Mathematics in Ancient Egypt

Two questions I have:
1. How did Egyptian surveyors make such accurate right angles with such simple tools?
2. How were these methods passed down without clear written rules?

One thing that surprised me:
I was surprised by how precise their measurements were, especially for pyramids and land after Nile floods.

Babylonian Algebra

How could one state a general mathematical principle in a time before the development of algebra and algebraic notation?
Before algebra, people explained general ideas using words, examples, and rules. Instead of formulas, they described steps that worked in many situations.

Is mathematics all about generalization and abstraction?
Not always. Early math was practical and based on real problems. General ideas came later, after people noticed patterns across examples.

Thinking about various areas of mathematical knowledge -- number theory, geometries, calculus, graph theory, etc. etc. -- how could you imagine stating general or abstract relationships without algebra?

Through diagrams, shapes, tables, and repeated examples. Relationships could be shown through patterns and procedures rather than symbols.

More on Babylonian Word Problems

The first thing that made me stop was learning that clay tablets with "problem texts" date back to 2000 to 1600 B.C. This surprised me because I didn’t think word problems went that far back. I always thought they were just an extra activity and not the main part. But thinking back to my own schooling, I remember they showed up early, like when I first learned addition and subtraction, and they’ve been a part of math ever since. The second thing that made me pause was the statement, "The impractical nature of these stories calls into question their referentiality in pointing to 'real world' situations and casts doubt upon the serious practicality of even those more plausible problems." This stood out to me because I used to think word problems made math more relevant. However, as we discussed in class, in real life there is often an easier way, like using a measuring tape instead of a formula. I learned that the point of word problems is not the scenario, but to practice a specific concept. The last thing that made me pause was the statement, "Høyrup's distinction is between Babylonian scribal school mathematics, which aimed to train students in methods available at hand rather than in an understanding of these methods, and Greek mathematics, which aimed to solve problems by extending mathematics and devising new methods." This made me question what we are currently doing in schools. As Skemp's article points out, we mostly teach instrumentally rather than relationally, and a change is needed, but it will take time.

Babylonian-style Base 60 Multiplication Table for 45

2           22,30
3           15
4           11,15
5            9
6            7,30
9            5
10          4,30
12          3,45
15          3

Interesting Resources on the (Babylonian-based) History of Time Calculations, Base 60 and Base 12

The first article, “Why is a minute divided into 60 seconds, an hour into 60 minutes, yet there are only 24 hours in a day?”, explains how early civilizations created and maintained systems for dividing time. It shows the behind-the-scenes process of how people determined how a day should be divided and how seconds and minutes came to be, leading to the system we use today. People measured time using tools such as sundials to divide the day, stars to mark what the article calls “periods of darkness,” or water clocks called clepsydras. The second article, "Babylonian Numerals," is really interesting to read since the Babylonian number system uses only two symbols and is in base 60. We learn how to understand these numerals, how they are separated and expressed in decimal notation, how spacing helped distinguish between two symbols that may look similar but have different values, how there was no representation of zero so an empty space was used, how fractions are represented, and why base 60 was chosen.

When I think about my own experiences, I realize that I still mark years by school calendars, since I spent the majority of my life in school and have only worked full time for 1.5 years. I also think of a year in terms of major events like graduation or trips. Seasons, even though they have official start and end dates, I mostly judge by the weather. Autumn, for example, begins at the equinox on September 22, but if it still feels warm, I think of it as summer. Each season has a different meaning for me. Winter is my favourite because of the festivities and my birthday. Spring feels like a fresh start. Summer is for holidays and vacation. Fall is back-to-school time. For months, I used to mark them by birthdays in school, but now I often think of them by my credit card statement. Weeks were tied to paycheques or assignment deadlines. I don’t focus on exact hours. It depends on my routine. For example, 8am to 4pm used to be my working hours, and the rest was free time. Even now that I’m back in school, I think of hours as commuting, being in class, eating, doing homework, or sleeping. I don’t think “12pm,” I think “time for lunch,” and 7am is when I’ll be on the bus. When it comes to the geometry of time, I see a year as a cycle. The start of the cycle depends on my life situation. When I’m in school, the start is in September, and when I was working, the start was in January. A day feels like a line of events, and a season feels like a segment with its own attributes such as weather or holidays.

Things that Surprised Me:

• Sunlight and dark periods were not considered part of the same day.
• The number 12 is important because it equals the number of lunar cycles in a year and the number of finger joints on one hand.
• The measuring system that used a set of 24 stars.
• A minute could have 60 or 61 seconds.
• That you can count to 60 using your hands. As there are 12 finger joints on a hand. Then using the other hand you can point to one of these 12 sections with one of the 5 fingers.
• A theory that suggests that the Sumerian civilization may have formed when two groups joined, one using base 12 and the other using base 5.

The Crest of the Peacock

The first thing that made me stop was the line about mathematical activity outside Europe being ignored, devalued, or distorted. This line really hooked me, making me curious about why this was the case. How will learning about this mathematical activity help me? What can be done to ensure that contributions outside Europe come to light? The second thing that made me stop was Figure 1.1, The 'classical' Eurocentric trajectory, and the paragraph showing how math in Mesopotamia, Egypt, China, pre-Columbian America, India, and the Arab world has been ignored. It made me reflect on my own knowledge and realize I hadn’t known how many civilizations contributed to math. The last thing that made me pause was Figure 1.4, The spread of mathematical ideas down the ages. As a visual learner, I felt it was a helpful way to explain the history of math. It was simple and easy to follow, and with so much history, a guide like this really helps the reader stay on track.

Why Base 60?

Speculative phase:

The first thing that I immediately thought of when asked why base 60 is time. There are 60 seconds in a minute and 60 minutes in an hour. I also know that 60 is much larger (6 times larger) than 10, and that 10 is a factor of 60. In our Monday class we discussed how there are 365 days in a year (except when it is a leap year) and that there are usually 30 or 31 days in a month (except February who loves to be special). From these observations we know that 365 is really close to 360 (a multiple of 60) and 31 is really close to 30 (a factor of 60). However, I wonder how this observation connects to base 60.

Research phase:

After doing research I discovered some cool facts about the significance of 60. Two of them I had already thought of are that 60 is a highly composite number and its role in time. I completely forgot about its role in circles (for example, a circle is divided into 360 degrees, which is a multiple of 60) and in the unit of degrees that are subdivided into minutes and seconds. I also learned about its role in languages. For example, in French there is a base 60 counting method (for example, 72 is called soixante-douze which translates to sixty-twelve in English). On the cultural side, 60 is significant in China and Japan because age 60 indicates a complete cycle. Since there are 12 animal signs and 5 elements, which means that there are 12 × 5 = 60 unique combinations. In these two cultures turning 60 is a time for celebration, and in Japan this celebration is called Kanreki. In Hinduism 60 is also important because it represents the completion of half the supposed human lifespan (120 years), and they have a celebration for it called Shashti Poorthi.

Why teach math history?

I would like to incorporate math history into my own teaching because I believe it would emphasize the importance of a concept and provide more context. Moreover, it would make the overall subject seem more ‘human’ and not so technical and mechanical. Also, there are some really cool stories associated with certain mathematicians and concepts. Off the top of my head, the easiest way I can incorporate math history into my teaching is by presenting a short paragraph about the concept, such as who created it and how it came to be, before starting the lesson. Another method I can think of is creating a project where students research the history of specific topics or mathematicians and present their findings to the class. This helps students develop important skills and adds a nice change of events in the class.

What first made me 'stop' was realizing how long incorporating history into math education has been advocated, along with the many objections to it. Most objections seemed logical, and I could see myself agreeing as a student. Though this makes me wonder if those objections explain why I had so little exposure to math history in high school. I can only recall one instance of learning it before the concept itself. This lack made university harder when classes focused more on history (problem, necessity, context) rather than what I used to call the ‘meat,’ the solution. One line I loved from the article was “No mathematical idea has ever been published in the way it was discovered. Techniques have been developed and are used, if a problem has been solved, to turn the solution procedure upside down." The objection I disagreed with was (O10) lack of assessment. I don’t think math history should be graded. Its purpose is to create motivation and engagement, and testing it might cause students to memorize facts. I remember in Grade 12 when my teacher told the story of Gauss and the sum of numbers. I was able to relax and listen closely because I knew it would not be tested. One part I still question is the reconstruction step, where history enters implicitly or explicitly. I understand the explicit part, but not the implicit part, and I would like an example. If we are looking at ‘the historical development from the current stage,’ how do we introduce past history?

Hello World!

This is my first blog post! Here is a picture of my absolute favourite Barnes & Noble location. This location is in New York City. Oh, and the best part, it is four stories!