Monday, December 01, 2014

The Rite of College Visitation

Given the tremendous importance of the decision involved, it's crucial to conduct purposeful college visits prior to acceptance of any offer of admission.

Students and parents should be convinced that the choice of institution is the best available for the particular student and family involved, and no amount of online data or guidebook information can take the place of personally setting foot on campus, examining facilities, talking with students and professors, taking the tour, checking out the dorms, sitting in on classes, and tasting the food.

The Princeton Review has published an excellent short article covering the basics of the rite of college visitation. Brennan Barnard, Deerfield's Director of College Counseling, offers his two cents on the subject in a NYT article, here.

As if higher education weren't already expensive enough, now you've got to fly out to trod the quad in person? Yes, indeed, you do.

College trips before matriculation are an essential part of the cost of college nowadays, helping to ensure that the major investment of time, money, and energy required will be most productively spent.


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

Saturday, November 01, 2014

Two Dynamite Questions

Dave Denman, an expert educational consultant and long-time colleague, gave me a tip many years ago that was one of the best study nuggets I’d ever heard.

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 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 your fellow classmates.


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

Wednesday, October 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 © 2006-present: Christopher R. Borland. All rights reserved.

Monday, September 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 should enable them to achieve critical mass for themselves and all their 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 doubled 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 in charitable endowments while doing almost $50 million in philanthropic work (in real 2018 dollars) by that time.

Starting at age 25, 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 and legacy goals. Makes working and saving more fun!


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

Friday, August 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 various factors driving the college admissions race.

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 prowess, and serving the public good.

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

Rankings by are widely cited and well regarded.

Business Insider's list of 50 Best Colleges in America emphasizes graduation rates and early-career earnings. See also BI's list of Top 25 Business Schools.

The London Times 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.

Other lesser-known lists warrant consideration, also (see here and here).

In the hyper-competitive early 21st century higher-ed arena, getting in to college – and then getting out, successfully – is arguably 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 © 2006-present: Christopher R. Borland. All rights reserved.

Tuesday, July 01, 2014

The MacTutor History of Mathematics Archive

Mathematics teachers, students, and other math fans and 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, behind Oxford and Cambridge, 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 © 2006-present: 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 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 even-numbered exercises can typically be inferred from those given in the manual for corresponding 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 © 2006-present: 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 © 2006-present: 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 natural 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 equations will generate Pythagorean Triples for any such x, y:

a = x^2 – y^2

b = 2xy

c = x^2 + y^2


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