Think Three Things

Beliefs to Build a Better Brain

Amy E. Barr, The Lukeion Project

We humans are our own worst enemies in too many ways to count but, more specifically, we torment ourselves when we want to learn new things. We ask a lot from our brains daily but never more so when we are students. The truth is that healthy humans should always be students of something, even when we are not strictly enrolled or controlled by a schedule of assignments and a system of grades. Even healthy humans struggle to have a good learner’s mindset. We often try to talk ourselves out of our own success. Negative thinking, perhaps we suppose, is a way to soften the blow when success is temporary or diminutive.

What can we do to help ourselves out? Start by making three observations about yourself.

I have…

If you are reading these words, you already have a good track record at learning tricky things. You have already developed the mental power and sustained the practice necessary to master reading and writing, two skills that fewer than 1 out of 20 people possessed 500 years ago. Today, your literacy still puts you in a position for ongoing success. Unlike most of the population of the world thus far, you are likely also skilled in the basics of math, music, science, and even managing a computer device. Well done.  

Your track record proves you can learn what is set before you because you’ve already done so much already.  As you look at challenges ahead, remember to say to yourself, “I have already been very successful at learning difficult things in the past.”   

I can…

A growth mindset, the belief that a person's capacities and talents can be improved over time, is the best predictor for future intellectual success. The opposite, if there is such a thing in the real world, is a fixed mindset, a belief that dictates that you are born with a certain skill set which can’t be expanded much, even with great personal investment.

Nobody celebrates the poetic, athletic, or musical abilities of a newborn. Logically, we all know that the skills we have at birth will develop over time. It makes no sense that we spring into this world fully finished yet some of us still lament “I’m just not GOOD at ________” (fill in the blank).

You aren’t good at something that you have only just begun to learn. A growth mindset simply acknowledges that learning new material and acquiring new physical skills take time.  The first time you walked, you did a terrible job (even while your family cheered!). The first time you read a book, you did a terrible job: it took you a long time and you had to take a lot of breaks. The first academic paper you write? Terrible. The first speech you’ll make? Rotten. The first time you play the piano? Ugh.  The first time on a basketball court, dance floor, theater stage, or football field? Not so hot!

The first several times you do anything, you won’t rocket to stardom in doing it (unless your friend gets a funny video when you weren’t looking). But the hundredth time looks much different. Do something with a positive viewpoint enough times and improvement is guaranteed. Your hundredth try means…that’s a good speech, a great paper, a solid game, a proper recital! As you face future challenges ahead, remember to say to yourself, “I can improve with effort, practice, and time.”

I will…

Negative thoughts rob our time. Allow a few negative thoughts to grow and they’ll develop into fatalistic excuses like, “I’ll never be that talented,” or “I’ll never accomplish everything necessary to achieve my goals” or “I’m just not that …” (smart, coordinated, athletic, beautiful, handsome, popular, wealthy, etc.).

Once you start wallowing in negative thinking, like a pig loves a good mud puddle, you’ll tend to stay there until you realize what you’ve done to yourself. Your future is unknown both to you and everyone else. Your negative thoughts hold no truth whatsoever. There is no virtue in having them and they serve no purpose. Negative thoughts are random weights that you collect until the weight of them forces you to stop everything. They are lies you tell yourself about yourself. Once you make a habit of negative thinking, you’ll guard your pile of lies like a dragon guards his gold.

One by one, recognize each of these lies you tell yourself and replace them with improved action plans. Instead of thinking you can’t, won’t, or shouldn’t, you must say “I will!” or “I can!” Remember that you already have a record of achievement, you are perfectly capable of continuing this success, and you will overcome the challenges that life puts in your way.

 

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.

Risk is Part of Life

Give it the Old College Try

Amy E. Barr, The Lukeion Project

According to Grammarist, “give it the old college try,” an idiom that means “to put forth one’s very best effort to achieve something with a high risk of failure,” came not from colleges but from baseball. The old saying was framed around the notorious enthusiasm of an amateur athlete playing for his college team. The passion of a young energetic person attempting to impress his coach and peers might compel a player to catch an impossible fly ball or to risk his good looks by sliding face forward for a home-run.  

Which risks seem worth “the old college try” to you? Maybe you are so risk-adverse that you plan to stay in your basement to safely hide from life’s troubles. Perhaps you enjoy taking risks. You might consider parkour, exotic pets, adventurous foods, and walking a slack-line across vast chasms as worthwhile activities. Good for you! Take pictures for the rest of us.

Risk is unavoidable. Much of it can be the spice of life. Some of it is not so spicy. Most of us are middle-of-the-road when it comes to both risk-taking and our level of enthusiasm for those risks. The average human maintains comfort zones carefully but will occasionally venture outside of favored boundaries to attempt something new, difficult, or even risky (especially to impress).

If you’ve ever been accused of over-thinking, this means you spend quality brain energy vacillating and oscillating between the virtues of remaining in your comfort zone vs taking on challenges. Over-thinking things is what humans have done for centuries. Who or what decides the outcome of ventures tempted?

In Virgil’s Aeneid, Trojan refugees are led by rex Aeneas. They are driven by fate to undertake harrowing trials. The first half of the Aeneid follows Aeneas as he tries to get home to his new territory in Italy. Virgil made his chief character as a Roman parallel to Greek Homer’s Odysseus. Unlike swashbuckling Odysseus, Aeneas piloted past monsters, shipwrecks, and even the underworld to fulfill his patriotic duty to secure the safety of his fellow Trojans and then found the Roman race. Virgil weaves the Fata, the Fates, into his heroic account because despite risks and massive effort, Aeneas and his people will not fail. Aeneas’ success was set in stone by The Fates and by a poet retelling the tale 1000 years later.

Cicero said in his partly surviving work On Fate, “What I want to know therefore is … if there were no such word at all as fate, no such thing, no such force, and if either most things or all things took place by mere casual accident, would the course of events be different from what it is now? What is the point then of harping on fate, when everything can be explained by reference to nature and fortune without bringing fate in?”

Estimating probabilities of success and failure did not really get going until 1494 when Luca Pacioli, a monk in the Franciscan order, posed a puzzle that laid the foundations for modern risk calculation when inn 1654 the Pacioli Puzzle was finally solved. Blaise Pascal and Pierre de Fermat had to get involved. Around this same time an Italian doctor, Girolamo Cardano, estimated the likelihood of different outcomes of rolling dice. Galileo would come to the same conclusions that Cardano did but didn’t make a fuss because he had his hands full with galaxies at the time.

For most human historical record, people viewed risk as something dictated either by fate or divine providence. Even Aeneas feared the fierce interference of an angry goddess intent on drowning what little remained of the Trojan nation, despite the dictates of The Fates (of whom there were three). The will of the Roman gods were not quite as powerful as destiny but often the lines were blurred for we poor mortals.  

What would you risk doing if you knew you could not fail? What if you believed that success—at whatever challenge is ahead—is something that fate has preordained? You would also have to acquiesce to the reverse: failure—at whatever challenge is ahead—is equally dictated by fate. This was the primary outlook of most ancient civilizations. Cicero (On Fate) detailed the problems we would face if we truly believed everything that happens is purely thanks to fate. He forecasts for such a person “the entire abolition of action from life.”

While Cicero continues to grapple with ideas of fate, providence, causation, and free-will, his worst forecast is for those who truly believe that fate is set: “If it is fated for you to recover from this illness, you will recover whether you call in a doctor or do not; similarly, if it is fated for you not to recover from this illness, you will not recover whether you call in a doctor or do not; and either your recovery or your non-recovery is fated; therefore there is no point in calling in a doctor.”

Aside from insurance companies and investment firms who now know how apply their best mathematical calculations to spread-sheets and apps to determine risks and possibilities in the form of fractions of standard deviations, most people still do their own personal calculations about how often they are ready to go all-in on something that seems risky, dangerous, or just labor intensive.

We modern souls are no further along than Cicero as he tries to calculate the role of fate, free-will, divine interference, or (as he says) “the unexplained swerve of atoms” as we navigate life, choose when and where to work, when to rest, and in figuring out what is “meant” to happen to us. Freewill resides in our own not knowing, either way. We wouldn’t want to see our fate nor that of our loved ones. We simply couldn’t bear the knowledge of our own destiny. So, the only logical alternative is to press on with all things as though we cannot and must not fail.

Cicero. De Fato. Trans. R. Rackham