May 442 Blog

Please read this piece and blog on three quotes from the article that made you stop and that surprised you in some way. one thing is that "in the trivium of the medieval university. Logistic was practical and utilitarian, a study for children and slaves; logic was a liberal art, a study for free men." eh.... yah. It seems like the theoretical/intellectual enjoyment are solely for the free men. There is a distinction between the classes. Well, I think this is kind of true nowadays too. Some people learn math or science just for a living. while some people learn it for the enjoyment. but it comes to the individual thinking and habits though, mostly, I think , for today's population. since most of the people are not starving.  Also one of the mathematicians "wrote much on number mysticism, a sort of theology of numbers. Numbers were identified with the various gods. He considered the odd numbers to be male and the even ones to be female." This is similar to the num...

For your homework blog writing, please read the article linked here and comment on three things that stopped you in your reading or that struck you about the relationship of mathematics and Euclidean geometry to body, Land and movement. Your writing is due on your blog on Tuesday November 17 at 9 PM. One thing that struck me was the concept of being the proof oneself.  "we somehow enter the page. We imagine ourselves to be like the two dimensional characters in Abbott’s Flatland [1]. While dancing the proofs adds a temporal dimension to Euclid’s original representation, the positionality of the dancers and audience (in the same plane) involves some loss of the third spatial dimension." I think this is quite interesting because when one embodies the proof, one should never forget it. If we let our students do the proof using this method, they would hardly forget it.  As said, " The process of choreographing the dance proofs — making decisions, practicing, memorizing — bot...

 *NEW* Read through all your own posts from the course, and reflect on what you've learned from the course, how your ideas have developed, and suggestions for ways to improve the course for next year. This is my first course on the history of math. This course gives me some new perspectives on how to teach math. Using the knowledge of math history and learning about the perspectives of ancient mathematicians gives me some new insights on the mathematical world. For example, some theorems are not developed totally by Europeans. I always thought that most of the theorems or concepts came from Europe. Moreover, the assignments gives me some idea of using the history of math in my own teaching practices. The first assignment is about solving puzzles using the ancient method. If we use this in our own classroom, it could still be valid, as long as we keep the puzzle easier.  It would allow the students to search for information themselves and then come up with some solution themsel...

 *NEW* Personal reflection on what you learned and take away from Assignment 3 after you present  I learned the history and wisdom of a traditional game Tangram and we found it fascinating. It's so useful in the mathematical realm and it's quite easy to use in the mathematical classroom. Almost all grades could figure it out. For example the grade 8 's will be learning areas and volumes of different shapes. We could give the tangram shapes to them and let them figure the areas out using their existing knowledge about the areas of certain shapes. The parallelogram example in the video is an example. Also, even for surface areas we could let them use tangram to build things and figure out how many 2-d shapes we have in a 3-d shape. (how we add those areas of the 2-d shapes to get the total surface area of the 3-d shape) In a thinking classroom. the kids could totally get this themselves.  They could even figure out areas of strange shapes such as trapezoids and many other p...

 Final Art Presentation Google Drive Link https://docs.google.com/presentation/d/1_dVG3csojlbZqqCm7hGdqxALpLiw4IEm29FPVQ9jCRw/edit?usp=sharing Video: https://drive.google.com/file/d/1DxnumrK28pBO3lP703NO6FFz5uCCi6z0/view?usp=sharing

One thing that stopped me is that: " The other is his suggestion that the idea of number needed to be enlarged to include a new kind of number, namely ratios of magnitudes. For example, in ‛Umar’s view, the ratio of the diagonal of a square to the side, or the ratio of the circumference of a circle to its diameter (π), should be considered as new kinds of numbers." Also, "parallel postulate follows from the other Euclidean postulates" I never realized that it is the Muslim people who introduced the idea of positive real numbers. I always thought that our current math system probably came mostly from Europe, especially the algebra notations and definitions. and it's also very cool to challenge the Euclid's Element and say that the postulate actually follows from the other Euclidean postulates. It shows the spirit of the mathematician. another thing is that:  However, a culture’s acquisition of intellectual material from an alien culture is a complex process a...

  1. Our final project will be working the art of tangrams and trace back its history to ancient China 2. References                                                     Read, R. C. (1965).  Tangrams: 330 puzzles . Courier Corporation. URL: https://books.google.ca/books?id=80yRBQAAQBAJ&lpg=PP1&ots=EzOW0rhuWt&dq=history%20of%20tangrams&lr&pg=PP1#v=onepage&q=history%20of%20tangrams&f=false   Russell, D., & Bologna, E. (1982). Teaching Geometry with Tangrams.  The Arithmetic Teacher,   30 (2), 34-38. Retrieved December 6, 2020, from http://www.jstor.org/stable/41192134   Siew, N. M., Chong, C. L., & Abdullah, M. R. (2013). FACILITATING STUDENTS'GEOMETRIC THINKING THROUGH VAN HIELE'S PHASE-BASED LEARNING USING TANGRAM.  Journal of Social Sciences ,  9 (3), 101.   T...

• Consider the so-called Hardy-Ramanujan number 1729, and the story of Taxicab Numbers, retold on Wolfram Mathworld at the link above. Hardy is quoted as saying of Ramanujan that "each of the positive integers was one of his personal friends". What do you make of this in terms of Major's paper?   In this story , when the number first came into his mind he thought of it as a rather dull number,  adding that he hoped that wasn't a bad omen . However, as he thought about it, he found that “it is a very interesting number. It is the smallest number expressible as the sum of two cubes in two different ways" . In the paper, it is stated that each number has its own personality and people give them associations with good/bad fortune, heavy/light senses. Also, different numbers stimulate activity in a different area in the brain. So there’s biological reasons associated with our feelings about the numbers as well. For Ramanujan, probably he just had some different arrang...

Euclid was the founder of geometry. His Elements is one of the most influential works in the history of mathematics, serving as the main textbook for teaching mathematics (especially geometry) from the time of its publication until the late 19th or early 20th century. I'm not good at interpreting poems but the first poem seems to talk about how Euclid found beauty in geometry findings, in shapes of shifting lineage, and this helped him experience the "flow" that releases him from dusty human bondage to luminous air. His  soul and his whole being  was lifted by this beauty, which gets him away from the human state and into a higher state of being. The next poem seems to disagree... and say that beauty is clothed, and say the first poet is an idiot. but I don't really get what he was objecting to .....and what he proposes instead!

Here is a link to our powerpoint :  https://docs.google.com/presentation/d/1infhLZ4Ow7wofXp-yRHw7PSjB_5yOkyrOllgYZgh5pY/edit#slide=id.g9a6108e0dc_10_32 In this project we looked at the fractals in art pieces and Indigenous designs,  and then developed a way to teach fractals using diagrams. I was in charge of the hypothetical class plan. I found the picture and labelled it. I didn't want to make it too hard for the first part of the class so I used a simple version of fractals. I asked them to measure the length and calculate the ratios.  I believe the first diagram is easy to manage/ understand. Then I led a class discussion to link the observations to the concepts. After that I led the lesson to a more complicated version of fractal tree. To make sure they understand the latter one, I asked them to draw a fractal tree themselves.  In this project I learned about how to incorporate art pieces into the math classroom. We could use graphics to teach certain concepts, ...

  4 3 8 9 5 1 2 7 6     To add to 15: 1+6+8 3+5+7 2+4+9 2+6+7 2+5+8   First fill in 2,5,8 as the diagonal Then fill in 6 and 7 at the bottom. since 7+8 exceeds the limit, use 6 for the column with 8 Using the two numbers present, fill in the rest. It works.       8   5   2 7 6

 Research online the significance of the Eye of Horus and unit fractions in ancient Egypt. What was most interesting to you in your findings? The Egyptian Unit fractions are based on the Eye of Horus. Horus was the ancient Egyptian sky god who was usually depicted as a falcon, most likely a lanner or peregrine falcon.[7] His right eye was associated with the sun god, Ra. It was believed by the Greeks and Romans that an evil heart could get to the eye. The thought to be powerful effects of eyes and optics created the myth that the energy-producing power of the eye had the ability to cast evil spells with just a glance. Because the ancients believed the evil eye could be counteracted with a 'good eye', myths about Horus arose.[9]  The eye of Horus was often used to symbolise sacrifice, healing, restoration, and protection.  I think the most interesting thing here is that the eye has so many connotations and it is something almost sacred in Egyptian culture. The an...

Adam gave Sam 45 dollars to buy groceries. He requires that the number of apples should be three times the number of bananas. The number of bananas should be double the number of oranges. suppose apples cost 2 dollars and bananas cost 1 dollar, and oranges cost 1 dollar, how many of each fruit should he buy? modern solution:  let the number of oranges be x.  x +2x+6x*2=45 15x=45 x=3 false position method: let x be 2  2+4+6*2*2=30 we need 45, which is 45/30=1+1/2 as big, so x must be 2 times 1+1/2 , which is 2+1=3 check: if x=3, x +2x+6x*2=45

 I think it is important to acknowledge non-European sources of mathematics because it promotes respect for other cultures. When only European sources of math are acknowledged, students tend to develop an understanding that only Europe is the origin of all creations in math/science. They would have unnecessary ego for their own ethnicity. When we acknowledge other cultures’ mathematical discoveries, we tend to have some level of humility, especially when some of the theorems claimed to be European are actually from China. For instance, “The method of the double-false-was first described in Jiu Zhang suanshu,indicating the method was known in China around 50 AD. Subsequently, the method was transferred to Muslim mathematicians at some point and then on to Europeans sometime during the Middle Ages. It then was brought back to China by the Jesuit missionaries, who claimed the method as their own.”   Also, “accounts and facts of Greek work during this time tilt more on the s...

  These ideas do rely on our familiarity with contemporary algebra because for Babylonians they don’t have a distinct separation of pure and applied mathematics. Their word problems are both practical and theoretical : “many look like real-world problems at first; but as soons as......the complete artificiality of the problems is revealed...has been disconnected from immediate practice” The Babylonian mathematics was based on problems that seemed practical but were actually “’pure’ in substance” They are used to train students in the use of ‘methods at hand’, which are essentially generalization or abstraction of the real world. Mathematicians use abstract methods and concepts to solve real world problems. There is certainly a stronger connection between the practical world and mathematics concepts in the Babylonian era than today. Nowadays there is a distinction between pure and applied mathematics and problems are not all based on real life situations. A part of math is based on ...

  1)     My pre-reading ideas was that math history is an important aspect of mathematics education because it shows where the theorems / concepts came from and gives us a better understanding of the process of mathematics exploration. It could be incorporated in to my math teaching by showing the students how a theorem was discovered (the experiments/ steps mathematicians took to figure out the theorem). We could also show them ancient ways to solve the problems which differs from modern methods. In this way they could realize that math is not only just the formulas they see in class but a result of logical thinking process and repeated experiments by mathematicians.  2)     one thing I agree is that history may be tortuous and confusing rather than enlightening. students may not enjoy learning the history of mathematics even though it might be beneficial. however, it is still crucial to integrate math history into the curriculum. Another thi...

  One thing that struck me was how the Pythagorean theorem took such a long time to complete. I never thought that such a theorem would be so hard to get. It leads me to think about how we are privileged to have so many theorems at our disposal when dealing with problems. We only need to learn the theorems and never thought of how hard mathematicians might have tried to figure that out.                                                Another one is how math is a universal language of the world which can be spread across countries. Arabs helped spread the Indian numerals and their associated algorithms to Europe as well as the trigonometry. India and China also exchanged knowledge about the concepts of mathematics, such as kuttaka and qiuyishu. It is obv...

In this reading we see that the Babylonians are using concrete words to substitute variables we have today. For a mathematical principle, one would have to use those words (such as ush for x, square for x^2)  to represent the varaibles.  “=” sign would be “Result” and any square root sign would be replaced by the word “square root”. An simple modern equation would be substituted by multiple steps described with words. I believe that math is all about generalization and abstraction. without these, math would not be able to represent / solve real life problems. We take these problems and transform them into the “math” form and solve them(using mathematical rules) , then we take it back to the real world. Without generalization and  abstraction, we could not do this.  For more abstract areas, such as number theory, geometries, calculus, graph theory, etc., it would be hard not to use algebra. If we don’t have algebra, number theory could hardly be represented, let a...

3     15 6     7, 30 8     5 ,37,30 12   3 ,45 16   2,48,45

  Speculative phase:   (1) Think for yourself why 60 might be a convenient, significant or especially useful number to use as the base for a number notational system. What is special about the number 60? How is it different from 10?   60 is much bigger than 10, but maybe it is easier for the Babylonians when dividing/using fractions. 60 is divisible by 2,3,5,6,10,12... It contains many factors. Hence, base 60 might be easier to use than base 10 in ancient times when calculations are not as convenient as nowadays.   (2) Then think for yourself how we still use 60s in our own daily lives, in Canada, and across cultures if you have knowledge of other systems (like the Chinese zodiac and time-telling system, for example.) Why is 60 significant in so many situations involving time and/or space?   In China, one day is separated into 12 sections. One year is separated into 12 months. 60 is a multiple of 12. In western cultures, one day is separated into...