Sing, Dance, Grow, Talk

A Note to Home Educating Parents

By Amy Barr with The Lukeion Project

As my youngest child prepares for his college graduation in a couple of weeks, I look back at the sum of our family activities, chores, achievements, and low points over the last 28 years since my first child was born. We celebrate that all three of my home-educated children finished their college degrees even as my middle child completes his graduate medical degree for yet another year. (That’s not a humble brag but a full-on boast, for those keeping score). All three of my adult children are hardworking, successful, lovely people that we enjoy being around every chance we can find.

I present here some things I’ve learned. Take the ones you like. Discard the rest. Find what resonates. What would I have done differently and how I could have improved matters under my control as a home educator and mom? What would I recommend to families who are still in the educational trenches?

As a professional educator, some of you might assume I will suggest families up their academic game for their offspring. Maybe more AP courses and more CLEP exams? Maybe harder classes and higher grades? Perhaps some academic summer camps mixed with year-round academics? On the contrary!

Approaches that worked for my family:

1. Provide a rich buffet of opportunities and tools in topics that interest your student (Let your musicians have instruments. Let your artists have art supplies. Let your problem solvers play games.)
2. Settle for a checkmark of completion in topics that don’t.
3. Make travel a big part of your family’s experience to expand the assortment of things that interest your student.
4. Let your child’s personal enthusiasm guide your budget of academic energies. Get requirements quickly finished and then lavish energy on topics that bring excitement.
5. If you conscientiously educate at home, be assured your child will exceed expectations at college.
6. If you foster time management skills, personal responsibility, plus the confidence to both endure and overcome challenges, your student will greatly surpass his peers at college.
7. Don’t overdue AP classes, CLEP, and dual enrollment opportunities. There’s no replacement for the friends and connections your child will make at college if they do so from the start. Too many frugal moves to skip the first few years of college will result in a failure to make social connections and then a failure to form essential professional ties with peers and professors. These connections are as important (if not more important) than the expensive diploma at the end.
8. Thinking, writing, reading, and understanding are topics taught best at the dinner table, garden, library, and barn rather than through a formal curriculum or workbook. Just talk to each other as adults, even if your adult is only 10 or 13… then see point number 4 when it comes to spending time or energy on classes.
9. Let each of your children become an expert in areas she loves and then trust her advice or let him guide the rest of the family in that topic. Respect and trust will enhance development but also it is nice to have somebody become a computer specialist, or chess master, or the best at making gnocchi or making a nice avocado toast, or maybe the go-to guide for Medieval weaponry.
10. Be firm with boundaries regarding work time for everyone. Adults and children alike need to trust that dinners will be together, Saturdays will offer some family time, Sundays are set aside for rest and ritual, etc. At the same time this becomes more difficult, it becomes more essential.

Things I wish I’d done more of:

1. Forgive each other more directly and more immediately. Shouting, “stop bickering!” might bring a short but artificial peace. Issues can fester rather than gaining resolution just because conflicts were misinterpreted as being “merely” temporary or childish and then brushed aside as bothersome noise. Talk through resolutions while issues are manageable so that calloused resentments are never formed.
2. Music (all kinds at all opportunities)
3. Dance (all kinds at all opportunities).
4. Turn off the TV
5. Grow to celebrate wins more often. Don’t let Etsy or your overly festive relative make you think you need to buy special decorations or put on massive party events before you can celebrate. Start at dinner. Focus as much as possible on thankfulness and then take joy in even small victories daily.

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.

 

The History of Zero

By Dr. Kim Johnson, The Lukeion Project (Check out Dr. Johnson's class Counting to Computers: Math from the Dawn of Time to the Digital Age)

It turns out that one of the greatest accomplishments in the history of mathematics is---nothing.

From where we sit today, the tools of mathematics seem so obvious that they hardly bear thinking about. It might surprise you, then, to learn that one of these everyday tools took thousands of years, deep philosophy and even controversy to become part of our everyday tool kit. What is this development that we use every day without even thinking about, and yet was so difficult for the most brilliant and advanced mathematicians in ancient times to grasp? The number zero.

Before Zero

In the mists of time, mathematics seems to have been used primarily for counting or measuring things. Whether you are counting the number of animals in your flock or the distance to the next town, the concept of assigning a number to “no animals” or “no distance” seemed unnecessary and overly complicated. After all, “zero cows” is the same thing as “zero apples,” while “one cow” and “one apple” are decidedly different from each other.

This is not to say that ancient civilizations never used any concept related to zero. The Egyptians had massive building projects which required sophisticated engineering: they used a number to refer to the ground level. They also had a concept of a “zero balance” when no money was owed. The symbol for these was called nef, which also meant “perfect” or “complete.” 

The Greeks had an even harder time with the concept of zero than the Egyptians. Pythagoras declared, “All is number,” but what he really meant was “All is ratio.” Numbers to him represented the ratio between two quantities or lengths. This is useful in geometry (recall the concept of similar triangles) and in music (the ratio between lengths of string that produce an octave is 1:2) but terribly difficult for the concept of zero--a ratio of zero is almost nonsensical. Aristotle, whose philosophy reigned over the western world through the Middle Ages, stated that, “Nature abhors a vacuum.” Not only was zero not a number, but philosophically speaking, it was impossible. The paradoxes of Zeno of Elia (490-430 BC) point out the difficulties that the existence of zero would imply.

The absence of the number zero became a problem for Babylonian computation. Unlike the Egyptians, Romans, and Greeks, the Babylonians had a positional place value system where each number meant different things depending on where it occurred. In our system, where the positions represent powers of 10, the digit 2 in 2,528 means either 2,000 (the first 2) or 20 (the second 2). In the Babylonian system different positions represent different powers of 60. The leftmost 2 would mean 2x(60x60x60) while the rightmost means 2x60. This is fine unless you need a number which contains a zero. For the Babylonians, 11 could mean 101, 1001, 110, and so on, depending on context. They eventually developed a place holder to mean zero, but it only worked in the middle of a number, not at the end. They could distinguish 202 from 22, but not 220 or 2200.

The Discovery of Zero

Fortunately for mathematics, something changed in about 300 AD. Indian mathematicians began developing the symbol sunya which referred to the number zero. Rather than just a place holder, sunya (first a dot, then a ring) was used as a number. It makes sense that zero should be developed in the context of eastern rather than western philosophy. The concept of nothingness or emptiness, anathema to the Greeks, plays a much larger part in Indian religious philosophy.

 

In the 7th century, the great mathematician Brahmagupta (598-668 AD) created a whole system explaining how zero worked with other numbers. Some of his rules included the fact that a fortune, take away zero, is a fortune; or a fortune multiplied by zero is zero. He did commit the mathematical sin of dividing by zero, but that concept remained difficult for mathematicians well into the modern era. Brahmagupta also talked about negative numbers with more comfort than western mathematicians would into the 18th century but that’s a blog post for another day. 

 

After Zero

It seems obvious to us now but zero took a while to catch on in the rest of the world. The first people to adopt zero—and see how useful it was—were the Arabs. Most notably, Al Khwarizmi (c. 780-850 AD) used Hindu-Arabic numbers as he was developing algebra at the House of Wisdom in Baghdad. He described zero, which he called sifr, in his book Algoritmi de Numero Indorum as “The tenth figure in the shape of a circle.”

Several individuals were instrumental in introducing zero to Europeans. One of the earliest, Pope Sylvester II (c 946-1003) had studied the Hindu-Arabic numerals when he went to Muslim controlled Spain to study mathematics and science. Because of his facility with these numbers (and probably for other reasons) he was accused of consorting with the devil.   

The next great advance in using this Hindu-Arabic numeral came when Leonardo di Pisa—better known as Fibonacci (c.1170-1240) wrote his great work Liber Abici. In addition to problems about rabbits, he introduced the numerals he had learned about when he studied in North Africa among Muslim mathematicians. He also introduced algorithms for doing basic mathematics including 28 pages about how to do long-division. We still use his method today.

 

 

“These are the nine figures of the Indians 9, 8, 7, 6, 5, 4, 3, 2, 1. And so, with these nine figures, and with the symbol 0, which is called zephyr in Arabic, whatever number you please can be written, as is demonstrated below.”

Even after zero came to be used in the west, it wasn’t immediately accepted by the public. Financial institutions in Florence banned the use of zero in 1299. They felt that there were too many opportunities for fraud because someone could use place value to change the values on the checks that were being written. Even today we write “seven hundred fifty and no/100 dollars” on our checks---perhaps as a deterrent to adding zeros and increasing the value of your check.

The medieval philosophy textbook Margarita Philosophica (Gregor Reisch, 1503) shows a picture of a contest between someone computing with the old counting board on the right, and the new-fangled Hindu-Arabic numerals on the left. You can see who is happier. The Hindu-Arabic number system eventually won out, including its use of zero.

Zero has proven useful beyond allowing a more efficient number system. Zero eventually became the star of the number line and of the Cartesian plane, invented by Pierre de Fermat and Rene Descartes, and used to visualize algebraic functions and equations. Zero is essential in the invention of calculus. Newton and Leibnitz use infinitesimals to find a tangent line to a curve, quantities that are almost zero but not quite. Don't forget that zeros make up about half of every binary computing system.

The history of zero spans thousands of years, multiple civilizations and continents---and yet it is something so simple that we use it every day.

If this story of the connection between math and culture piques your interest, you might want to sign up for the course Counting to Computers: the history of math from the dawn of time to the computer age. This course is appropriate for strong pre-algebra students through high school students taking higher math.