Exploration of Babylonian Algebra: Stating mathematics without algebra

As demonstrated in the section of Babylonian algebra from Crest of the Peacock, the Babylonians stated mathematical principles using concrete words in place of algebraic notation. In Example 4.8, they defined ‘ush’ as length, and ‘sag’ as width. Multiplying ush and sag gives ‘asha’ which means area. In this example, it was demonstrated that the Babylonians understood multiplying length and width gives the area for a rectangle. However, as a result of using concrete words to explain mathematical principle, blocks of text become wordy and it becomes harder to follow information being presented.

I agree that mathematics is all about generalizations and abstraction. For example, the generalization of Fermat’s Last Theorem: a^x + b^y = c^z for a, b, c, x, y, z are positive integers not necessarily all equal, could be a generalization for some quadratic equation or some system of equations. Formulas and equations in mathematics can be generalized to explain a pattern or relationship. Similarly, with geometric shapes, we can generalize them into other shapes; tangrams are an excellent example of this. Abstraction in mathematics reveals a deeper connection or pulls ideas from other areas to assist in solving mathematical problems. For example, Andrew Wiles proved Fermat’s Last Theorem; a number theory problem, using elliptic curves which is a branch of algebraic geometry.

I think some mathematical knowledge may be easier to explain without algebra. For example, the Fundamental Theorem of arithmetic can be explained as every number greater than 1 can be written as a product of one or more primes. In a college level number theory textbook, this theorem is usually stated like, “consider some integer n > 1 with two different factorization into primes: n = p₁ p₂ … p_s = q₁ q₂ … q_t , where p_s and q_t are primes with p₁ ≤ p₂ … ≤ p_s and q₁ ≤ q₂ … ≤ q_t.” This college explanation uses more algebra and makes reading the theorem complex yet generalizes the prime factors. Alternatively, trying to explain something like the generalization of Fermat’s Last Theorem without algebra will require significantly more lines of text, rather than how it was presented in paragraph 2. I recall Skemp who argued that mathematics has an “over-burdened syllabi” where a single line of math contains many concentrated ideas (1976). I imagine stating more general mathematical principles and abstract relationships without algebra to be very wordy and complicated.

Source:

Joseph G.C., (1991). The Crest of the Peacock The Non-European Roots of Mathematics. Princeton University Press. Princeton and Oxford.

Skemp, R. Relational understanding and instrumental learning. Mathematics Teaching, 77, 20-26, (1976)

Babylonian style base 60 multiplication table for the number 45

 

Babylonian style base 60 multiplication table for the number 45. 

The Crest of the Peacock, the non-European roots of mathematics

A quote by Joseph summarizes my thoughts on this reading really well, "How we see ourselves is shaped by the history we absorb, not only in the classroom but in films, newspapers, television programmes, novels and even strip cartoons" (Joseph, 1991, p.1). Everything we see is influenced entirely by what we are allowed to see. This reading argues that many contributions of colonized civilizations and people were ignored as part of the subjugation and dominance of Eurocentric view of mathematics. Recent case studies in India, China, and parts of Africa indicated the existence of scientific creativity and technological achievements long before the invasion of Europe (Joseph, 1991, p.2). I have no doubts that the inhabitants of these countries were intelligent and made many discovers which would contribute immensely to mathematics but were lost due to forgotten knowledge or omittance. It surprised me to learn how much knowledge from India, China and the Hellenistic world influenced mathematical developments Arabs advancements and the interconnectedness of these countries. 

Figure 1.3 from Joseph shows the transmission of knowledge to Western Europe (1991). However, it was discovered that this knowledge dates back even further to Egypt and the Mesopotamians, shown in Figure 1.4. 

I learned that India influenced the Arabs by knowledge in Indian numerals and algorithms, Indian trigonometry, and solutions of equations in general and indeterminate equations. I also learned that westward flow of technology from China in 15^th/16^th century introduced ideas like the wheelbarrow, crossbow and gunpowder but also mathematical ideas like the Chinese Remainder Theorem used in number theory. These are all important contributions to mathematics, but it is disheartening to know that there are possibly many more just forgotten knowledge and not passed on through the dominance of Eurocentrism. 

 

Source: 

Joseph G.C., (1991). The Crest of the Peacock The Non-European Roots of Mathematics. Princeton University Press. Princeton and Oxford. Pp. 1-24

Base 60

 

Speculative phase:

I think the Babylonians chose to use base 60, known as the sexagesimal numeral system, with the discovery of time. As there are 60 seconds in a minute or 60 minutes in an hour, they chose to follow that same measurement for numerical calculations so that they can relate it back to time. 60 has prime factors 2, 2, 3, 5 and 10 has prime factors 2, 5. Their greatest common divisor is 10 and their lowest common multiple is 60.

In my daily life, 60s is used often, particularly through rates of change. Rates of change such as speed and acceleration are used all the time when operating a motor vehicle as they tell information as to how far something is relative to time (eg. Kilometers per hour). When driving 50 km/h, this means that I will travel 50 kilometers for each hour I drive (holding all else constant). I also rely heavily on time to manage my daily schedule between classes, cooking, chores, and resting. My average class length is about 2 hours long, or 120 minutes, or 7200 seconds.

Research phase:

According to Numberphile, the Babylonians used 60 because it was a number that divides nicely into 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60. It is the smallest number that divides 1, 2, 3, 4, 5, 6. We use base 10 because we have 10 fingers. Numberphile notes that the Babylonians used knuckles on one hand to count to 12 and grouped them into 5 sets with the other hand. Babylonians were great astronomers and found that it took around 360 days for Earth to go around in a circle. They divided a circle into 360 degrees, which could be divided into 60. Today, we still use the sexagesimal system when measuring angles and coordinates. In Chinese cultures, the yearly calendar is used heavily to determine when optimal harvesting seasons.

Sources:

Numberphile. (2012). Base 60. Retrieved September 17, 2020 from https://www.youtube.com/watch?v=R9m2jck1f90.

Sweeny J.F. n.d. Why Base 60. Retrieved September 17, 2020 from https://vixra.org/pdf/1407.0062v1.pdf.

Why teach math history?

I think integrating math history in my classroom would be a great opening to a topic or motivation to students prior to learning a topic. For example, if I was to teach a chapter on Pythagorean theorem, introducing how Pythagoras discovered the Pythagorean theorem and first used it would provide students a nice connection between history and current mathematics. I would also be interested in incorporating a small project at the end of the year providing students an opportunity to research any math history or run a math history club.

One thing that immediately grabbed my attention while reading this paper was on the last 4 objections to teaching history of mathematics: lack of time, lack of resources, lack of expertise, and lack of assessment. Given the situation public schools face with budget cuts and packed curriculums, it would be difficult to formally integrate this material in a classroom. When I was a student in high school, I felt like each year of math from grades 8 to 12, the amount of content we covered was so much that we had literally no time to lose. If we even missed a day of material, we would already be behind. This forced the teacher to shorten or skip some sections. In grade 12 however, I was in an accelerated math program which covered Pre-Calculus 12 and Calculus 12 in the same school year. We did have time for a year end project which essentially provided us an opportunity to research the history of any math topic we learned from kindergarten to grade 12. Therefore, it is possible to integrate the history of mathematics in a classroom, but this has some constraints such as student interest and time left at the end of the year. Another part that made me stop and think was when the paper mentions how history can be used to bridge math between other subjects. Seeing a connection between math and economics/physics/engineering was integral in my mathematics learning experience so I could see application and usefulness beyond the textbook. This is an extremely large and valuable benefit for students to learn math history, as it provides the opportunity for them to see how the founder of the equation saw a need and application. Towards the middle of the paper, the authors present an example of the area of triangle in Egypt dating back to 2^nd century BC. I would be interested in learning about how the Egyptians deduced or came to the discovery of such a formula. I learned about the Egyptians on a very basic level in grade 7, but never thought about the mathematics they used back then.

After reading this piece, I have been extremely motivated and excited to think about ways to integrate learning of mathematics history in my future classrooms. The last few sections suggest different forms of ways to do this such as re-enacting life of mathematicians or famous arguments in plays, view math history films, or spend some time in outdoors finding patterns in nature or architecture. These three different mediums are of great interest to me because I feel that students will be able to have fun and enjoy learning about math through other methods besides the traditional practice questions and answers.

 

Paper: John Fauvel, Jan van Maanen (eds.), History in mathematics education: the ICMI study, Dordrecht: Kluwer 2000, pp. 201-240.  

What's 12 + 7?