Not Dead Yet

Latin & Greek are Still Practical

By Amy Barr, dir. Latin at The Lukeion Project

I spend a lot of time at home school conferences talking about the benefits of Latin and Greek during the high school years. People always want to know why they should bother with “dead” languages when live ones seem more exciting or at least more…modern. All things being equal, Spanish, French, or Italian seem pretty interesting and the restaurant field trips can be downright tasty.

Just a few decades ago people seemed to intuit the value of Classical languages.  Then a good education, as had been the case for hundreds of years, depended on a firm foundation in Greek and Latin to ensure a literate and logical mind. At the turn of the last century, for example, even an average high school graduate could handle Cicero, Vergil or Caesar with enough finesse to make a professor proud. Today, even the most ambitious high school student will rarely feel motivated to study these languages for more than two years. Have Latin and Greek fallen out of style?

Let’s look at why Latin and Greek have been foundational since Alexander the Great was just average. When a student first tackles these languages at an advanced level (say, age 13 and older), he will have to experience transformative intellectual changes before he can successfully decode the foreign sentences in front of him. This is a fancy way of saying that Greek and Latin students must jump into the deep end grammatically. There shouldn’t be much time learning how to order food in Latin or driving directions in Classical Greek. Since these languages are primarily learned for the purposes of reading (wherein lies the most benefit), students jump in at a relatively more complex level and advance quickly as they learn decoding skills. Though there are “natural” spoken-Latin programs out there, before moving to Golden Latin authors, a student must master Latin grammar no matter what.

In only chapter 3 of Wheelock, for example, students learn Seneca’s wise observation that nulla copia pecunia avarum virum satiat, “No amount of money satisfies a greedy man,” and therefore modum tenere debemus, “We ought to maintain moderation.”1 Are these lifechanging ancient ideas? Not necessarily but they are also not quick and easy exchanges of basic ideas as is common with learning modern spoken languages.

A Classical language student must always read everything syllable by syllable. This means her analytical skills will increase a thousand-fold as she practices her powers of deduction to decode the ever-changing language puzzles at hand. Progressing more quickly than she would in any spoken language, she rapidly learns to apply language mechanics and analysis to everything from math to music, English to exegesis, and calculus to composition. Learning Latin and Greek will make a student analytical and logical by necessity. She becomes a person who reasons.

Look at how Cicero explains why we humans are different from animals: “But man—because he is endowed with reason, he understands the chain of consequences, observes the causes of things, comprehends the relation of cause to effect and of effect to cause, draws analogies, and, connecting and associating the present and the future, he easily surveys the course of his whole life and makes the necessary preparations for its conduct” (de officiis, 1.11).

Becoming logical, analytical and well-reasoned is just one beneficial side effect. These subjects certainly have an impact on the quality of writing and composition skills, vocabulary, speech, and comprehension. Estimates of how many words have entered English from Latin and Greek start with a conservative 60%. Those who are legal, medical or scientific professionals might say it is closer to 80%. The study of Classical language used to be the primary lens through which we could better understand the mechanics and vocabulary of English. This is why previous generations were so much better at our own language.

Studies conducted by the Educational Testing Service show that Greek and Latin students consistently outperform all other students on the verbal portion of the SAT based on data from the past decade.2 These same studies show that Classics majors tend to have a higher GPA at the college level and have accelerated performance in nearly all other subjects such as math, music and history. According to the Association of American Medical Colleges, students who major in Classics have a better success rate getting into medical school than do students who concentrate solely on science.3 Classics majors also have the highest success rates of any majors in law school.

Latin is now on the decline in all public schools and most private schools. Good luck finding more than a couple of years in most programs. Classical Greek is nearly extinct except at The Lukeion Project. Since 2020, even the small offerings of these topics have stretched so thin that students rarely accomplish more than the basics even as schools extend the period of those basics from two years to three. The Lukeion Project has maintained the traditional high school pace that was the norm 20 years ago.

Consider the results: reasoning, logical, independent learners, thinkers, auto-didacts. Such nonconformists are poorly valued by a world that prefers cookie-cutter educations followed by lack-luster jobs. Home educators and some subsets of conventional students not only grasp the value of these skills but make every effort to achieve them.

Latin and Greek are still fundamental and Cicero was right about anybody who can think analytically: “he who is endowed with reason understands the chain of consequences, observes the causes of things,…easily surveys the course of his whole life and makes the necessary preparations for its conduct.” Golden Latin and Classical Greek aren’t dead. They are still bringing to life the brilliant intellects that we desperately need to frame our future world.    

1 Wheelock, Frederick M., and Richard A. La Fleur. Wheelock's Latin. New York: Harper Collins, 2011. 29. Print.

2 Annual reports from College-Bound Seniors — A Profile of SAT Program Test Takers from years 1999-2005, 2007 available at http://www.bolchazy.com/al/latadv.htm#sat

3 http://www.princetonreview.com/Majors.aspx?page=1&cip=161200

Kim Johnson

Lukeion Faculty Interviews

What do you teach at The Lukeion Project?

I am the resident mathematician at Lukeion---the classicists make me welcome, even if I don’t always get their jokes!  I teach Logic and I have taught the History of Mathematics.  I also lead mathematical workshops, including Bizarre Ancient Numbers and Art and Math.

How did that subject first inspire you and what kind of education do you have as you develop and teach your beloved subject(s)?

There are two stories: first, how I came to study math and, second, how I came to teach at the Lukeion Project.

The high school I went to had Latin, and although I didn’t take it, many of my friends did. My senior year I took a class in Classical Literature, and based on that class and what my friends had told me about Latin, I planned on majoring in Classics and Philosophy in college.

Then came the reality of the actual college classes! My first philosophy class was about the theory of knowledge and how do we know things? It was a fascinating class, but I found that I was always missing something in my arguments. I got average grades on my papers, but I didn’t know how to make them any better.

I enjoyed my first Latin class very much. We used Wheelock and covered the book over two semesters. The grammar came easily to me because it was just like math!  But the nuances of vocabulary sometimes seemed out of my grasp. I wasn’t enjoying my chosen majors as much as I had anticipated.

Then in the second semester of my freshman year I took a calculus class. Every class felt as though I was going on a new exciting adventure. We were studying some of the most difficult topics in the first year of calculus such as the techniques of integration, but they seemed like a walk in the park. Doing my calculus homework was a joy. It was not that I always got the right answer but rather that I was able to tell by the work itself whether I was right or not. I didn’t need an instructor to tell me. I majored in mathematics and went on to get a PhD in mathematics.

Fast forward about 30 years when I discovered Lukeion from our local homeschooling mailing list. I wanted some good courses for my middle schoolers that I didn’t have to drive to, and Lukeion offered the high quality, interesting, and challenging classes I was looking for.  I listened to all three of my students as they took Witty Wordsmith, Barbarian Diagrammarian, and Latin 1 and 2, and I was extremely impressed with the way the instructors made the classes come alive despite not being on screen and even though students only typed responses in the chat. My kids would be yelling at the computer to try to answer questions---it was just as well that there was no audio! They were constantly engaged and therefore learned a lot, even during the relatively short weekly class sessions.

My son was happy to join Lukeion for Logic when it was first offered in Fall 2019 with instructor Michael Haggard. Unfortunately, that spring Dr. Haggard was unable to continue teaching, and I wanted to help. Although I had never taken a formal logic course, I had been following along with my son as he took the course. I had also taught a unit on Formal Logic in a Discrete Math course as I taught as an adjunct after receiving my PhD in math. I used logic in mathematics all the time both in creating arguments and in understanding and critiquing arguments that other people had made. To help Lukeion out, I volunteered to be a Teaching Assistant to one of the other Lukeion instructors. My thought was that I would grade homework and answer emails, both of which seemed interesting and doable.

The Barrs came back and asked if I would be willing to teach the class. I swallowed and said yes. There were a few 60-hour weeks as I learned the curriculum from scratch and prepared to teach engaging and interactive lessons, but the years of listening and absorbing the Lukeion method for online teaching made the format clear. In addition, the topic was fascinating. Although I was not familiar with the specifics, the principles were the same as those I used in math throughout my career. 

The in-person classes I had been teaching that spring were cancelled due to Covid. My kids were growing older and were becoming more independent.  It seemed like a good time to try something new. I was honored that Lukeion asked me to return the next year.

What do you wish everyone knew about this cool topic?

The history and development of logic is fascinating.  We study two branches of formal logic at Lukeion: Categorical Logic and Propositional Logic.  The formal study of Categorical Logic was founded by Aristotle when he wrote the Organon: not a book, but a collection of books about logic. A classic example based on Categorical Logic is the following syllogism:

All men are mortal.
Socrates is a man.
Therefore, Socrates is mortal.

Although it may seem that this type of argument is inflexible and only useful in obscure philosophical arguments, it turns out that many of the arguments we make every day can be rewritten as syllogisms.

Aristotle’s logic was taught to students for millennia. Then, starting in the 1800s, mathematicians developed a new type of logic intended to unite the fields of formal logic and mathematics. One of the pioneers trying to formalize how mathematics worked was George Boole, the son of a shoemaker. He used mathematical ideas like addition and multiplication to combine propositions, revolutionizing logic.

Then in the 1930s, Claude Shannon added electricity to logic. When he was a graduate student at MIT he maintained an analog computer which used switches and relays. His great insight was that you could use the digital switches to encode logical arguments using Boolean algebra. Thus, in the most influential masters’ degree thesis in history, the modern digital computer was born.

How would this class help you if you are a student with limited time?

Some areas of study are obviously connected to logic. All mathematics, whether geometry and formal proofs or applications in finance or physics, relies on making assumptions and reaching conclusions based on reasoning rather than guessing. Computer science is literally built on logic. Both computer hardware and software rely on the foundation of propositional logic. If a student is interested in either of these two topics, logic will strengthen their skills.

However, logic is ubiquitous. Analyzing words and sentences in Witty Wordsmith and Barbarian Diagrammarian relies on logic. When you are doing a Latin translation and are figuring out how the nouns and verbs fit together in the sentence, you are using logic. When you write a thesis statement for history or literary analysis and you support it, your argument must be logical, or your instructor and audience won’t believe you. Even rhetoric, which also uses tools relating to emotion or character, without logic, your argument can be easily disproved. Being aware of logic, its structures and rhythms, can help students in all subjects understand topics and communicate their ideas more effectively.

In college, writing was not my favorite subject. If there was an awkward way to write a sentence, I would find it. I could have used Lukeion’s courses in my high school! However, I found that the logic I used in writing proofs and making mathematical arguments gave me power when writing papers. In essence, a paper is an argument. You are trying to convince someone that your conclusion is true. If you do this convincingly enough, your argument stands on its own without needing any teacher or professor to tell you whether you are right.

Tell us a cool story from your teaching experience in this subject.

One of the disappointing things about mathematics and logic is that you don’t need to travel to any foreign lands to study it, you can literally do logic anywhere and anytime. Its very transportability means that logic is everywhere.

In my class students complete a project called “Logic in Real Life.” Some of the places my students have found logic used are:

• Insurance documents: when is something covered by insurance, and when it is not
• Historical arguments: why did politicians and military forces act the way they did?
• Current articles: do the conclusions follow from the premises the writers use?
• Advertisements:  The conclusion of an advertisement is almost always, “Therefore you should buy our product.”  But does their argument make logical sense?
• Books: Characters make decisions based on logic, to some degree, or the story doesn’t hold together.
• Comic strips often use unstated premises or conclusions to create humorous or ridiculous situations.

Students find a huge variety of examples of “Logic in Real Life,” often relating to their interests. I love seeing how they take what they have learned in the classroom and apply it in real situations.

 

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.

All About Arguments

How to Win

By Dr. Kim Johnson with The Lukeion Project

Of all the topics and ideas we cover in logic, the argument is certainly the star. Students may dream of outarguing their opponents in the courtroom, or winning an important political debate, or even proving their parents wrong about curfew, homework, or allowance.  Unfortunately, because of the nature of logical arguments, showing that someone has reached the wrong conclusion is difficult.

What makes an argument?

When logicians talk about “argument,” they don’t mean the typical immature argument between siblings (“Did too!” -- “Did not!”). The technical definition of a logical argument is “a set of premises which imply a conclusion.” That means that each argument has at least two statements, one of which is a conclusion.

Seems easy enough, right? Unfortunately, even figuring out which is the premise, and which is the conclusion, can be difficult. Consider the argument in the Declaration of Independence. Is the conclusion, “All men are created equal”? Or is it that King George had established tyranny? Or is it that it was necessary for “one people to dissolve the political bands which have connected them with another”? To discuss an argument, you need to know what is being argued.

To add insult to injury, sometimes parts of the argument are left unstated including even the conclusion. Arguments with unstated propositions are called enthymemes, from the Greek for “to keep in mind.”  Advertisements and politics are some of the worst offenders: “Open a Coke, Open Happiness.” Why?  Because Coke (according to the advertising company) is happiness. “My opponent hates cats. You would never vote for someone who hates cats.” Conclusion: Vote for me!

When faced with an enthymeme, your duty is to interpret the argument in the most favorable light for your opponent. The first reason for this is simple: treat others the way you would like to be treated.   The other reason is practical.  Responding to a command to “Surrender!” by saying, “You wish to surrender?  Very well!” is silly and soon no one will want to discuss anything with you anymore.

What makes a good argument?

Once you have found an argument, we can analyze its validity. In Lively Logician 1, we use the principles of categorical logic developed by Aristotle to analyze syllogisms, a specific form of argument with 3 terms, 2 premises and a conclusion. The logician’s favorite syllogism is as follows:

All men are mortal.

Socrates is a man.

Therefore Socrates is mortal.

In Lively Logician 2, we branch out into truth tables and use the laws of inference and replacement to judge whether an argument is valid. If each of the steps in the argument is made with valid principles, then the entire argument is valid.

Validity means that if the premises are true, then the conclusion must be true. The tricky thing is that validity refers only to the form of the argument, and not to the truth of the conclusion. Here is a valid argument (of the same form as above) with a conclusion that is obviously false:

All jellybeans are made of sugar.

All dogs are jellybeans.

Therefore all dogs are made of sugar.

What’s the difference between the argument with the false conclusion and the original one above? In the second argument, the minor premise (“All dogs are jellybeans”) is clearly false---so the conclusion is not guaranteed to be true.

Not only can valid arguments have false conclusions, but having a true conclusion and true premises doesn’t mean that the argument is valid. Here’s another syllogism:

All canines are mammals.

All dogs are mammals.

Therefore all dogs are canines.

The conclusion and premises are all true---but the argument is bad. If you’re trying to prove that “All dogs are canines,” you might want to get a different argument.  

Saying an argument is valid is a bit like showing that the structure of a house is solid and will stand. It makes no claims about the decorations on the house or the painting job or the light bulbs. Having a valid argument just means that if the premises are true, the conclusion is necessarily true as well. As we’ve seen, the implication doesn’t go the other way. A sound argument is much better.

Sound arguments start with true premises. This is not a requirement for valid arguments.  The argument in a sound argument is also valid. Thus, the conclusion of a sound argument is guaranteed to be true. If your opponent’s argument is sound, there’s not much you can do to counter it.

How can you prove someone is wrong?

This brings us to the goal: Show that your opponent (or parent, or friend) is wrong. Unfortunately, formal logic can only go so far with this goal. We can find fallacies in their arguments: ad hominem attacks, or circular reasoning, for example. Logic doesn’t have many tools to show that someone’s conclusion is false. They might need a better argument, but their conclusion could still be true.

The best way to vanquish your opponent is to create your own, sound argument including true premises and a valid argument plus a conclusion that is the opposite of what your opponent has stated. The best way to win arguments with friends and parents is not to prove them wrong, but to make your own, better argument.