Assignment 3 and Course Reflection

The project on Tower of Hanoi was interesting since I was able to learn why the puzzle was named that way and the stories that revolved around it. I was surprised that the puzzle was only created a couple of centuries ago by Lucas in the 1800s, but nonetheless it fascinating to see what influenced the nature of the alternate names to this puzzle. I was fortunate to take a class on combinatorics and went over this puzzle in the recursion unit. It looks tricky to solve at first, but once what is behind the scenes is understood, it won't be as mindless/tricky to solve. I may use this as a fun project in the geometric series topic. First introduce it, talk about the steps behind on how to solve it and later deduce the closed form formula, which is a geometric sum. 

With respect to the course I really liked learning about the ancient Babylonian sexagesimal based 60 arithmetic. Playing around with fractions on the boards in the orchard garden were fun times. To understand different number bases is important, like binary in computing. There is also good reasons why 60 is a nice base number to work with. Getting in touch with the mathematical findings and work during the Islamic golden age was curious too. My least memorable moment is probably focusing too much on ancient Babylonian math, as most of the blog posts were on it. I think some suggestions is to do more hands on work and solving more problems for the future.

Assignment 3: Exploring a math history topic through making/the arts

Assignment 3 Tower of Hanoi

Nov 27 Medieval Islam Mathematics Reflection

The House of Wisdom is the first point that made me stop. While I was doing my presentation on historical integration, I came across Alhazen (Ibn al-Haytham) who lived over a thousand years ago. He had major contributions to optics and integration and was involved with the House of Wisdom. Initially I thought it was a group of collective individuals coming together to work on..discovering new things, though in fact it was primarily a library and a place for translation and research. The people in there would study ancient Greek mathematics/astronomy and further built on these topics. The one who established the House of Wisdom was Caliph alRashid. His son, Caliph al-Ma’mun, was the ruler who made the House of Wisdom so important. This caught my eye since the word "caliphate" means "an Islamic state, especially one ruled by a single religious and political leader". This word might've came from the two prominent aforementioned father/son. It also referred to the Islamic world during the medieval period, when Europe was in its dark ages while science and technology grew in the middle east. Furthermore it was interesting to how algebra and geometry flourished during the Islamic world which resulted in beautiful artwork captivating the eyes. A lot of the constructions with shapes inscribed within each other were mystifying too. Some circle geometry were shown as well with procedures I have never encountered in an elementary Euclidean geometry class. The good news is that many of these constructions can be replicated by hand with easy to follow instructions. This makes it friendly for school students to use so that they can create fascinating math works.

Nov 20 Maya Numerals

For the number "1729", Ramanujan noted that it was interesting because it was the smallest number expressible as the sum of two cubes in two different ways: Major's paper might discuss the role of intuition in mathematical discovery or the personal relationship that mathematicians have with their subject. Ramanujan's work often relied on deep intuition, which could serve as a prime example of this relationship. The collaboration between Hardy and Ramanujan, who came from very different educational backgrounds, could also be used to highlight how diverse perspectives can lead to significant mathematical advancements. If I would introduce this to secondary math class, mentioning these stories can be a captivating introduction to several math topics and would likely engage the students. The stories connects math to real world contexts, making it accessible and interesting. It shows that mathematics isn't just about solving abstract problems but can be playful and intriguing. For advanced students, exploring the proofs behind these math properties can deepen their understanding of mathematical rigor and methodology. In terms of numbers/letters having personalities, some might see the number 2 as a generator because it is even and small. It is also the smallest prime number. So many concepts seem to trace back to this number. The number 7 as more mysterious due to its associations in literature and folklore. The number 3 appears a lot as well and it's important since 3 is the smallest number of edges to form a closed polygon as well as having a triad stand to support a seat. In many languages the number 4 is dispised since it sounds like death. Dates such as Friday the 13th have had superstitious stories revolving around it which makes some people uneasy to write an exam at that time! 

Nov 8 Trivium and Quadrivium Article Response

The article introduced some interesting things. To begin, this article has a Latin title and it is separated into two parts, one of which has 3 topics and the other has 4. I initially saw this as a distinction between what is now known in modern day as the Humanities and STEM. This practice has been around since the medieval times. Either way, both of these parts required people to be literate and well versed cultured citizens. About 15%-30% of the population were like this back then. So there weren't that many opportunities for serfs and labor people to grow in their knowledge in these areas. However, looking at it today, most people can read and write, so the literacy rate has gone up. Many people are engaging in areas of both fields. Kids are learning a variety of everything in their young elementary years, and begin to specialize more and more as they get older. Another point that stood out was the different methods of division. There was the abacist method which was complex and cumbersome so it wasn't that favored, thus it got the name "iron division". It required a good understanding of the abacus and the ability to manipulate the beads to represent the division process physically. I personally had a hard time learning the abacus (mine was the Chinese suanpan) since there were some things to keep at the back of your head when doing the computations. On the other hand, the algorist method was simpler and more straightforward, thus it got the name "golden division". It used written calculations and was based on the Hindu-Arabic numeral system, which included the use of zero and place value, so it was more efficient to do the steps physically on paper. To get to this point wasn't that easy initially, as switching from the previously used Roman numerals to Hindu-Arabic numeral system took over two centuries. This newer method also allowed calculating the date for Easter much easier than before. I never knew why the date for Easter always changed annually, but it definitely is dependent on something, just like how the Spring Festival is dependent on the lunar cycles. Seeing now that Easter's date is determined by the Golden Number and the Dominical Letter and all the math involved behind it made me realize the complexity of the arithmetic people centuries ago had to perform. This, blended in with the fact that the clerics should know how to do the calculations emphasized the importance and significance this key date played in their faith.

Nov 6 Dancing Euclidean Proofs Reflection

This whole concept of using dancing to display Euclidean geometry proofs in multisensory ways does help us understand it in new ways. It is visual and unique. I’ve never seen math proofs done using dance before so watching it for the first time was interesting. There is harmony to it and seeing how the human body, as intricate and complex as it is, is able to do what a compass and straightedge can do - and much more! Watching the dances on screen created a zen atmosphere so it was quite relaxing. This integration could help the general audience appreciate the beauty of mathematics. I have seen folk dance being used in a computing science course when I took it at the UofA. The lessons were on sorting algorithms and there were some supplementary videos that showed us a row of dancers moving around like how objects would move around depending on which sorting algorithm was used. Here is the link to the playlist! Another thing that made me fathom was that a dot on a page is just as inaccurate of a way to physically represent a point as a dancer’s body. The dot on the page takes up some two dimensional space in reality, so it isn’t a totally accurate physical image of a point, just like how a sheet of paper isn’t really 2D, but is 3D even though the thickness of it is super thin. The dot and the body are both representations of the true abstract concept. Dancing allows one to connect with that abstract concept through embodying it with phyical movements. I think it is nice to have students up and moving around since students around high school sit for a long time in front of a computer screen to do homework, research, watch shows, or play games. Having these body motions allows students to remember certain concepts better and associate its value to oneself. Also there is no technology used during this process of dancing with Euclidean proofs, so this practice would have also been implemented historically too. However I do see that not every student is into dancing since they may not feel comfortable with it. Maybe for traditional reasons they can’t have a partner to create these proofs visually with the body (from the video there were two people “dancing out the proofs”). Other students might just not care or don’t enjoy it. I probably won’t do dancing fully, but maybe some physical actions moving arms and hands around (maybe a bit of the legs). I think unless they really don’t want to, they don’t do it, but I encourage those who can (who may just be somewhat indifferent or unamused by it) to move around and try it out.

Oct 18 Reflection on Euclid's Works

I don’t remember when was the first time I heard of Euclid’s moniker, but there was a math contest from the University of Waterloo named after him. People who were applying there [for math/CS/engineering] in grade 12 were recommended to take that exam. Younger feverish students also took it for fun (or for fame/rewards). The early bits were doable, but the difficulty got harder as the questions progressed. There were some very interesting geometry questions later on that I’ve never seen before. The fact that this contest (which was a supplementary material for evaluating application forms for UWaterloo) was named after him shows how lasting his name is embedded in our academia institutions. I later studied Euclidean geometry in university, a topic I haven’t revisited since grade 9. Euclid’s book (The Elements) has indeed left a great mark on mathematics and the way we think about abstract concepts. His methods are quite systematic. He presented a subject in a logical manner based on definitions, axioms, and theorems. This systematic approach became a model for subsequent mathematical works and the scientific method in general. Starting with small lego blocks, larger things could be built from those blocks which in turn build into even larger things. Geometric proofs are also quite visual, so those who are more visual thinkers would have a better time understanding them than say a proof from real analysis. The geometric proofs are also presented in a clear manner. This clarity has made it a favored choice for educators over time. Overall it lays out the foundations of plane geometry nicely and succinctly. Although beauty is quite subjective, it often appears with elements of simplicity, symmetry, clarity, depth, and harmony. When applied to abstract concepts like mathematics, beauty often emerges from the elegance of a certain topic, the depth of insight it offers, or the surprising connections it makes (how well things go together). Because of this, many find beauty in the Euclidean axioms. Some of the proofs and constructions that use those axioms are thought to be beautiful, like some sort of minimalist perfection where every step feels necessary with no redundancy. Beauty isn’t all about a physical image that is captivating to the eyes, but it’s also about mathematical notions working together and fitting together nicely like jigsaw pieces.

Oct 16 Was Pythagoras Chinese? Reflection

    I think it’s important to acknowledge non-Euro math sources because it allows people to better understand the universality of this field of study. That groups who were not in contact with each other all discovered the same concepts using different methods. The complexity and ingenuity of the ways the ancient peoples/cultures used act as a bedrock to what we are learning today. People now can’t begin to come up with ideas on how certain problems were solved with no technology. A lot of theorems and findings are already done from the past, we are so used to just using them in our daily lives. From ancient China, the significance of these foundational texts, especially the "Jiu Zhang" and "Zhoubi suanjing," persisted for millennia. Around the time these works were completed, the Chinese civil service grew under the Eastern Han dynasty. So China’s prosperity didn’t come from using knowledge gleaned from Euro math sources. If I mention and give credit to how the ancient Babylonians, Indians, Egyptians, Chinese, and Myans, it displays integrity to those groups. Seeing from the lens they derived some rules from opens our minds and makes us scratch our heads to think in new directions. It also gives a breath of fresh air to know that new ideas aren't confined to one area of the world.

    The "Zhoubi suanjing” focused on astronomy and mathematics related to land surveying and construction. It introduces the gou-gu theorem, which resembles the Pythagorean theorem. The notion that of a right-triangle the sum of the squares of its legs equals the square of its hypotenuse isn’t solely a Pythagoras (or ancient Greek) finding. The ancient Babylonians centuries before Pythagoras also listed Pythagorean triples, which is not a trivial task. Other mathematical terms like Pascal’s triangle fall into the same category. Pascal wasn’t the first one to discover it. It is also known as the Staircase of Mount Meru which pays tribute to Indian mathematician Pingala. Clear evidence/sources from ancient China mentions it as Yang Hui’s triangle. It’s the same triangle we know today but is depicted using rod numerals similar to zongs and hengs. The reason why we use Pythagorean theorem and Pascal’s triangle is because it’s well embedded into the literature and there is less ambiguity when talking about them. People will know automatically what they mean, whereas if I say the Staircase to Mount Meru, others might think I’m referring to some attraction site. So it’s helpful to mention these things.

Oct 11 Assignment 1 Reflections

Researching about how the ancient Babylonians solved this difficult problem on generating reduced Pythagorean triples was no easy task. Lots of rich information was lost from the primary source as the left part of the tablet was broken off. In the end people had to resort to proposals on how the ancient Babylonians actually did it based off of their other works and discoveries. The proposal I used was the reciprocal pairs method, however it used a lot of modern day algebra to deduce what happened, and didn't utilize much of the notations and numeral systems the ancient Babylonians used. It was nice to revist the cuneiform script in sexagesimals (base 60) again, however I didn't explicitly use them. I still think it was neat to know that they knew such big Pythagorean triples with the limited amount of tools and knowledge they had. I worked well with my group mates. Each person had their own task and it was a nice transition between us presenting. I felt we could've spent more time on interactive activities, but due to time constraints we briefly brushed through them. Giving the participants/audience more time to think and go through what was happening would've been helpful too. I am glad to present on a topic that I am familiar with, something I had a fun time learning about when I took number theory one year ago. I saw the intricate details of proving the Pythagorean triples theorem. Boiling it down to a simple fun history math presentation was a mild way to express the interesting aspect of this topic.

Oct 11 Two nice visual proofs of the Pythagorean Theorem

I like Eddie Woo's teachings and character and below are two of his videos on visual proofs of the Pythagorean Theorem:

Visual proof of the Pythagorean theorem

Another Visual proof

Edward Woo is an Australian secondary school teacher and writer best known for his online mathematics lessons published on YouTube.

Oct 4 Assignment 1 Solutions and Extension

To find positive integer solutions to Pythagorean triples, it is proposed that the ancient Babylonians used the reciprocal pairs method: Given s (some fraction greater than 1, which they wrote in sexagesimals) and 1/s we take their sum and difference, s+1/s and s-1/s. Next we take the difference of their squares: (s+1/s)^2 - (s-1/s)^2 which is always equal to 4 = 2^2. This was a great finding since the equation can be rearranged to (s-1/s)^2 + 2^2 = (s+1/s)^2. However this doesn't necessarily give postive integer solutions to x^2 + y^2 = z^2. To see how what could be done, since s is some fraction, write s as p/q where p>q>0. Then (s-1/s)^2 + 2^2 = (s+1/s)^2 becomes [(p^2-q^2)/pq]^2 + 2^2 = [(p^2+q^2)/pq]^2. Multiply both sides by (pq)^2 to get (p^2-q^2)^2 + (2pq)^2 = (p^2+q^2)^2. Given this formula now, positive integers p and q can be chosen such that x=p^2-q^2, y = 2pq, z = p^2-q^2. The problem is that this won't always generate a reduced Pythagorean triple. If the GCD of x,y,z is not 1, it'll have to be divided out from each of the three in order to get a reduced triple. Several observations/conditions were gleaned regarding the conditions for p and q to get a reduced triple, but no proof has been made on those claims. Modern solutions use more areas in known number theory to derive a proof on how to generate reduced Pythagorean triples. The conditions are that p and q are coprime, and exactly one of them is even and the other odd. This is a necessary and sufficient condition in order to have reduced triples. 

Reduced Pythagorean triples satisfy the Pythagorean equation x^2 + y^2 = z^2, where x,y denote the length of the two legs of a right triangle and z its hypotenuse length. The Pythagorean theorem, which applies to right triangles, is a special case of the cosine law, a^2+b^2-2abcos(C) = c^2, where the angle opposite to the hypotenuse, c, is 90°. Since cos(90°) = 0, it boils down to just a^2+b^2=c^2. The cosine law is useful for finding an angle measure when given all side lengths. It is also useful for finding a missing side when given the other sides and one angle measure. A great proof video on the derivation of the cosine law can be found on Khan Academy.

Oct 4 Comparing Ancient Egyptian and Ancient Babylonian Number Systems

The most notable difference between the two numeration systems is that the ancient Babylonians, as we have first studied, used a system in base 60, sexagesimals. 60 had many interesting properties and was helpful in concepts like time/representing fractions. The ancient Egyptians used the base 10 system which is what many are familiar with today. The symbols the Egyptians used to denote numbers are also more complex in looks as they are more pictorial. For the Roman numerals, it also uses a base 10 system. 1 is I, 2 is II, 3 is III. However there is a special symbol for 5, V. 4 is IV since it is one before 5, and 6 is VI since it is one after 5. 7 is VII, 8 is VIII for the same reason. But 10 is X, and 9 is IX, one before 10. Roman numerals have special symbols not only for powers of 10 but also for the powers of 10 times 5, i.e. 5 is V, 50 is L, 500 is D. Roman numerals are also more efficient in writing and carving onto blocks. In elementary, my social studies class went on a field trip. We learned that when European traders came to Canada, they etched Roman numerals on the trees since it was easy to carve lines instead of the curvyness of the Arabic numerals. A nice affordance for the Egyptian system is that it uses base 10 which we are familiar with. The Babylonian system uses base 60 which is helpful in representing fractions and time. A constraint for both systems is that it is slow since you have to write down literally 9 symbols to represent 9. The Egyptian system’s symbols are also complex which take more time to write out. Those pictorial symbols do have associations for the numbers they are representing though. It correlates with the quantity of/ level of importance associated with the symbol drawing.

Oct 4 Ancient Egyptian Land Surveying Response

The article mentions the symbolic connection of the surveying rope with the ram's head of the ancient Egyptian god Khnum, which may indicate the sanctity of measurements. I wonder what religious beliefs were attached to surveying in ancient Egypt. Were there rituals/ceremonies performed at the start or completion of important surveying tasks, especially given the importance of these measurements for agriculture and construction? The article also mentions the length of a remen which is half of √2 cubits long. I wonder if the ancient Egyptians had a way of calculating its decimal expansion to some place value. There is a trick to find square roots of numbers and giving an approximate decimal expansion, but I have not been taught about it at school. I only saw kids working on it while I was helping out at Kumon. I’m wondering if they knew about rational and irrational numbers as well. If they did, did they know that √2 was irrational, and what was their proof? One particular thing that surprised me was that the annual inundation of the Nile necessitated the need for consistent remeasurement and redistribution of arable land. This impacted not only on the practice and development of surveying but also on taxation and governance. The Nile was a source of life, but its yearly inundation would also erase boundary markers, making it vital for surveyors to redefine land divisions and boundaries. Such a consistent activity would eventually lead to advancements in surveying techniques.

Sept 27 Russian Peasant Multiplication and Ancient Egyptian Multiplication

It was interesting to see the video on Russian peasant multiplication. Numberphile videos never fail to amuse me. I had to scratch my head when trying to relate it to the ancient Egyptian multiplication method. The aforementioned method uses doubling, binary, and sums based on the multiplicands as shown in class. It's an efficient method that is more intuitive for students to understand how to multiply large numbers together. The Russian peasant multiplication method uses a different algorithm but is similar to the ancient Egyptians. For example:   

With the two examples above, the left side uses the Russian peasant method and the right side uses the ancient Egyptian method. I realized that the number of times I half each round for Russian is the same number of times I double each round for Egyptian. There was a consistency there. I also saw that writing [the left] multiplicand in binary, the rows with 1 for Egyptian corresponded with odd numbers for Russian, and the rows with 0 corresponded with even numbers. Therefore the rows with even numbers in the Russian peasant method are not used. The rows with 1/odd are highlighted in teal, and the rows with 0/even are highlighted in red. In essence, with a chosen multiplicand, one method is doubling starting from 1 and reaching the largest power of 2 that is less than or equal to that multiplicand, and the other method is starting from the multiplicand and halving (ignoring the remainder if odd) until 1 is reached. 

Sept 27 Ancient Egyptian Multiplication Example

Sept 20 Babylonian Word Problems

It is quite interesting to see that many ancient cultures, Babylonians, Egyptians, Greeks, Indians, and Chinese were able to think about and devise mathematical problems without any of the discoveries/technology we have today. Indeed their problems were devised based on practicality so it may prove useful to them. Other times it wasn't necessarily how realistic/practical it was, but using ideas that we can abstractly generate in our minds that could work in a different world. The problem is that if it doesn't make sense realistically (shooting an arrow off the moon) it gives people an impression that such knowledge is useless, so it is better to apply it using more appropriate examples instead (shooting an arrow in a space with no air friction). The point is to exercise our minds and train our brain to think in different ways. The same mathematical concepts were discovered by cultures all over the world that didn't have interactions with teachers back then, hence math is considered a universal notion. For me, pure math is proof based using rigorous axioms and theory build ups, whereas applied math is using what is learned and constructing them into problems we face daily or into problems that are imaginable. Conceptualization of these ideas doesn't require what we know about algebra per se. Even young kids can think about what situations math is needed without formal education in the subject. In essence we cannot look at the ancient peoples and say they weren't as smart as we were just because they didn't have the sufficient amount of resources. Instead they were the first ones to discover a lot of mathematical properties that we build off of today. The foundation is important and we should give credit to them for it.

Sept 20 Babylonian Algebra

The statement can be used using physical properties of spatial objects, eg. length, width, height, square, cube. One may want to know the dimensions of a particular field or construct a table of certain measurements. The ancient Babylonians were able to use word problems to describe different situations, which are now modeled more easily with the algebraic expressions we have today (for instance, buy and sell, property related questions). They also had a sophisticated number system which allowed them to perform arithmetic in meaningful ways with respect to time, distance, and volume. Mathematics isn't all about generalizations and abstraction. Linking it to physical properties is important. For calculus, it is used in finding volumes of shapes, used in engineering, and physics. Calculus also links to rates of change and accumulation of quantities. Graph theory requires not much algebra as it is simply a mathematical way of understanding vertex and edge relationships using some sets, but mostly diagrams. This connects with social relationships, such as for networking sites like LinkedIn. Graph theory also helps deal with transmission of virus linkages from a positive source, and there was a nice video with Eddie Woo and Poh Shen Loh talking about it.

Sept 18 Babylonian-style base 60 multiplication table for the number forty-five

Susan gave an example of 2 and 22,30 whose product is 45 in base 60

Below are five pairs of numbers whose product equals 45 in base 60:

- 4 and 11,15: 4 times 11 is 44, but 4 times 11 and a quarter (,15) is 44 + 1 = 45

- 6 and 7,30: 6 times 7 is 42, but 6 times 7 and a half (,30) is 42 + 3 = 45

- 8 and 5,37,30: 8 times 5 is 40, so to do 0.625 in sexagesimals, 0.625*60 = 37.5, so the tenths place is 37, and the hundredths place is the value that represents half of 60 which is 30

- 12 and 3,45: 12 times 3 is 36, but 12 times 3 and three quarters (3* ,15 = ,45) is 36 + 9 = 45

- 20 and 2,15: 20 times 2 is 40, but 20 times 2 and a quarter (,15) = 40 + 5 = 45

Sept 13 Response to Crest of the Peacock

Many information/discoveries made by European expanders/explorers came to be over the past few centuries and that what we learned about these information/discoveries are from their observations. It’s as if they weren’t readily available before any encounters the Europeans had. We all know that isn’t the case. Many mathematical concepts and techniques often attributed to European scholars were known and used centuries earlier in other parts of the world. So for me, getting to observe how the ancients were able to do calculations and mathematical observations to such a fascinating degree where they could build the pyramids, roads with no pot holes, and the sewage system of the Forbidden Kingdom is truly remarkable (and no current European system can replicate such a feat). I never learned that a lot of these fathers of ancient mathematics from Greece (whose foundation became bedrock for modern Euro-mathematics cultural dependencies) actually interacted heavily with other ancient cultures from Egypt/Mesopotamia/India. That entails that indeed the honor and respect for these other cultural observations on math findings are important and somewhat embedded in math teachings today. I say somewhat due to the fact that there is so much more that could be extracted/implemented/used/taught from them, but also because the credit for them is hardly ever mentioned. So if a student (who learned their math history) came from one of these countries to North America and sees many concepts being credited to whites in relatively more recent times compared to the great mathematicians from their culture, would they not feel dejected and subverted? The numeracy skills and systems developed in those places back then were also observed to be more efficient, and in some places used in areas of our daily lives like in computer languages and time telling. So, to be flexible is important and to use and appreciate how vast this whole topic truly is will lead to a more inclusive and diverse field of expertise.

Sept 13 Why Base 60?

60 is a number with many factors: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60. Contrasting it with 10, which only has 4 factors. The large number of factors makes base 60 calculations more flexible in many situations, especially when it comes to divisions or fractions. For example, it's much easier to express a third in a base 60 system (as 20) than in base 10 (which results in an endless repeating decimal: 0.333...). So getting rid of the notion of repeated values for some rationals may allow people to think more clearly and focus more on the value of a third, rather than the repetitions. 

The factors of 60 are used in many places in time and geometry. The number of degrees in a circle is 360, which is very similar to the number of days in a year 365). 60 degrees is also the angle of a special right triangle. The number of days in a year is defined by the orientation of the star constellations so defining time with base 10 would be difficult. In many cultures around the world, there are 12 months in a year and are often represented with creatures real or mystical (like the Chinese and Greek zodiacs),  Time within the day is measured using hours, minutes, and seconds. There are 60 seconds in a minute and 60 minutes in an hour. There are two 12 hour half days in a day. For time, sometimes in English 7:15 will be stated as a quarter past 7, since 15 is a quarter of 60. 7:30 will be stated sometimes as half past 7, since 30 is a half of 60. 

After doing some research, I’ve found out that there are more uses for 60 in the [Babylonian] numeration system. For example the harmonic connotations. The number 60, being a multiple of 12, might have cultural or religious connotations in societies that revered the number 12. The Babylonians used base 60 for their calculations, and many of their mathematical/astronomical findings were foundational for subsequent civilizations, which lead to 60 being used instead of 10. The number 60 is also the lowest common multiple of all the numbers 1 through 6. This makes it easy to integrate or switch between different bases, such as base 2, 3, 4, 5, 6 within a base 60 framework. This property cannot be offered with base 10.

Sept 11 Integrating history of mathematics in the classroom: an analytic survey

    Math is a universal subject and its rich history of discovery across the world is worth mentioning. Humans are critical thinkers and math arouses topics that explain how things are around us. Different discoveries around the world lead to new findings, and applying how the people in the past figured things out without the use of technology is astounding. They understand that the math mysteries are deep within nature. A certain concept is called differently throughout history. The credit should be given to anyone who independently came up with the idea. I like to not only call it Pascal’s triangle, but also the Staircase of Mount Meru or Yang Hui’s triangle. How the information came to be in our school books doesn’t come in a blink. It is through eons of discovery and trials, and it gives the evidence of the first forms of applications/usage.

    Math is displayed as a learning subject that evolves throughout time. I see how math notations and instructional styles taught 300 years ago during Newton’s time are not the same as today. How students learn math around the world is also unique, so knowing the universal history of math allows a better context for the entirety of the subject’s exploration. It’s like many flowing streams branching out of a large basin. Why study and row in one stream when it's connected to all other streams? The deeper awareness of math in its intrinsic and extrinsic nature reminded me of the Philosophy of Mathematics course I earlier took. How to analyze math per se in its rigor and asking questions on the realistic or abstract meanings of mathematical concepts will allow students to think deeper and openly.

    The article allowed me to better know the benefits of having a deeper mathematical foundation through learning its rich history of discoveries. I know I will mostly bring up a lot of topics done differently in grade school compared to university (eg. the notations, change of definitions), so it is fascinating to see why that is. Finding and setting some time aside to implement some history in the classroom initially seems complicated, but bringing forth some historical problems which lead to what is being studied gives a background context for why certain topics are being studied in the classrooms.