Babylonian Algebra and Mesopotamian Scribes - Thoughts

Before the development of notation, it is clear that math was still being done rhetorically, in the fashion of Babylonian algebra using words such as breadth and length to explicitly write out what we now abbreviate to symbols like ‘l’ and ‘w.’ When I think about how this can apply to more principle generalizations, I feel this may actually ease the transition from the specific to the general, since the principles of math themselves are often written in words to describe the key concept. For example, we may know the formula for the area of a rectangle using the symbolic shorthand for the sake of utility, but in stating a general principle on how we ought to view the area of a rectangle and how we can find it, we would certainly state this in terms of things like the rectangle's width and length and their relationship, rather than saying the rectangle's "l and w."

This could tie into how students learn in our classrooms today, since we may now see why it is actually rather ineffective or confusing for students to first introduce this short hand or symbolic methods of algebra and expect them to conclude from here the general principles which are free of the symbols and instead speak to the greater concepts and relationships that actually drive why things work. Perhaps this should inform how we teach, in that students likely could develop firmer/deeper understandings of math, rather than just the ability to apply memorized symbolic formulas, if we moved them from rhetoric, to syncopated, and only lastly on to symbolic. This way, they learn algebra in plain words which they already know, then begin minor substituting or mixing of words and symbols, until they are fully comfortable moving to symbols entirely, as it is now simply a representative shorthand for a broader idea they have already learnt.

I find this also speaks to why mathematics is not all about generalization and abstraction, even if these categories help to cover or sum up much of the broad concepts of math. Generalizations and abstractions often have the key benefit of being highly transferable within the broad topic and they help us make connections between different areas as well. However, mathematical application is an important part of math that lies in clear contrast to abstraction. Additionally, highly specific proofs can be very important for understanding or 'proving' areas of math, even though they are far from being general or transferable topics. For other areas of mathematical knowledge, like geometry and number theory, I therefore think it is still possible to state general or abstract relationships without algebra, as we can see that even many proofs in these topics are already done only in words, drawings, and ideas, rather than symbols and algebra. For example, many geometry proofs and principles can be discussed without algebra, in that we can construct the ideas instead with basic methods of compass and straight edge. 

Base 60 multiplication table for 45

 

Left out 9 x 5, though it is a duplicate. From further investigation, 8 should also be possible using more fractions, since 45/8 = 5.625. This would require 8 x (5, 37, 30), given that 0.625x60=37.5, i.e., 37, 30. 

Reflecting on how we measure time

 While reading the articles, I related with the Egyptian usage of the sundial as a way to measure time and divide the day into smaller parts in a circular format. In particular, when I picture time, such as hours or minutes, I always have the mental image of the round, analog clock in mind. Additionally, I even picture the year split into months and seasons in this circular form, split into 12 sections. However, in my conceptualization, September is at the top in the 12-o-clock position. This is likely influenced by many years in school, causing me to view September, the start of the academic year, as the ‘top’ or starting point, rather than the typical new year in January. My conceptualization of the seasons is simply layered on top of this mental image, by portioning off 4 quadrants aligning with the quarters of the hour. In my mind, these months and quadrants also have a general color assignment, influenced by the physical colors or mental color hue (due to association with heat v.s. cold) I perceive in the world around me at the time of each season. 

When I compare the two articles, I did notice there may be an inconsistency in the discussion of the historical usage of base 12. While the MacTutor site states no major civilization used base 12, the article written by Lombardi for Scientific American states that civilizations, like the Egyptians, used the duodecimal system rooted in base 12, likely in connection to their usage of the sundial. While I knew of the use of the cesium atom to define the second, something that surprised me was the need to have a small handful of minutes each decade that have an extra second, to maintain the agreement between atomic and astronomical time. This made me reconsider how we measure/view time, as it contradicts the idea that time is something completely objective and unalterable. Requiring this manual addition of a second to maintain the agreement shows that even time has its exceptions and is ultimately still an arbitrary assigning of rules and definitions by people. 

The Crest of the Peacock reading notes

One thing I was surprised by in this reading was the quote suggesting that math simply lay dormant after the decline of Greek civilization until the Europeans were able to revitalize it. I knew the tendency of Eurocentric writers to attribute significant responsibility for mathematical development to the early Greeks and later Europeans alone. However, the rather blatant and outright ridiculous suggestion that other areas on the globe simply did not engage in math or develop it when the Greek's were unable was quite shocking from a fairly contemporary writer. 

Such claims are additionally ironic when considering the Greeks themselves seemed happy to acknowledge the role of other areas' scholars in teaching them or contributing to their work. In particular, I was surprised to learn how some scholars from Plato's academy found refuge in Jund-i-Shapur (in modern day Iran) to continue their study of astronomy and mathematics and that Pythagoras may have also ventured as far as India in his own search for knowledge. Both examples seem to speak to the idea that the actual Greek scholars of the time were more than open to knowledge from these other sources and thus likely would have attributed any findings to the places where they gathered it. 

One other point that surprised me and that I found insightful was on the flow of Chinese inventions toward Europe. While I was already aware of some of the specific inventions, such as paper, the compass, or the segmental arch-bridge, I had not previously thought about it in the context of the flow of math knowledge as well. In particular, the idea that certain mathematical ideas would likely have been transferred with this technology because of its role in inventing them, thereby highlighting this direction of knowledge transfer into Europe from outside, was new and interesting to me. 

Integrating history of mathematics in the classroom: an analytic survey - Response

Before reading the article, my general perspective is that math history should be incorporated into teaching, as I feel it provides students with context for the content being learned, which may help those who wonder why seemingly unconnected, abstract math topics matter or relate. For example, the appearance of new problems and solutions, or even entire branches of math, were often consequences of both successful and unsuccessful attempts at solving previous problems in older areas. As such, I feel it could be worth incorporating some math history with each new unit as it comes, and explaining each subsequent unit’s history in reference to any already discussed. This can help students trace the topics through time and purpose, which may stimulate interest or give the topics more real-world value. However, I am curious about more engaging ways to include this history, beyond just describing some of it with each unit's content. 

One point that I paused on while reading was the idea that historical contextualization of math is made difficult if students lack general understanding of historical time periods. I do, in part, agree, given that the history of math is hugely intertwined with the history of related disciplines, such as philosophy or the sciences, all of which were influenced by social and political aspects of the time periods as well. Given that many of the key figures were polymaths themselves, it may also be difficult to discuss the history of their mathematical work without some reference to their other pursuits or ideas. However, if student learning is being supported from all angles and their other classes, like the sciences, are also making this effort to discuss a subject’s history, this would instead be an incredible opportunity for students themselves to make connections between the history of various disciplines and the key figures they are learning about. 

I also paused to study the in-depth example given of a possible historical package for grade 8 students studying Pythagorean theorem. I particularly liked the idea of using history as a basis with which to present various proofs of the same theorem, as I feel that this combined history with another useful part of math education: displaying multiple methods of arriving at the same answer. This has always been such a fundamental idea in math, that my own teachers always encouraged, and showing that it even applies to the theorems that students were taking as absolutes could be a powerful reminder. The subsequent step of bringing in the works of various cultures and discussing the historical narratives surrounding the theorem then flows seamlessly from the above. I was struck by how this portion has the added benefit of characterizing math as a human endeavor and possibly even helping students feel more personally represented in the discipline. 

The main change from my prior reasoning on the need to teach math history is the added idea of using history to provide role models for students through learning about the human aspects of math. My reasons before were more clinical, with the desired outcome of simply furthering students’ knowledge. However, the article helped bring to light how showcasing the real people and their trial and error efforts behind what students may be viewing as rigid, abstract, absolutes, could be inspiring for the student or at the very least comforting to know that the process of math has always consisted of failure and re-trying. I also appreciated the suggestion of incorporating math history through group research projects on self-selected topics of interest. One of my pre-reading questions was on interesting ways to incorporate history into classes. This type of interest-guided group project could be seen as more fun/engaging by some, while also helping students understand that math can and should be a collaborative subject. 

Hello World

Hi again everyone!

Looking forward to becoming math teachers alongside you all!

Mt. Rigi near Lucerne.