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.

Why Is Time So Weird? (And Why Are We Still Counting Like Ancient Babylonians?)

Whenever I think about a year, I visualize a giant oval track, summer sits at the top, winter is down at the bottom, and I'm just jogging around the loop as the months pass. An hour looks like a round pie cut into slices, while a month feels like a straight row of calendar blocks.

What’s crazy is how much we treat our time system as if it's just basic common sense. In school, we use decimal (base-10) math for almost everything - money, science, counting on our ten fingers. But the second we glance at a clock, our brains instantly flip to base-12 and base-60 without breaking a sweat. Reading these articles made me realize that my mental geometry of time isn't natural at all, it's just a set of habits passed down from civilizations thousands of years ago.

Comparing the two articles turned up some pretty clear contradictions in how experts explain where these numbers came from.

1. Who actually counted finger joints?

• Scientific American credits the Egyptians with using finger joints to count to 12 (three joints on four fingers, using the thumb as a pointer), which is why they split the day into 12 daylight hours.
• MacTutor article, on the other hand, uses that exact same finger-joint trick, counting 12 joints on one hand while keeping track with 5 fingers on the other hand (12 x 5 = 60), to explain how the Sumerians and Babylonians came up with base-60. 

So... who actually started the finger-joint counting trend?

2. Math neatness vs. human messiness

• Scientific American claims base-60 took off because 60 is super convenient for fractions, since it divides evenly by 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, and 30.
• MacTutor calls this fraction argument "too scholarly a reason". The author points out that ancient societies didn't sit down in a committee to pick a neat math base. Instead, MacTutor argues base-60 came from messy, real-world stuff, like physical finger counting or two different ancient tribes merging together (like a base-12 group trading with a base-5 group).

Reading through the history of time measurement completely changed how I look at my watch:

• Hours used to stretch and shrink with the seasons: For most of human history, an hour wasn't a fixed 60-minute block. Early Egyptian sundials just split whatever daylight existed into 12 parts. That meant summer hours were way longer than winter hours. A Greek astronomer named Hipparchus suggested equal-length hours, but people ignored him and kept using seasonal hours until mechanical clocks showed up in Europe in the 1300s.
• Minutes and seconds were astronomy math, not clock units: Terms like "minute" and "second" come from Claudius Ptolemy subdividing degrees of latitude and longitude in his astronomy book Almagest (partes minutae primae and partes minutae secundae). Nobody even had minute hands on regular clocks until the late 1500s!
• Some minutes have 61 seconds: Today, time isn't measured by Earth's rotation anymore; it's defined by atomic vibrations of cesium atoms. Because Earth's spin speeds up and slows down slightly, scientists have to throw in "leap seconds" about eight times a decade, meaning roughly eight minutes every ten years actually last 61 seconds.

It's wild to realize that whenever I set an alarm or check my phone, I'm using a mix of ancient Egyptian star-gazing, Babylonian base-60 math, medieval mechanical clockmaking, and modern atomic physics. We treat time like an unchangeable law of nature, but it's really just a 4,000-year-old group project.

Babylonian-style base 60 multiplication table for 45!

Here is how my experimentation of the Babylonian base 60 system table for 45 turned out. Please feel free to comment any suggestions if any of them is not right!

Reflections on The Crest of the Peacock

Honestly, going into this reading, I figured math history was just a straight line from ancient Greece to Western Europe. Reading the introduction to George Gheverghese Joseph's The Crest of the Peacock completely threw that assumption out the window. Here are three specific things in the reading that really caught me off guard:

1. The ancient Greeks explicitly admitted they learned their math from Egypt.
   I was surprised to find out that people like Aristotle explicitly called Egypt the "cradle of mathematics". Key figures we always study, like Pythagoras, Thales, and Eudoxus, actually traveled to Egypt and Mesopotamia to study arithmetic, geometry, and astronomy. It changed how I view where "Western" math actually came from.
2. The Maya created zero and crazy accurate calendars with no clocks or glass.
   I knew zero was invented in India, but I didn't realize the Maya independently developed a base-20 place-value system with zero completely isolated from the rest of the world. What blew my mind even more was their astronomical accuracy: without optical lenses, glass, or any mechanical clocks, they calculated the synodic period of Venus (584 days) with an error of less than 0.08 days. Seeing what they achieved purely through observational patience was wild.
3. "Algebra" originally meant setting broken bones.
   I knew algebra was an Arabic word, but I had no idea about its literal medical root. The word comes from Al-Khwarizmi’s book Hisab al-jabr w'al-muqabala. In Arabic, jabr means "restoration" or "reunion" (moving negative terms across an equation), but historically it referred to setting broken bones. Joseph even mentions that in medieval Spain, barbershops hung signs reading "Algebrafista y Sangrador" to advertise bone-setting and bloodletting! Connecting my high school math class to medieval bone-setting was easily the weirdest detail in the whole reading.

Article response: Why teach math history?

Before reading this article, I definitely looked at math mostly as a finished product, just a bunch of formulas, theorems, and steps to memorize. I always figured that focusing strictly on the procedural mechanics was the best way to keep the class moving efficiently. To be honest, I thought of math history as a bit of a luxury or an afterthought. I had this assumption that you have to teach the actual math first before you can even touch its history, and I honestly worried that bringing up old, outdated methods would just confuse students rather than help them.

Reading through the text, a few specific things really jumped out at me and made me stop and think. First, Freudenthal’s quote about how math discovery gets turned "upside down" to turn a "hot invention into icy beauty" completely clicked with me. It made me realize that by only showing students the polished final product, we hide all the actual motivation and doubts that make the concept make sense in the first place. Second, the section on Japanese "San-Gaku" bulletin boards in temples really surprised me. It was cool to see how ordinary people treated geometry as a shared, community hobby, rather than it just being some elite academic subject. Finally, reading about Hamilton struggling for years with spatial rotations until he finally decided to just drop the rule of commutativity was a great reminder that hitting walls and making mistakes is just a normal part of doing math.

This piece definitely changed my perspective on how to actually bring history into my own teaching. I used to think it just meant throwing in a random biography or a timeline, but the idea of a "genetic approach" makes way more sense. It showed me that math teachers can use history implicitly by designing lesson plans around a sequence of older, motivated problems. That way, students can see exactly why a concept was invented to solve a real-world problem in the first place. I also realize now that historical errors aren't just fun facts, they're actually super useful tools to help teachers anticipate the exact same roadblocks our students are going to hit today.

 

Welcome to my Math History blog!

Hello fellow teacher candidates,

Welcome to my blog!