Helping vs. Hindering

Seek Balance when Students Need Help

By Amy Barr with The Lukeion Project

A new class has your child flummoxed. So many new things to sort out, assignments to turn in, links to click, passwords to remember, and study skills to sharpen. Most parents are eager to lend a hand. Did the instructor explain all of this in class? Is there a handout or homework? Are there difficult problems to solve? Parents always wonder how much help truly helps and hope to hit that delicate balance between supportive and “doing your child’s work.”  Where is the line between helping and hindering?
Knowing that there is a line which should not be crossed is the first chore. When parents take a substantial role in “helping” by completing a student’s course work or reading all their assigned passages so they can summarize for them, the student will soon learn the benefit of pretending to be helpless at the same time they internalize the idea, “maybe I really can’t do any of this.” Very soon that same student will actually be helpless, no pretense required. High marks on those first quizzes will diminish as the foundations were never built, the basics never learned. If the student has been reduced to a mere observer, the line was crossed and obliterated.
A student who takes no real role in necessary problem solving, detail work, or research assigned for a class while having a parent do all the heavy lifting (mentally) is doomed. In especially egregious cases, the student internalizes helplessness, and never fully recovers even as an adult. A few skirmishes are won but the war utterly lost.
The pendulum might swing too far in the other direction for some families. Students really can benefit from a little support when getting started in the foreign territory of a tricky new class or, even in subjects that are comfortable in a subject, they need a little help working out a winning study regimen. Parents with a fully hands-off approach in every area may wish that their student had felt more comfortable asking for academic guidance in, say, college when it has long term implications.
The right balance between helping and hindering must change every year of a child’s education. What is appropriate in first grade is inappropriate in 6th grade and disastrous in 12th grade. If the student has any special struggles, the variables increase but the dangers remain the same. Handing out too much accommodation convinces a student that she will not ever be able to handle the tough stuff. Offering too few might be damaging as well. Add siblings to the mix and, well, you get the idea. Parenting is tricky.
As a lifelong teacher, I have seen trouble at both ends of the spectrum, but I have seen worse damage done by the overly “helpful” parent who does not step back enough when the time is right. When in doubt, less is more. Here are five suggestions to help you get a little closer to that balance each year of your student’s education.

1.    Be Socratic

According to Plato, the ancient Athenian philosopher Socrates was best known for knowing nothing. In Socrates’ interactions with students, bystanders, and fellow philosophers, he would simply keep asking them questions. Instead of giving information and facts, somebody using the Socratic method asks students open-ended questions. This dialogue helps a student discover what she does or does not know or might only partly understand. When your child asks for help on academic work, do not give answers but do ask questions. The student is still responsible for the intellectual maturation needed to master a subject but can be guided through areas that need work past areas that are already understood. More importantly, once they have understood the way forward, they will not unnecessarily turn to you as a short cut. A Socratic parent guides, discusses, looks for trouble, but offers no shortcuts.

2.    Do not Offer Easy Answers

So, the answer is clearly 12 or 1962 or laudant or any other easy-to-you response. Your child suspects she can shave 5 minutes off the homework session just by asking you for that fast answer. You might be tempted to give that fast answer because you need the dining table cleared up for dinner. Instead of a fast answer go slow. Return to my first suggestion and be Socratic. A student that really needs help will be guided toward a better direction this way. A student who is more motivated to finish quickly will not develop the expectation that the parent is there to do the work but will be there to offer a little guidance about how the student can find the answer.

3.    Supply the Right Tools

When prepping for a new school year, do not forget essential time management and “executive function” supplies. Chief among such tools is an academic schedule/calendar. I recommend a paper version rather than a digital app. A phone or tablet already offers an abundant assortment of distractions, so it does not make sense to seek a remedy to being distracted on a distraction-device. A paper version should allow a student to see a whole week at a time as well as a whole month. The student needs to plot out a whole semester. Being good at time management starts with fully understanding what must be done, how much prep time is needed to get it done, and managing the amount of time allotted to the task.
Students must learn coping mechanisms to manage their own personal quirks such as being able to concentrate better at certain times of the day or managing any level of ADHD that might trouble them. No formal ADHD diagnosis? No worries! Almost everyone has “popcorn brain” and our collective attention span has shrunk down to the length of a TikTok video. Pushing out one’s attention span and focus into longer and longer periods takes time and intention.
Every successful day starts with a few minutes of reviewing what is coming up, how long a task requires, and what things need priority over others. Parents can help discuss the difference between tasks that require our best time of day by helping to sort through anything that is quickly accomplished. Any chore that can be knocked out in only a few minutes needs to be at top of the list. After that, pick the top three things that must be done that day and then take a do-or-die approach to all three.  
After your student hits middle school years, 6th or 7th grade and up, they must have the primary role. Grab some Post-it notes or a whiteboard to help track priorities. Spending quality time getting (and staying) organized always saves time.      

4.    Check in not out

As our children mature, we must give them more academic autonomy. That is normal and healthy. Students must discover they can do challenging things without somebody stepping in to save them every time assignments become challenging. By the time a student is ready for the end of their high school years and the start of their college years, parents should function exclusively as an advisor not a manager. This does not mean you need to check out. Be Socratic even in this area by being genuinely interested in what is going on in classes. We always accomplished this at the dinner table with discussions about a project they have been working on, that big exam that was last week, or the long research paper that needs more research. You do not need a full inquisition but just check in.

5.    Celebrate wins

Finding the right balance between doing too much and too little for our children as they mature into adults is tricky. Your child will not feel especially great about a top score if they did little of the work themselves. On the other hand, offering just the right amount of support without crossing that line gives the gift of true victory and confidence in personal academic success.  When that happens, celebrate. Sometimes that is just a pat on the shoulder to acknowledge your pride in their success. Let your student overhear your pleasure and confidence in their abilities, sure, but especially their tenacity and their hard work.
Getting organized and staying there is not easy even for adults. Everyone prefers to do enjoyable tasks first leaving less pleasant things for last, rather than prioritizing. Run through several techniques that will help your student stay organized but also produce ways to reward getting the tough stuff done. For my youngest son, a reward was getting to spend twenty extra minutes playing the piano. For my oldest, that meant she got to do something creative in her sketch book. My middle son would climb the tallest mountain if it means he got to spend a little extra time on a game he enjoyed. Rewards can simply be the liberty to do what a student already loves.

The Assignment Sheet

Instructions Should be one of Your Best Friends

 By Randee Baty with The Lukeion Project

Students, when a professor gives you an assignment sheet, he isn’t just repeating what he told you in class. The sheet will have lots of useful information and essential details that must be observed. Use it well to your benefit. Never adopt a “too long, didn’t read” approach to assignment sheets. That never pays off.
Professors put quite a bit of time into creating clear assignment sheets with everything he or she thinks you need to know to finish the assignment just as it needs to be done. Nevertheless, even good students continue to submit assignments that violate clear instructions and therefore lose points or miss out on learning lessons the teacher wants you to learn.
Along with the parameters of the assignment, the assignment sheet will have additional information, possibly including the rubric the professor will use for grading, issues they want you to specifically avoid, tips for completing the assignment correctly, and what they will look for as they grade. They expect assignments to be completed in a specified format and they always expect it to be published to the instructor using a particular file type or link. Guess what? This is vital information if you are concerned about your grade on that paper.
Here is my suggestion to students when using an assignment sheet. Read the sheet a minimum of three times. I don’t mean read it three times before you start. I mean read it at three different points in the writing of the paper.  
Obviously, read the instruction hand-out carefully before you start. Read it however many times it takes for you to understand what the professor wants. Once you think you get the idea, read it from start to finish again. Make notes on it if you choose. Underline and circle important instructions or tips. Email the professor at once if there is anything you don’t understand. Then start the paper. There’s nothing worse than finding out you’ve taken the wrong approach right before the deadline. Most of us can't afford to lose hours of work.
About halfway through the paper or maybe when you finish the first draft, pull out the assignment sheet again. Re-read it from top to bottom, beginning to end. Is there anything the instructions mentioned that you may not have done yet?  Anything they told you not to do that you are drifting into? Are you staying relentlessly on topic? Are you avoiding fluff and padding? Do you have citations in place if needed? It’s a good time to self-check before you spend a lot of time on something and then realize you had neglected a basic instruction or reminder.
Once the paper is finished and polished, read the instruction sheet again. Don’t hit “submit” (or click that submission link) until after that third reading session. The process of writing and polishing can drive other factors from your mind, and you might send in a paper that that took an immense amount of work but isn't at all what was assigned. If you ignore the instructions, there are not fairy-tale endings in which your professor is suddenly OK with a submission that is way off the mark, even if you spent many hours perfecting it.  A perfected error is still just an error. A big part of your grade is accounted for simply by following instructions.
When you are concerned about getting an assignment right, the assignment sheet is your best friend!  Hang out with it, spend time with it, get to know and understand it. Mark it up with notes and underlines. Your professor and your grade transcript will thank you for it.

The Summer Brain

A Season for Creativity and Inspiration

By Amy Barr with The Lukeion Project

Now that we are ending summer months in North America, we can all either reflect on a pretty good summer or we can wish for better chances next year. Depending on each student, family, academic approach, and region, summer means different things. For our brains, summer can be sustained all year, depending on how we allow it to engage with the world. Even for families that choose to “do school” all year ‘round, students should learn how to enjoy a summer brain. What's different about summer?

Engage with Nature

This used to be easy advice for everyone except those who suffered involuntary limitations. For my generation, engaging with nature used to be firm parental instructions to go outside and stay busy “until the streetlights come on.” Threats of backbreaking household chores used to be sufficient to keep younger people from venturing inside before the time was right.
Based on unscientific polls in my own classes, around half of my students have a strong desire to stay inside all year long. Aside from those that enjoy team sports and outdoor practice, only a small percentage of the rest feel compelled to go outside and enjoy (or suffer) all that the great outdoors has to offer unless (and until) they are deprived of one thing: technological entertainments. What a sad situation we are in. More people than not prefer to view other people’s lives in short fake (but entertaining) videos on social media. This means it might take some motivation to exchange indoor habits for outdoor ones.
Engaging with nature doesn’t need to be strenuous or painful nor, necessarily, buggy. Just find a good spot outside every day, get some sunshine (yes, you need some), and find some fresh air. Once you get good at the basics, expand the amount of time you are outside.

Engage with the Present

We spend most of our lives scrambling to get the next thing done. Especially as a person matures and experiences more complex periods of their education and then early adulthood, making and then completing lists becomes all consuming. There isn’t enough time in the day to finish most lists. Sometimes we get to set aside to-do lists for just a little while for a holiday or vacation. When your brain is overly stressed, try engaging with the present for at least a little while. The best way to do this is to try something creative like playing an instrument, creating art, completing a project (garden, woodworking, sewing). These activities force us to focus exclusively on what is at hand which, just like summer break, serves to refresh our brains by enjoying something that is exclusively for now rather than later.  

Engage with Exercise

The best part about summer is that you can find exercise that requires no special skills, tools, equipment, or anything else. Grab a friend, dog, or family member for a walk. Grow some glowers or a garden. Join an informal or formal team for some friendly competition. Learn some dance moves. Mow the grass. Go build something. Nobody feels good if they spend all their time sitting on a couch. A summer brain knows that the best way to get unstuck physically, emotionally, or even academically is to get moving.

Engage with Friends

In the last five years something truly unique in human history has happened. While it is historically important, it is socially devastating. Humans are no longer maintaining a variety of healthy relationships. Even during typically very social periods of our lives, extroverts are becoming more introverted. Introverts are becoming isolated. Academically focused students excuse their lack of social interaction by insisting they have their studies but to their detriment. Friendships and relationships are even more important than a nicely filled out transcript, though one need not preclude the other when accomplished cleverly. Humans must find their tribes or suffer from loneliness.
Kasley Killam, author of The Art and Science of Connection, suggests the 5-3-1 rule. Engage socially with at least five people each week. Intentionally foster friendships with your favorite three people by connecting with them a minimum of once a week. Spend at least one hour a day enjoying a quality connection with others.
What this could look like for you is that maybe you get involved with a sports team, a band, a dance class, or maybe a club in which you have quality time interacting with at least five other people. Parallel play (like watching a movie) or sitting on a bench staring at your phone next to a friend doesn’t really count. Find a way to interact cooperatively with at least five others.
Intentionally foster friendships with at least three people each week. Hanging out is best, phone calls are good, interacting on social media ranks low but will do in a pinch. These minimum of three will be your support system. Yes, siblings count! If you are an extrovert, you’ll have a hard time keeping it to three. If you are introverted, you’ll be hard pressed to find three. Shoot for three as a goal because that’s how many people most of us can manage well.
Finally, tally up at least an hour of socializing daily. It doesn’t need to be formal (though it can be). Maybe just a good discussion at the dinner table is enough to feel connected with other humans. If you don’t have the luxury of a family that takes meals together, don’t forget to link up with a bestie or sibling or relative daily just to talk things through. You’ll feel better when you do.  

A Blank Page

Educational Choices Are Actually Yours

Amy Barr, The Lukeion Project

The start of a new academic year has arrived. Some hate getting back to academic work as they are forced to abandon the relative freedom of summer activities. Others are keen to get back into the swing of things as they look forward to new challenges. Most see their yearly academic schedule as inevitable, something that must be done, boxes that must be checked, and time that must pass one way or another. Which kind of student are you?

I have found through over 30 years as educator that students who have some educational autonomy will have a better attitude towards each new academic year. Conversely, students who feel they are locked into a parade or generic classes and subjects tend to have less enthusiasm about the work at hand.

A person who works a factory job completing a repetitive chore may accomplish much in a single day but seldom looks forward to his task. Even if he manufactures something he loves, his enthusiasm for it will not last long. Humans quickly get bored with repetition and with a lack of personal challenge. Most humans avoid undertakings that offer no room for improvement, no challenges, no changes.

When I was in high school, I worked for a nice pizza restaurant. I picked that job because I loved the pizza there and thought it might be fun. Wishing to earn enough money for college, I spent most of my evenings for a full year making or selling that pizza. I saved carefully and did indeed pay for my own education with that money. I am glad I had that job, but I went from being a pizza-loving teen to somebody that still dislikes pizza many decades later. You can have too much of a good thing.

Most of us gravitate towards projects that allow us to improve and eventually excel if we choose to put in the work. Good health (both mental and physical) depends on the ebb and flow of seasons, changes, and challenges. If everyone gets the same grade no matter how hard they work or if every day looks the same no matter what you do, boredom will lead to apathy. Apathy will lead to resignation. Looking back at my high school pizza job, I know that my boredom was not just eating too much pizza but the constant sameness.

As you start a new academic year or semester, you might feel you have been given few choices about your classes because certain topics must be covered in the short span of a few years. That transcript is not going to finish itself! You may begin to see what is in store and learn to view it through the lens of sameness and respond with a bored yawn.

You may also see what is ahead and realize that your education is a blank page that you get to illustrate and fill out how you like. Yes, you need an English credit each year of high school. NO, that English credit does not need to be the same one that everyone else completes. Foreign language? Math? History? Science? Writing? Yes! Everyone needs those. Learn to tick those boxes and finish those requirements in ways that challenge you.

Your academic plan is all your own. Prepare yourself for a future that interests and engages you. Even if you do not feel you have a ton of choices over the types of things you are expected to study this year, retain control on how you prepare, the amount of effort you expend, and the level of mastery you achieve. You have more control over your current circumstances and your future than you might imagine. Even small investments towards excellence now will fill out the blank page of your education in ways that only you can control. Everyone has the autonomy necessary to become truly excellent at many things. Start today.Your education is not something done to you or for you but by you.

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!
 

What the Romans Did for Us

Thinking about Rome?

By Amy Barr with The Lukeion Project

“Rome,” is now apparently a common answer to the question, “what are you thinking about?”
I agree that anyone hoping to better understand the modern world should look at Rome. The parallels are clear. Comparisons will lend insight to students of history. The best takeaways still serve us well today. Her worst ideas continue to echo through current events. Take five minutes to think about Rome. I’ll keep things brief.
The Romans were famous for adapting and adopting good ideas from other nations and then turning them into great ideas. Architecture, for example, was given a tremendous boost by the ancient Greeks. The Romans enjoyed their artistry and then tweaked the Greek post-and-lintel system by adding the arch. Adapting arches into barrel vaults and groin vaults allowed architects to craft massive structures like the Colosseum and Pantheon. If they tucked those vaults underground, they crafted effective sewer systems that allowed towns to become massive cities. Further adaptations meant Romans had fresh water year-round. This lead to all sorts of luxurious spa experiences that would fetch high prices today though they were usually free in ancient Rome.
The list of Roman contributions is long, and this blog is brief. Highlights include roads (many still used after 2000 years), Latin (still used today in all academic fields worldwide), cement, plumbing, reliable water pressure, good sanitation, heated floors, and hot tubs the size of swimming pools, and unforgettable sports venues. Rome got close to a global currency that worked just about everywhere (we have found Roman coins in India, China, and throughout Europe. Their legal system was sensible, reliable, and included basic human rights applicable to citizen and foreigner alike.
Neutral contributions include the introduction of fast-food restaurants with take-out, massive free public entertainment events that would last for days of blissful vacation time, and satire as a national pastime. The Romans had a relaxed lifestyle with delicious and readily available food but still loved to crack jokes about all of it. Such is human nature.
The Roman Republic was first established in response to the emergent need to topple a violent and dysfunctional monarchy. As is often the case in historical matters, the Romans had to quickly find a good way to prevent tyranny’s return through a decent political and legal system with plenty of checks and balances on her politicians (they considered term limits a great idea, for example).  Early American founders had much in common with early Romans. They too sought a better future for our nation when it became clear that none of us needed kings nor queens.
The Roman Republic worked very well for a little more than 400 years. It served the people and senate of Rome in practical ways while Rome grew from a swampy village to city-state, to nation, to world power, to mega power. Things went well until her politicians discovered loopholes in Rome’s systems of checks and balances. The last century of the Republic left the city in tatters and too much of her population dead. A lengthy period of civil war marked the end of the system. Her politicians had learned how to do whatever they wanted whenever they wished. Once laws are gone, tyranny is the result.  
Chaos diminished under the very strong leadership of Rome’s first emperor. In a series of astounding turns of events, the 19-year-old successor to Julius Caesar would overthrow his competitors and, after having united Rome for the first time in a century, he would rewrite the Roman constitution and then quit. The senate begged their new commander-in-chief to remain and power while they gave him a new name, Augustus. In a little over a decade he would change everything, all while heralded as a patriotic and traditional man-of-the-people. He did love Rome and her empire. He was quick to rebuild, reunite, and expand. Augustus had tremendous organizational abilities, but he also had two big problems. First, he couldn’t live forever. Second, though he was long lived, by the time he did die nobody could remember living under any other system.
Wanting a proper heir, Augustus promoted a disappointing stepson who lacked interest in Rome or her people. Under Tiberius, the whole system could have reverted to a Republic, but the Romans bypassed many opportunities because they were too busy attending sports events. They pressed on and installed a long series of truly dreadful emperors who were easily pressured and conned by those that worked for them. The Roman equivalent of the secret service/FBI/CIA (the emperor’s Praetorian guard) learned that they could easily manipulate any emperor who served. The Roman empire was barely beyond infancy when the Praetorian learned how to obtain whatever commands they placed on Rome’s weak rulers.
The most successful emperors learned how to appease the Praetorian guard with lavish perks and extra power, but it was difficult to maintain positive momentum during transitions to a new ruler. Very rarely was an emperor’s son a worthy candidate so success was limited to emperors chosen for practical skills and intellect rather than legacy.
The Roman empire became massive while her enemies became clever and powerful. They would learn to pick and peck at her boundaries while Rome slowly died of a thousand cuts.   
What caused the fall of Rome? That’s the subject of many books and theories. Rome fell when she outlived her own success by a couple of centuries. Once there were no more major money-making-wars to fight, she turned her attention to micro-managing the empire. Too much governmental oversight was disastrous. Towards the end, the powers governing Rome split her into quarters and made her rulers and sub-rulers bustle around in mobile command units as needed (or even not as needed). Rome as a city lost her identity as capital while new city-centers grew in importance for the sake of convenience.
Perceiving that the leviathan formerly known as Rome was aging and weakening, her enemies kicked at all her doors and windows until they were eventually invited in. Under the pressures of disease, overextended military forces, squandered taxes, under-performing leadership, and a disinterested population, Rome invited her enemies to join their rat race. They were invited to serve in her military and guard her boundaries. Such soldiers had no interest in a Roman way of life. Rome fizzled, sputtered, and died.   

Think-Os

Stupid Mistakes

Dr. Kim Johnson, Logic (Lively Logician) and Counting to Computers with The Lukeion Project

I cannot tell you how many times as a mathematics teacher I’ve seen a student react to a question they’ve gotten wrong on a test or quiz by saying, “Oh, that’s just a stupid mistake.”  It seems as though calling something a “stupid mistake” or “careless error” allows students (and sometimes me) to dismiss the mistake as a onetime occurrence and go on to think about more pleasant things.

Celebrating your wins on tests or essays is crucial. But don’t dismiss the stupid mistakes out of hand---sometimes there is more to them than just carelessness.

Correcting “Think-Os”

A stupid mistake can also be called a thoughtless error. Sometimes I call them “think-os”. Like a typo, a think-o is something that you knew or understood well but wrote or typed incorrectly. Most often they stem from a momentary lapse in concentration.

Given that they are so fleeting, why should we not dismiss them and move onto something that is more important? The truth is, sometimes thoughtless errors turn out to be very important.

In 1999, scientists launched the Mars Orbiter with hopes that it would expand knowledge of the climate on Mars. Their hopes were dashed when, instead of exploring Mars, the orbiter crashed into the surface. The European scientists on the project used the metric system, the Americans used the English system, and so their calculations didn’t’ match up. This was truly a thoughtless mistake because every scientist in that group understood the difference between metric and English measurements. This thoughtless error cost millions of dollars.

“Fine,” you might say “…but I am not in charge of million-dollar projects, I’m just taking my Latin exam, doing some math homework, writing an essay, or taking a history quiz. A stupid mistake in these situations does not have any dire consequences. So why should you care?”

You should care because sometimes a seemingly innocent mistake hides a deeper misunderstanding. For example, if the instructor asks for one form of a verb and you give another, or if you are expected to give a number and you give a function, perhaps you never fully understood the difference and need to clear something up. It’s just a simple mistake---but it indicates that there is something you need to understand more fully before moving on.

Another type of worrying mistake is one that you make over and over. In working with students, I’ve noticed that each student has their own personal set of repeated mistakes. My personal set of mistakes is to lose negative signs and copy down digits in the wrong order. One friend misspelled the word “their” nearly every time he wrote it. Some students have particular facts that they always get backwards.

The best part about these mistakes is that you are not doomed to make them forever.

How To Avoid Careless Errors

Obviously, if you have a gap in your understanding of some part of the topic, work to figure it out!   Honestly appraising your work is crucial in clearing up misunderstandings. Without evaluating the mistake, you might think there is nothing left to think about and move on. The great thing about fixing mistakes which come from misunderstandings is that by fixing them early, you eliminate a whole raft of mistakes later. It’s kind of like killing a mosquito in the spring will eliminate thousands of mosquitos later in the summer.

Fixing mistakes that you make again and again can be slightly more difficult. The problem with trying to fix repeated mistakes is that they indicate a type of momentary blindness and inability to notice errors.

The first and obvious thing to do is to check your work. This involves a few things. First you must write something down to check. Have you ever heard your teacher tell you to “show your work?” This is why! Although an abacus or a counting table is a powerful tool, written work wins, hands down, when you are trying to catch errors.

Second, coming back to your work after a day or at the end of the test is another good habit. Even a few minutes can give enough distance so that mistakes become visible instead of seeming to be correct. When checking my work, I think of it like a “hidden picture” puzzle, or a “spot the difference” puzzle---each difference should be deliberate, not an oversight.

You can develop a habit of going over your work looking for your most common mistakes. For example, my friend would search his papers for all instances of “their” spelled in his particular way, and I watch for lost negative signs. You can learn what errors you tend to make and look specifically for those mistakes. By focusing on and analyzing what you first dismissed as “stupid mistakes” you can find them and even get better at avoiding them.

Another strategy for catching foolish errors is to check with someone else. When mathematicians are working, they often work together. This is in part to generate more ideas, but it also allows them to catch each other’s momentary lapses in thinking. Even the most accomplished writers have editors. Software engineers review each others’ code to catch bugs. We can help each other reveal thoughtless errors.

Sometimes this is not allowed because of an honor policy, but even then getting another person involved can help. One of the best ways to catch errors is to explain the work to someone else. By going over your work out loud you introduce another sense---your ears can catch what your eyes do not. Pets are some of the most patient listeners. Even though your dog will never have to factor quadratic equations of conjugate Latin verbs, they will listen patiently and never judge when you find a mistake.

Get Over It

Finally, not all mistakes are signs of misunderstanding or part of a pattern. Sometimes a stupid mistake is just that. If that’s the case, your most potent tool is to just leave it behind and move on. Dwelling on mistakes takes mental energy that would be better put to use in learning new material and analyzing new ideas. Try pretending that you are looking at someone else’s work as a teacher or a scientist. Then you can judge more accurately whether the mistakes require further action or if they are not matters of concern.

Mistakes will definitely come, that is certain. Making mistakes is part of being human. Learning to deal with them appropriately---with honesty, attention, and grace---makes us better humans

Autodidacticism

You Have Everything YOU Need For a Good Education

Amy Barr with The Lukeion Project

The English word “education” comes from the Latin word educare, a verb that can refer to the process of bringing up children to adulthood or providing a wholesome place for animals or crops to grow. Modern usage would have us think that education is a commodity that we might purchase in the form of a book, a program, a class, or whole degree program. When we are done with one or all of these, we proclaim ourselves “educated.” The Latin word does not refer to any of these things, per se. Instead, one’s education is a matter of having a favorable environment or fertile soil for somebody (or something) to grow productively. Education, for much of human history, has been a personal choice and under our own control.

In the Roman world, those that worked on a subject very diligently might eventually be considered doctus by others, meaning they were well-informed, experienced, and clever. Doctus is a passive participle of doceo, a verb which means “teach” (a doctor was a teacher not a medical expert). Somebody who is doctus has been “taught” but not necessarily by others and certainly not in some established program. Remove your modern sensibilities about being taught. The English words we use to define doctus or docta have more to do with how one treats oneself as one builds one’s own learning environment. A well-informed, experienced, and clever person takes the initiative to self-educate regardless of one’s station in life.

The most brilliant men and women in history have been autodidacts who, until very recently, had zero framed diplomas hanging in their offices. Degrees, as such, didn’t factor into anyone’s career qualifications until the far more recent past. The notion of placing a measure (and certificate of completion) on one’s education was not something that most people thought about until public education compelled us to think in terms of something that could be measured, quantified, qualified, and marketed to the human race.

Greek philosopher Plato once said, “The first and best victory is to conquer self.” Aristotle, Plato’s student, listened carefully to his master’s words for 20 years and then reversed his opinion on most of Plato’s teachings to form his own thoughts. He famously said, “Through self-discipline comes freedom.” As he too would become an educator of famous people like Alexander the Great, Aristotle would also say “What lies in our power to do, lies in our power not to do.”

Education today is not something our Classical fore-bearers would recognize or condone. Today we view education as a set of stairs. We climb up some of those steps and celebrate when we’ve arrived at a certain stage while we earn a slip of paper that some institution has issued to say as much. Leaving behind all that was accomplished before, we climb a new set of slightly more difficult stairs and look for new affirmations of completion until, armed with seemingly enough certificates, we are deemed qualified. Those that didn’t climb those particular stairs are not authorized to form opinions or advice in that area. Such people are merely amateurs, a word that comes from the word for love (amare) in Latin. 

Modern education is a lucrative system that is jealous of all forms of competition. We are told we aren’t qualified to do our own research and we aren’t qualified to educate ourselves. Those who are home-educated must simulate a public education by jumping through similar hoops in hopes of being issued a graduation certificate which, despite numerous studies that prove that home education is the far superior method of learning, is often considered sub-standard by many secondary institutions. Many top universities are slowly understanding how valuable our home educated autodidacts are! The top skill mastered in home education is how to educate oneself, a skill that becomes a game-changing advantage in college, graduate school, career, and life. Those that don't want any part of that type of conveyor belt system are benefited as they teach themselves how to live the kind of life that interests them most. 

Autodidacticism (self-education) is and has always been the most reliable method to become well-informed, experienced, and clever. Our own interests and passions are a powerful tool to compel us to excel. Autodidacts are never sated with mastering a single body of knowledge. Having learned one and then two, self-educators are driven to learn by their own interests. They choose to master more and still more. While such people often go on to collect a variety of degrees because they are still constrained by our modern system, they will always continue their own learning process provided they have life and breath.

As Plato points out, your first task is to master as an autodidact is self-discipline. Rising above your own lazy instincts to sleep more or play more means you’ll have the ability to push on, to persist in your efforts to master some topic or skill. If you have enough self-discipline to persist, there are no limits to your goals or ambitions. This is true freedom. The choice falls on you. You have control over your own timeline, effort, and skill. Degree programs might require a set number of years and certainly a prearranged fee, but a self-disciplined autodidact has control of his own time, her own effort, and the quality of all outcomes. Armed with far more educational materials and opportunities than history’s brightest minds, the best education today can be free, or nearly so.

 

 

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.