Did Mesopotamian scribes have algebra? (And how can we recognize 'rhetorical' and 'syncopated' algebra?) - Response

To be honest, it is weird seeing a math problem written out without any x or y variables. In school, we get so used to just plugging numbers into formulas that it’s easy to forget people were doing high-level math long before modern symbols were invented. I am still amazed to see that Babylonians' IQ level was very high. 

Below are my response to the questions:

Math rules/ideas without algebra

Before equations even existed, people basically had to use story-style descriptions or pictures:

• Writing it out in plain English: They explained rules like a narrative recipe. Instead of writing A = someone would write: "To find the area of any triangle, multiply the length of its base by its height, and then take half of that total."

• Using word placeholders: Instead of writing x or x^2, they used real words like "length" or "the square".

• Showing one example: They will solve one problem using real numbers—like 3 and 4—and explain the steps so clearly that you could find answers to the questions by just swapping the numbers. 

• Number Theory: They used words to describe the number theory. For example, an even number plus an even number is always even.

• Geometry: Can be expressed through pictures and diagrams or even using papers. For example, fold a square piece of paper diagonally from corner to corner. The crease cuts the right angle in half, instantly showing that a square's diagonal splits 90 degrees into 2 angles of 45 degrees.

• Calculus: Can also be expressed through geometry, motion. Example - Changing a rectangle into a square can describe how limits work. 

• Graph Theory: Can be expressed through diagrams. Example - draw dots for cities and lines for roads to find the shortest delivery route.

Is math all about abstraction?

Not really, Math is not all about formulas. It’s about real, physical things such as counting money, dividing bread, measurement etc. Abstraction is a kind of shorthand so that people can save time and they don't have to write an essay. Symbols were created to make math easier.

Do our students go through similar stages of rhetorical and syncopated algebra as they move toward learning and doing  symbolic algebra?

Yes, I think they go through similar stages of algebra. We can definitely see that when we move from elementary to middle to secondary school. 

Rhetorical: Elementary school uses plain words and physical stuff—like counting blocks or sharing snacks.

Syncopated: Middle school uses empty boxes like  _ + 3 = 9 to hold space for unknown numbers.

Symbolic: High school swaps the empty box for x. For example, splitting an$80 restaurant bill equally among four friends turns into 4x=80. It's still just finding what each person owes, but written with a letter. 

When teachers jump straight to variables without explaining them in plain words or pictures first, math just feels like memorizing pointless rules. Building up through these stages is what gives the symbols actual meaning.