Math ...History?

A Course on How Humans Have Used Math Through the Ages

At The Lukeion Project, we offer a unique course which covers the history of math. Students considering taking a math history course sometimes wonder if the course is a mathematics course or a history course. As with most good questions, the answer is complicated. 

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!
 

What the Romans Did for Us

Thinking about Rome?

By Amy Barr with The Lukeion Project

“Rome,” is now apparently a common answer to the question, “what are you thinking about?”
I agree that anyone hoping to better understand the modern world should look at Rome. The parallels are clear. Comparisons will lend insight to students of history. The best takeaways still serve us well today. Her worst ideas continue to echo through current events. Take five minutes to think about Rome. I’ll keep things brief.
The Romans were famous for adapting and adopting good ideas from other nations and then turning them into great ideas. Architecture, for example, was given a tremendous boost by the ancient Greeks. The Romans enjoyed their artistry and then tweaked the Greek post-and-lintel system by adding the arch. Adapting arches into barrel vaults and groin vaults allowed architects to craft massive structures like the Colosseum and Pantheon. If they tucked those vaults underground, they crafted effective sewer systems that allowed towns to become massive cities. Further adaptations meant Romans had fresh water year-round. This lead to all sorts of luxurious spa experiences that would fetch high prices today though they were usually free in ancient Rome.
The list of Roman contributions is long, and this blog is brief. Highlights include roads (many still used after 2000 years), Latin (still used today in all academic fields worldwide), cement, plumbing, reliable water pressure, good sanitation, heated floors, and hot tubs the size of swimming pools, and unforgettable sports venues. Rome got close to a global currency that worked just about everywhere (we have found Roman coins in India, China, and throughout Europe. Their legal system was sensible, reliable, and included basic human rights applicable to citizen and foreigner alike.
Neutral contributions include the introduction of fast-food restaurants with take-out, massive free public entertainment events that would last for days of blissful vacation time, and satire as a national pastime. The Romans had a relaxed lifestyle with delicious and readily available food but still loved to crack jokes about all of it. Such is human nature.
The Roman Republic was first established in response to the emergent need to topple a violent and dysfunctional monarchy. As is often the case in historical matters, the Romans had to quickly find a good way to prevent tyranny’s return through a decent political and legal system with plenty of checks and balances on her politicians (they considered term limits a great idea, for example).  Early American founders had much in common with early Romans. They too sought a better future for our nation when it became clear that none of us needed kings nor queens.
The Roman Republic worked very well for a little more than 400 years. It served the people and senate of Rome in practical ways while Rome grew from a swampy village to city-state, to nation, to world power, to mega power. Things went well until her politicians discovered loopholes in Rome’s systems of checks and balances. The last century of the Republic left the city in tatters and too much of her population dead. A lengthy period of civil war marked the end of the system. Her politicians had learned how to do whatever they wanted whenever they wished. Once laws are gone, tyranny is the result.  
Chaos diminished under the very strong leadership of Rome’s first emperor. In a series of astounding turns of events, the 19-year-old successor to Julius Caesar would overthrow his competitors and, after having united Rome for the first time in a century, he would rewrite the Roman constitution and then quit. The senate begged their new commander-in-chief to remain and power while they gave him a new name, Augustus. In a little over a decade he would change everything, all while heralded as a patriotic and traditional man-of-the-people. He did love Rome and her empire. He was quick to rebuild, reunite, and expand. Augustus had tremendous organizational abilities, but he also had two big problems. First, he couldn’t live forever. Second, though he was long lived, by the time he did die nobody could remember living under any other system.
Wanting a proper heir, Augustus promoted a disappointing stepson who lacked interest in Rome or her people. Under Tiberius, the whole system could have reverted to a Republic, but the Romans bypassed many opportunities because they were too busy attending sports events. They pressed on and installed a long series of truly dreadful emperors who were easily pressured and conned by those that worked for them. The Roman equivalent of the secret service/FBI/CIA (the emperor’s Praetorian guard) learned that they could easily manipulate any emperor who served. The Roman empire was barely beyond infancy when the Praetorian learned how to obtain whatever commands they placed on Rome’s weak rulers.
The most successful emperors learned how to appease the Praetorian guard with lavish perks and extra power, but it was difficult to maintain positive momentum during transitions to a new ruler. Very rarely was an emperor’s son a worthy candidate so success was limited to emperors chosen for practical skills and intellect rather than legacy.
The Roman empire became massive while her enemies became clever and powerful. They would learn to pick and peck at her boundaries while Rome slowly died of a thousand cuts.   
What caused the fall of Rome? That’s the subject of many books and theories. Rome fell when she outlived her own success by a couple of centuries. Once there were no more major money-making-wars to fight, she turned her attention to micro-managing the empire. Too much governmental oversight was disastrous. Towards the end, the powers governing Rome split her into quarters and made her rulers and sub-rulers bustle around in mobile command units as needed (or even not as needed). Rome as a city lost her identity as capital while new city-centers grew in importance for the sake of convenience.
Perceiving that the leviathan formerly known as Rome was aging and weakening, her enemies kicked at all her doors and windows until they were eventually invited in. Under the pressures of disease, overextended military forces, squandered taxes, under-performing leadership, and a disinterested population, Rome invited her enemies to join their rat race. They were invited to serve in her military and guard her boundaries. Such soldiers had no interest in a Roman way of life. Rome fizzled, sputtered, and died.   

Think-Os

Stupid Mistakes

Dr. Kim Johnson, Logic (Lively Logician) and Counting to Computers with The Lukeion Project

I cannot tell you how many times as a mathematics teacher I’ve seen a student react to a question they’ve gotten wrong on a test or quiz by saying, “Oh, that’s just a stupid mistake.”  It seems as though calling something a “stupid mistake” or “careless error” allows students (and sometimes me) to dismiss the mistake as a onetime occurrence and go on to think about more pleasant things.

Celebrating your wins on tests or essays is crucial. But don’t dismiss the stupid mistakes out of hand---sometimes there is more to them than just carelessness.

Correcting “Think-Os”

A stupid mistake can also be called a thoughtless error. Sometimes I call them “think-os”. Like a typo, a think-o is something that you knew or understood well but wrote or typed incorrectly. Most often they stem from a momentary lapse in concentration.

Given that they are so fleeting, why should we not dismiss them and move onto something that is more important? The truth is, sometimes thoughtless errors turn out to be very important.

In 1999, scientists launched the Mars Orbiter with hopes that it would expand knowledge of the climate on Mars. Their hopes were dashed when, instead of exploring Mars, the orbiter crashed into the surface. The European scientists on the project used the metric system, the Americans used the English system, and so their calculations didn’t’ match up. This was truly a thoughtless mistake because every scientist in that group understood the difference between metric and English measurements. This thoughtless error cost millions of dollars.

“Fine,” you might say “…but I am not in charge of million-dollar projects, I’m just taking my Latin exam, doing some math homework, writing an essay, or taking a history quiz. A stupid mistake in these situations does not have any dire consequences. So why should you care?”

You should care because sometimes a seemingly innocent mistake hides a deeper misunderstanding. For example, if the instructor asks for one form of a verb and you give another, or if you are expected to give a number and you give a function, perhaps you never fully understood the difference and need to clear something up. It’s just a simple mistake---but it indicates that there is something you need to understand more fully before moving on.

Another type of worrying mistake is one that you make over and over. In working with students, I’ve noticed that each student has their own personal set of repeated mistakes. My personal set of mistakes is to lose negative signs and copy down digits in the wrong order. One friend misspelled the word “their” nearly every time he wrote it. Some students have particular facts that they always get backwards.

The best part about these mistakes is that you are not doomed to make them forever.

How To Avoid Careless Errors

Obviously, if you have a gap in your understanding of some part of the topic, work to figure it out!   Honestly appraising your work is crucial in clearing up misunderstandings. Without evaluating the mistake, you might think there is nothing left to think about and move on. The great thing about fixing mistakes which come from misunderstandings is that by fixing them early, you eliminate a whole raft of mistakes later. It’s kind of like killing a mosquito in the spring will eliminate thousands of mosquitos later in the summer.

Fixing mistakes that you make again and again can be slightly more difficult. The problem with trying to fix repeated mistakes is that they indicate a type of momentary blindness and inability to notice errors.

The first and obvious thing to do is to check your work. This involves a few things. First you must write something down to check. Have you ever heard your teacher tell you to “show your work?” This is why! Although an abacus or a counting table is a powerful tool, written work wins, hands down, when you are trying to catch errors.

Second, coming back to your work after a day or at the end of the test is another good habit. Even a few minutes can give enough distance so that mistakes become visible instead of seeming to be correct. When checking my work, I think of it like a “hidden picture” puzzle, or a “spot the difference” puzzle---each difference should be deliberate, not an oversight.

You can develop a habit of going over your work looking for your most common mistakes. For example, my friend would search his papers for all instances of “their” spelled in his particular way, and I watch for lost negative signs. You can learn what errors you tend to make and look specifically for those mistakes. By focusing on and analyzing what you first dismissed as “stupid mistakes” you can find them and even get better at avoiding them.

Another strategy for catching foolish errors is to check with someone else. When mathematicians are working, they often work together. This is in part to generate more ideas, but it also allows them to catch each other’s momentary lapses in thinking. Even the most accomplished writers have editors. Software engineers review each others’ code to catch bugs. We can help each other reveal thoughtless errors.

Sometimes this is not allowed because of an honor policy, but even then getting another person involved can help. One of the best ways to catch errors is to explain the work to someone else. By going over your work out loud you introduce another sense---your ears can catch what your eyes do not. Pets are some of the most patient listeners. Even though your dog will never have to factor quadratic equations of conjugate Latin verbs, they will listen patiently and never judge when you find a mistake.

Get Over It

Finally, not all mistakes are signs of misunderstanding or part of a pattern. Sometimes a stupid mistake is just that. If that’s the case, your most potent tool is to just leave it behind and move on. Dwelling on mistakes takes mental energy that would be better put to use in learning new material and analyzing new ideas. Try pretending that you are looking at someone else’s work as a teacher or a scientist. Then you can judge more accurately whether the mistakes require further action or if they are not matters of concern.

Mistakes will definitely come, that is certain. Making mistakes is part of being human. Learning to deal with them appropriately---with honesty, attention, and grace---makes us better humans