquently than I could ever say it, but I 100% agree. "Lack of time", I also agree with this, highschool math already has a ton of concepts packed into it, I feel that adding the history of math would increase that "knowledge overload" that I discussed above. "The active predisposition towards mathematics", I agree that this point is interesting. Math should be seen as ever-evolving and should inspire students to question everything and maybe even discover new are better ways to approach math. A few hundred years ago math looked a lot different than it does now and in a few hundred years with will look different again.  

Conclusion

I think that some very valid points were made for both sides and I now definitely now see the benefits of teaching the history of math to students more than I did prior to reading this. However, my main doubts remain and I think some of them were even strengthened and confirmed by the article. 

Alice Major on Mayan and Other Numbers

Is this something that you might want to introduce to your secondary math students? Why or why not?

I likely would not teach this to secondary math students. My reasoning is, although this is interesting, I'm not sure it would help the students understand math better. In high school, when students are learning concepts like pre-calculus and calculus, math can be very hard to grasp. I wouldn't want to complicate things any further

If you would use these ideas in your math class, how might you do so?

I think that these concepts can be applied and useful to a different age group. For example, in primary school when students are first learning to use addition and subtraction I could see this playing a role. The concepts and personalities of the numbers would obviously be simplified from what the Mayans used, to apply to primary students. I think that a video, showing numbers as personalities and how they interact could be extremely fun and useful to these primary school students. For example, showing a pair of fives as young and energetic twins, then showing a ten as a more mature and larger version of those fives, showing that two of those fives equal to that ten. This is a very rough example, but hopefully, you get the point. 

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?

I have never thought about it like this, but now that I am thinking about it, definitely. I saw a meme a while back that said "I can't explain why, but the number eight, the day Thursday, and the month October all are the same", and I couldn't agree more. I tried dissecting the reason for it and I think it's all about being an even number and being close to the end of their respective spectrums. Eight is an even number and is very close to ten, October is an even-numbered month (tenth) and is close to the end of the year. I know technically Thursday is the fifth day of the week, with Sunday being the first but I think most people think of Thursday as the fourth day of the week and Monday as the first, which would make it an even-numbered day and close to the end of the week. Additionally, I see five and ten as being preppy and perfect, likely because of being seen as "round numbers". I see numbers 2,4,6 and 8 as being related because they are even. Three and nine are seen as the "rebels" to me, somewhat related to six, but a little rough around the edges. Then that leaves one and seven, I see one as being the baby of the group (self-explanatory) and I see seven as being the hipster. I think this personality comes from people choosing a number between one and ten, somehow seven is always the most common choice. This is likely because it doesn't fit into any of the usual categories (even numbers, multiples of five, multiple of three, close to zero, or close to ten), it is just kind of its own thing and I respect seven for that. After reading this story, I will definitely look at numbers differently. 

Assignment 1 Reflection

 Pythagorean Triples have been a favourite of mine for many years, since we learned about them in high school. It just seemed to be an incredibly simplistic solution. In high school, we didn't learn anything about the history of Pythagorean Triples though, so this was a cool opportunity to dive deeper. For example, I didn't know that Pythagorean Triples went as far back as Babylonia, this shows the depth and complexity that the Babylonians were able to accomplish, very impressive. It was discovered that the Babylonians knew a form of this theorem because it was found on a tablet and the numbers were so large that it stands to reason that trial and error could not have been used. It is a shame that one side of the tablet was cut off however, it would provide great context on how and why this tablet was used. Aside from the history of Pythagorean triples, this assignment was also a fantastic opportunity to put myself in the shoes of a teacher. I am one of the few non-teaching majors in the class so this experience likely taught me more than it did most of the other students. I was able to learn about a subject, then think about how it could be most effectively taught, which was a lot of fun for me. Being a finance major, excel is always my go-to-method, but this also made me think about how other students would want to solve the equations. Overall, working with my teammates on this assignment was an extremely fulfilling experience, and one I hope to have again (excited for assignment 2!). 

Dancing Euclid Proofs

Euclid is iconic not only for the collection of proofs that he has combined into a series of textbooks, but also for the beauty of his work. The first thing that jumped out to me was just how easy it was to communicate Euclid’s proofs into dance form. The simplicity of Euclid’s proofs makes it so easy to understand and therefore easy to communicate. There are not many other mathematicians that come to mind when thinking about interpretive dance, which I think really speaks to the artistry behind his work. Secondly, I was really taken by the way these shapes symbolized connections between two people. Despite the size of people’s arms not being the same, the dancers didn’t let that stop them from using their limbs to create connections in this dance. When I see lines of a triangle, I have always thought of them as just that, lines, now I will be able to see the symbolic connection of two people. Lastly, this article and video opened my eyes to the math and patterns that surround us in nature. This can be in the form of trees, shells and waves, all things that I have never thought of as being math related but truly, they are. Altogether, this article and video has made me think of math more as a living, breathing thing and less like numbers on a board. 

Euclid Alone Has Looked on Beauty Bare

 Admittedly, I have never been very good at understanding or writing poems. As far back as middle school, poems have never made a ton of sense to me. This poem was no different, it wasn't until I did further research that I began to understand this poem. Euclid was a Greek mathematician, often referred to as the "founder of geometry". His elements are often thought of as one of the most influential works in the history of mathematics, often being used as the main textbook for teaching math, especially geometry. 

My interpretation of these poems is that Euclid has seen beauty in shapes that no one else at that point was able to see. His geometric proofs have given beauty and grace to shapes that were once thought of as just "shapes". In "The Euclidean Domain", David Kramer questions this notion, asking "Has no one else seen hide or hair?". This is basically Kramer saying "we have all seen shapes, how is Euclid the only one that sees this beauty?". Kramer may have a point, during the current day and age I'm sure a lot of people see the beauty and complexities of geometry. Is this only because of the discoveries and proofs of Euclid though? I guess we will never know, but I think even Kramer would acknowledge how important Euclid's proof are still to this day.   

The Eye of Horus and Its Significance in Ancient Egypt

I felt that the above diagram was very helpful to explain the Eye of Horus and its significance with unitary fractions.  Each of the "sacred unit fractions" were attributed to six parts of the eye of the god Horus. These fractions, all with powers of two in their denominators, were used to represent the fractions of hekat, which is the unit measure of capacity for grains. The most interesting aspect of the story for me was that according to legend, the pieces were lost in a battle and were restored by the god Thoth. 

For special numbers in my life the closest I can relate to is my hockey number. I wear the number 8 because my birthday is February 8, 1998 and my favourite player growing up was Alexander Ovechkin who also wears the number 8. 

The Magic Square

As a finance major, I am most comfortable doing math and solving puzzles on excel. That is why it was an easy decision to tackle this problem with my trusty friend. With no rhyme or reason, I decided to start with trial and error. For optimal problem solving, I made sum formulas at the end of each row to see if I was on the right track and a formula at the bottom right that would tell me if I have solved the puzzle or if I should keep trying. For attempt one I tried to start with the largest number (9) then fill in the adjacent squares with the smallest numbers, then go from there. This attempt eventually proved to not work. On my second attempt, I chose a diagonal approach with the three largest numbers, knowing that no two of them could be on the same row (horizontally or diagonally). I then chose to fill in the empty corners with the numbers that made the most sense to me. After that step, I filled in the boxes accordingly and think I got extremely lucky. I enjoy puzzles and because of that I enjoyed this blog post the most out of all the homework assignments thus far. 

The History of Babylonian Word Problems

 I used to think that word problems were a new concept that had been developed to assist children in learning math. Little did I know that these word problems have been used for hundreds of years. The word problems look pretty similar to current day word problems, with the exception of some obvious differences, like the use of drachmas and the purchasing of livestock as shown in the problem below. 

That is where practicality comes in, back then in 900 AD that was a practical way of learning math, these are actual situations that someone could be caught in. These days word problems look somewhat different and are possibly less practical. Now this question would likely involve dollars and common grocery store items. The difference is that in current-day calculators are incredibly accessible, being on every smartphone. This means for the most part that the majority of calculations do not need to be done by hand. This also means that word problems are less practical and necessary in my opinion. 

Then there is the issue of "pure" vs. "applied" mathematics. Pure mathematics is in its own world, a world that has its own language and can be extremely daunting at times. Applied mathematics takes these over-complicated concepts and applies them to real-world problems that need solving. As a child or a student of math, I also believe it is easier to understand when it is placed in real-world situations. For example, as a child, it is easier to divide twelve apples between your three friends than just dividing twelve by three. As a person that is not overly fond of math, I appreciate it when applied mathematics simplifies a concept for me. 

Assignment 1

1) When x=100, there are 30 solutions

2) When x=210, there are 54 solutions

3) When x=420, there are 162 solutions

4) When x=35, there are 18 solutions

Extension: See if you can find out the number of solutions for x=5 within the next two minutes

Answer: When x=5 there are 2 solutions; y=12, z=13 & y=0, z=5

Babylonian Algebra

After seeing how the Babylonians solved algebra I have become very thankful for modern methods. The Babylonian method was very convoluted and quite frankly hard to follow. When thinking about the question that was posed to us on the class blog "How could one state a general mathematical principle in a time before the development of algebra and algebraic notation?" and I was quite perplexed. I had a very hard time coming up with a solution. The Babylonian method reminds me a lot of the way I used to do math when I was younger, I was never a very attentive student but I somehow got by in math. The teacher would teach the class the "correct" way of doing things, but since I never paid attention I would just come up with my own method. Depending on the year and teacher, they would either care or not care about me using my own method. For example, in grade three we learned long division, but I just decided on my own method. The teacher was fine with it because I got the correct answer, but as time has gone on division questions I was given got harder and harder until my method didn't work anymore. I never learned the proper way to do these questions. I see the Babylonian methods like this, there was no way to answer these questions, so they made one up. Their method does work, but the modern method is the easier option and I view it as the correct one.

"Is mathematics all about generalization and abstraction?". Yes, I believe that it is. Generalization is important for a ground-level understanding of math, knowing these general rules is important. Examples of which are "BEDMAS", which stands for Brackets, Exponents, Division, Multiplication, Addition, Subtraction. This is the rule for the order at which to follow for algebra. In real life however, not every situation is cookie cutter. The abstractions and intricacies of math are important for this. 

While thinking about different areas of mathematics it is difficult to think of ones that can be explained without algebra  (especially as a finance major, we use a lot of algebra!). Even geometry, at its most basic core of calculating the area and edges of a rectangle uses algebra to fill in the missing gaps. 

My conclusion is that the Babylonians did an extremely impressive job explaining mathematical concepts, given their limited resources. Also, I never thought I would be so thankful for algebra!

Babylon-Style Table

 

Column 1	Column 2
2.5	18
3	15
3.36	12.60
4	11.15
5	9

Base 60

 I must say, it has never occurred to me to use base 60 as opposed to base 10 or 100. When giving it some thought however, it does make a lot of sense. The most obvious use of base 60 is time, 60 seconds in a minute, 60 minutes in an hour, but then 24 hours in a day? Seems interesting to me. It appears that this time system was invented by the Babylons, who derived this system from the Sumerians, who used it as early as 3500 BC (according to the Guardian). When reading about this, I wondered to myself, why not 100 seconds in a minute or ten hours in the AM and PM? It turns out that using twelve hours for morning and twelve for the afternoon or night is much more useful. Twelve is divisible by two, three, four, (not five), six, and itself. Ten on the other hand only has three divisors. Sixty also has twelve divisors. Sixty and twelve both have more divisors than any number below them (many of these facts are also courtesy of the guardian). It turns out my first thought that base 10/100 was superior to base 12/60 isn't correct. In fact, I'm glad that civilizations after the Babylons adopted this system, it seems to work pretty well.