Assignment 1 Solution and Extension

Surveying in Ancient Egypt Reponse

One thing that surpised me is the resourcefulness of Ancient Egyptian surveyors given their limited resources and knowledge at the time. It surprised me that they used simple tools such as measuring rods, plumb bobs, and ropes with knots for precision in their projects, architecture, and agriculture. It amazes me that a civilization with limited technology still had human beings who were just as smart as us, and were just as capable of problem solving and critical thinking. It made me ponder that maybe we do take for granted what we have today. Us Gen Z's are living too comfortably in a world of convenience and technology that a lot of us has becomed conditioned to ignore all issues that require thinking, problem soving, and application to real life. Instead, we decide to call and pay a professional. Reading this article has really shed some light on modern day gratitude and emphasizes the importance that we continue to critically think and train our students to do so as well.

Here are two followup curiosity-driven questions I have as follow-up to reading the article:

1) How did the surveyors deal with the challenge of maintaining accurate and precise measurements over long distances, given the tools they had? (ropes and knots, etc.)

2) How were Ancient Egyptians educated and trained for precise work in that era?

Ancient Egyptian mathematics: Numeration

 What differences do you notice between ancient Mesopotamian/ Babylonian numeration and ancient Egyptian numeration systems?

Its also similar to the mesopotamian method of counting, in the sense that they count objects. For example, 9 would be 9 sticks.However, it is clear that the biggest difference is that the Babylonian system uses base 60 while the Egyptian method uses base 10, like we use today. Another difference is that the Egyptian method does the ones first, then increases in powers of base 10, while on the other hand, the Babylonian system does its highest power/"digit" first and then decreases accordingly.

(And if you are familiar with Roman numerals -- which we will visit later -- what similarities and differences do you notice?)

They are very similar in the sense that they both use base 10 and counts objects. However, I believe that the roman numeral system is more modern, and thus has a sign for 5, allowing numbers to more condensed (instead of writing 9 sticks). They are different in the order. Egyptian numeration starts with the ones digit on the left and goes up powers of 10, while Roman numerals are the other way around.

What affordances and constraints do you notice for the Egyptian system? For the Babylonian system?

For both systems, it is a hassle to write large numbers. Imagine writing 99999. The symbols would compile and would take up a large amount of space. In addition, there is no symbol for 0, as the number 0 has been discovered by India. As we know, the number 0 however is a very important one, so this sets limitations to the mathematics they can do. For the Egyptian method in particular, I am also unsure how they do fractions. For the Babylonian system, fractions are certainly easier, but with base 60, numbers like 59 and 119 might be a hassle to write. 

Russian Peasant and Egyptian Method Multiplication

 

After watching the Numberphile video on Russian multiplication, I was shocked that it worked, especially with the disregard for fractions. However, after doing a problem or two, I realized that they are equivalent. For starters, the numbers on the right column are exactly the same. In addition, both methods revolve around writing the multiplicand in binary, but the Russian method is just the Egyptian method backward. Both methods also spell out the multiplicant in binary, but just in opposite directions. Overall, this was pretty cool and I definitely learned something new today.

Egyptian Multiplication & Division

 

Babylonian Word Problems

Reading this article has got me to reflect on the word problems today. It was interesting to read that although some word problems in ancient Babylon were practical, others were not. Instead, the impractical ones were used for critical thinking, problem solving, and as a way to extend mathematics into a more abstract, pure, and generalized form. As a math major myself, I am always seeking a good challenge. There is nothing more satisfying than solving a problem that you were stuck on for a long time. Similarly, the ancient Babylonians were probably looking for challenging word problems as well that would push scholars in their education.

Overall, this article makes me think of the praacticality of modern day problems. There are still many modern day problems that have no application and are worded weirdly and poorly. Some of them give quantities that you would never actually find or need in the real world. I can see how these types of questions may push the students who are more interested in abstract math, but it does not help the students who already struggle in math and are finding it hard to enjoy it. Those students are the ones who need to see that math is in our everyday lives. Reflecting back to teaching practices, it is important for teachers to choose questions wisely and know who our "audience" is. The types of questions we assign can either motivate students or demotivate them.

Babylonian Style Algebra

How could one state a general mathematical principle in a time before the development of algebra and algebraic notation?

Mathematics principles were often described verbally or through pictures and diagrams. 


Is mathematics all about generalization and abstraction?

Generalisation and abstract are very important areas of mathematics as it helps people come up consistent with ways to solve recurring problems. However, it isn’t all about generalisation and abstraction. Generalisation and abstraction are just tools used to create a general case for everyday problems. However, there are many problems, especially those in fields like economics, engineering, and physics, that may not be so easy to generalise. Mathematics is more about problem solving and critical thinking, as generalisation does not apply to all problems.

Thinking about various areas of mathematical knowledge -- number theory, geometries, calculus, graph theory, etc. etc. -- how could you imagine stating general or abstract relationships without algebra?

Again, most mathematical principles  were stated verbally or pictorially. For example, the oythofrean theorem might be stated as “in a right angled triangle, the square of the hypotenuse is equal to the sum of squares of the other two sides.” Stating mathematical concepts back then obviously was more work/writing, but there were no global and abstract symbols used back then.

Create your own Babylonian-style base 60 multiplication table for the number forty-five

 

Crest of the Peacock Introduction

Prior to doing the reading, I already knew that mathematics outside of Europe was advanced and that it was ignored and undervalued. I knew that Ancient China and India were both advanced civilizations with the Chinese being one of the first to use modular arithmetic and the Indians to invent the number 0 as well as the numberical system. What I found interesting from the reading were that some areas were isolated and still developed math (ex. Maya of Central America). This shows the use of math in every day life and its widespread importance across cultures. However, I believe that such countries develop mathematics at a slower growth rate than others? I'd assume that isolated (and perhaps more poverish) civilizations only developed what was necessary for every day life. 

Another thing that I found interesting to see is the overall evolution of mathematics. I did not know that there may have possibly been an exchange of ideas between ancient Mesopotamia and Egypt. We often hear of all these famous western mathematicians, but rarely talk about how math has got there at the first place. What was the incluence that took place? How did the educational systems that taught mathematics come about? 

Lastly, I found it interesting how different civilizations were able to develop their own version of a numerical system, with all of them using the same ideas but being different. It was interesting to read that there were certain parallel developments in geometry and algebra. This simply alludes to the importance of math in our every day lives. It seems like mathematics back then was more than just numbers and formulas, but it seemed like it was also an essential means of communication.

Why Base 60?

 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?

(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?

Answer to (1) and (2): 60s are used a lot in circular objects. The number/base 60 is used in clocks, degrees, and various circular objects. I think that another reason 60 is widely used is its divisibility by so many numbers. Among the numbers less than 120 that have the most factors (12 different factors), 60 is the smallest one to do so. Since a lot of things have to be split and proportioned, 60 (unlike 10) is a good base number and a denominator because it can give pretty clean fractions.

Research phase:

(3) Finally, do a bit of research via the internet and/or the library to find out what others have learned about the significance of 60 in Babylonian numeration systems, in our contemporary world, and possibly across cultures.

Answer to (3): My intuition was roughly correct regarding the Babylonian's usage of base 60. I read that 60 is a good number to use as a base for fractions, especially with its divisibility by the number 3. This is useful because with ancient Babylon having no decimal system, doing fractions by base 10 would be complicated and inefficient due to its lack of factors. This applies to our contemporary world as well, especially in clocks and circular objects. The only difference is that we have a decimal system to compare different fractions, which helps us make better sense of numbers.

Why teach Math History?

Prior to reading the article: I honestly do not see the importance of teaching math history before college/university. As a math major, we covered some math history and talked about various mathematicians and the theorems they proved/discovered, but I personally was not introduced to math history at the secondary level. In addition, I do not yet see the relevance of math history at the secondary level, especially with classes like Pre-Calculus and Calculus solely being based on algebra and graphing. That being said, I am looking forward to what the article has to say about why math history is important.

The article mentioned that math history spurs creativity and encourages students to come up with their own questions. In university, I can agree that learning math history and how certain theorems are proven develops better problem solving skills and creativity. I also can see how learning math history in high school can be engaging and can stir up interest. When I learned math history in university, I would come up with ideas and follow up questions for further theorems (that may or may not already be solved). However, my concern with math history at the high school level is that there is not sufficient time in the curriculum (this is from my experience). In addition, I felt like it deviates from the more important content that students need to focus on to prepare for university (i.e. algebra, graphing, functions).

Looking into the authors' thoughts on how math can be integrated into the classroom, I believe that I may have subtly incorporated a little bit of mathematics history at the secondary level, but just not a lot. For the most part, math history was only explicitly discussed with the students who were interested and sought a deeper understanding of math (for example, I showed Monty Hall Problem to a student who indirectly asked about conditional probability). These were mostly the students participating in the CEMC Waterloo math competition. But for the most part, the math history I incorporated in the classroom were the brief snippets of historical information in the textbook that served as "motivations" for the students to ponder on (intrinsic nature of math activity).

All in all, I cannot really say that this article has changed my perspective of math history's irrelavance in the classroom at the secondary level. The authors made some good points, but apart from spurring interest in the brighter students, the authors' points seemed more relevant for students doing math at the post-secondary level. For example, I taught the quadratic formula in my Pre-Calculus 11 class. Later on, we went on to factor easy cubics and quartics with the factoring by grouping method. One of my students asked me if there is a closed form for cubics and quartics. I briefly explained that there is for both, but there is none for quintics. I googled the closed forms of the formulas for him, and told him that he will learn more about it in university if he ever decides to take a course called Galois Theory. I guess me explaining it to him may have spurred some additional interest in the subject, but my point is that I couldn't go into detail nor answer his question based what was learned at the secondary level.