The Ageless Confluence: 4 Math Strategies from the Ancient World
When we search for strategies in dedicated curricula, we are often left wanting and disillusioned not just in the system itself, but in the very capacity of humanity. We, as a collective of civilizations, have been exploring and refining mathematics over thousands of years and THIS is the best we have to give to students?
The answer, quite generally, is NO. That book is NOT the best. It may be the most familiar to modern educators, but the methods presented in a textbook are the "vanilla" of mathematical engagement: nothing inherently horrible about it, and plenty of people enjoy vanilla, but it's there because it's the most palatable to the most people if there's nothing else that's appealing.
As a 6th grade teacher in a self-contained classroom, I also taught an Ancient Civilizations curriculum that promoted the contributions of past societies, and the mathematicians cited caught my attention as a potential alternative to what I could offer students. As I discovered these, I initially fumed ("Why aren't any of THESE in our textbooks?"), then cooled to, "If it worked back then, why wouldn't it work now?"
In this post, I share four of my favorite ancient mathematical strategies and how they can be applied an utilized in our 21st century instruction.
The Mesopotamian 60
We, as a species, have globally settled on base-10 as our standard for a counting system, yet many of our measurements utilize multiples of 12 (months in a year, hours in a day, inches in a foot) or 60 (degrees in a circle, minutes in an hour, roughly days in a year). These conventions have existed for thousands of years, but it's not as if people had more fingers or toes to count back then.
As Marcus du Sautoy highlighted in his uniquely-fascinating BBC series "The Story of Maths", the earliest counting systems in ancient Mesopotamia did not utilize 10 fingers, but 12 knuckles.
Modern application: being able to count to 60 on two hands is a ceiling-smasher for those exploring early number sense, but using knuckles as a pattern standard provides a helpful benefit for those expanding fact fluency. Using one hand to track multiples frees up the other hand for counting up AND allows for both sides of the brain to play an active tactile role in preliminary calculations.
Babylonian Method for Square Roots
Similarly, the Babylonians held a remarkable penchant for recognizing the impact of square roots. Roughly a millennium before the development of the Pythagorean Theorem, mathematicians in Babylon developed a method to generate a most precise approximation of the value of a square root.
Generally, your approximation will be accurate to the thousandth within your second average.
Modern application: The process is time-consuming and eventually easily substituted with a calculator. However, it allows for a more involved process than simple approximation, where students may not have the enthusiasm for multiplying multi-place decimals after a square root estimate. It also provides a historical context for the "What headache does this solve?" approach: This is how they USED to do it, and because it wasn't the most efficient they developed a newer way.
Chinese Remainder Problem
This classic problem originates from the third century AD:
Now there are an unknown number of things. If we count by threes, there is a remainder 2; if we count by fives, there is a remainder 3; if we count by sevens, there is a remainder 2. Find the number of things.
Work out a solution, then reflect on the method you utilized: what did you uncover?
The idea is that if we have an unknown divisor, but know some dividends and their remainders, then we can calculate their divisor without initially knowing their quotient.
Modern application: The remainder problem resonates today as a rich math task for students around 4th grade or beyond. Without getting into technical-ese, the problem introduces students to modular arithmetic, which can greatly benefit students who depend on patterns to make sense of values, or are interested in coding or engineering as modular is frequently used as a key operator.
2400 years ago, the Greek mathematician Euclid dedicated an entire book in his landmark work, "The Elements" to the proof of a method for finding greatest common factor (GCF) using only subtraction.
Modern application: Euclid's Algorithm is a game-changer for students who either struggle with equivalent fractions (the simplified kind) or ratio. Generally, this struggle arises from a gap in multiplication/division fact fluency, which can especially hamper students as they begin to explore proportional relationships in 6th grade. Euclid's Algorithm allows for students to easily find a GCF while still refining fact fluency, and removes fact fluency entirely as a prerequisite.