Why Rhetorical Algebra Belongs in the Modern Classroom!

Reading about Babylonian Algebra from The Crest of the Peacock, I got really stuck on how we actually define algebra and its history. The text looks at some early methods, specifically comparing Babylonian procedures to the later Diophantine method. Diophantus was doing things that look a bit more familiar to us. But the translation of a Babylonian tablet from Senkereh is what really caught my eye.

Instead of our usual variables, they literally just used words: ush for length, sag for width, and asha for area. It states relationships through completely narrative descriptions like, "I have multiplied ush and sag, thus obtaining the area (asha)." It makes you wonder: how do you state a general mathematical principle before modern algebraic notation even exists? The Babylonians did it through rhetorical algebra.
It’s a good reminder that we don't strictly need symbolic algebra to express complex or abstract relationships. If we think about math areas like geometry, number theory, or graph theory, we can communicate massive, abstract ideas entirely through spatial reasoning, visual models, or descriptive language. The Greeks did this constantly with geometry. Instead of writing a symbolic formula like a^2 + b^2 = c^2, the Greeks expressed the Pythagorean theorem through pure descriptive geometry: stating that the area of the square built upon the hypotenuse of a right triangle is equal to the sum of the areas of the squares built upon the other two sides. This really pushes back on the assumption that mathematics is entirely about generalization and abstraction. While abstraction is obviously a core part of modern math, the discipline's roots are incredibly concrete. It was originally about measuring actual fields, distributing resources, and solving tangible problems.

Looking at this historical progression from rhetorical math (full words) to syncopated (some abbreviations) and finally to symbolic algebra, I can't help but think about how our own students learn. Do they go through these exact same stages? I think they absolutely do. When kids first encounter algebraic thinking, they're reasoning through word problems and drawing pictures (rhetorical). Then they might start using a quick shorthand, like writing "L" and "W" (syncopated), before fully understanding and accepting abstract variables like x and y.

As a teacher, this historical context reinforces why concept-based math is so important. I think we sometimes rush the symbolic stage because it's the "standard" way math is done in high school. But when we force abstract notation before students have a firm grasp of the actual relationships, it just breeds math anxiety. I tend to view teaching through the lens of a gardener, we cannot force the plant to grow faster, but we can prepare the soil so it has the right environment to thrive. Letting students sit with rhetorical and syncopated algebra a bit longer, encouraging them to explain relationships in their own words before handing them a formula, is how we prepare that soil. It gives them the solid, concrete grounding they need before the heavy abstraction sets in. Basically, here is why we need to let students hang out in the flower math stage a little longer before hitting them with the variables.