Response: More on Babylonian Word Problems

 I enjoyed Susan's perspective on Babylonian word problems being more about exercising methods that the Babylonians had developed rather than for practical problem solving purposes. It gives me a sense that some Babylonians were simply very excited to share what they'd learned! I agree that the presence of infeasible numbers in some of the real-world examples seems to imply that the word problems were more about doing the math and thinking through the process rather than obtaining a solution and using it to do something useful in the physical world. Thinking about the overlap between pure and applied mathematics made me consider the occasional times where I've applied some of my math knowledge outside of the classroom. Whenever this happens, it's often because I see something with potentially interesting combinatorial structure and ponder what questions I could ask about it. Then, I try to see how I can use the methods I've learned to answer these questions, usually about how many different ways there are to arrange some group of things. For examples like these, I'm doing the same process of seeing places where I can do math for the sake of doing math rather than truly needing the answer to some question for a practical reason and then using math to give me the solution.

I was also fascinated by Høyrup's suggestion that Babylonians did lots of word problems because they didn't really have any other "non-applied" ways to exercise these methods, since no formal system of algebra was around yet. I'm reminded of the questions we've seen about solving for the side length of a square where you are given a sum in terms of its side length and area (which are really just about solving quadratic equations.) I think it's interesting that what would normally be expressed as simply an algebraic equation nowadays had a much more geometric interpretation back in Babylonian times through these word problems. While they may not all be extremely practical, I still think that word problems are quite useful in getting students to be problem-solvers and think about ways in which they might be able to apply what they've learned in different areas.

Homework for Sept 17: Multiplication Table for forty-five

 A multiplication table for 45. Arabic numerals on the left side, and the corresponding Babylonian numerals on the right.

Homework for Sept 15: History of Time Calculations

I found both of these articles to be interesting reads about the sociological aspects of choosing a base for a number system. The first article, I found, had more of a focus on the practical applications of using certain numbers to divide up time in different ways. I found the section about how sundials could only be used to track time in the day to be quite thought-provoking. I hadn't given much thought to this issue, so it makes sense in this context why day and night were considered "two opposing realms" rather than part of the same day. It's also interesting how this lead to hours being longer on days with more sunlight, since an hour was just a 12th of the sunlight on a given day. Getting around the issue of sundials not working at night by tracking stars in the sky I thought was very clever, and reminded of how many early cultures had remarkable understandings of the movements of celestial objects while only having very limited tools.

The second article seemed to be more interested in the question of why the Babylonians chose to use base 60 over another base, and had some interesting remarks about how they may have gotten there. I agree that it was unlikely that the Babylonians collectively "chose" to use base 60, and rather that it slowly became common practice over time with repeated use (like a how a new language develops). I found the theory that it may have arisen as a combination of a base 12 and base 5 to be quite interesting, but I am skeptical that this is true because of the more natural combination of 6 and 10 seen in the tally marks of the numbering system. I did find the mixed method of counting 12s on one hand and 5s on another to be neat, though. The article also claimed that no major civilizations used base 12, which seems to contradict the first article's claim that the Egyptians used duodecimal. Perhaps Egyptians really only used base 12 for time-keeping? Either way, the article provided some interesting questions to think about regarding how a culture would decide on a numbering system.

Response: Crest of the Peacock

Sorry this is late!

I enjoyed learning more about the history of math from a non-Eurocentric point of view. I wasn’t aware of much of the European revisionist history of the origins of mathematics, but I can’t say I was entirely surprised to hear that information obtained from colonized countries was either ignored or claimed to be a wholly original discovery (or from the Greeks). The idea of scientific progress being a “uniquely European phenomenon” made me uneasy, especially after reading the two Eurocentric quotes from Rouse Ball and Kline.

There were a couple other tidbits mentioned about the math used in various cultures that I thought were interesting. I was intrigued to learn that the Babylonians had observed and used the Pythagorean Theorem, but seemingly did not prove it. I wonder if they had in fact proved it but their proof just hasn’t been found, or whether they had simply made enough observations to be confident that it was true. If not, I wonder what their possible attempts at a proof may have looked like. As well, I noticed that when interpreting figure 1.3 as a graph, there are no “source vertices”, i.e. all of the nodes have at least one arrow pointing towards them (In contrast to figures 1.1 and 1.2, which do). This demonstrates the interconnectedness of math history; notably how many different cultures came up with their own math as well as borrowed from others.

Why Base 60?

Full disclosure; I have already done a good bit of digging into the potential benefits of using different number bases. Not that it would really be possible to implement, but I am personally an advocate for using a base 12 number system for similar reasons as base 60.

60 is a convenient number because of how nicely it factors. It is equal to the product 2*2*3*5, which makes it the smallest number which can be divided by 2, 3, 4, 5, and 6 without leaving a remainder. Because of this, if you have a group of 60 things, it's very easy to split them up evenly in many different ways. For the same reasons, division also becomes much cleaner in base 60. For example, 1/3 becomes 0.(20) and 1/6 becomes 0.(10) in base 60, where as both of those decimal expressions repeat infinitely in base 10. The fraction 1/4 becomes 0.(15) as opposed to 0.25, which means it can be written with one less digit (here, the number in brackets is considered to be a single digit.)

Two of the main areas in which we use 60s in our lives are in telling time and for rotation. We have 60 seconds in a minute and 60 minutes in an hour, which means we can easily divide up segments of time into thirds, quarters, fifths, etc. without requiring any decimals. There are also 360 degrees in a circular rotation, which is 6*60. This similarly allows us to express divisions of the unit circle with angles without requiring decimals. In general, it is useful to be able to express divisions of units such as time and rotations simply, which 60s help us with.

Looking into the number 60 deeper, it is what is known as a "superior highly composite number", which means that 60 has a lot of factors relative to its size and the number of factors of other numbers of a similar size. Some other numbers in this category include 2, 6, 12 (a dozen of something), 24 (the number of hours in a day), and 120. It's no wonder that these numbers also appear in places where you would want to divide things into equally sized groups. For example, having a dozen donuts means you can split them evenly between a group of either 1, 2, 3, 4, 6, or 12 people!

Response: Why teach math history?

Prior to reading the article, I admit I was skeptical about the benefit of incorporating math history in the classroom. I figured that math was a fairly rigid subject that could be taught and motivated without historical context. I assumed that there was already so little time in a class to cover all the relevant concepts that it would be difficult to spend a significant amount of time on the historical aspects as well, and that this side of math might bore some students that would otherwise just enjoy learning the concepts. The article ended up addressing both of these qualms, which I appreciated.

I enjoyed the article's explanation of how adding historical background to certain topics in math could act as a motivator for students. A common question students will ask is "when will I use this information in the real world?" Like is described in the article, I think it would be powerful to be able to point to the specific origins of some of these concepts to show students the possible benefits of learning the material. In addition, I agree that it could be very useful to pull back the curtain on some areas of math and how this information arises. Prior to university, the subject of math always felt like a bit of a black box to me in that I didn't know where certain formulas came from or how one would possibly go about proving them. I think it would be helpful for students to at least get an idea of what the process of discovering new math looks like to make the subject less mysterious. As the article mentions, it could also make the subject more inviting to learn that the process of finding these results is often much, much messier than the presentation of the results in a classroom.

Reading the article has made me reflect on the importance of historical context on math and how it can be used as a motivator to aid the learning of students. I agree that the history of math belongs in a math classroom, and I hope to be able to share some of the many interesting stories from math history with my students where it would benefit the class. More generally, it has also made me think more critically about how I can best inspire students to think about where the math I teach in a classroom comes from, and how they can use these ideas to tackle interesting problems they find outside of class.

Hello World!

Happy September :)