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.
