Babylonian Word Problems

    I do not like the binary view of pure and applied mathematics, but instead, I prefer to see all math as applied math but with different time frames: applied math as math with immediate practical applications and pure math as math with future applications. The foundation of my math education was learned relationally with an emphasis on mathematical modelling, so my interpretations of anything in mathematics rely heavily on contemporary algebra. My interpretation of practicality is in regard to a close connection to the realities of the world. This does not, however, restrict numbers to be small and manageable, but is linked to the context of the problem. As such, the numbers look very different when calculating the amount of paint you need for your new accent wall versus the economic cost of shutting down a country’s economy for two weeks but are nonetheless practical applications of math. My interpretations of abstraction focus on the ability of the math to transfer to different contexts, often requiring contemporary algebra for representation. 

    Gerofsky (2004) suggests that Babylonian mathematics was practical in nature. Without a prior foundation to draw on, the Babylonians’ derivation of math from the natural world does not come as a surprise that the supposed lens through which they viewed mathematics was practicality. Greek and other mathematicians, now had a foundation to build off, allowing for the pursuit of pure mathematics. However, this does not happen unless civilizations meet their basic physiological and safety needs. As is the case today, funding for mathematical research is driven largely by economic factors. 

    As a grade K-12 student, I believed that math did not have value unless I perceived it to have a practical use for me at that time or eventually in the future, which led to my deep appreciation for a good word problem. I defined good word problems as being closely connected to real-life scenarios and whether it was able to enlighten me about parts of how the world operates. It was common practice for my teachers to reserve word problems as the final step up the learning hill and I often found myself frustrated by the contrived nature of the word problems which eventually led to my disinterest in learning math before taking my first calculus course. As for Babylonian word problems, I appreciate their contextualization of the problems to things relating to their livelihood as a civilization, like agriculture, commerce, law, and the military. I would like to believe that was a motivating factor to learn mathematics and use math to contribute to society. 

 

References

Gerofsky, S. (2004). Chapter 7: The History of the Word Problem Genre. A Man Left Albuquerque Heading East: Word Problems as Genre in Mathematics Education (First Edition, pp. 113-120). Peter Lang.

Crest of the Peacock

    The first thing that surprised me was the etymology of the word algorithm. The modern use of the word is so heavily tied to computational mathematics that its origin is hard to believe: a poor translation of al-Khwarizmi’s Algorithmi de numero indorum leading to the misattribution of the Indian number system to al-Khwarizmi, where “algorithm, or later, algorithm” came to refer to any scheme using Indian numerals (Joseph, 1991, p. 11). I find this to be a particularly interesting case in archaeolinguistics due to the technical nature of math and the importance of definitions in modern mathematics. 

    As a non-European, I am aware of the eurocentric bias of historical narratives. However, in my ignorance, I have rarely been critical enough to look past the eurocentric lens. I was delightfully surprised by including Arab contributions to science contrasted with an incorrectly attributed European counterpart (Joseph, 1991, p. 7). There are high levels of reverence in the western scientific canon for Darwin’s groundbreaking ideas on evolution and Newton’s discussion on gravity, both of which can be traced back to ibn Miskawayh and al-Khazin, respectively. The recognition of these facts has provided a strong motivating force for me to continue educating myself on other scholars' contributions to math and science.  

    The third thing I found surprising is the general disregard for math and scientific advancements made during the European dark ages by the Arab world (Joseph, 1991, p. 1-22). I found this to be increasingly jarring to read as the chapter progressed. It is such a clear example of the erasure of non-European history, and it made me think about how my high school is still using the same dated textbooks that I used, and how teachers are continuing to reinforce, in students, the ideas of European supremacy. I think math is often considered an objective discipline of study, but reading this chapter has so clearly revealed that it is not.  

References 

Joseph, G.G. (1991). The History of Mathematics: Alternative Perspectives. The Crest of the Peacock: The     Non-European Roots of Mathematics (Third Edition, pp. 1-22). Princeton University Press. 

Base 60

    Using 60 instead of 10 as for a number notation system would be convenient in situations where space was a major issue since the number of “digits” would represent 60n and not 10n. Additionally, compared to 10, 60 has 3 times as many factors: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60. Though most notably, 60 is divisible by 3, avoiding the common floating point decimal notation of ⅓.  

    In our daily lives, the most apparent use of 60 is in how we tell time: 60 seconds in 1 minute and 60 minutes in 1 hour. 60 is also embedded in geometry concepts, like circles to equilateral triangles. A less obvious place it shows up, perhaps, is its involvement in retirement in Canada. For most Canadians, 60 is the earliest at which you can start receiving CPP payments. It is possibly an apt analogy for those privileged enough to do so, reaching the age of 60 can be viewed as entering the next stage in your life. 

    Something common in Chinese culture which was taught to me by my parents is a method of counting with your fingers that adds up to 60 instead of 10. Using your thumb to point and keep track, your remaining 4 fingers have 3 segments each, or 12 in total. Once you’ve reached the last segment, or 12, you start over from the beginning while incrementing using your other hand, so that when 5 fingers are raised, it represents 60. 

    Perhaps 60 is significant in so many situations involving time and space because of the Babylonian's cultural legacy, revealing their development or their influence on other cultures in the development and advancement of astronomy and timekeeping. 

    According to Conner and Robertson (2000), 60 could have had some relation with the number of days in a year, as 360 is close to 365. A note of interest is that the hexagon inscribed in a circle can be divided into 6 equilateral triangles, the fundamental geometric building block of the Sumerians. Another note of interest is the theory that the Babylonians happened upon two earlier cultures that used base 5 and base 12 and combined the two systems to create a base 60 number system. I find this last point to be particularly interesting since it has a connection to the alternative finger counting technique I mentioned earlier: counting up to 12 on one hand and up to 5 on the other. 

Lamb, E. (2017, September 12). The joy of sexagesimal floating-point arithmetic. Scientific American Blog Network. Retrieved September 11, 2022, from https://blogs.scientificamerican.com/roots-of-unity/the-joy-of-sexagesimal-floating-point-arithmetic/

O'Connor, J. J., & Robertson, E. F. (2000, December). Babylonian numerals. Maths History. Retrieved September 11, 2022, from https://mathshistory.st-andrews.ac.uk/HistTopics/Babylonian_numerals/ 

Why Teach the History of Mathematics?

    My current teaching philosophy is grounded in project-based learning and constructivism. I think incorporating math history can provide a necessary holistic view of mathematics, one that is built around student knowledge and intuition, rather than being overly prescriptive with seemingly obscure axioms and theorems, as it is commonly taught. For example, topics in geometry can be introduced to students via navigation activities using measurement tools and astronomical objects. Through the activity, students can think through problems and construct meaning from their observations and reflections. Relating the activity to its historical contexts and formalizing language can occur afterward when students encounter problems with communication while using different names to refer to the same thing. Ideally, his process would be able to show that math is a collaborative effort to come to an agreement.

    Math, as it is often taught, exists as a decontextualized set of rules and algorithms, regardless of teachers’ well intentions, so it is not difficult to agree that integrating the history of mathematics in the classroom is indispensable for its understanding (Tzanakis et al., 2002, p. 201). I believe this is one viable solution to the multi-faceted problem of math education. Placing problems and ideas in their historical contexts can provide a much-needed narrative for students to anchor their learning and develop their skills in critical and creative thinking, problem-solving, and collaboration. For example, I struggled with the game theory concept of pure and mixed nash equilibria until we discussed the ‘Battle of the Bismark Sea’, which involved an interactive activity grounded in historical significance and a logical narrative that explained the decisions of the US and Japanese militaries. Game theory benefits from its comparatively recent development as a mathematical discipline and its rich repository of historical real-world applications, so I am excited to see student research projects on “The early development of game theory” (Tzanakis et al., 2002, p. 215) in a high school setting. I find research projects based on math history to be an intriguing idea since it provides the flexibility of student choice, while also allowing students to explore the process and development of mathematical ideas.

    After reading Tzanakis et al. (2002), I have come to the conclusion that integrating math history in the classroom can take a multipronged approach. Instead of designing entire units rooted in math history, I can also use other approaches like worksheets and historical snippets when appropriate and relevant.


Tzanakis, C. et al. (2002). Integrating history of mathematics in the classroom: an analytic survey. In: Fauvel, J., Van Maanen, J. (eds) History in Mathematics Education. New ICMI Study Series, vol 6. Springer, Dordrecht. https://doi.org/10.1007/0-306-47220-1_7