A Course on How Humans Have Used Math Through the Ages
It’s a Math Course
In Counting to Computers (C2C) we discuss many of the topics mathematics students encounter from the beginning of their math career to the end of calculus and beyond. Despite covering so much content, because these concepts developed in a natural way over time we cover them in a format accessible to students in algebra or are advanced prealgebra.
At the “dawn of numbers,” the only tools prospective mathematicians had were their brains and their fingers. It took thousands of years to build up the tools to do calculus and computing. In C2C, we begin where all children begin, with counting. Even counting turns out to be not so simple!
Over the centuries people have come up with many different number systems which have different characteristics and strengths. The Egyptians developed fractions, but they only used 1 in the numerator (with a few exceptions). Instead of writing ⅔ you would have to write ½+⅙. The two forms are equivalent, but the second seems much more complicated. On the one hand that seems like a lot of work for a simple concept! On the other hand, for solving certain types of division problems, Egyptian fractions were much superior to the our simpler fractions.
In another example of crazy counting Babylonians developed a base 60 system for doing arithmetic. At first glance, base 60 seems much more complicated than our good old familiar decimal system. Why would they make things more complicated? As it turns out, when we use minutes in an hour or degrees in an angle, we actually pay tribute to the Babylonian system. For all its faults, the Babylonian system was miles ahead of Roman numerals.
Some of the problems we do in the course come directly from problem sets developed to train ancient scribes and mathematicians. We can do problems from the Ahmes papyrus (1650 BC) and decipher cuneiform tablets found in the trash pile from a Babylonian school. Fibonacci’s Liber Abaci, originally written to demonstrate Hindu-Arabic numerals to merchants used to cumbersome Roman numerals, becomes a source for puzzles for homework.
Gradually the math that we discuss becomes more powerful. Even students who are a long way from trigonometry can look at the ratios created by Hindu mathematicians to talk about astronomy, or practice interpolation to find difficult values for sine and cosine using prealgebra techniques. We don’t have to do calculus problems, but we can practice the methods of Archimedes which were precursors of those developed in the 17th century by Newton and Leibniz. The key to thinking about these more advanced concepts is that they developed gradually and not all at once. By following along with the history, we can do the math along side the ancient mathematicians.
It’s a History Course
C2C is also a history course. At its most basic, any history course is a discussion of past events. We discuss events in the history of math such as the development of zero and the conflict between Leibniz and Newton over the development of the Calculus. We show how the development of mathematical symbols over hundreds of years makes the calculations we can do easier and more powerful.
However, there is more to math history than just the events in the subject of mathematics. We discuss how world events led to the export of the modern Hindu-Arabic number system to Europe. The development of the printing press helped spread trigonometry across Europe and made it useful for the scientists discovering how orbits worked and how the stars seemed to move. In addition, understanding cultures and their motivations makes clear why the mathematics developed by one culture differs from others. The differences in cultures michte explain why the Greeks valued rigorous proof, while Indian mathematicians were able to see the concept of zero, and Islamic mathematicians developed the idea of algebra. When we study the origins and history of mathematics, we see that mathematics didn’t arrive full blown in an Algebra 1 textbook, but was a gift from many places and times.
In addition to the events in history, we discuss the character and biographies of mathematicians. The people who do math turn out to be as interesting as the ideas they discover. We discuss what is known about the biographies of great mathematicians and why they became involved in mathematics, from Hypatia to al-Kwarizmi to Leonhard Euler. The personalities and culture of these mathematicians influenced what they thought of as important, and therefore what mathematics they developed.
It is a Mix of Both
The power of studying history and math together is that, no matter your background or preferences, there is something for every student. If you would rather read an article on ancient history than solve a single linear equation, this course is for you. If you love math and can’t get enough of it but really hate thinking about people, places and things, this course is for you.
Here are two comments from past students, first from a more advanced student (post geometry) and then from a student who is closer to the beginning of her math journey.
"The history was the most useful part, because while I knew quite a bit of the mathematics, I knew very little of the history of those who discovered it. The brief overviews of the mathematics in the form of the homework was also incredibly valuable, because it helped refresh and cement mathematical principles and even teach me a few new things."
"I enjoyed the connections between math and history and the new math concepts like calculus and logarithms."
Mathematics informs history, and history informs mathematics. There is no reason to be satisfied with just one or the other when you can get both in one course!
