Thursday, January 01, 2015

Return of the Daily Quiz

A regimen of low-stakes testing in class – and self-quizzing (reciting notes from memory) while studying – vastly improves learning outcomes and makes high-stakes testing far less daunting.

So says professor Henry L. Roediger III, author of "Make It Stick: The Science of Successful Learning" and an intriguing NYT article on the subject.

This certainly reflects my own experience as a student and teacher. The effort required to recall information does seem to exercise critical brain functions, improve intelligence, and promote academic success in a number of important ways. Well-established learning models like SQ3R and Cornell Notes are classic implementations of this idea.

Though out of fashion pedagogically for some time, memorization is an essential element of learning and an important part of what students should be doing during school hours and while studying at home.



Copyright © 2014 Christopher R. Borland. All rights reserved.

Monday, December 01, 2014

Two Dynamite Questions

Dave Denman, of Dave Denman Associates (with offices in Sausalito, California: 415-332-1831), an expert educational consultant and long-time colleague of mine, gave me a tip many years ago that was one of the best study tips I’d ever heard. I’ve had the opportunity to pass it on to many of my students over the years.

Here it is:

During or after completing your homework in a given class (say, history), think of two "dynamite questions" to ask your teacher the following day, about the general topic you’re currently discussing in class. Make these questions interesting, not run of the mill, but as thought provoking or even controversial as possible. Jot these questions down in your notes.

Then, at an appropriate time during class the next day, be sure to ask your teacher these two questions!

Asking thoughtful questions is one of the best ways to make the class more relevant and interesting to other students, and doing so will certainly get you noticed by your instructor. In addition, coming up with these questions in the first place gets you thinking more deeply about the material you’re learning in class and therefore tends to make it more interesting and memorable to you. Teachers always greatly appreciate and respect students who ask great questions, especially those who make a habit of doing so on a regular basis. These students are obviously taking a special interest in the class, and nothing makes a teacher’s day like students who consistently put forth the effort to ask insightful, probing questions.

Although it only takes a couple of extra minutes to do so, recording and then asking dynamite questions really livens up class time, makes your teacher’s job a whole lot easier and more enjoyable, and goes a long way to making any class more fun and interesting for you and for your fellow classmates!



Copyright © 2008 Christopher R. Borland. All rights reserved.

Saturday, November 01, 2014

Memorize The Quadratic Formula ... In Seconds

Ah ... the venerable Quadratic Formula:

Beginning algebra students everywhere have to memorize this well known formula for solving quadratic equations of the form:

ax² + bx + c = 0.

Many students in early algebra courses lose critical test and quiz points just because they haven't yet fully committed the Quadratic Formula to memory.

I recently came across a brilliant way to do so almost effortlessly ... a memory trick so sublimely ridiculous I couldn't help but think of the famous line: "This is so crazy ... it just might work."

(Sung slowly to the tune of "Pop Goes the Weasel")

"X is equal to negative B

Plus or minus the square root

Of B squared minus 4AC


Over 2A."



Try it!



Copyright © 2007 Christopher R. Borland. All rights reserved.

Wednesday, October 01, 2014

Eternal Prosperity for Your Posterity

Assumptions and Definitions:

Individual X and spouse will raise two children each of whom will have spouses and raise two children (four grandchildren for individual X and spouse).

Individual X and spouse and all descendants successfully attend college around age 20, complete grad school and begin productive work or start and run a business from 20-30, marry for life someone of similar means and have two healthy children at age 30, raise their families successfully and are able to save modestly till retirement age, retire at age 60, live and retire comfortably (but not lavishly) the entire time, and die at age 90.

Undergrad education costs $200,000, and grad school or starting a business costs $100,000.

All such individuals live with spouse and minor children in a $600,000 home that required a 20% downpayment to purchase at age 30 with a 30 year fixed mortgage at an affordable interest rate and can invest money till age 60 at 7% real growth and 3% thereafter (i.e. nominal rates of 10% and 6%, respectively, above 3% inflation) taxed at reasonably low average rates.

Prosperity: Having such a life.

Protoparents: The original two spouses who begin the process of eternal prosperity for their posterity.


Given the miraculous effect of exponential growth (compounding interest), there exists a minimum "critical mass" number N for which Protoparents could fund an endowment for each grandchild at birth that would allow each of these grandchildren to live prosperously and do the same for each grandchild, and so on, ad infinitum … effectively guaranteeing eternal prosperity for their posterity.

What is the value of N?

Turns out, N is about $80,000 in 2018.

Following are some key milestones along the way (attainment of any one of these by each prospective protoparent will enable them to achieve critical mass for themselves and all posterity):

Age 0:

Grandchild G is born, whose grandparents establish a trust fund in the amount of $80,000.

Age 10:

G’s trust fund balance is $160,000.

Age 20:

G’s trust fund balance is $320,000, and G is in college. $200,000 is paid out to cover undergraduate tuition, leaving a balance of $120,000.

Age 30:

G’s trust fund balance is $240,000. G has completed graduate school and begun work or started a business, has married and has two children, and owns a home with spouse that required G and spouse each to contribute $60,000 toward the down payment. $100,000 is paid out to repay grad school or business loans and $60,000 to cover G’s contribution toward the down payment on the home, leaving a balance in G’s trust fund of $80,000.

Protoparent Goal: $80,000 + $80,000 + $60,000 = $220,000
Protoparent Spouse’s Goal: $80,000 + $80,000 + $60,000 = $220,000

Age 40:

G’s trust fund balance is $160,000.

Protoparent Goal: $440,000
Protoparent Spouse’s Goal: $440,000

Age 50:

G’s trust fund balance is $320,000.

Protoparent Goal: $880,000
Protoparent Spouse’s Goal: $880,000

Age 60:

G’s trust fund balance is $640,000. Parents of G and spouse all die at age 90, leaving G and spouse each $640,000. G now has liquid assets equal to $1,280,000. Since G and spouse were of similar means, G’s spouse now also has liquid assets totaling $1,280,000. Together, G and spouse have $2,560,000 in liquid assets. G and spouse establish $80,000 trust funds for each of their four grandchildren, at a total cost of 320,000. This leaves $2,240,000 total liquid assets for G and spouse, together, which provides them with a safe 4% real return of $89,600, with no mortgage payment. Assuming their home has doubles in value, subtracting 1% annual property tax leaves them with a comfortable $77,600 real annual income, with no housing expense, for the rest of their lives.

Protoparent Goal: $1,760,000
Protoparent Spouse’s Goal: $1,760,000

Age 90:

Total liquid assets for G and spouse have remained at $2,240,000 during their entire 30 year retirement, since they’ve only withdrawn real annual returns equal to 3% over inflation. G and spouse die with $2,240,000 in liquid assets, leave, and leave $960,000 to charity and $640,000 to each of their two children … and the cycle recurs.


The above assumes that neither G nor spouse saves any of their own money from work. If together this couple can save an extra $10,000 per year on average for 35 years (age 25-60), this would add $1,400,000 to their total liquid assets at retirement, raising their annual retirement income to $145,600, enabling them to establish a nearly $2.5 million family charitable foundation at their deaths, which after 90 years (3 generations), assuming all descendants followed suit, would grow to almost $25 million while doing almost $50 million in charitable work (in real 2018 dollars) by that time!

Starting at age 25, with average annual savings of $20,000 would grow to approximately $440,000 by age 40 at 7% real interest, achieving the indicated protoparent savings goal for age 40. This is an achievable initial goal for many aspiring protoparents (double this average annual savings figure would be required for single-earner households).

Naturally, life rarely conforms to such rigid schedules and round numbers as these. Nevertheless, I think it's useful to have rough long-term financial goals such as the above milestones. Makes saving more fun!



Copyright © 2018 Christopher R. Borland. All rights reserved.

Monday, September 01, 2014

College Rankings Conundrum

Love them or hate them, college and university rankings are an important feature of the American higher education landscape.

In articles addressing the trouble with college rankings, John Tierney and Malcom Gladwell opine on factors driving the college admission frenzy.

USNews College Rankings are focused on big name prestige and are usually the most quoted; but they're not necessarily the most valid or useful.

Washington Monthly College Rankings focus on the other side of the coin: social mobility, research, and serving the public good.

Forbes publishes a well-known annual list of America's Top Colleges.

Business Insider's list of 50 Best Colleges in America emphasizes graduation rates and early-career earnings.

The London Times also provides yearly World University Rankings and U.S. Liberal Arts College Rankings.

In this day of quarter million dollar undergrad degrees, the USNews lists of Best Value Universities and Best Value Liberal Arts Colleges get lots of well-deserved attention.

Numerous other lesser-known lists warrant consideration, as well (see here and here).

In the hyper-competitive early 21st century higher-ed arena, getting in to college – and then getting out, successfully – is probably more important than where one matriculates.

Ultimately, success for graduating high schoolers will depend mostly on the factors that have always mattered most: relationships, talent, and tons of hard work and persistence.



Copyright © 2018 Christopher R. Borland. All rights reserved.

Friday, August 01, 2014

How To Fail A Test With Dignity

Sometimes, you just can't win. In that case, there's no harm or shame in surrender.

So why not have a sense of humor about it?

These students tried and failed, but succeeded in turning their loss into laughs.




Copyright © 2010 Christopher R. Borland. All rights reserved.

Tuesday, July 01, 2014

The MacTutor History of Mathematics Archive

Math teachers, students, and other math enthusiasts will absolutely love the MacTutor History of Mathematics Archive (MHMA) published online by the School of Mathematics and Statistics of the University of St. Andrews, Scotland (the third oldest university in the English speaking world, founded in 1413).

This comprehensive site allows users easy access to thousands of biographies of important mathematicians throughout the ages, informative articles on hundreds of math history topics, treatises on dozens of famous mathematical curves, fascinating time lines of mathematicians' lifetimes and key events in math history, a helpful glossary of mathematical terminology, indices of female mathematicians and math educational history, and links to other noteworthy math history sites.

Math fans can spend many enjoyable and informative hours on the MHMA. The articles and other resources it contains combine rigor with accessibility as scholarly works that nevertheless remain perfectly intelligible, interesting, and useful to the lay reader lacking a mathematics degree. It's no wonder the site gets two million hits per week and almost one million unique visitors per month.

One of my favorite sections is the quotations index, which lists pithy quotations by famous mathematical figures.

Here are a few gems I lifted from this area of the site:


George Pólya

The first rule of discovery is to have brains and good luck. The second rule of discovery is to sit tight and wait till you get a bright idea.

To teach effectively a teacher must develop a feeling for his subject; he cannot make his students sense its vitality if he does not sense it himself. He cannot share his enthusiasm when he has no enthusiasm to share. How he makes his point may be as important as the point he makes; he must personally feel it to be important.

A mathematics teacher is a midwife to ideas.

Look around when you have got your first mushroom or made your first discovery: they grow in clusters.

A GREAT discovery solves a great problem, but there is a grain of discovery in the solution of any problem. Your problem may be modest, but if it challenges your curiosity and brings into play your inventive faculties, and if you solve it by your own means, you may experience the tension and enjoy the triumph of discovery.

Bertrand Russell

The method of "postulating" what we want has many advantages; they are the same as the advantages of theft over honest toil.

The desire to understand the world and the desire to reform it are the two great engines of progress.

Although this may seem a paradox, all exact science is dominated by the idea of approximation.

The whole problem with the world is that fools and fanatics are always so certain of themselves, but wiser people so full of doubts.

Boredom is a vital problem for the moralist, since at least half the sins of mankind are caused by the fear of it.

Men who are unhappy, like men who sleep badly, are always proud of the fact.

A habit of basing convictions upon evidence, and of giving to them only that degree or certainty which the evidence warrants, would, if it became general, cure most of the ills from which the world suffers.


Check here for the MHMA biography of Zeno of Elea, here for a discussion of the epicycloid curve (here for the Java-enabled version), here for a index of chronologies of important discoveries in mathematics, here for an index of time lines of mathematicians, here for the quotations index, and here for topics in the history of mathematics education.

If you like numbers, and you enjoy history, you'll love MacTutor History of Mathematics Archive.

See you there!



Copyright © 2009 Christopher R. Borland. All rights reserved.

Sunday, June 01, 2014

Get The Solutions Manual

While textbooks required for demanding math and physical science courses generally provide explanations and examples designed to help students review covered material or make better sense of teachers' lectures, the sample exercises included are often too limited or few in number to paint students a complete picture of the topics presented. Answers are usually given for odd-numbered homework problems, but oftentimes this just isn't enough. The issue for many students in these tough classes is how to get those particular answers.

Wouldn't it be nice if there was a companion book that would actually list the step-by-step solutions for each odd-numbered problem in a given math or physical science textbook ... not just the answers?

In most cases, there is such a book. It's called the "solutions manual," and possession of one of these miracle volumes can sometimes make all the difference between ease and stress, progress and frustration, success and failure.

The solutions manual is the perfect compliment to the required text for any important math or science course. When stuck in the mud on a particular homework exercise, students can simple appeal to the solution shown in the manual, allowing them to answer their own questions and continue their forward progress with the assignment. Instead of pulling out their hair over a particularly vexing problem, students can use the solutions manual to quickly figure out what they've been doing wrong, and then apply the techniques illustrated in the manual's solution to other similar exercises, saving valuable time and energy (e.g. solutions to corresponding even-numbered exercises can typically be inferred from those given in the manual for odd-numbered problems). In this way, the solutions manual acts as a virtual tutor, enabling students to make much more efficient and effective use of time spent doing homework or studying for exams.

I always recommend that students about to take a difficult math or science class invest in a copy of the solutions manual for the textbook required for that class. The best way to find a copy is to visit the publisher's web site or do a Google search using the full name of the text followed by the author's name and preceded by the phrase "solution manual" or "solutions manual" (without quotes). Once you have an ISBN number for the appropriate solutions manual, you may be able to find a less expensive copy by searching for it on amazon or eBay

(Note: the term "solutions manual" usually refers to the "student solutions manual" containing solutions to odd-numbered problems; the "instructors solutions manual" shows solutions to all problems in the text, but is available only to course instructors).

Don't pound the table, become discouraged, or fall behind ... just get the solutions manual!



Copyright © 2009 Christopher R. Borland. All rights reserved.

Thursday, May 01, 2014

What NOT To Bring To Show And Tell!

Most of us have fond memories of grade school "Show and Tell" times, when we as youngsters could bring "cool stuff" to share in front of the class.

"Show and Tell" fosters confidence, introduces little ones to the art of public speaking and presentation, and injects welcome variety (and uncertainty ...) into the daily classroom routine. Thankfully, this old-fashioned pedagogic technique has not gone the way of the Dodo, and is still practiced in many early elementary school classrooms throughout the country.

Nevertheless, some "cool stuff" really ought to stay at home.

An elementary school near Dallas was evacuated recently when a second-grade student innocently brought a deactivated hand grenade to present during his class's "Show and Tell" period.

The principal took the prudent step of emptying out the entire school until police could determine that the neutered explosive device posed no threat. Though it contained a pin, the grenade was empty and harmless, and ultimately no one was hurt.

Can you imagine the lecture this kid got? My ears burn just thinking about it.

An article in Yahoo! News documents the incident.



Copyright © 2009 Christopher R. Borland. All rights reserved.

Tuesday, April 01, 2014

Pythagorean Legends, Pythagorean Triples

The Pythagoreans were a secret society in ancient Greece, a mystical brotherhood dedicated to mathematical study. Their motto was "All is number." Esoteric groups such as the Freemasons and Rosicrucians claim descent from the Pythagorean Brotherhood.

The central assumption of the Pythagorean brotherhood was that the entire universe could be explained by careful application of whole numbers or their ratios (i.e. fractions, or rational numbers). Legend has it that when a member of the cult of Pythagoras proved that even something as simple as the diagonal of a square could not be measured using whole numbers or their ratios, he was summarily executed to avoid insulting prevailing Pythagorean doctrine.

Nevertheless, Pythagorean theories have been applied over the centuries to such diverse problems such as building the great pyramids of Egypt, establishing musical scales, and mapping the orbits of planets.

The good ol' Pythagorean Theorem is one of the most fundamental formulas in all of mathematics. Falsely attributed to Pythagoras of Samos, this famous rule defines the relationship between the lengths of the sides of all right triangles:

a^2 + b^2 = c^2.

Pythagorean Triples are trios of whole numbers that conform to the requirements of the Pythagorean Theorem, and therefore measure the three sides of a right triangle: the two "legs" forming the right angle, and the "hypotenuse" (not "hippopotamus") opposite the right angle.

Here's a fun and simple way to generate Pythagorean Triples from any given natural number (indeed, all Pythagorean Triples can be generated this way):

1. Pick a whole number larger than two. Call this number n.

2. If n is even: let y be 1 and let x be half of n; otherwise (if n is odd) first subtract 1 from n, then let y be half of this new number and let x be one more than y.

3. The following expressions will generate the numbers in a Pythagorean Triple:

x^2 – y^2


x^2 + y^2



Copyright © 2008 Christopher R. Borland. All rights reserved.

Saturday, March 01, 2014

Car Talk Puzzlers

NPR's uproariously funny Car Talk could be the single funniest thing on radio.

On more than one occasion it was all I could do to maintain control of my car as I listened to the hilarious rantings of Tom and Ray Magliozzi, better known as "Click and Clack ... the Tappet Brothers," during their weekly automotive question and answer show (to get an idea, check out the current list of Car Talk staff members, presumably vetted by the show's ace legal firm: "Dewey, Cheetham, and Howe").

Far from being among the dimmer bulbs in the lamp, these grease monkeys are actually scientists with MIT degrees, no less. A featured part of each broadcast is the weekly "Puzzler," and a fair number of these fabulous brain twisters are mathematical rather than automotive in nature (here's where the guys show their MIT roots).

Click here for an archive of Puzzlers from past shows ... or here to buy a recently published collection of favorite Car Talk Puzzlers at for as little as $.01 plus S & H!

To listen to Car Talk (online, NPR affiliate station, Sirius Satellite Radio, podcasting, etc.), click here.



Copyright © 2008 Christopher R. Borland. All rights reserved.

Saturday, February 01, 2014

Finite Simple Group Of Order Two

Whoever said math isn't musical has never heard The Klein Four.

If you've got a couple of minutes, take a break, give yourself a treat, and check out this hilarious, cleverly written, well performed a cappella composition:

Finite Simple Group of Order Two

Although knowledge of group theory isn't at all required to enjoy the performance, the brilliant plays on words that comprise the lyrics will make real sense only to those with some knowledge of higher mathematics.

Here they are:


Finite Simple Group of Order Two

The path of love is never smooth
But mine's continuous for you
You're the upper bound in the chains of my heart
You're my Axiom of Choice, you know it's true

But lately our relation's not so well-defined
And I just can't function without you
I'll prove my proposition and I'm sure you'll find
We're a finite simple group of order two

I'm losing my identity
I'm getting tensor every day
And without loss of generality
I will assume that you feel the same way

Since every time I see you, you just quotient out
The faithful image that I map into
But when we're one-to-one you'll see what I'm about
'Cause we're a finite simple group of order two

Our equivalence was stable,
A principal love bundle sitting deep inside
But then you drove a wedge between our two-forms
Now everything is so complexified

When we first met, we simply connected
My heart was open but too dense
Our system was already directed
To have a finite limit, in some sense

I'm living in the kernel of a rank-one map
From my domain, its image looks so blue,
'Cause all I see are zeroes, it's a cruel trap
But we're a finite simple group of order two

I'm not the smoothest operator in my class,
But we're a mirror pair, me and you,
So let's apply forgetful functors to the past
And be a finite simple group, a finite simple group,
Let's be a finite simple group of order two
(Oughter: "Why not three?")

I've proved my proposition now, as you can see,
So let's both be associative and free
And by corollary, this shows you and I to be
Purely inseparable. Q. E. D.



Copyright © 2008 Christopher R. Borland. All rights reserved.

Wednesday, January 01, 2014

Study Buddies!

One of the best things you can do to improve your chances of success in a given class is also one of the most fun.

"Study Buddies" are people you meet and with whom you agree to form informal academic partnerships in each course you're currently taking. Not only will these relationships help you in making sure you make it through the course successfully, they can also add spice to your social life!

As a new semester begins, find two people in each class, fellow students you like who seem to share your particular standards and interests, and exchange contact info (phone, email, etc.) with them. Now, if you miss class and need information about homework, an upcoming test, lecture notes, etc., you have two people you can call (if one isn't available, the other probably will be).

Study Buddies may even choose to go beyond occasional contact to form more formal study groups that meet before important tests to go over course material and help answer each others' questions, work together on group projects, etc.

Forging and utilizing these alliances can result in better grades, reduced stress, and enhanced involvement and participation in class, while providing a safe and productive way to make contact with other students you'd like to meet and get to know better.



Copyright © 2008 Christopher R. Borland. All rights reserved.

Sunday, December 01, 2013

The I Hate Mathematics! Book

I came across this little gem back in the early days of my private teaching career nearly 30 years ago (I still have my original copy), and it remains one of my very favorite books for students young or old who struggle with math phobia or maintain an abiding hatred for the subject.

In a wonderfully accessible style, with engaging illustrations, jokes, activities, and explanations, the authors manage to show the "other" side of math ... the creative, beautiful, quirky, fun side hidden from all but the most fortunate students.

This is not an instructional text, not at all, but rather a funny, fascinating exploration aimed to crack the fearful shells of the math phobic and blow the minds of math haters everywhere.

The I Hate Mathematics! Book should be required reading for all young students and others who wish math would just "go away." After any serious reading of this short work, it's difficult to believe that math is hopelessly beyond one's grasp, or to retain a complete lack of interest in the subject (much less an thorough contempt for it). Parents should buy this book, read it themselves, and then read and discuss it with their youngsters who display any signs of serious aversion to mathematics.

The only improvement I could suggest would be to strike through the word "Hate" in the title and replace it with the word "Love," as I did with my original copy – such is the unexpectedly powerful, positive effect this book can have on its readers.

Recommended for ages 9-12.

Find it here.



Copyright © 2008 Christopher R. Borland. All rights reserved.

Friday, November 01, 2013

Don't Let Your Studies Put You To Sleep

Once students enter high school, and certainly after starting college, late night study sessions become at least an occasional necessity. As a red-eye session grows longer and the body and mind increasingly tire, the law of diminishing returns eventually sets in, reducing the benefit of additional study. Before fatigue and stress combine to turn your brain to cardboard, it's good to learn a few key tips about how to avoid falling victim to "sudden brain shutdown syndrome."

In essence, the rule is to do that which is most likely to put you to sleep as early as possible in the day. If you've got a lot of boring reading to get out of the way, or need to complete some hideous math homework that makes you want to do almost anything else instead, be sure to do these things while you're fresh and wide awake, resisting the temptation to put these off until the wee hours of the evening when you can barely keep your eyelids up as it is. Uninteresting, yawn-inducing activities most likely to put you to sleep should be done as far from sleep time as possible, saving more engaging or physical activities like lab experiments, model-building, or group collaboration for later in the evening when extra interaction or increased interest will help you stay awake.

If there's just no way to avoid dull, passive activities late at night, consider moving your studies temporarily to a coffee shop or similar gathering place where the buzz of people around you will help keep your eyes open and prevent "zombie brain" without so distracting you that you can't hear yourself think.

It's a good maxim to live by, anyway: do the stuff you don't like first, then reward yourself with the cool stuff afterward. The willingness to delay gratification yields even greater benefit for busy, semi-exhausted students who must find a way to get an impossible amount of work done and yet remain conscious and functional while doing it all.



Copyright © 2007 Christopher R. Borland. All rights reserved.

Tuesday, October 01, 2013

Arithmetic, Algebra, And Mathemagic

Arithmetic can be called the study of "known" numbers, or calculation.

Algebra, then, is the study of "unknown" numbers.

Arithmetic is easy. All it takes is a good teacher, and sufficient practice.

Likewise, since all real numbers (whether known or unknown) obey the same rules, algebra is easy – provided the student is well taught and well practiced.

Knowing only simple algebra empowers one to do some pretty interesting and impressive things, including all kinds of "mathemagic" tricks, like the one below (trust me ... it's just simple algebra).

Give it a try!

Here goes:

1. Start with the number of doors in your home.

2. Multiply by 2.

3. Add 5.

4. Multiply by 50.

5. Add the number of legs on a normal moose.

6. Subtract 335.

7. If you’ve already had your birthday this year, add this year; otherwise, add last year.

8. subtract the number of days in July.

9. Add 85.

10. Subtract the year you were born.

11. Add 29.

12. Subtract the number of ears you have.

The three or four digit number you now have reveals the number of doors in your home followed by your age.



Copyright © 2007 Christopher R. Borland. All rights reserved.

Sunday, September 01, 2013

Follow A Daily Study Schedule

One of the most important study habits for students to establish and maintain is that of following a daily study schedule.

Without setting aside regular times each day to get homework, project work, and test prep done on time, it's just too easy to procrastinate and not get the job done. Many students study only haphazardly, in fits and starts and at odd times, often waiting till "crunch time" to frantically complete work in a mad rush just before deadlines pass. Quality suffers as stress and pressure increase, and as a consequence, grades diminish (to say nothing of students' enjoyment of the learning process).

By contrast, forming the habit of "clocking in" and "clocking out" at particular times in a well-planned daily study schedule helps ensure that homework and projects will be completed consistently and on time, and will meet or exceed acceptable standards of quality. Eliminating the feeling of "overwhelm" that comes from being chronically late is just one of the many benefits associated with adhering to a strict daily study routine. In addition, maintaining regular study hours often makes it possible to get important long-term work done early, allowing extra time to improve quality and polish the final product.

As soon as children are given homework on a regular basis parents should provide them with a daily study schedule to follow to make sure they're able to get their school work done as soon as possible after coming home from school. After a suitable break to unwind after a long day at school, three consecutive periods of study should begin, roughly equal in length, with a short break in between: "Homework Time," "Project Time," and "Extra Time."

Homework Time is for completing short-term assignments due the next day (i.e. homework), with the most important or difficult assignments done first. Project Time is for long-range assignments, projects, or papers generally due more than one day in advance (including upcoming tests and quizzes). Extra Time is to complete any work not finished during the first two daily study periods, or to review or polish work already completed.

Each period should be of a reasonable fixed length (e.g. one hour for high school students, 45 minutes for middle school students, and proportionally less for younger children). Students should habitually "switch gears" at the end of the allotted time and move on to the next period of study (even if work remains from the previous period), but not before. On particularly light days, it's permissible to finish the first two periods early, if all work has been accomplished, but not the last; Extra Time should always last the full time allotted, if only to do extra review, focused reading, skills practice, or similar study. If sports or other after school activities prevent commencing the study schedule just after the school day ends, it should begin as soon as possible after these other activities have concluded.

Here's a sample schedule for a high school student who gets home at 3:30 P.M., with no after school activities during the week:



Homework Time: 4-5pm; Project Time: 5-6pm; Dinner and relaxation: 6-7pm; Extra Time: 8-9pm.


Additional time as needed to keep academic skills, assignments, and projects on track (i.e. ahead of schedule), and to be prepared for upcoming tests and quizzes.


Making a habit of working on both homework and long-term projects and/or upcoming tests and quizzes each and every day makes it difficult for too much unfinished work to pile up. Nevertheless, pile-ups can occur, and should be dealt with as quickly as possible over the weekend to prevent them from dragging on or growing worse. Likewise, it's a very good idea to occasionally use free time on weekends to get out in front of especially difficult or complex projects before they can turn into problems; investing extra hours to get ahead in this way is like "saving money in the bank," and is an extremely profitable and wise use of one's time.

It really is by far the best approach, if possible, for children to make a habit of completing all school work each day right after getting home from school. Nothing should be allowed to interfere with a student's work during study periods (no phone calls, television, music, internet, etc.). But under NO circumstances should a student's study schedule be allowed to interfere with adequate sleep; all study periods MUST be completed before bedtime, preferably well before (nothing ruins health and wrecks academic performance like sleep deprivation).

Committing to a regular study schedule is a sacrifice – but a far less painful and much more profitable one than defaulting to the helter-skelter "study when I feel like it" method adopted by so many unfortunate young people. Parents interested in the academic progress and happiness of their children should insist they follow a daily study schedule to maximize the probability of their success in school.



Copyright © 2007 Christopher R. Borland. All rights reserved.

Thursday, August 01, 2013

Money Flow

To optimize the flow of money in one's life, it’s helpful to use the metaphor of cascading “money buckets.” Just as a series of linked buckets store and direct water, one into the next, money buckets hold and control the movement of money in your life.

You’ll need four basic money buckets, each flowing into the next, in the following order.

1. Checking Account

The top-most bucket is your household checking account into which paychecks and other income is deposited and from which bills and other expenses are paid.

This bucket should contain an amount sufficient to pay at least one but no more than three months of essential expenses. For example, if your essential monthly expenses total $2500, your checking account balance should be maintained between $2500 and $7500.

Once this bucket is full, extra money should be transferred into the next bucket:

2. Emergency Savings

It’s prudent to have an emergency cash fund equal to 3-6 months of essential monthly expenses. For an extra benefit, this money can be stored in a savings account linked to your household checking account, providing overdraft protection to the checks you write. For example, if your essential monthly expenses total $2500, this fund should contain between $7500 and $15,000.

This is your “attitude money.” These emergency funds allow you to have a “positive attitude,” knowing that, should an emergency arise, you’ll be well able to handle it.

Likewise, once this bucket is full, extra money should be transferred into the next bucket:

3. Retirement Savings

Always take full advantage of employer contributions toward your retirement savings! For example, if your employer will add an extra 3% if you commit 6% of each paycheck toward 401K or other retirement accounts, be sure to do so. This is free money!

It’s generally a good idea to maximize tax-advantaged retirement savings opportunities (adding a Roth or Traditional IRA, etc.) – but you’ll also want money to flow into general investment accounts, since retirement investments are hard to access until you hit retirement age. Set a sensible maximum amount for yearly contribution toward retirement funds, based on your income, in consultation with a tax accountant or other trusted advisor who understands your situation and applicable tax laws.

Again, once this bucket is full, extra money should flow into the next bucket:

4. Investments

Investment accounts should be further split, in roughly equal amounts, into passive and active investments.

Passive investments include lower risk, highly-diversified vehicles like index funds/ETFs and the like. The strategy is to buy and hold, no matter what, and take advantage of the market’s 10% average annual growth to virtually guarantee you’ll make huge profits in the long run. You’re not trying to beat the market, here; you’re simply joining it. Most or all retirement savings should be held in passive investments vehicles.

Active investments include particular hot stocks, precious metals, REITs (real estate funds), and even commodities and futures contracts. The strategy here is trying to beat the market. This is gambling, essentially. Sometimes you’ll win, sometimes you’ll lose. With study, and some good luck, your big wins will more than make up for your losses, and you’ll have fun trading in and out of these positions, swinging for home runs.

It’s a good idea to put more into passive investments and less into active investments, the more you dislike risk. But if you have the interest and can afford the time to study various investment opportunities, the rewards can be more than worth the risks, especially if you’re young and have plenty of time to earn back lost money.


It’s mathematically impossible to beat the market over a long enough period of time. Gamble intelligently. Put at least half your investment money in less risky, well-diversified passive investments.

Don’t become too euphoric when markets are going up, and don’t worry much when markets go down. Just keep buying and holding your passive investments, following a methodical investment plan. Invest rationally, not emotionally.

As your investments rise and fall in value, it’s important to look at your entire investment portfolio periodically and rebalance it (at least 1-4 times a year).

Once your investments have grown, use them to fund goals that bring you joy in life: owning a home, travel and vacations, buying cool toys, giving to others, making the world a better place, etc.



Copyright © 2018 Christopher R. Borland. All rights reserved.

Monday, July 01, 2013

Elephant's Memory

Hypercard was a simple but ingenious software development program included free with every early Mac computer that single-handedly ushered in the hypertext revolution responsible for the internet and modern computing as we know them today.

10 years ago, one HyperCard stack called "Elephant's Memory" made quite a profound impression on me. Essentially, Elephant's Memory taught simple techniques that allowed readers to unlock a phenomenal ability to memorize and recall information. By taking the user through a few simple exercises, it showed that anyone can have a truly amazing memory by learning to employ basic memory association techniques that act as "hooks" on which to hang information.

Mnemonics (pronounced "new-MAWN-iks") are clever semantic associations that help a person remember something by linking it mentally to something else that's already in his permanent memory, or to a wild, humorous, "off the wall" (i.e. memorable) new image created specially for the occasion. Like using training wheels on a bike or crutches while a broken leg heals, memory devices like mnemonics act as a bridge between short and long term memory, a way to "get by" until true, permanent memory has been achieved. Often amazingly simple and yet unbelievably effective, these methods offer a welcome alternative to the slow, painful "brute force" memorization technique most people use to memorize uninteresting things like lists, formulas, etc.

For example:

Suppose you wanted to memorize the countries in Central America. You could do this by brute force, that is, by sitting down and repeating over and over again the names of these countries until you finally succeed in remembering them. Or, you could create and use a mnemonic, like this:

"Great Big Elephants Have Never Conned Pennsylvanians."

Now, with such a zany, unforgettable image in one's mind, it's easy to remember that the countries of Central America are (more or less north to south):

Guatemala, Belize, El Salvador, Honduras, Nicaragua, Costa Rica, and Panama.

Being from California, I always had a hard time remembering how to correctly place the states of New England on a map ... until I created the following mnemonic to help me get them straight once and for all:

"Most Newborn Vampires March in Correct Rows."

By conjuring up such an outlandish sentence that gives rise to an indelible mental picture (who could forget newly born vampire babies marching in neat, tidy little rows), it's easy to recall that these states are, (north to south):

Maine, New Hampshire, Vermont, Massachusetts, Connecticut, and Rhode Island.

You can forever remember the first twenty digits of pi by using a "weird and silly story" as a memory device (similar to mnemonics, weird and silly stories make it fantastically easy to remember even very long lists of otherwise dissociated items):


"THREE days ago, ONE panda bear went to make a phone call with FOUR giraffes. Each giraffe wore FIFTEEN scarves having NINETY-TWO stripes and SIXTY-FIVE polka dots. After THIRTY-FIVE minutes, EIGHT HUNDRED NINETY-SEVEN drops of lemonade rain fell and NINETY-THREE fairies flew out of the phone. TWENTY-THREE of them proceeded to sing the Star Spangled Banner EIGHT times, until FOUR of the fairies collapsed from exhaustion."

So, naturally:

π = 3.1415926535897932384...


It would only take a bit more work to create five different stories such as the one above (or, five "chapters" of the same story) to easily enable the average person to repeat from memory the first 100 digits of π in just a short while, something most people would ordinarily consider impossible. By reviewing the linked stories, and practicing this feat several times, one could then permanently establish the ability to recall at will the first 100 digits of π (in general, memory devices like these need only be rehearsed successfully a handful of times to produce the desired result). Quite impressive at parties.

Mnemonics and memory stories are easy to make up on the fly, as you need them. Just remember to make them as outrageous, funny, and visually bizarre as possible.

Now ... go memorize something.

Have fun!

(For a short summary of simple memorization techniques, click here.)



Copyright © 2007 Christopher R. Borland. All rights reserved.

Saturday, June 01, 2013

Best Source Of SAT Essay Prompts

When it comes to the SAT, nothing is more important than practice in achieving an outstanding essay score. And nothing is more important in maximizing the effect of SAT practice essays than the quality of the essay prompts utilized in their creation.

Although a google search for "sat essay prompts" will return nearly 100,000 hits, most of these will not yield a large number of SAT-like essay prompts useful to average students preparing to take the SAT.

By far the best online source of SAT essay prompts I've seen is found here. (Look near the top of the page, under the heading "SAT Essay Prompts.")

This assemblage contains essay prompts taken from actual SAT tests given since March, 2005, and is large enough to give even the most ardent SAT student more than enough essay outlining and writing practice.

As such, this collection is perfectly suited to the needs of those getting ready to take the SAT!



Copyright © 2008 Christopher R. Borland. All rights reserved.