Monday, July 01, 2019

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 loss into laughs.



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

Saturday, June 01, 2019

Cosmic Eye

This epic science video takes viewers from the realm of everyday human experience to macrocosm, microcosm, and back, through the magic of exponential growth and logarithmic zooming.

Barely three minutes long, it's an excursion through Reality that could make your day.

This is why I love science.

Wednesday, May 01, 2019


You are a ghost driving a meat-covered skeleton made from stardust riding a rock floating through space.


[My daughter's photo of a sign stuck in the ground at Outside Lands, circa 2013.]

Monday, April 01, 2019

What’s Going On in this Graph?

Reading – i.e. the ability to understand and derive meaning from textual information – will always an essential academic skill.

But today we live in the age of data, and basic numeric and statistical proficiency are also likely to be de regueur throughout the 21st century.

Academic goals and curricula are already changing to reflect this trend. Among the most pressing of these new number-based skills is the ability to understand charts and graphs, to wean pertinent information and draw relevant conclusions from visual information (e.g. ACT science scores depend on this skill).

In partnership with the American Statistical Association, the New York Times is aiming to make its own contribution to the data literacy of America’s collective student body by presenting a new feature throughout the 2018-19 academic year:

What’s Going On in This Graph?

Each week, a new professional-grade NYT infographic will be presented with questions to aid in analyzing, understanding, and questioning the information it illustrates. The stated purpose of the educational project is to “teach students how to read, interpret and question graphs, maps and charts,” and is intended to support math and stats teachers across the country.

Information presented graphically is going to become more and more a part of daily life as time marches on. Data visualization, and visual communication generally, are on the rise as essential academic and life skills.

This effort by the NYT and ASA is to be applauded. I look forward to checking out the featured infographics each week, and hope you will do so, as well.

Click here for the latest graph in the series.


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

Friday, March 01, 2019

White House Fails English

The recent experience of retired English teacher Yvonne Mason echoes the exasperation felt by many of us who spent the entire first two years of high school English doing nothing but arcane grammar exercises out of a workbook.

It may be too much to ask in this age in which English teachers no longer feel the need to teach grammar (and English majors aren't even required to study it), but shouldn't we expect better than this from the office of the highest governmental official in the land?

Argh. [Facepalm.]

I might suggest these English language resources.


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

Friday, February 01, 2019

Until you declare a major, take highly-rated professors, not classes!

Think of as Yelp for college professors. Search for your school (or prospective schools), and get reviews and ratings by real students of instructors in all departments. Use the site to find life-changing teachers and avoid duds.

As with Yelp, some reviews are more helpful than others, and doesn’t tell the whole story. But with quantified measures like "Overall Quality" and "Level of Difficulty" (among other indicators) it’s a whole lot better than having no idea at all as to which teachers are likely to be golden and which should probably be avoided like the plague.

Most/all established professors at are listed at each institution, and university-wide averages give you some idea as to the quality and collective personality of various faculties.

Schools are rated by students according to other important factors, as well (e.g. reputation, happiness, food, facilities, location, social life, etc.), providing useful comparative data.

As an example, click here to see data for Wesleyan University.


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

Tuesday, January 01, 2019

Math-as-Art Blog

I had planned to store the overflow from the "Math as Art" project in a single post on this blog, but it quickly became apparent that the project would require a site of its own.

Here it is:



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

Saturday, December 01, 2018

101 Things to Do Before You Graduate

Life Coach Julian Gordon has published an indispensable book for college freshman:

101 Things to Do Before You Graduate.

Listing 101 truly useful and important academic and personal goals to fulfill as an undergrad, Gordon gives callow freshmen a roadmap to follow to make the most of their college years.

Those four undergraduate years will be over before you know it, and with them will disappear once-in-a-lifetime opportunities. Don’t waste some of the most important and potentially profitable and enjoyable years of your life!

This book makes a great gift. Should be required reading for every college-bound high school senior.

UPDATE: Great TED talk given by Jullien on the 4.0 GPA That Really Matters.


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

Thursday, November 01, 2018


Everything you wanted to know about the Mandelbrot Fractal. A great gift for the math nerd in your life.

Posters are 24 x 36 inches or 54 x 36 inches.

Buy one here.


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

Monday, October 01, 2018

The Engine of the World

I remember when tenor Andrea Bocelli sprang out of nowhere as the heir apparent to the great Luciano Pavarotti.

For audiences, it always seems like that; suddenly this great supernova explodes into view, changing the sky forever. But of course it's not like that at all.

Fame may happen overnight, but the artistic growth and development of a phenomenon like Bocelli is often excruciating slow and painful. The degree of hyper-focus and self-centeredness required of great artists makes intimate relationships all the harder to create and maintain. No wonder addiction and madness are familiar attendants to those having regular intercourse with the Muses.

Yet somehow the fuse finally gets lit, the coming eruption only a matter counting down. Those that survive ignition share us with glimpses of other worlds, uncommon truths and ephemeral realities, souveniers of Heaven.

Dear Veronica, my dear children, 
Every life is a wonderful story worthy of being told. Every life is a work of art, and if it does not seem so, perhaps it is only necessary to illuminate the room that contains it.
The secret is never to lose faith, to have confidence in God's plan for us, revealed in the signs with which He shows us the way.
If you learn to listen, you will find that each life speaks to us of love. Because love is the key to everything, the engine of the world. Love is the secret energy behind every note I sing.
And never forget that there's no such thing as happenstance. That's an illusion lawless and arrogant men invented so that they could sacrifice the truth of our world to the laws of reason
Andrea Bocelli (2017 biopic: The Music of Silence) 

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

Saturday, September 01, 2018

2, 800, 44

Concerning the SAT Math Subject Test, the three most important numbers are:

2, 800, 44.



You should take the SAT Math Subject Test, Level 2 (SAT2M2). There’s virtually no reason for anyone to take Level 1.

Here’s why:

SAT Subject Tests allow students to pick the two subjects at which they most excel and then show off by getting very high scores on those tests. A very high score on Level 1 of the SAT Math Subject Test is meaningless, because it begs the question: “If this student is so good at math, why didn’t she take the SAT2M2?”

And that would be a good question. If you’re so good at math that you’ve chosen mathematics as one of your two subject test areas, you should certainly be taking the most advanced level of the math subject test, which is Level 2.

Opting for Level 1 of this particular test defeats the purpose of choosing this particular subject area in the first place.



A perfect 800 is the score you want on the SAT2M2. Fully 20% of students who take the test get this score. A much lower score, once again, kind of defeats the purpose of electing this subject in the first place.

By choosing math as one of your two subject test areas, you’re declaring yourself to be a hotshot math student. You should therefore be able to score in the top 20%, which means you want an 800 on this test, or darn close to it.

The good news is that this isn't that hard to do, as long as you’re truly good at math, make the best possible use of your calculator, and work hard to prepare (i.e. take lots of practice tests, assiduously critique the results, do careful error analysis, and review regularly).



This is the number of right answers you need out of the 50 questions presented on the SAT2M2.

Notice that you don’t need a perfect score to get a perfect score. Not even close. On average, you only need an 88% (44/50) to score a perfect 800. No one can tell the difference between someone who gets 88% of the answers right and someone who gets 100% of the answers right; on the SAT2M2, both students receive the same scaled score: 800. So you’re shooting for 44. Anything above that is nice, but superfluous.

What about the other six questions?

You get a pass on these six. You can skip them all, get them all wrong, or skip some and get some wrong. It doesn't matter.

Here’s why:

According to the most recently released official SAT2M2 practice tests, on average, you actually only need a raw score (correct answers – .25 * incorrect answers) of 43 to get a perfect 800 scaled score. If you skip all six questions, your raw score is 44. If you answer all six incorrectly, your score is 42.5, which rounds up to 43. If you skip three and get three wrong, your raw score is 43.25, which rounds down to 43.

No matter what you do with the other six questions, as long as you get 44 right answers on this test you’ll likely receive a perfect 800 as your SAT Subject Test score.


You can probably do this.

Anyone who could score 700 on their own just sitting down and taking the test with no prep at all (requiring a raw score around 33: e.g. 35 right answers, five skips, ten wrong) can score 800 with strong preparation, plenty of practice, good calculator skills, and the right calculator programs.

And once again, if after a year of high school precalculus you can’t just sit down and get a 70% on this assessment (35/50), you might want to pick another subject test.

For more SAT2M2 advice and resources, click here.


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

Wednesday, August 01, 2018

The Matchless Enthusiasm of Martin Gardner

The hyper-competitive struggle for survival with the old Soviet Union made American math and science education a top priority during the Cold War decades of the 20th century.

Many Baby Boomers (like me) remember with fondness the fascinating articles published monthly in Scientific American magazine. One of the most interesting features of the magazine in our time was Martin Gardner’s column on recreational mathematics, which ran for 25 years.

Among the many necessary qualities of truly great teachers, enthusiasm might be listed first. An instructor’s genuine, overflowing enthusiasm is that which excites students' souls and convinces them that the required academic work and sacrifice will be amply rewarded. The etymology of the word “enthusiasm” (en-theos: literally, "in God") points straight at the Divine, and no one could excite the soul with the beauty of mathematics like Gardner could.

A 1998 article by the master preserves for modern readers the flavor of Gardner’s contagious enthusiasm and gold-medal exposition that so characterized his column, presenting to Gardner fans and neophytes alike the pure noetic joy that accompanies deep dives into the realm of creative mathematics.

Reflecting the timelessness of the subject, the article reads as if it were penned yesterday, fresh and new. It’s not long, and is well worth a bit of your time:

A Quarter-Century of Recreational Mathematics.


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

Sunday, July 01, 2018

Calculator Policies: SAT and ACT

SAT and ACT policies regulate the types and models of calculators allowed for use in solving math problems on standardized tests administered by each organization.

For a while, the SAT had a more or less unrestricted calculator policy whereas the ACT had a tightly controlled one, allowing only those user-installed programs comprising 25 or fewer lines of code. Aside from being unduly onerous, the old ACT policy was obviously unenforceable, and has recently been changed to essentially match the unrestricted SAT rule.

In a nutshell, here's what you need to know:

All features, apps, and user-installed programs are permitted for use, without restriction, on any model of the Ti-84 graphing calculator family (including the powerful Ti-84 Plus CE model) on the SAT, the SAT Math Subject Test (Levels 1, 2), and the ACT.

Below are links to current policies:

SAT Calculator Policy (updated: 2023)

ACT Calculator Policy (updated: 2023)


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

Friday, June 01, 2018

Fractal Coloring Book

Adult coloring books have become popular in recent years.

Coloring Mandalas for Meditation was a hit with my daughter and her friends at Wesleyan who needed to periodically take a break from the stress of studying.

Adult Coloring Book: Fractals by Ben Trube not only presents beautiful fractal designs to color, but also teachers about fractals and the math behind them.


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

Tuesday, May 01, 2018

The Game of LIFE

No, not the board game … the singular computer program!

LIFE is a coding creation released by mathematician and programmer John Horton Conway in 1970.

This fascinating program graphically mimics life's basic processes in an utterly simple yet strikingly lifelike way.

Hypnotic to watch, LIFE happens like life happens. Sometimes it seems to go on for ever; other times, it’s over in a flash. Chaotic and patterned at the same time. Incalculably complex, but real and finite.

Check out the links below to learn more and to experience LIFE for yourself:

Wikipedia: Conway’s Game of Life


Stanford: A Discussion of the Game of Life

Download: Golly

Conway’s Game of Life FAQ


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

Sunday, April 01, 2018

Official GRE Prep Materials

Official materials produced by test makers form the backbone of all legitimate standardized test prep efforts.

The GRE is produced by Educational Testing Service (ETS), which also administers the TOEFLPraxis, and other standardized tests.

ETS offers several useful tools for those preparing to take the GRE. These materials are listed below, followed by short articles offering suggestions on how best to use them.


PowerPrep Online and PowerPrep Online Plus

ScoreItNow! Online Writing Practice

The Official Guide to the GRE General Test

Official GRE Super Power Pack


PrepScholar: How to Utilize Official GRE Materials

9 Tips: GRE PowerPrep

How to Use the GRE Official Guide


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

Thursday, March 01, 2018

Teach Your Child to Read

There’s no excuse for illiteracy, and this book proves it:

Teach Your Child to Read in 100 Easy Lessons (by Siegfried Engelmann, et al).

Virtually any educated adult can teach virtually any interested child to read at a comfortable second grade level in four to six short months (one 20-minute lesson a day, six days a week) using this classic phonics-based home learning tool. If desired, one can take up to a year to complete the 100 lessons (supplementing learning sessions with additional reading materials, extra writing exercises, etc.).

Instructions for the parent-teacher are crystal clear at every stage, and super easy to implement. Students should be able to recognize upper and lower case ABCs before starting, and it helps to know the main sounds each letter makes (but this isn't necessary).

Using this system our daughter learned to read at age four, the proverbial child with her nose stuck in a book. She went on to be a happy, successful student, and a voracious life-long reader. Very precocious children who want to learn to read could start even earlier. Most kids would probably do well to begin at four to five years of age.


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

Thursday, February 01, 2018

How Hard Would You Work for an Extra $1 million?

It's no secret that a top-notch college education generally confers valuable special advantages to those fortunate, talented, and diligent enough to obtain one, including: superior instruction; well-connected networks of professors and alums; preferred access to coveted internship, research, and employment opportunities; etc.

Let's assume that landing a seat at a top school rather than an average school means you're able to increase your month salary $1000, on average, over the course of your working lifetime, and that this allows you to save an extra $500 per month, on average, for 40 years at 6% real growth.

That's an extra $1 million in present value purchasing power once you hit retirement. And these dollar numbers are conservative. You could quite easily double them, in the right jobs and fields.

That's a completely different life, not only for you, but for those who come after you.

Now are you motivated?

In general, the best thing you can do to increase the probability of admission to top schools is standardized test prep and college application essay prep. Assuming you're already taking the toughest classes you can and are getting the best grades you can, time spent maximizing your SAT/ACT scores and nailing the various common app, personal statement, and supplemental essays you'll write for your college applications will do far more to improve your College Application Marketability than anything else you could possibly do.

Once again, there are no guarantees. A hot diploma doesn't mean anything by itself, and students unable to gain entrance to a top 30 school can make up for most/all of that advantage with extra grit, hard work, dedication, and perseverance.

Still, it's worth it to aim high. Go for 100%! Then celebrate the result, whatever it is. Use failures as feedback, as opportunities to recalibrate the machine.

Werner Erhard taught that life is a summit-less mountain to climb. Every time you reach the top, there's a taller peak not too far off in the distance. As he would often say, "Best to learn to love climbing!"


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

Monday, January 01, 2018

College Application Marketability (CAM)

The following is a rough approximation of the relative importance college admission committees give to basic elements of a typical application:
  1. Coursework (grades and rigor): 40%
  2. SAT/ACT test scores: 30%
  3. Essays (Common App, personal statements, supplements, etc.): 15%
  4. Everything else (extracurriculars, leadership, legacy, ethnicity, etc.): 15%
Let’s assume that an average honors or AP-level high school course requires about eight hours of work per week (four hours in class, and four hours outside class). Assuming six classes and 36 weeks of school per year, that’s 1728 schoolwork hours per year for the typical college-bound high school student. By the end of junior year, that’s 5184 academic hours. Just to be safe, let’s round down to 4,800 hours. That’s 120 hours per CAM point.

Suppose a student spends an average of two hours per week over the course of 12 months preparing for the SAT. This requires a total investment of 104 hours. To be safe, lets round up to 120 hours. That’s only four hours per CAM point.

Similarly, time invested in planning, drafting, editing, and polishing college application essays is hugely profitable! Let’s assume the average student needs to write one 1000-word Common App essay and eight 500-word supplemental essays and personal statements; that’s nine college application essays totaling 5000 words. To do an outstanding job on these critical pieces of academic work might require 75 hours. That’s only five hours per CAM point.

The takeaway:

Making a serious commitment to long-term standardized test prep and to putting in the time and effort required to write great college essays is highly intelligent!

Yes, of course, you should do all you can to take the toughest courses and get the very best grades possible. You do need to show sincere interest in your favorite schools, committed involvement and initiative in pursuing extracurricular activities for your own enjoyment and in service of others, and accomplishment of notable NTA's ("non-teenage activities").

And, of course, there are only so many hours in the day. Nothing on Earth is more important than maintaining good mental and physical health, properly balancing work and play, and getting enough rest and sleep. You can't do your best work if you're sick, unhappy, or exhausted.

Notwithstanding these important considerations, the fact remains that work on test prep and college essays is up to 30 times more productive than anything else the typical high school student can do to maximize CAM and boost the odds of admission to a top school.

Imagine that you’re taking an additional half-course called “CAM Class” throughout junior year and during the first semester of senior year. The content of this independent study course will consist mainly of your own research into “good fit” colleges, standardized test prep, college essay work, and general college application planning and preparation. You’ll put far less time into CAM Class than you would into any ordinary course – but your devoted participation here has the potential to do far more for your chances of gaining entrance to the college of your dreams than do all the other courses you’re currently taking put together!

This is an incredible opportunity for those committed students willing to step up, embrace the challenge, and make a relatively small sacrifice of time and energy in exchange for the excitement, fulfillment, and future success that only a great college education can provide. It’s smart to commit yourself to building the best college application package you can by working diligently to maximize your score on the SAT or ACT and nail your college application essays.

Make a plan to investigate various colleges and universities that match your goals and fit your personality, prepare thoroughly for the SAT or ACT, write and finely polish your college essays, and complete and fine-tune your college applications well ahead of time. If you can, find a qualified test prep coach and private college counselor to help you along the way. If this isn’t possible, you can do quite well working entirely on your own – without paying for any outside help at all – simply by reading good books on these subjects, researching online, putting in the time, and becoming a test prep wizard and college application expert through self-study.

Summing up:

There’s no better investment than the time and energy required to earn an “A” in CAM Class.


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

Friday, December 01, 2017

MOOCs are Coming of Age

At their inception several short years ago, MOOCs (Massive Open Online Courses) were an unproven concept with passionate advocates on opposite sides of a great debate. MOOCs were going to revolutionize higher ed, or destroy it. No one could tell which it would be.

A decade later, top MOOC providers like Coursera and EdX have grown and prospered. Top-notch course offerings by the best universities in the world have attracted millions of students world-wide. Legions of online pupils of all ages have completed courses, some earning coveted professional certificates and even fully-accredited graduate degrees online. Low cost has made high quality higher ed available to a much wider, world-wide audience.

Though forms are still evolving and the precise roles to be played by MOOCs are still uncertain, both the radically new concept and the traditional educational landscape have survived and even thrived as a result of the introduction and mainstreaming of MOOCs.

Today, MOOCs and associated certificates/degrees are legitimate educational alternatives.

See links below for further info:

Massive Open Online Course (WikiPedia)

By the Numbers: MOOCs in 2017

The Future of MOOCs


Coursera Professional Certificates

Coursera Undergrad and Grad Degrees


EdX MicroMasters Certificates

EdX Professional Certificates

EdX Series Programs

MIT Open Courseware

Stanford Online


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