Assignment 1 Reflection

I had a nice time working Clementina and Juma on this project. We worked well together and did a good job of organizing our responsibilities. We each worked on the individual parts of the project (modern, ancient, and extension solutions) so that we would be knowledgeable in each; and then split up the work so that each of us would present a different part of the work. I think splitting up the presentation gave us all an opportunity to demonstrate our teaching skills, and I thought both of my partners did an excellent job communicating their sections effectively to the class.

I regretfully made a bit of an error during the demo section of the presentation. I asked people to work on a simplified version of our problem, but provided the wrong numbers to work with on the board. I think I was able to recover well enough and get people back on the right track, but it wasted a bit of time and I felt a bit embarrassed about it. I think I did a good job of relating the mistake to the actual problem later though (I was accidentally making people come up with a solution that involved non-integers, which is what you had to do in the actual problem.) I think this was a good reminder that it's very easy to get tunnel visioned during teaching and miss when you say something erroneous. This makes sense, since it's quite a challenge to focus on both your presentation skills and also the math behind what you're presenting at the same time. The Vice Principal at the high school I recently visited told us a story about how he taught a whole class without realizing the lights were off, and it reminded me a bit of my mistake: it can be easy to miss obvious things when you're overly focused on something else. I think it's also a good reminder that it's ok to make mistakes though; as long as you can recover without causing too much confusion (and potentially get students to analyze your mistake and think about why it's wrong!)

Assignment 1 Solutions

 Group Members: Chris, Juma, Clementina

Problem 10 (Loaf-Sharing Problem): https://drive.google.com/file/d/1wLGmLuxA6N8R7Os46QbEEJtRo1s8qWoa/view

Modern Solution:

The modern method for solving this problem involves labelling an arithmetic progression with variables a and d, setting up a system of linear equations in terms of a and d, and then solving to obtain the arithmetic progression.

Ancient Solution:

To solve this problem with ancient Egyptian methods, we used the method of false position and Egyptian division.

EDIT:

Extension:

As an extension to the problem, we considered a generalized version of the problem in terms of 4 variables: number of loaves, number of people, and then two more variables from the statement "the sum of the first k terms is equal to (1/m)th of the sum of the remaining terms." Here, we provide the general solution for a and d in terms of these variables.

Homework: Market Scales Puzzle

Response: Tomb of Menna

Some things I thought about regarding the painting before the reading:

-Many of the people in the painting are holding what looks like a modern day ruler; with a dot on top. What could this be? Could it be something equivalent to a ruler in Egyptian society? Most people holding them do not seem to actually be making measurements. Could they have a different use?

-What is the long rope people are holding in the top panel? Why does there need to be so many people holding onto it? Could they not just have two people using it on either end of the rope? Are they measuring something, or using it for a different purpose? They seem to be near a field of wheat. Could they be making a measurement about the land?

-It seems like there is a group of people holding the same "rulers" and also writing something on papyrus in the lower panel. They seem to be sitting near what looks like a large pile of grain. Could they be trying to calculate the amount of grain present? What measurements with the "ruler" might they have made to help with their calculations?

Further questions I had after the reading:

-What's the origin of the two different types of cubit (royal vs. short)? Might multiple people have spread the concept of measuring from elbow to middle finger tip, but with different arm lengths, leading to divergent systems of measurement across Egypt? Or might multiple people have independently come up with similar systems that converged? (Think convergent vs. divergent evolution)

-I checked on my own arm, and I agree that 28 finger (widths) is roughly the length of my elbow to middle fingertip. However, my palm seems more like 5 or 6 finger widths than 4. What was the basis, then, for calling a seventh of a cubit a "palm"? Is my hand just weirdly shaped or are they considering a very specific part of the palm? Also, why did they choose a system that divides the cubit into sevenths as opposed to something like twelfths? (Ok, they probably didn't "choose" the system so much as naturally developed it over time. Dividing by twelfths, like in a foot, is nice for all of the base 6/12/60/360 reasons we've discussed before.)

One thing that surprised me:

-I was shocked to see the use of √2 in the measuring system for the remen! I would not consider using irrational numbers for a measurement system in the current day, since it would lead to a lot of long decimal expansions in measurement conversions. However, I can see the practicality of being able to double the area of a square (or, say, a plot of land for crops) using them. I think they're a very clever tool for this reason, and I could see how they would make more sense in a time when math was understood much more geometrically rather than just numerically.

Response: Babylonian Algebra

 I've been curious about how the Babylonians handled algebra, since I keep hearing about how they more or less "didn't have it." Whenever I come across an ancient problem, it feels a bit like cheating to use modern algebraic techniques, since they had to focus a lot more on geometric and text-based explanations. I think that after seeing how they used algebra in this reading, I'm left somewhat unsatisfied with their version of algebraic manipulation. Most of the steps in the ancient method are more or less the same manipulations as in the modern methods, except it feels like they're missing justification. Modern algebra feels very satisfying to me, because you set up one or more equations related to the question and simply manipulate those equations through a series of valid moves which leave the underlying structure of the equations unchanged. Eventually, this puts the equation in a form that gives you a desired answer (e.g., solving a quadratic equation.) As long as you set up your initial equations correctly with respect to the question, you can have full confidence that your answer is correct. 

Babylonian algebra, on the other hand, feels very sketchy to me. In the Babylonian version of solving the quadratic, it feels like numbers from the question are manipulated in obscure and seemingly arbitrary ways. I suppose the methodology for solving a quadratic would have been figured out and made into a repeatable format by the Babylonians, but the justification for the steps taken and which numbers to manipulate in which ways is not at all clear. In this way, I suppose Babylonian algebra is a bit like a mysterious function, algorithm, or table which you just have to trust spits out the right answer. It feels like this type of algebra lends itself to having specific setups for specific types of problems rather than the inherent flexibility and universality of modern algebraic techniques, which is a shame. It's difficult not to be a prescriptivist about Babylonian techniques, but to me, it really does feel frustrating and restrictive to think about solving problems in this way.

I think it is a real challenge to make many general mathematical statement without algebraic notation. The Babylonians and other ancient cultures place a lot of emphasis on geometry and other related subjects; and it is apparent that the language they have access to supports inquiry of these topics. Take, for example, how the use of the word "square" when Babylonians talk about quadratic equations is much more explicitly about the shape of a square rather than the process of multiplying a number by itself. They effectively mean the same thing, but the wording almost forces a geometric interpretation of the problem. For similar reasons, I think it would be very difficult to try to make statements about different areas of math with fewer geometric interpretations, since they more or less hadn't developed the appropriate words and language to communicate ideas about these topics. Attempting to state many analytical results, such as the formal definition of the limit of a function, would be a nightmare.