Babylonian Algebra

 

While the use of algebra, or the use of single letters to represent unknown quantity, makes problem solving easier to present and understand, I think it is not strictly necessary to do so in order to state anything mathematically precise. For instance, I can state Pythagorean’s theorem as the sum of the square of the two shorter side lengths in a right-angled triangle is equal to the square of the longest side length of the same triangle. This would be more compactly represented using a diagram and labelling the sides with letters like a,b,c as commonly done. Using words specific in language to represent math might bring trouble when communicating with others outside of your region, but perhaps that wasn’t a problem when people were mostly confined to their local region without effective transportation.  

As for whether math is all about abstractions and generalization, I would say that abstraction is a tool that allows people to work on problems presented to them without having to consider details that may be crucial to the problem but not to the solution. There are problems that are very real, going from splitting candies between a group, collecting crops and paying taxes; they can be presented as algebra questions suitable for high school students, but often students feel detached about doing some algebra problem that they couldn’t care less of, likely because they didn’t find solving those real problems fascinating to begin with. Word problems are likely an attempt to introduce problems as worthwhile things to solve for, but perhaps it becomes yet another type of problem to grind through to collect some points.

Stating principles and abstractions without the use of algebra would be a worthwhile challenge to try. For instance, I am thinking about the Chinese Remainder theorem, and stating it in terms of division in words would not be too hard. Saying something like if I divide the number of words in this response by 7, I have remainder 2. If I divide by 29, I get 5. The solution is unique upto addition of multiples of 203. They certainly sound wordy but for teaching purposes the concise notation is usually followed by the “wordy” explanations anyways.