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.