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.

Combining author Marilyn Burns's witty writing with engaging illustrations, jokes, activities, and explanations, this wonderful book celebrates the "other" side of mathematics – the creative, beautiful, quirky, fun side hidden from a great many unfortunate students.

This is not an instructional text, at all. Rather, it's 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 on amazon here, and archived here.


Copyright © 2006-present: 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 © 2006-present: Christopher R. Borland. All rights reserved.

Tuesday, October 01, 2013

Arithmetic, Algebra, 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 simple algebra empowers one to do some pretty interesting and impressive things, including all kinds of "mathemagic" tricks, like the one below (involving just basic 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.

For more, see: Mathemagics: How to Look Like a Genius Without Really Trying (pictured above) by math wizards Arthur Benjamin and Michael Shermer.


Copyright © 2006-present: 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 Study 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 © 2006-present: Christopher R. Borland. All rights reserved.

Thursday, August 01, 2013

Money Flow

To optimize control of 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 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.

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.

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 the fences.

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 of active investing 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 (perhaps 1-4 times a year).

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


Copyright © 2006-present: 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 © 2006-present: 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 © 2006-present: Christopher R. Borland. All rights reserved.

Wednesday, May 01, 2013

2 = 1

All kinds of falsehoods can be "proven" true if subtle errors in reasoning are allowed to go unnoticed. I get a kick out of debunking these arguments, and thought I'd share one with you.

See if you can find what's wrong with the well-known "proof" that 2 = 1, outlined below:


a = b

Multiply both sides by a.
a² = ab

Add a² on both sides.
a² + a² = a² + ab

Simplify the left side.
2a² = a² + ab

Subtract 2ab from both sides.
2a² - 2ab = a² + ab - 2ab

Simplify right side.
2a² - 2ab = a² - ab

Factor 2 out of each term on left side.
2(a² - ab) = a² - ab

Divide both sides by a² - ab.
2(a² - ab)/(a² - ab) = (a² - ab)/(a² - ab)

Which "proves:"
2 = 1


Can you find the error?

Come on ... don't look ahead until you at least give it a try!

All right, here's the mistake:

Look at the eighth step. If a = b (given), then a² - ab = 0. Division by zero is impossible, and therefore not allowed. Breaking the "never divide by zero" rule breaks the proof.


Copyright © 2006-present: Christopher R. Borland. All rights reserved.

Monday, April 01, 2013

Budgeting for Lazy People

Successful budgeting can be summed up in a single key principle:

Live below your means.

One often hears how important it is to live “within one’s means,” but this isn't good enough. Such a level of spending makes saving impossible. In order to save, you must learn to be happy living below your means (i.e. spending less on fun than you are actually able to spend, given your income).

You can’t save if you don’t earn! Until you begin your career, make sure to set your sights on well-paying  employment and/or side hustles that will enable you to live comfortably and happily well below your means.

Until you earn average local income, save at least half of each paycheck after taxes and essential expenses are paid (your “disposable income,”) and spend the rest on whatever you want.

The Rule of Fourths

Once you’re earning above average income, budget roughly one fourth of your gross earnings (total income, before any deductions) into four major categories:

1. Taxes

2. Living Expenses

3. Savings

4. Spending (fun)

The amounts committed to each area probably won’t amount to exactly 25% each, but do your best to approximate that number.

The savings category should be split in roughly equal amounts between tax-advantaged retirements savings (401K, IRA, Roth IRA, etc.) and general investments (those with no particular tax advantages). Be sure to reinvest all earnings from your investments, to take advantage of the power of exponential growth (compounded earnings).

The spending category includes travel, toys, entertainment, and anything and everything else. Have fun spending this money any way you wish. You’ve earned it. Enjoy life and make memories!

Once you earn enough that your monthly expenses comprise roughly half of the budgeted 25% of your gross income (about 15% or less) … congratulations! It’s time to move up and incur higher expenses (e.g. move to a nicer place, buy a better car, get premium cable tv, etc.).

Once your household income is above average, observing the Rule of Fourths is a simple, effective way to sensibly balance four key requirements of a good life: living lawfully, prudently, comfortably, and joyfully.


Copyright © 2006-present: Christopher R. Borland. All rights reserved.

Friday, March 01, 2013

ADD: Harmful Disorder Or Helpful Variation?

Thom Hartman has extensive experience working with and writing about various aspects of the ADD/ADHD debate.

Hartman's Web site can be found at:

and some of his writings about ADD/ADHD at:

Harman has many interesting things to say about a number of important topics, not the least of which is the topic of education in America.

I strongly endorse his views, and I urge you to visit his site, read his thoughtful and interesting articles, and go on to some of the other links he lists there.


Copyright © 2006-present: Christopher R. Borland. All rights reserved.

Friday, February 01, 2013

Can Your Computer Read Your Mind?

The first time you visit this site, you may think your computer has come to life and has found out how to read your thoughts.

You might suppose our machines are ready to take over the world (bringing to mind “Hal,” the infamous on-boight supposeard computer in Stanley Kubrick’s masterpiece film “2001: A Space Odyssey”).

Try it a few times:

You won’t believe it. Seems impossible. Take a minute or two right now and see if this doesn’t make you scratch your head in amazement.

But have no fear ... it’s all just simple algebra!

Here’s how it works:

Any two digit number has a ten's digit called "t" and a unit's digit called "u." The value, v, of any two digit number (and therefore of the original two digit number you pick from the puzzle page) is 10t + u. The sum of the two digits, s, is: t + u. You are instructed to subtract the digit sum from the original number; let's call this result "r."


r = v – s = (10t + u) – (t + u) = 9t.

Since t can only be a whole number from 0 through 9, n = 9t can only be a multiple of 9 less than 90: 0, 9, 18, 27, ... 81. So, no matter which two digit number you initially select, you will end up looking up a multiple of 9 in the chart! The writer of the program on this site has been careful to place the same symbol next to each multiple of nine in each version of the chart shown to viewers, and has instructed the program to expose that symbol in the "crystal ball" when the user clicks on it.

Go back to the puzzle page. No matter how many times you open a new and totally different puzzle page, you will notice that, on each new page, all multiples of 9 less than 90 are always associated with exactly the same symbol (the other numbers will never come up, and so they are assigned random symbols).

Here's an entertaining variation of the game:

First, send someone a number/symbol chart together with a letter the game's instructions. Then, after this person tells you he's received the letter, has calculated his "magic number," and has found and focused on their "magic symbol," you mail him another letter in which you tell him which symbol you "psychically" picked up (of course, this is the symbol associated with each multiple of 9 on the rigged chart you sent).

For even more fun:

After your victim receives the initial letter from you, you might like to place a harmless bet with the victim (say, for lunch at the winner's favorite restaurant), that you will, in fact, be able to "perceive" his magic symbol through your superior psychic powers. When you successfully "intuit" his symbol, you’ll not only get a good laugh out of it, but a free lunch with a good friend as well (unless your friend is unusually adroit with numbers, he will totally miss the pattern underlying this trick).


Copyright © 2006-present: Christopher R. Borland. All rights reserved.

Tuesday, January 01, 2013

Easy Bibliographies

Ever had trouble figuring out how to put together a proper bibliography for a research paper?

You need an automated bibliography composer!

A detailed bibliography, with complete and correctly cited entries, is an essential part of any good work of academic research. Without one, the reader cannot easily assess the credibility of your work or the thoroughness of your research, and even after taking the time to read your paper is left at a very unsatisfying kind of "dead end."

The following free online service makes the process of organizing, properly formatting, and printing a polished MLA or APA bibliography fast and easy:

A premium bibliography service on the same site ("MyBib Pro") costs $8 per year, and offers online storage of bibliographies (making them available from any computer), MLA footnotes, parenthetical citations, etc.


Copyright © 2006-present: Christopher R. Borland. All rights reserved.