Math as Art (Part 1)

Mathematics has gotten a bad reputation for being dry, stodgy, and decidedly uncreative – a very left-brained activity. This reputation is undeserved, however. In fact, math is art!

A recent article on the Apple Web site highlighted the work of Michael Trott, a pioneer in the effort to popularize the creative/artistic side of pure mathematics and the world’s most advanced Mathematica user.

The article begins:

“Fifteen years ago, Michael Trott, a theoretical physicist at a small university in Germany, carried his department’s only available computer, a Macintosh IIfx, from his office to the lecture hall every week to share knowledge about Stephen Wolfram’s groundbreaking mathematics application, Mathematica.”

Read the full text of the article here:

http://www.apple.com/science/profiles/trott/.

You can view examples of some of Trott’s beautiful mathematical art, and works of other Mathematica users, here:

http://www.borlandeducational.com/page10/page10.html.

A Google search for “mathematical art” will lead to innumerable other sites and resources through which to explore this under-appreciated aspect of mathematics, the language of science.

(Click here to go to part 2.)

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

"Honor Society" Rip-off Alert

You know those slick "honor society" membership enticements you get in the mail once your high achieving teenager gets about halfway through his or her sophomore year? Tempting you to provide your child membership in a "selective and prestigious" national honor roll that may boost chances of admission to top colleges just by joining? You know ... offers that seem perhaps too good to refuse?

Refuse them.

Writing in the Education section of the New York Times, high school guidance director Robert Bardwell exposes the fraudulent nature of these seemingly legitmate offers:

I NEVER recommend that a student pay to join any organization that will supposedly have an impact in the admission process. Even free honor societies available to high school students (i.e.: National Honor Society, Pro Merito and individual subject area societies) have little or no effect, especially at highly visible institutions. 

If a student wants to join such an organization for the other benefits of membership, that is fine, but not to help get a leg up on his admission chances.

If there is a hint that admission chances would be improved as a result of his being a member, I would suggest you report it to the Better Business Bureau as it may constitute fraud.

You can read Bardwell's entire post here.

As usual, too good to be true is neither true nor good. It's amazing how deeply infected with manipulative, deceptive commercial spin American culture has become in recent decades. What appear to be beneficient invitations may in reality be academic industry shakedowns, and any unfamiliar but official-sounding scholastic organizations that solicit enrollment by mail should be regarded with strong suspicion and thoroughly investigated before raising your hopes or sending any money.

Bardwell is also the president of the New England Association for College Admission Counseling, and provides Times readers with additional advice about the daunting college admissions and financial aid processes here.

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

Cite It Right

Properly formatting the information one presents in an academic work is important. Doing so is probably required, and in any case, it's easy ... with a little help!

To the unprepared, however, conforming to particular writing and formatting styles in high school and college research papers can be a real hassle, and many students find this requirement irritating, time consuming, and distracting. APA, MLA, and Chicago styles all have different requirements for presenting and citing information. Strict and correct adherence to the style favored by the instructor is absolutely necessary, and even minor deviations may result in a lowered grade for the project you're working on.

Dealing properly with headings, tables of content, page numbering, bibliographies, footnotes, etc. can be troublesome. But full compliance with standards of correct formatting is non-negotiable to many teachers, and patient submission to rigorous formatting requirements gives your work a professional look and feel that noticeably and significantly improves its value.

Luckily, software that interfaces with popular word processors to easily produce correctly formatted written projects with a minimum of fuss and bother is available for both Mac and PC platforms!

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Reference Point Software Templates cost only $28, work with both Mac and PC operating systems, and allow the user to compose properly formatted works in APA or MLA style using Microsoft Word, Microsoft Works, or Corel WordPerfect (although I don't have personal experience with this product, numerous customer testimonials tout it's effectiveness and ease of use.

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Various online sites exist to help students with proper formatting and citation. The following are two you might like to visit:

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Dr. Scribe's Guide to Research Style:

http://www.docstyles.com/;

and an excellent writing help site produced by Purdue University:

http://owl.english.purdue.edu/owl/.

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Various, now you have no excuse.

Cite it right!

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

Fear And Loathing In Academia

It seems that neocon strategists have been hard at work on a new project: to use the left-leaning political biases of most university faculty members as evidence that liberal indoctrination of students is occurring at these institutions.

As it turns out, graduating from college actually correlates with conservative, not liberal, political affiliation. The entire notion that left-leaning professors need to be replaced by right-leaning ones is a solution in search of a problem – another deliberate, politically motivated deception by those on the far right.

A recent editorial in the (conservative) Wall Street Journal brings the issue into the open:

http://www.opinionjournal.com/extra/?id=110009058.

Apparently, there is something dangerous afoot – but not from an imagined specter of liberal professorial brainwashing. Imagine this: conservative ideologues are now actually creating traction with an idea to ban instructional content that any student claims to be "offensive."

That's right! In order to eliminate a non-existent threat of liberal ideological indoctrination at the nation's colleges and universities, far-right activists want to allow anyone offended by any information presented in any class to have that information stripped from course syllabi. Not taught. Not discussed. Just gone. In true Orwellian tradition, the plan is called the "Academic Bill of Rights (ABOR)."

Let's see ... you find murder offensive, so we can't teach Shakespeare's Hamlet. I'm offended by the notion of public entitlement programs, so we can't talk about Roosevelt's New Deal or compare capitalism to socialism. Someone else takes offense at the notion of evolution, so half the biology courses have to be rewritten to exclude any mention of it. A depressed student is offended by the color blue, so we strip it (along with purple and green) from the color wheel.

Aside from the surreal kookiness and unworkability of such an idea, and the obvious totalitarian implications, what amazes me the most is that this plan is actually gaining support at prestigious institutions such as Princeton and Temple University, which have already adopted versions of the ABOR.

Have we all lost our collective marbles?

Is it just me ... or is this one of the worst ideas ever concocted?

Like many things born on the political fringes, this whole thing would be laughable if it wasn't so scary.

Click here to read an excellent Washington Post article that provides all the details.

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

SAT Prep Planning (Part 2)

A host of nation-wide testing companies (Princeton Review, Kaplan, Ivy West, Revolution Prep, etc.) offer pre-packaged test prep courses. These courses are sometimes quite good – but inevitably they're pricey and usually require a huge commitment of time and energy. School-sponsored courses offered by local firms or coaches are often less expensive and time consuming, but can also be much less effective than those administered by established companies. 

[Buyer beware: prep course "guarantees" are not normally straight money-back guarantees, but instead merely allow dissatisfied students to retake the course one more time for free or at a reduced rate.]

For students who wish to prepare on their own, there are a plethora of ACT/SAT study guides and workbooks available commercially. While not as effective as working with a good private coach or investing considerable time and money in a top-flight SAT course, disciplined study with a good work book along with plenty of practice before test day can make a big difference, and is certainly much better than no test prep at all! 

Unfortunately, most self-study test prep books are decidedly inferior in quality; very few include reliable, comprehensive advice and realistic, truly useful practice tests. The majority are a waste of time, at best, and should be avoided. 

For my recommendations as to the best current self-study materials for the ACT, SAT/PSAT, and SSAT, click here, here, and here.

By far the best way to prepare to for important standardized tests is to hire an experienced private test prep coach. For many students, private coaching offers the best value, combining superior instruction, flexible scheduling, and maximum cost-effectiveness. Top quality professional test coaches in private practice with decades of experience generally charge from $150 to $250 per hour (or more). 

However, it's possible to pay a lot more for a lot less! 

The Princeton Review, for example, charges more than $300 per hour for their best tutors (those with as few as five years tutoring experience), in large blocks of hours, and then pays these tutors a small fraction of that hourly fee. For this reason, the best test coaches do NOT work for agencies, and instead run their own private practices. So while paying absolutely top dollar, you could be getting just an average quality coach. 

Click here for information on how to find the right coach.

Below is a sample medium-term ACT/SAT prep plan (adjustments should be made for the particular needs of individual students):

• Months 1-2: Strategy coaching and instruction (weekly 90-minute sessions, with timed or untimed practice between sessions.

• Months 3-4: Two full-length, timed official practice tests each month. Each practice test should be followed by at least one 90-minute coaching session to critique and correct the student's performance. In addition, students should take another complete practice test per month one section at a time (timed or untimed, as needed).

• Month 5-6: One complete, timed, official practice test per week, taken full-length or in sections (as needed), each followed by a 90-minute coaching session to critique and correct the student's performance, scheduled to conclude just before test day.

Click here for recommended test prep schedules (ACT, SAT, PSAT, SSAT, ISEE, HSPT).

Planning and preparation are the keys to unlocking a winning score on high-stakes standardized tests. The probability of success correlates most directly to the type and quality of instruction and the amount of test taking practice and performance critiquing students complete before test day. While it's certainly possible to dramatically improve one's score in a short period of time, a long-term plan allows average students sufficient time to do required practice, discover and fix patterns of error and flaws in their approach, and gain the confidence, competence, and experience necessary to perform optimally on test day. Stronger students general require less prep time.

A long term prep plan offers the vast majority of students the best chance of a successful outcome. Although it's unlikely that most students will be able to complete enough practice testing and critiquing in just a few months or weeks to maximize their final scores, even a little high quality test prep and a few well-critiqued, full-length official practice tests can make a significant difference.

Get a coach or take a course, read some books, plan your work, and work your plan.

The worst plan for test prep – by far – is no plan!

(Click here to go to Part 1.)

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

SAT Prep Planning (Part 1)

What is the ideal length of time to devote to systematic ACT/SAT study and preparation in advance of test day? What's the best way to plan a given student's ACT/SAT prep?

Although the answers will vary considerably depending on the particular student and family in question, most students should plan to spend at least one year thoughtfully and diligently preparing to take the ACT/SAT.

Whether one's plan involves hiring an coach, taking a course, or utilizing commercial prep books, three essential ingredients combine to enable successful performance on standardized tests like the ACT/SAT: subject knowledge, strategy knowledge, and test taking practice. Each one of these factors plays a crucial role in maximizing a given student's score, the ultimate goal of any prep plan. Without sufficient development in all elements of this triad, maximum performance is unlikely to be achieved.

To do very well on the ACT/SAT, it's (of course) essential to be academically strong and know as much as possible about the particular content tested on each of these important assessments! Superior subject knowledge and basic skills, however, require years of disciplined effort and excellent classroom instruction to obtain; they simply cannot be acquired or noticeably improved any other way, certainly not in the short run. The only way to strengthen this element of the triad is to take challenging classes and work hard in school over the long term.

Knowledge of effective test taking strategy, on the other hand, can have a dramatic positive effect on students' test scores and be very quickly learned and trained (in hours, rather than years). Many unfortunate students receive artificially low scores on the SAT that give an unfairly negative impression of their academic abilities – not because they aren't capable students, but simply because they don't know how to correctly approach and take standardized tests! Without realizing it, these students make fundamental strategic errors that have a disastrous effect on their scores. Fortunately, it is possible in a short period of time to “learn the ropes” of standardized test taking, for students to adopt simple, powerfully effective test taking strategies and learn simple skills that allow them to maximize their performance on test like the ACT/SAT. This “strategic approach” enables students to make the most of the knowledge and skills they already have, and can mean the difference between disappointing scores and real success. Utilizing the "strategic approach" is the only feasible way for most students to quickly impact ACT/SAT scores in a meaningful way, and hiring an experienced academic coach who specializes in teaching the strategic approach to ACT/SAT prep is the best way to ensure the success of any test prep plan.

The third key element in the triad is perhaps most important. One would never take piano lessons without practicing at home and expect to become a much better piano player. Likewise, students with advanced subject knowledge and strategic skill who nevertheless fail to practice run a high risk of falling short of their maximum potential score. Practice is the most critical factor in predicting large score improvements on tests like the ACT/SAT. There's just no way around it – substantial score improvement requires lots of practice! The long-term plans I recommend allow students sufficient time to be thoroughly trained in test taking strategy, take plenty of timed and untimed practice tests, and fully critique their practice work – and thereby maximize the odds of obtaining optimal test scores.

Just as an athlete always warms up before a sporting contest to ensure his body is primed and ready to function at maximum strength and efficiency, it is critical for academic competitors to "warm up" before any important testing experience. No serious musician would consider hitting the stage without first warming up her fingers and reviewing the music to be performed; so too must the ACT/SAT student awaken and warm up her test-taking "muscles" before each test rehearsal or performance to guarantee the best possible results. For at least 10-20 minutes immediately before dong any SAT practice test (30 minutes before any actual SAT test), students should diligently review all notes and earlier practice work, and intently visualize themselves putting to work all they've learned in the upcoming test they're about to take.

Taking practice tests at regular intervals (say, once each month) over a long period of time (at least several months), mastering and implementing the strategic approach, and carefully scoring, critiquing, tracking, and reviewing practice work is the best way to optimize scores on the standardized tests. Though it's possible to implement a successful plan that comprises only three or four weeks'' preparation, such a tight time frame makes it that much harder for most students to fit critically important practice testing into their already full schedules. Generally, it just doesn't happen – and without sufficient practice, students lack the experience needed to ensure the best possible result, and normally fall short of their goals.

Still, some preparation is much better than none! Some students will see remarkable improvement in test scores in just a few weeks by combining quality test coaching (emphasizing the strategic approach) with as much practice as time allows. While not as effective as the long-term approach I advocate, medium or short-term prep heavily focused on the most important test taking strategies, some carefully critiqued practice testing, and disciplined review can nonetheless lead to very pleasing results.

(Click here to go to Part 2.)

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

Are Our Boys In Crisis?

Growing up has never been easy, but the evidence indicates that it's been getting a whole lot tougher in recent years for boys. While the feminist movement has brought much needed attention to issues facing girls and young women in our culture during the past two decades, a similar commitment to critical issues facing boys and young men has failed to materialize.

One result is that from 1995 to 2005 males 15 to 24 years old were more than five times more likely to commit suicide than were females of the same age.

According to William S. Pollack, director of the Centers for Men and Young Men at McLean Hospital (Harvard Medical School), psychological, educational, and other problems affecting male youths have worsened to such a degree that the word "crisis" may well be required to adequately describe them.

Rachael Rettner recently penned an informative and sobering article on this subject for LiveScience.com (excerpts follow):

"... it tends to be boys whose deeper problems are not looked into, and for whom programs that exist are not funded ... that’s absolutely true."

... compared with girls, American boys have lower literacy rates, lower grades, less engagement during school and higher drop-out rates. Boys also have higher rates of suicide, arrests and premature death.

... From 1995 to 2005, the rate of suicide among 20 to 24 year-old boys was 20.7 suicides per 100,000, while the rate for girls was just 3.5 per 100,000. Among 15 to 19 year olds, the rates were 12.5 per 100,000 for boys and 2.8 per 100,000 for girls.

An expert quoted in Rettner's article brings attention to the alarming increase in the gender gap with regard to suicide rates. In 1933 young males killed themselves 1.54 times more often than did young females, but by 2005 the rate among males had grown to 4.63 times the female rate. It's a profound and deeply disturbing fact that in just over 70 years, the suicide gap has fully tripled in size.

These numbers are a real punch in the gut.

How are we so favoring one gender and ignoring the other in so many critical areas that such brutal, heart-bruising statistics could come to reflect reality in our society?



Certainly, our girls and young women deserve all the help and dedicated, devoted attention we can give them. But we've got to start earnestly and deliberately addressing the emergency affecting our young men, who we love no less, and who at present are, apparently, in even greater need of rescue.

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

“Provide The Answer” vs. “Pick The Answer”

Understanding the essential difference between “Provide the answer” and “Pick the answer” tests – and the radical differences in preparation and approach that they require as a result – is critical to maximizing one’s performance on standardized tests like the SAT1/2, PSAT, and SSAT.

On “Provide the answer” tests (the vast majority of tests you’ve taken thus far in school):

• your task is to provide the correct answers;
• your focus should be on precision and thoroughness (accurate, complete answers); 
• hard questions are worth more points than easy ones; 
• 90% (or more) of your score depends on subject knowledge and 10% (or less) on test taking strategies; 
• you DO need to answer every question on the test to score well. 

On “Pick the answer” tests (like the SAT1/2, PSAT, SSAT):

• your task is to pick the right answers (i.e. fill in the right bubbles);
• your focus should be on approximation and speed (general, fast answers);
• hard questions are NOT worth more than easy ones;
• 50% of your score depends on subject knowledge and 50% on test taking strategies;
• you DO NOT need to answer every question on the test to score well (in fact, most students will score much higher by deliberately avoiding the most difficult questions).

Think about it ...

If you’re earning one point on every question, would you rather spend your time answering four very hard questions or eight easy ones (it will take roughly the same amount of time in each case)?

You’d earn twice as many points by skipping the four super-difficult questions (if necessary) in order to get the eight easy questions right, than you would by skipping the eight easy questions just to answer the four hard ones (even if you're able to answer all the difficult questions correctly – which is doubtful, at best)!

So ...

On standardized tests, get all the “easy points” on the scoreboard right away. Do the easy questions first, and skip the worst for last (or not at all). If you don’t get around to the really tough questions, you’ll still have earned the highest possible score you were capable of that day, without the risk of wasting valuable time and energy on impossibly difficult, unproductive questions.

And that, of course, is what it’s all about!

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UPDATE 6/1/19:

Most standardized tests calculate raw scores equal to the number of correct answers given. On these tests, it's important to answer EVERY question, even if you have to guess. However, most students should still avoid the hardest questions! The technique I recommend here is called "Skip Guessing:" Simply eliminate answers, if possible, and then quickly guess (I recommend the "Last Letter" strategy: after eliminating, choose the answer closet to the end). Whatever you do, be sure to answer every question; you'll get some of your guesses right, just by luck.

As of this update, some standardardized tests (e.g. the SSAT and SAT Subject Tests) still deduct a fraction of a point for each question answered incorrectly. On these tests, it's best to skip entirely (i.e. do NOT answer) any difficult questions on which you cannot confidently eliminate at least two out of five (or one out of four) answer choices.

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

ACT’s Financial Aid Need Estimator

Ready to do some initial research, and get an idea as to where you stand with respect to college costs and federal financial aid?

Use this service to estimate both your “expected family contribution” and “eligibility for federal need-based financial aid."

It’s free!

Click on this link to go there:

http://www.act.org/fane/index.html.

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Contact Chris

borlandeducational.com

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

Distance Learning for High Schoolers

Truly, many roads lead to Rome ... and to successful receipt of a diploma.

I myself am no stranger to "distance learning," a popular buzz-phrase signifying the earning of full academic credit for structured, supervised independent study work done outside the physical walls of a major educational institution. In fact, I earned my undergraduate degree back in the early 1980's from the State University of New York at Albany while enrolled in the Regents External Degrees Program, which was at the time one of only two fully accredited external degree programs in the United States. I remember with fondness teaching myself calculus and studying the history of mathematics though college-level correspondence courses administered by the University of California at Berkeley (UCB) and Brigham Young University (BYU).

In the old days, this meant learning on one's own from an approved textbook, completing assignments on paper, mailing them to the instructor for grading and comments, and then ultimately taking a proctored final exam to receive credit for the course. Nowadays, distance learning courses are typically conducted online, since the miracle of the internet makes possible a much richer and more efficient flow of educational content and course work between student and instructor. But the idea is essentially the same: independent study, followed by submission of course work, followed by assessment and finally the granting of credit.

Over the course of my 29 year career as a professional academic coach, tutor, and mentor, I've occasionally been hired to help high school students raise an unacceptable grade or earn additional course credits by completed various accredited distance learning courses offered through prestigious institutions such as UCB and BYU. These courses offer many advantages over typical high school classes taken during the academic year at school (e.g. being able to work at one's own pace, polish and perfect one's course work before submission and grading, study subjects outside the standard high school curriculum, etc.), and generally grant full academic credit to students completing them. Nevertheless, it's very important to first obtain written approval from your school before enrolling for credit in any distance learning course, to be sure that credits and grades earned will be fully accepted by your school.

Serious independent study work such as this both demands and fosters greater than average self-discipline, motivation, and organization, and requires real commitment of the part of all concerned ... but can be an excellent way to supplement or improve one's academic resume and develop qualities of independence, pro-activity, and responsibility so critical to success in college and beyond. Nevertheless, it may still be a good idea to secure the services of an academic coach or other capable adult to supervise distance learning courses to make sure that timely progress is being made and important deadlines are met. You've got to be committed and follow through ... but if you're a motivated, self-starter who reads well and wants or needs extra academic credit or stimulation, online or other distance learning courses may be just what you're looking for!

Below are links to accredited distance learning opportunities offered to high school students by UCB and BYU:

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UC Berkeley Extension Online

http://explore.ucxonline.berkeley.edu/highschool/body_hscourses.html

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BYU Independent Study

http://ce.byu.edu/is/site/courses/select.cfm?&TYPE=hs&SUBJECT=75

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borlandeducational.com

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

When Math Is Uncool, Other Nations Eat Our Lunch

Since I attended high school in California in the 1970's, things have changed in so many ways. Not the least of these changes has been the gradual draining of rigor from math and science instruction in America's elementary and secondary schools.

As we faced the threat of Soviet domination in critical scientific fields, American schools (and particularly those in California) were second to none, and produced the most reliable stream of super brains the world had ever seen.

Today, national academic goals and priorities have shifted dramatically to raising the academic floor while virtually ignoring the needs of gifted students. The unhappy result is that the full potential of our best and brightest young minds is going to waste.

Below are excerpts from an article on the subject by Sara Rimer appearing recently in the New York Times:

The United States is failing to develop the math skills of both girls and boys, especially among those who could excel at the highest levels, a new study asserts, and girls who do succeed in the field are almost all immigrants or the daughters of immigrants from countries where mathematics is more highly valued.

... in China math is regarded as an essential skill that everyone should try to develop at some level. Parents in China, he said, view math as parents in the United States do baseball, hockey and soccer.

“There is something about the culture in American society today which doesn’t really seem to encourage men or women in mathematics,” said Michael Sipser, the head of M.I.T.’s math department. “Sports achievement gets lots of coverage in the media. Academic achievement gets almost none.”

In America today, intelligence and academic achievement are not only generally underrated; it has now actually become fashionable to deride intellectual prowess and accomplishment. Increasingly, the doltish are hailed as heroes while those with the sharpest minds are ignored or shunned as undesirable.

One needs look no farther for crystal clear evidence of the victory of mediocrity over excellence in our culture than the recently concluded 2008 presidential election, in which a gifted scholar and brilliant orator was framed as "elite" in contrast to inarticulate intellectual super-lightweights who were touted as "true, genuine Americans."

Sarah Palin and Joe the Plumber are the new models. Bushisms and mangled communication have become acceptable. Smart just isn't cool, anymore. Math is for Asians ... not for "real" Americans.

Raise the floor, yes, by all means. But don't forget that raising the ceiling is just as important.

Other countries understand this. We ingore it at our peril.

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

Thought Boxes Revisited

Last month, I presented a puzzle that I asserted a bright first grader would find easier to solve than a bright well-educated adult.

One would think that education and experience would be advantages that should make solving any puzzle easier, not harder.

Not necessarily true.

Such is definitely not the case with the "OTTFFSSEN" puzzle, which is more easily solved with an open, uncluttered, "beginner's mind" than with a trained, sophisticated, educated mind.

One of the characteristics of human intelligence is that as it becomes more and more educated, more and more experienced, it "learns" to craft specialized shortcuts (generalized assumptions or "boxes") that increase the probability of quickly finding dependable answers to questions and viable solutions to problems.

Normally, this is a good idea. Sometimes, however, it isn't.

This is one of those times.

An adult assumes unconsciously that the "OTTFFSSEN" puzzle I presented last month must have a complicated solution, since if it didn't, it wouldn't merit consideration or attention in the first place. Unfortunately, this puzzle has a very simple solution, so looking for a complex one guarantees much needless frustration, at best. The adult's unconscious aversion to thinking "outside the box" dooms him or her to failure.

A bright young child, however, makes no such assumption of necessary complexity, and since he or she has only really studied two things thus far in school, letters (early spelling) and numbers, easily notices the pattern in the given letters:

One
Two
Three
Four
Five
Six
Seven
Eight
Nine
(Ten)
(Eleven)
(Twelve)

The answer to the puzzle, therefore, is of course:

TET.

(Click here to go to Part 1.)

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