Babylonian ‘algebra’ from Crest of the Peacock

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I think generalizations before the use of algebra would have been similar to the descriptions found in the reading. For example, the part stating, "I have subtracted the side of the square from the area..." uses words in place of variables. To generalize this further, the answer could simply be "the result," meaning the whole equation would be "Subtract the side of the square from the area to obtain the result" to represent modern day x² - x = y. 

I don't think math is all about generalization but there are benefits to both methods. For example, concise/algebraic notation allows us to focus on the calculation. The downside to this is we can lose sight of what it is we're actually doing through those calculations (which brings to mind the Bollywood song lyrics that translate to "There's confusion everywhere, and no solution in sight. Now, I've finally found the solution but who knows what the question was"). If the question isn't translated back and forth from algebra correctly, we can end up with incorrect results. If we use descriptions or key words while solving, we have a concrete understanding of what we're doing at each step, but the extra writing can become cumbersome and make the actual calculation difficult. The additional clutter can also be distracting.

I think descriptions could be used as alternate methods to algebra in lower level math, but eventually using keywords would be preferred. For example, in Pythagoras, a description could look like:

[one of the smaller sides of the right triangle squared] added to [the other smaller side of the right triangle squared] gives [the longest side of the right triangle squared]

To replace this description, we could have

[leg squared] added to the [other leg squared] results in [longest side squared] 

This would be slightly more manageable, as long as "leg" and "longest side" were defined correctly and universally understood. Something like calculating slope is still sort of taught this way with "rise over run." It's not overly descriptive but still uses word substitutions to obtain understanding before introducing Δy/Δx. 

As concepts get more difficult, however, descriptions and even keywords would get harder to balance. I think the excess clutter would make things harder to understand. Thinking of partial differential equations, it would be very difficult to keep everything in order without a compact notation. It would be longer and create more confusion when trying to decipher what calculations were taking place between each line of work.