Solution to: The market scales puzzle & connections with ancient Egyptian mathematics

 

Rethinking Word Problems (For Sept 28th)

The word problems at the beginning of the reading immediately caught my attention. They looked interesting, so I started trying to solve them, although I got stuck on the second one. At first, I thought I could approach some of the questions without advanced mathematical knowledge. However, knowing the operations involved did not mean I immediately knew how to find an answer. This made me wonder how much of the difficulty came from the mathematics itself and how much came from understanding the situation and organizing the information.

I was also drawn to the connection between Babylonian word problems and the society around them. The problems included situations involving agriculture, trade, law, and military administration, which seemed closely related to the work that scribes were preparing to do. At first, I liked that their learning connected to the world around them. As I continued reading, however, Dr. Gerofsky’s discussion challenged that first impression. Some problems involved unrealistic measurements or questions that people would probably never need to answer in their daily work. A familiar setting did not necessarily make a problem practical. These situations could also provide a way to explore mathematical ideas beyond their everyday uses.

Ainley’s point that problems can have only a superficial connection to the real world especially resonated with me. Growing up, I often felt that our textbooks were outdated. Many of the situations seemed to belong to the 1980s and had little connection to my own life. When I encountered a word problem, I thought about what we had recently learned in class and tried to apply the method that seemed to fit. My attention was mainly on recognizing the expected approach. I felt that I was just solving math word problems, with little sense that I was solving a problem connected to real life outside the classroom.

This experience connects closely to the questions at the end of the reading: do we use word problems to help students understand mathematics, or mainly to practice methods they have already been taught? It also made me reconsider my first thought that I could improve problems simply by updating their settings. Replacing an old situation with something familiar to students might make a question more appealing, but would it change how they approach it? If students are still trying to guess which formula the teacher wants them to use, a more modern story may not make much difference.

In my own teaching experience, some students have struggled to find a starting point when a word problem looks different from the examples they have done before. I wonder whether this is partially related to how we teach them to approach these questions. If most of their practice involves recognizing a type of problem and repeating a procedure, developing their own approach may feel unfamiliar. A familiar context could help them picture the situation and suggest strategies, but that connection alone is not enough. I would also want to give students space to discuss their ideas, try different approaches, and explain why a method makes sense.

Problem wording matters as well. Unclear language can make it difficult for students to understand what is happening or what they are being asked to find. This reminds me of the AP Calculus free-response questions I have worked with, where I appreciate the care taken to describe the information and the task precisely. Still, clear wording does not automatically make a question meaningful. In my future teaching, I want to think more carefully about the purpose of each word problem. Whether I am asking students to try a method, explore a mathematical relationship, or investigate a practical situation, I want the problem to support that purpose and help them understand what they are doing.

(I used ChatGPT to help translate my ideas from Chinese into English. The personal experiences and reflections are my own. )

Algebra Before Letters and Symbols

Even after two weeks, I am still amazed that the ancient Babylonians used a sexagesimal system and solved algebra problems without the symbols we use today. They used words such as “length” and “breadth” to describe unknown quantities and explained their methods through instructions. This made me realize that people could communicate general mathematical ideas through words, diagrams, and step-by-step examples that could be applied to other problems. However, I actually find these written explanations much harder to understand. In Example 4.8, I struggled to follow the wording and the text solution, but when I looked at the modern notation, the relationships became clear immediately. Perhaps this is because I am so used to symbolic algebra. Although symbols aren't necessary to express mathematical relationships, this example made me appreciate how much they help me organize my thinking and write more easily.

I do not think mathematics is only about generalization and abstraction. It is also about solving practical problems, exploring ideas, and enjoying a challenge. During class, Susan mentioned how Babylonian grain measurement reminded me that mathematics connects to everyday needs, such as figuring out how much food is available. I also enjoyed the suggestion that some problems might have been a way of “showing off” mathematical skills. It makes the people behind these ancient texts feel more human to me. At the same time, practical applications can involve abstract thinking. A problem about grain can build relationships that also apply to completely different situations.

There are many ways to express general relationships without algebraic symbols. In number theory, we can say, “Adding two odd numbers always gives an even number,” and explain why by arranging objects into pairs. In geometry, we can use a diagram to show that a triangle has half the area of a parallelogram with the same base and height. In calculus, we can describe the idea of instantaneous change by imagining two points on a curve getting closer enough together. In graph theory, dots and lines can show connections, and we can explain that each connection contributes once to the number of connections at each of its two ends. These examples remind me that an idea can still be general or abstract when expressed through words and pictures.

I can also see similarities with how students learn algebra. A student might first explain a pattern by saying, “Double the position number and add one.” Later, they might use shortened words mixed with mathematical signs before writing a rule entirely in symbols. This resembles rhetorical, syncopated, and symbolic algebra, although students may move between these forms rather than follow fixed stages. My experience with Example 4.8 reminds me that familiarity matters: symbols feel clear to me, but they may feel confusing to a beginner. As a teacher, I want to give students opportunities to explain ideas in words and drawings, then help them connect those explanations to symbols so that the notation has meaning for them.

Babylonian Table for 45

The divisors can only be numbers whose prime factorization includes only 2, 3, and 5.

 

 

How Ancient People Made Sense of Time

After reading these two articles, I started to think differently about something I normally take for granted: time. One thing that surprised me was that ancient people could use their fingers and even their finger joints to help with counting. I was also fascinated by the Babylonian base-60 system and its influence on how we measure time today—60 seconds in a minute and 60 minutes in an hour. Since 60 can be divided evenly by many numbers, it is very useful for fractions and measurement. What I found especially interesting is that we still do not know exactly why the Sumerians developed base 60 in the first place.

What surprised me even more was how different cultures developed similar ways of thinking about time, even when communication was much more limited than it is today. This immediately reminded me of traditional Chinese timekeeping. Ancient Chinese people also used sundials, and a day was divided into twelve shichen(时辰), with each one roughly equal to two modern hours. There are also the 24 solar terms, which connect the year with changes in seasons, weather, and agriculture. (https://www.travelchinaguide.com/attraction/beijing/forbidden-city/miraculous-sundials.htm) For example, Jingzhe (惊蛰), or “the awakening of insects,” describes the time when nature begins to become active again. (https://global.chinadaily.com.cn/a/202403/04/WS6222aebfa310cdd39bc8a814.html) I have always found it amazing that when this solar term arrives, you can actually start noticing small insects appearing around you. It makes me wonder how ancient people, simply by observing the Sun, stars, seasons, animals, and weather over long periods of time, were able to recognize these patterns. I find it fascinating that numbers like 12 and 24 appear in different cultures and different periods of history.

The readings also made me think about the geometry of time. I used to imagine a year mostly as a straight timeline, moving from January to December. But now I think a circle might make more sense. Seasons repeat, the 24 solar terms form a cycle, and days themselves are connected to the repeated movement of the Sun. A clock is also circular, divided into 12 sections, which creates another interesting connection between time and geometry. The Babylonian base-60 system makes this connection even stronger. A circle has 360 degrees, and angles are measured in degrees, minutes, and seconds: one degree contains 60 minutes, and one minute contains 60 seconds. I had always learned these as angle units in geometry, but I had never really stopped to think about why “minutes” and “seconds” appear in both angles and time. Now the relationship between 60 minutes in an hour, 60 seconds in a minute, and 60 minutes or seconds in angular measurement feels much more meaningful. It makes me see time, astronomy, and geometry as historically connected rather than as separate mathematical ideas.

The readings also made me curious about ancient Chinese mathematics. Babylonian mathematics was recorded in cuneiform on clay tablets, which reminded me of Chinese oracle-bone inscriptions written on animal bones and turtle shells. I did some searching after the reading and became interested in how ancient Chinese mathematics developed around decimal counting, while Babylonian mathematics used base 60 much earlier. It makes me want to learn more about the mathematical history of China and perhaps bring some of these historical and cultural connections into my future mathematics teaching.

Finally, I realized that doing calculations in base 60 would actually be quite difficult for me. If I had to multiply or divide Babylonian numbers, my instinct would be to convert everything into base 10, calculate it, and then convert it back—which was obviously not how Babylonian mathematicians worked. They had their own methods and calculation tables. Now I actually want to learn how they did multiplication and division in base 60 and experience a little bit of the joy—and probably the challenge—of doing ancient mathematics myself.

Overall, these readings made me realize that measuring time is not as “natural” or obvious as I once thought. The systems we use today carry traces of how people from different cultures observed nature, discovered patterns, and mathematically organized their world.

Discovering the Global History of Mathematics

The first thing that surprised me was learning about the idea of the “Dark Ages.” Before reading this article, I did not know much about this period or really understand why it was called the “Dark Ages.” What interested me was learning that this term largely reflects a European perspective on history. While some parts of Europe were experiencing significant changes, mathematics was still developing in other parts of the world, including India, China, and the Islamic world. This made me think about how the same period in history can look very different depending on whose perspective we use. I found the question “Dark for whom?” especially interesting because it made me realize that even the way we describe history can shape how we understand the development of knowledge.

The second thing that surprised me was learning about the connections and exchanges between different cultures, especially between ancient China and India. As a Chinese person, I had always known a few names of ancient Chinese mathematicians and some of the mathematical ideas associated with them, but I had not really thought about Chinese mathematics as part of a larger international history. I found it fascinating that China and India were not simply developing their knowledge separately, but that connections also allowed mathematical and cultural ideas to travel between them. This made me realize that the history of mathematics is much more connected than I had imagined. It also made me want to learn more about ancient Chinese mathematics and its connections with other cultures. I think I still have a lot of my own cultural history to explore more deeply.

The third thing that interested me was the idea that mathematics is also connected to culture and history. Before this reading, I mostly thought about mathematics in terms of concepts, formulas, and problem-solving. Now I am thinking more about the people, cultures, and historical circumstances behind the mathematics we learn today. As a future math teacher, I would like to bring some of this history into my classroom. In particular, I would love to introduce students to aspects of Chinese mathematical history and culture, while also showing them how mathematical ideas have travelled between different cultures. Since China is part of my own background, this could also give my students an opportunity to learn something about where I come from and the history and culture that have shaped me. More importantly, I hope it could help students see mathematics as a human and global story, rather than something that belongs to only one culture.

Rethinking the Role of History in Mathematics Teaching

Before reading this article, I thought the history of mathematics could be useful in teaching, but I mostly saw it as something extra that could make math lessons more interesting. As a math teacher, I usually focused on the concepts and methods that students needed to learn, especially for exams. I did not really think about why a certain method was developed or how it changed over time. Looking back, I focused mainly on the mathematical side of math and didn't pay much attention to its historical, social, and cultural sides.

One thing that really made me stop was the discussion about teachers’ lack of historical knowledge and confidence. I have experienced this myself when students asked questions like, “Why do we use this method?” or “Where did this come from?” Sometimes I honestly did not know how to answer. The article made me realize that knowing some math history is also part of becoming a better math teacher. I was also interested in the idea that understanding the historical development of mathematics can help us understand modern mathematics more deeply. It made me think about our class assignment where we used historical methods to solve math problems. Before, I often taught a method simply because the curriculum or an exam required it. Now I think it is also important to ask why we use that method and how it came to be. I also realized that mathematics is not completely separate from society and culture. The article shows that mathematics has connections with areas such as philosophy, art, science, and the humanities, and that the society and culture of a particular time can influence how mathematics develops. This made me see mathematics not just as a collection of formulas and methods, but also as part of the cultural heritage of different civilizations.

After reading, I see the role of math history differently. It is not just about telling students interesting stories about famous mathematicians. It can help students understand why mathematics developed in the way it did and give them a deeper understanding of mathematical ideas. The practical examples in the article also gave me some ideas about how I could bring history into my own classroom. I would like to try some of these approaches during my future practicum and teaching.