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!
 

The Mathematics of Grades

How to Focus on the Things that Matter

Dr. Kim Johnson with The Lukeion Project, educator for Lively Logician

Did you know you can flunk a final and still perhaps get a B in a class?  What matters more to your grade: quizzes or homework? Grades are a part of life for almost everyone at some point in their schooling. They seem simple: it’s just a number (or letter) that tells how you did in a class. Because teachers don’t want to fall victim to favoritism or arbitrariness, we turn to mathematics to help us assign fair, transparent grades. However, many students (and some teachers) don’t understand the mathematical implications of the way they grade.

The Grading System

For most teachers, the goal of a grading system is to provide a symbolic representation of learning. The tool we use is called the grading system. Grading systems reflect what the instructor finds important for you to know and to be able to do.

A grading system assigns weights to various aspects of the course. Typically, a demonstration of knowledge, usually in essays, papers, or quizzes, is the most important part of your grade. Also important is a demonstration of behaviors that lead to better a understanding of the material through participation, attendance, and homework completion. You can see what parts are most important to the instructor by what weight the instructor gives to that section. It is interesting as well to see what parts have equal importance in the instructor’s mind.

Some examples of grading schemes:

1.  

Item	Percent
Attendance and participation	10%
Homework credit	20%
Vocabulary quiz	20%
Grammar quiz	50%

 

2.  

Item	Fraction
Quiz average	1/3
Paper average	1/3
Participation	1/6
Final exam	1/6

 

3.  

Item	Percent
Quiz average	70%
Homework completion	20%
Attendance and participation	10%

 

4.  

Item	Percent
Quiz average	35%
Homework completion	15%
Fallacy homework	10%
Attendance and participation	10%
Logic in Real Life project	10%
Final exam	20%

 Mathematically speaking, note that all percentages and fractions add up to 100%.

Percentages do not reflect the number of assignments: for example, in chart B the final seems to be worth less than the quizzes---but there is only one final and 14 quizzes. In the last example, there are 7 quizzes each semester, so each is worth 5% of your grade. The final exam is worth 4 times as much!

Grading schema can be used in a calculator to figure out your course average at any point in the class. Plugging this calculation into an online calculator (such as Desmos) will let you easily replace potential grades in each area and see what your grades might be. You could enter a calculation of your final grade like this:

(Weight of area 1 *grade in area 1)  +  (weight of area 2* grade in area 2)  +  weight of area 3 * grade in area 3)

You should use the weights as percentages: remember that a 25% weight is equivalent to the decimal 0.25. Also note that your grades should be averages---if you get 45/50 on a quiz, that score should be written either as 0.90 or 90 (either way provided you are consistent).

A little algebra will allow you to see what grade you need in an area (such as the final) to get the grade you want. Your needed grade in area 3 is this:

Use grades out of 100. The weights should be percentages which add up to 1. Both of these formulas can be generalized to different courses with more or fewer areas.

Don’t Sweat the Small Stuff

The grading system helps us figure out some quandaries students and instructors have when figuring out grades. One of the most common grading questions instructors deal with is individual points on assignments, quizzes, or finals. It is surprising to most students how little effect an individual point has on their grade but it is not surprising to most of your educators who know that most point-wheedling is counterproductive in every way.

To begin with, from a grade perspective it is never worth arguing points with your instructor on an assignment graded for completion. If you completed the assignment on time and conscientiously, you got 100% credit for the assignment. No amount of argument will benefit your grade in any way.

Even on a quiz, arguing for a few points rarely makes a difference in your grade. If the quiz is worth 5% of your grade, changing a few points will only change your final grade by tenths or even hundredths of a percent. If quizzes are worth ⅓ of your grade, but you have 12 quizzes, each individual quiz is worth ⅓*  1/12=2% of your grade. If the quiz has 100 points on it, getting two more points will raise your grade 0.04%---going from an 85% to an 85.04%.

Perhaps you are one of the few people in the class whose grade is an 89.99% (an occurrence that is extremely rare) and you wish to get up to an A. Arguing for a few points on some assignment somewhere in order to increase your grade is not worth it---your time would be better spent making sure you fully understood the point of the question and studying for the next test.

When Small Stuff Makes a Big Difference

It is clear that niggling about points on a quiz will not mathematically change your final grade. But there are some small acts that have an outsized impact on your grade: completing homework on time and participating in class.

Suppose you are a decent student in a course and your grades on the quiz and final average is about 94%. You learn the concepts easily without having to turn in all the homework graded for completion: you only get about half of those turned in, and you sometimes don’t pay attention during class plus you tend to arrive late so your participation grade is lower than it should be. Here is your final grade computation if quizzes are worth 70% of your final grade, homework 20% and participation 10% (note: I wrote the quiz, homework and participation average out of 100):

 70% * 94 + 20% * 50 + 10% * 50 = 81.5%

Despite your grades on quizzes and the final, you only get 81% in the course! On the other hand, if you are a conscientious student and diligently complete homework and participate actively in the online class sessions, but only manage an 86% on your quiz average, the participation and homework grades will raise your grade in the course:

70% * 86 + 20% * 100 + 10% * 100 = 90.2%

Although it seems as though homework and participation are insignificant acts compared with doing well on quizzes, they can make a big difference in your grade. Of course, the real reason to turn in assignments graded on completion and participate in class is that instructors have noticed over the years that doing these things are a reliable way to improve your quiz scores and your understanding of the subject matter. It is worthwhile to be conscientious in these acts even if you think you understand it all perfectly.

Zero vs. 59%

There are multiple ways to fail an assignment. In students’ and parents’ minds, there is no difference between a zero and a 59% because both scores are failing. However, as far as the grade calculation is concerned, there is a big difference.

Suppose you have 5 quizzes which contribute equally to your quiz grade. Here is your quiz average if you get 85% on the first four quizzes, but don’t turn in the last one:

⅕  (85+85+85+85+0)=68%

On the other hand, if you do your best and turn in a bad quiz (along with an email to your teacher explaining your new improved study system) you will get

⅕  (85+85+85+85+59)=79.8%

Instructors know that things come up so in many classes, your educator will drop your lowest quiz score. Making a habit of not turning in quizzes that you might do poorly on will hurt your grade much more than turning in a quiz you “fail.”

Peace of Mind

One way the grading system can help students is to give them peace of mind before an assignment or the final. Even though a final exam or paper may be a large portion of the course grade, its effect on your final grade is mitigated by what you have been doing all semester. Suppose you are a student in the Lively Logician with only an 85% quiz average but have been diligently and conscientiously completing all the other assignments as well as attending and participating in class. Up until the final, your grade looks like this, with question marks for the last paper and the final exam.

If you are worried that an alien will abduct you and erase all your knowledge, or perhaps the wifi in that giant python (the one that swallows you on the last day of the semester) will be insufficient and you will not submit your final paper, you can plug these into a calculator and compute your grade so far. Even if the remaining 30% of your grade is not submitted at all (and gets a zero), the lowest grade you can get is a 64% (though you will still have to deal with residual complications from being swallowed by a python).

If space aliens only destroy half your knowledge (you get a 50% on the final and the paper) your grade goes up to 79%. If you get a few more points on the final (perhaps one of the questions is about aliens) and you get 53% but your course grade goes up to a B. Realistically speaking, your grade on the final is likely to be around 85% based on what you have done before. And so you can go into the final knowing that there is a very small chance you will get anything lower than a B. On the other hand, you can compute that if you can do slightly better than your earlier average and get an 89%, your final grade in the course may go up to an A.

The most amazing thing about these averages is how “sticky” they are. If 70% of your grade is already completed as a B, everything from a grade of 53% to a grade of 88% on the final assignments will get you a B---that’s 30 percentage points difference! This means that you can go into the final with confidence, ready to demonstrate your knowledge instead of panicking about a question here or there. It also shows that there is no point in arguing with your instructor about a few points here and there. Remember that your conscientiousness about homework assignments and participation really pays off in the end.

Here’s Lookin’ at Euclid

Why was Euclid so Great?

Dr. Kim Johnson at The Lukeion Project

Educators at The Lukeion Project include some great Classical authors in their courses: Cicero, Plato, Tacitus, Sophocles…the list goes on! But one Classical author stands above them all. More versions of his work have been published than any work other than the Bible. He’s been read almost continuously for more than 2,000 years. Knowledge of his work has been considered the mark of an educated person in Rome, Baghdad, London, and Athens. When Abraham Lincoln would enjoy circuit rides through the Illinois countryside, he would bring two books: the Bible and this work. This author was a topic of one Edna St. Vincent Millay’s sonnets. Who is this celebrated author?

He was a mathematician!  Euclid of Alexandria (Εὐκλείδης, which means "renowned, glorious"), 325-265 BC, wrote The Elements. This work not only changed the course of mathematics, but it also changed the way people think of logic and argument forever afterwards.

Who was Euclid, and what were The Elements?

The truth is that we don’t really know much about the personal life of Euclid. He came to work in the library at Alexandria around the time Ptolemy II, Hellenistic monarch of Egypt, was expanding it. We don’t know where he was originally from, although he is possibly one of the few Alexandria natives who came to fame at the library.

We have a few stories written many years after Euclid’s life. When he was teaching Ptolemy geometry, the king asked if there was a shorter way. Euclid responded scornfully, “There is no royal road to mathematics.” When one of his students asked, “When will I ever use this?” (a cry common to modern mathematics students), Euclid responded, “Give the boy some money, for he has to see some profit come from this work.”  Mostly, we have Euclid’s masterwork, The Elements.

The Elements is a collection of 13 books (really chapters) collecting most of the Greek knowledge of geometry up to Euclid’s time. Euclid himself did not discover most of the theorems in this book. He simply collected and organized them but the way he organized them was unique.

Previously, Greek mathematicians who wanted to prove something would list the assumptions they were making. They ran into the problem experienced by modern geometry students: what you can assume, and when? After all, you wouldn’t want to prove that the angles in a triangle add up to 180 degrees while inadvertently assuming that very thing. This would be circular reasoning and a fallacy. Euclid’s method avoided such problems by starting with almost nothing and assuming only things that he had previously proven.

The Elements’ starting point included definitions, common notions, and postulates. Definitions are important for any practitioner of logic: Before statements and arguments, one must make the terms precise. Common notions were meant to be clear without proof. For example, things which are equal to the same thing are also equal to each other. Postulates were slightly more complicated but self-evident. After making clear his assumptions, Euclid cataloged much of Greek mathematics. Each theorem was based on the theorems stated before. He included the steps of the proof and diagrams (including labels) to make his reasoning clear. He began with theorems about triangles and parallel lines (precisely where modern geometry students start) and ended proving that there are exactly 5 platonic solids, the primes are infinite, and some numbers are irrational.

This is not to say that Euclid was perfect in 200 BC. Over the years his proofs were corrected, and some of the corrected proofs became part of the canon. His postulates were not enough to prove all his theorems so in the 19th century more postulates and notions were added. The most stunning change came in 1832, when Janos Bolyai and Nicolas Lobachevsky proved that if you change the 5th postulate you get an entirely new geometry. Over all his theories held, and The Elements spread far and wide.

What came before Euclid?

Certainly, people did math before Euclid and without Euclid. The Babylonians had a number system in 2000 BC which would not be surpassed in Europe until after the 1200s when place value and the use of zero started overtaking inefficient Roman numerals. The ancient Egyptians clearly had a solid grasp on geometry. Just look at the pyramids which are still a wonder of the world! Indian mathematics predating Euclid were very advanced and included negative numbers, complicated computations, and zero was considered a number. Chinese mathematicians worked out theories of solving systems of equations, solving polynomials, Pascal’s triangle, and the Pythagorean theorem all without Euclid.

Although these mathematicians made great advancements, Euclid still stands out. These other mathematicians weren’t interested in making their mathematics accessible to the public. Perhaps they didn’t think it was necessary. Perhaps they wanted to keep their secrets to guarantee their future employment or the exclusivity of the scribe and priest class. But most of their work was published with the problem, the solution, and perhaps a method for solving it. Euclid was accessible to anyone willing to put in the work.

The other characteristic of mathematics in other cultures without Euclid was that they were interested in showing solutions to specific problems. Although an intelligent reader could generalize, the writings of mathematicians before and without Euclid would use specific values for the lengths of sides, radii, and other calculations. They did not show the general solution.

What changed after Euclid?

First, most of the old geometry books were destroyed. It’s hard to find any geometry books written before Euclid. Euclid’s work is the only one cited for basic theorems about geometry.

Euclid’s The Elements spread everywhere. For example, there are potsherds with references to his work found at a military outpost found near the Nile plus copies of his book from personal libraries in Babylon and Constantinople. Euclid’s work spread to Baghdad in the 700s, England in the 1000s, and eventually to India and China. Euclid’s work was not a secret but something that anyone could work through and discover for himself. Judging by the computations written in the margins of many ancient copies, some readers were more successful than others.

Euclid’s method was adopted for many other mathematical, scientific, and philosophical efforts. Think about the other variations on The Elements that have been written over 2000 years, including the Elements of Theology by Proclus in the 5th century AD to the Elements of Style by Strunk and White in the 20th century. When Descartes wrote “I think, therefore I am,” he was using Euclid’s method of eliminating all extraneous assumptions to reach the most basic statements we can affirm.

Reading Euclid and learning to think logically came to be one of the cornerstones of a good education. From when it became one of the four liberal arts in the Middle Ages until every schoolchild in England had to pass the “pons asinorum.” Knowing Euclid was essential to training one’s brain to think logically. This is why Lincoln studied Euclid and used his reasoning as an example in a debate against Stephen Douglas. This is why Edna St. Vincent Millay wrote, “Euclid alone has looked on beauty bare.”  Anyone who builds or measures, who makes arguments from first principles, or uses science built on any mathematics that came after Euclid, is ultimately referring to Euclid. That’s why Euclid is so great.

The Elements concentrated on building knowledge up from the smallest possible number of assumptions. He wasn’t just interested in knowing what was true, but how you could be sure that you knew it. In addition, Euclid didn’t just solve one problem, he solved them all. One of the previous mathematicians might notice that 3^2+4^2=5^2. Euclid proved that for all right triangles, the square of the hypotenuse was equal to the sum of the squares of the other two sides.

In The Lukeion Project course Counting to Computers: Math from the Dawn of Time to the Digital Age, we investigate the connections between the great mathematicians of the past and the mathematics we learn today. Whether you love, hate, or are indifferent to the standard course of math study, if you are interested in learning more about why we do things the way we do them today, how they were invented in the first place, and the people who were involved, come join us Wednesday afternoons this fall!