The Market Scales Puzzle and Connections with Ancient Egyptian Mathematics

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To help students understand mathematics more deeply, this puzzle could be extended by reversing the direction of the problem. Instead of having students find the weights that model the solution, they could be given a set of weights and asked to determine how many consecutive whole numbers, beginning with 1, can be measured using them. This would encourage students to identify patterns and develop a deeper understanding of why certain weights work.

This puzzle connects with number theory and bases that I am familiar with because, for the one-pan scale, each weight can either be used or not used. This is similar to the binary number system, where each digit is either 0 or 1. The weights also follow powers of 2 for the one-pan scale and bases of 3 for the two-pan scale. These bases are similar to what students know and practice in class, which forms connections between these answers and answers from previous math problems that involve the same bases.

Did Mesopotamian Scribes Have Algebra?

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After reading about Babylonian algebra, I found it interesting to see how differently mathematical problems were represented compared to how we solve them today. I was especially interested in the terminology they used, such as “ush” for length, “sag” for width or breadth, “asha” for area, “sahar” for volume, and “lagab” for a square. Since I am not familiar with these terms, I found the problems much harder to understand at first. I also found it interesting that unknowns were not represented by symbols such as x or y. Instead, the unknowns were described using words such as length or breadth. This shows how mathematical relationships could be communicated through words before the development of the symbolic algebra we use today.

This made me think about how mathematics does not necessarily need symbols in order to express general ideas or relationships. The Babylonians were still able to identify patterns, solve problems, and generalize mathematical relationships, even though they did not have the same notation that we have today. Their methods are an example of rhetorical algebra because they communicated mathematical relationships using words.

I also think this connects to how students learn algebra today. Students often begin by describing a relationship in words or working with specific examples before learning how to represent it using variables and equations. In this sense, students may go through similar stages of rhetorical and syncopated algebra as they develop their understanding of symbolic algebra.

Overall, I find it inspiring to see how mathematics can be interpreted and communicated in so many different ways. The Babylonian use of base 60, their notation, and their lack of a zero symbol are very different from the decimal system we are used to today. Although their methods can seem more complicated to us, they show that mathematical reasoning and abstraction can exist without the symbolic notation we rely on today.

History of time calcualtions, base 60 and base 12

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When relating my personal experiences and understanding of time to its history, I think of a year as 365 days, or one complete trip around the Sun. I view each season as approximately 1/4 of a year, or 1/4 of the Earth’s revolution. A month represents roughly 1/12 of a year, and an hour consists of 60 minutes, which I visualize as one complete revolution of the minute hand on an analog clock.

One inconsistency I noticed between the articles is that the Scientific American article discusses Egyptian timekeeping and how Babylonian base 60 mathematics influenced the divisions of time, whereas the MacTutor article focuses on the Babylonian number system and the uncertainty surrounding why the Sumerians originally adopted base 60. Although both articles explain the significance of this system, Scientific American emphasizes the usefulness of base 60, while MacTutor highlights that its original purpose remains unknown.

I was surprised by how many different mathematical and historical developments contributed to the ways we measure time today. I had never really questioned why we divide an hour into 60 minutes or why a day consists of 24 hours. These measurements have always seemed natural to me, and I’ve never thought deeply about them. Learning about the influence of different civilizations made me realize that the way we organize time is not simply based on nature, but is also shaped by mathematical systems and cultural practices. I also found it interesting that something as familiar as looking at a clock has connections to ancient number systems. These articles motivated me to reconsider how we use these measurements, considering how they developed over time and could have been interpreted differently.

Babylonian Tables

 

The Crest of the Peacock

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    While reading The Crest of the Peacock, I found the historical development of mathematics to be very intriguing, especially the neglect of many contributions from non-European cultures. I was surprised by the discussion of the neglect of Arab contributions to the development of European intellectual life in general and to mathematics specifically. This neglect is described as a serious drawback of the classical view of the history of mathematics. This surprised me because I had always learned about the development of mathematics as being largely centred around the Greek people, without knowing much about the contributions of Arab scholars. It made me question how much of what I have learned about the history of mathematics has been shaped by a singular perspective.

    I was also surprised by the idea that the Greeks believed mathematics originated in Europe. I found the discussion of early Greek travellers particularly interesting because it showed that the Greeks themselves had contact with other civilizations and were exposed to mathematical knowledge from outside of Greece. This made me think about how mathematical knowledge is not necessarily developed independently within one culture, but can be influenced by travel, communication, and the exchange of ideas between different civilizations.

    Additionally, I was surprised by the archaeological evidence showing both cultural and commercial contacts between Mesopotamia and the Indus Valley. This was surprising because it was also mentioned that there is no direct evidence of mathematical exchange between the two areas. This demonstrates how much uncertainty there can be when trying to understand the history of mathematics. We can have evidence that different civilizations interacted with one another, but that does not necessarily tell us exactly what knowledge was exchanged. It made me realize that what we know about the history of mathematics is based on the evidence that has survived, and there can still be different interpretations that we are unaware of.

Why Teach Math History?

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    I believe math history should be incorporated into my teaching because it highlights the importance of understanding how we arrived at today’s mathematics curriculum. One way to incorporate history could be to show students how a mathematical problem was solved in the past and then compare it to how we solve it today. This would allow students to see how mathematics has developed and how new methods have made problem-solving more efficient. However, before reading the article, I was concerned that there may not be enough time to cover historical content while also meeting the curriculum expectations, raising the question, “How important is it to teach math history?”

    While reading, one argument that stood out to me was the idea that history can sometimes be “torturous and confusing” rather than enlightening. I somewhat agree with this because historical mathematical methods can be multiple pages long, while modern methods may only take half a page. However, I think this could be addressed by presenting historical methods as a way to show students how mathematics was done in the past, followed by the modern method so they can compare the two. The 10th implementation, plays, also stood out to me because students could reenact historical mathematical milestones or arguments, making the learning experience more interactive and memorable. Additionally, the 12th implementation, outdoor experience activities, made me pause because it connected to our own experience of having classes in the Orchard Garden rather than a typical classroom. I am excited to explore how different forms, shapes, and areas found in nature can be connected to mathematics and provide an experiential learning opportunity beyond typical pen and paper activities.

    After reading this piece, I am more open to incorporating math history into my teaching because I learned that there are many interesting and engaging ways to connect history to mathematical content. Previously, I was concerned about not having enough time and worried that students might find the historical aspects boring. However, the different classroom implementation ideas showed me that math history does not have to be a separate lesson or take up a significant amount of class time. Instead, it can be incorporated through interactive activities, discussions, and connections to the curriculum, which I now believe can make mathematics more meaningful and enjoyable for students.

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