Joe has received recognition as an expert author in the field of educational and math articles. His extensive writings on this subject are read throughout the country and indeed the world. Joe's educational articles touch primarily on the many facets of mathematics and show how this subject is intricately woven into every aspect of our lives. His articles give a glimpse of the writing that can be found within his many ebooks. Indeed Joe's didactic approach and simplistic style make learning and understanding mathematics as easy as walking through the park.
Here in full force are the published articles that Joe has written on a broad array of topics. Focused primarily in mathematics, these articles wil take you on a journey through basic math, like learning the multiplication table, to some more advanced realms, like abstract algebra. Included here as well is a collection of articles touching on some everyday topics as well as some Christian themes. Come back often to find new ones as they are published by Joe. Also, be sure to sign up for our free newsletter.
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For those of you who have discovered article writing as a way to get worldwide internet exposure, take a bow. Welcome to the International Brotherhood of Article Writers. As part of this brotherhood, you should be aware of some article etiquette, particularly as pertains to supporting and encouraging your article brothers. One of these etiquettes has to do with rating articles.
Thinking of entering the teaching profession? Maybe you're even thinking of teaching a subject like mathematics. What a crazy thought! Why would you want to do such a thing? I mean, I can come up with at least a hundred other ways of frustrating the heck out of yourself. All joking aside, teaching has to be the most noble profession on earth, but one that should not be entered into on a wing and a prayer, nor after only perfunctory consideration. And if you are selfish, then teaching is definitely not for you.
So you’re tired too of dealing with your teenage son or daughter? You mean you’re walking on eggshells when he or she is home? Can’t get them out of bed in the morning? Having trouble getting them to follow any rules? Well, welcome to child rearing in the rip-roaring new millenium.
There is just something simply elegant about packing a ton of meaning into a fourteen line poem, which we know as the sonnet. I came to write sonnets as a result of having been asked by a student, whom I was tutoring over the internet via email, to help her with an English assignment. This task involved writing a sonnet. Having been a big fan of the Sonnets of William Shakespeare (having memorized at one time at least twenty-five or so, and having written a short book on the subject), I delighted in the task. The result was that I became an avid writer of this famous English poem.
When I first saw the quadratic formula, I was amazed that there existed such a thing no less a way to derive this elegant formula. For those who remember, this formula gives a sure-fire way of getting the solution to those things we call quadratic, or second-degree equations, in mathematics. For many students, this formula is a nightmare of grand proportions and its mastery seems no more probable than striking it rich with the lottery. However, with some novel techniques and some different approaches, the mastery of this formula—which provides a linchpin to understanding algebra on a deeper level—becomes a walk in the park.
So we all know the rap music industry is big business. How big? Assuredly in the billions of dollars. But is this rap music—and more specifically the genre referred to as gangsta rap—good for our children’s ears and moreover society? Common sense would dictate that the last question is quite a rhetorical one.
There are many interesting number tricks, and, if you’ve followed my articles, I’ve written quite a few about such interesting numerical wizardry. When we attach the name Einstein to a number trick, however, the idea takes on an entirely different dimension, both literally and figuratively. If you are interested in some fun with numbers, try out the trick below on some friends and family, and by all means teach it to your kids, if you have any.
Children who are difficult to manage are often poor problem solvers. Any parent who is experiencing trouble raising their teenage son or daughter knows how vexing a problem this can be. My wife and I experienced this firsthand and I can honestly tell you that the trials we went through were nothing short of nightmarish. After purchasing a specialized program to teach us how to deal with this problem, we learned that children who are behavior problems are often poor problem solvers. It was then that I saw a revelatory connection.
To continue with my Why Study Calculus? series, I discuss here some interesting applications of this branch of mathematics to numbers. Numbers and the operations on them are the key to mathematics, and all higher branches are one way or another intimately linked to their inherent properties. One nice application of the Calculus to numbers is the approximation of square roots. What is more, this technique can be done without a calculator and without even a knowledge of the underlying theory.
You know I thank God that I was given the ability to communicate math to the masses. I guess this talent comes from my struggles with this subject early on. My ability to convey the guts of this subject derives from my belief that if I can understand it, anyone can. Such is the case with calculus. Read on as I show you how this subject allows us to calculate the exact area of even bizarrely shaped objects.
Look. Just because a person is a mathematician and loves numbers and that sort of thing, doesn’t mean this person is an absolute weirdo. You know, if you meet me in person, I don’t really look like a mathematician, or someone who spends time reading about things like topological spaces or the construction of real numbers using Cauchy Sequences. In fact, I think I’m downright cool because I like this stuff, and this attitude is what enables me to bring this subject to others and make it understandable to the math haters.
Do you really want to get your kids off to a fast start in math? Then they need to know some basic facts. Teach kids their addition and subtraction facts as early on as possible. Once conquered, move on to multiplication facts. Finally, start plodding into the territory of division. With the four pillars of arithmetic conquered, your kids will be on the fast track to academic success.
There are times in one’s life when something so miraculous happens that it defies ordinary explanation. We all hear of amazing stories—of miraculous recoveries from disease or illness, of daring escapes, of angelic interventions—but when it happens to you, you don’t know quite how to explain it other than to say thank you God, and praise be Yours. Something like this happened to me today, and I describe it as the best Christmas gift I ever got. Here I relate the details.
We all know how annoying and vexing the common cold virus can be, particularly during the winter season. Nobody likes that runny nose or that constant urge to sneeze; nor is the coughing, headache, or scratchy throat anything to write home about. To avoid catching the common cold, we know that we should wash our hands often and keep from being breathed or sneezed on by those already infected. Yet despite all our efforts, we still seem to get that yearly cold. Maybe mathematics has some clues as to how we might lessen our chance to pick up that awful cold.
So you love to write and you found out about submitting articles on the worldwide web. You ask yourself whether you can actually get good exposure from spending time writing articles and whether it’s worth the effort. After all, like me, you’re a nobody from nowhere. But then again, at one time so were all the celebrities and people in the current media. Well, I started writing articles a little less than a year ago, and when I started, I asked the same questions. Now that the Fox TV interview has aired on television and on their website, I am thoroughly convinced that, yes, article writing is worth the effort. Read on.
Labor Day is fast approaching and you can almost smell the students at the front door of your classroom. Another year and another sixty or so Algebra I students to grapple with. And although the first few weeks might go fine, you know that as you hit the third chapter or so of the text, the kids will start having problems. You dread having to explain things over and over again, and you loathe grading those awful tests. You just can’t understand how the students don’t get it. To you it’s all trivial but to them it’s relativistic mechanics. Ponder this situation as you may, what you might not have realized is that if you slay the demons first, the rest is...well easy.
So do you think as a mom or dad that the prestigious status of whiz kid, accorded to children with outstanding mathematical abilities, is reserved for the other parent’s children? Well think again. My bold claim is that any child can be accorded such privileged status provided that he has an interested parent and that the process of acquiring this status is begun early, preferably during the early pre-teen years.
Mathematics has a way of taking us by the hand and not just leading us down the path of reason, as Pythagoras once said, but sometimes down the path of insanity. With all the beautiful truths that math can show us, there are also inherent contradictions of nature that this field forces upon our senses. Such seeming impossibilities are found within the world of fractals, those weird yet curious geometric objects that have caused us to look at nature in a whole new way. From the surface of a mountain to the head of a broccoli, fractals are being used to explain things that we normally take for granted.
When I contemplate the title of this article, I enter a reverie which catapults me back twenty-five years to my college days. Back then, I learned how hard mathematics was. Often I felt inferior to tackle such a difficult subject with my limited brain capacity. Little did I know then that in retrospect I would come to understand that indeed math is hard, but that life too is just as challenging.
Nothing smarts so much as when the world gives you a good butt-kicking. Nobody—not even the rich and famous—are immune to such occasional whippings. But the rich, famous, and highly successful have their riches, fame, and financial success to lean back upon. What do the regular average Joe’s and Jane’s have to lean on when the cold world turns its ugly head and starts its unrelenting chase?
Writing has always been a favorite pastime of mine. When I discovered article writing a year ago, the means for expressing myself as well as the ability to share my knowledge with others became possible. This open forum has created a dialogue with my readers, who although unknown to me, create the necessary feedback that drives me to write even more. Moreover, well written and interesting articles can pave the way for much exposure—even to the extent of news related stories and television interviews.
When I decided to become a mathematics major in college, I knew that in order to complete this degree, two of the required courses—besides advanced calculus—were Probability Theory and Math 52, which was statistics. Although probability was a course I was looking forward to, given my penchant for numbers and games of chance, I quickly learned that this theoretical math course was no walk in the park. This notwithstanding, it was in this course that I learned about the birthday paradox and the mathematics behind it. Yes, in a room of about twenty-five people the odds that at least two share a common birthday are better than 50-50. Read on and see why.
Calculus does have its limits. Indeed. In order to understand the pun of the first sentence, you need to know that calculus has two key branches: differential and integral. Although the concept of limit belongs to both branches and is an essential component to the understanding and mastery of this kind of math, differential calculus gets its name from the derivative; and such a creature depends entirely on the concept of limit. In fact, the derivative is nothing more than a special kind of limit.
As part of a subcategory within my “Why Study Math” series, I am pleased to introduce a new group or articles that will touch upon some of the higher branches of mathematics. The purpose of these writings will be to introduce people—particularly the lay person—to some of the more advanced reaches of this subject. Many people unfortunately never get to glimpse the fascinating beauty of the higher realms of mathematics such as the Calculus, and far too many are under the mistaken belief that this field is only reserved for the so called erudite and genius crowd. Not so I say. Give me a chance and I will be happy to accommodate you.
So you think math isn’t very sexy, heh? Well then you don’t know about the aesthetic appeal of the golden ratio, which has appeared in works of art and nature from eons past. The great Da Vinci even maintained that the human body has proportions which approximate the golden mean. So what is this golden number, and what the heck does it have to do with sex appeal? Read on.
There’s an old joke about two schoolboys that basically goes as follows: Johnny, one of the boys, would go off to school every morning, and as he left his mother would yell, “Johnny, don’t forget your lunch!” The other boy, Steven, would go off, and as he did, his mother would yell, “Steven, don’t forget your books!” Johnny grew up to be a successful restaurateur and Steven a world-renowned brain surgeon.
As students, we become comfortable with what we learn first. Of the four arithmetic operations of addition, subtraction, multiplication, and division, we learn to add first and for this reason are most comfortable with addition. If we apply the principle of thinking in terms of what we are most comfortable with, then subtraction need not be a difficult operation to master. Consequently, by applying addition principles to subtraction, we find our shortcut to mastery of this operation.
In this complex world of technological gadgetry, we run at a pace which would make our ancestors blush at the thought. Faxes are not even fast enough as we have email which is instantaneous. And no matter that you are not at your computer because many cell phones deliver email in a snap. The latest news from abroad—good or bad—the latest stock quotes to brighten or stress your day—it’s all there in a digitally quick transmission of electrons. Is it any wonder that we have little or no time for self-reflection, for a quiet moment far from the maddening crowd?
Probably the most important fundamental branch of mathematics is algebra. No matter what anyone tells you folks, if your kid is not good at algebra, there is no shot that he will excel in any of the higher branches. In fact, the famous calculus is really a form of “glorified” algebra and this is what I used to tell my high school students all the time. Therefore, if you want your kids to have a chance to study and enter the beautifully intellectual realm of the calculus, then be sure that they have the tools to master algebra.
So do you think as a mom or dad that the prestigious status of whiz kid, accorded to children with outstanding mathematical abilities, is reserved for the other parent’s children? Well think again. My bold claim is that any child can be accorded such privileged status provided that he has an interested parent and that the process of acquiring this status is begun early, preferably during the early pre-teen years.
The famous Greek mathematician Pythagoras (you know the one with that theorem) said, “Numbers have a way of taking you by the hand and leading you down the path of reason.” What Pythagoras was getting at, I think, is that numbers—by their very nature—permit us to do things which enable an understanding of the very universe and its intricate laws. Numbers have their roots in arithmetic, and a mastery of this field, particularly the operations of addition, subtraction, multiplication, and—yes that monster operation of division—will certainly pave a smooth road down that path of reason.
The New Jersey HSPA (High School Proficiency Assessment) Test is one which all juniors in New Jersey high schools must take and pass in order to receive their diplomas. For many, this test presents a challenge as it consists of two sections, one in English, and one in math. The English section requires that students demonstrate competency in English, both from a written and comprehension perspective; and the math section requires students to demonstrate competency in four areas in high school mathematics: number sense, concepts, and applications; spatial sense and geometry; data analysis, probability, statistics, and discrete mathematics; and patterns, functions, and algebra.
The name Singapore derives from two Malay words meaning “lion city.” So what’s the connection between this small South-East Asian country of four million and arithmetic mastery? Well for one, the Singaporean mentality toward arithmetic mastery as the stepping stone to higher mathematics is as well ingrained in their culture as apple pie and baseball is in ours. Indeed their lion-like ferocity toward taking no prisoners in insuring that their children master the mathematical basics of arithmetic—to wit, multiplication, division, addition, and subtraction—is the main reason that their country serves as a model in educational excellence, particularly in the mathematical and scientific fields.
What sets the exceptional man apart from the common man is the way in which he approaches most day-to-day dealings. The exceptional man knows that no good deed goes unnoticed nor does it go unreturned. Just as night must follow day, good must follow good, truth must follow truth, and justice must follow justice. This is something that the common man simply does not understand for he is plagued by the seven follies. What are the seven follies and why the to-do about this concept?
In Part II of this article, I discuss the other three follies of the common man: hate, spite, and unforgiveness. Clearly, these are three negative qualities, the first two being very strong negative emotions, and the last being a cancer-causing bitterness-arousing trait. These follies wreak havoc on a man’s emotional state of being and cause all kinds of disturbing conditions even to the point of making man sick. Let’s discuss how we can avoid the pitfalls into which these three follies plunge us and recognize ways to rid ourselves of these onerous states.
For many juniors in high school, the SAT (Scholastic Aptitude Test) looms as a big ugly monster on the horizon. Indeed for both parents and students, the anticipation of taking this test creates all kinds of unwelcome anxiety. The stress becomes enhanced for the competitive college-bound individual who wants to get into a better-than-average school. In order to achieve a respectable score on the SAT and to ward off excessive amounts of stress, students should give adequate time to prepare for this examination. For those parents who can afford it, private tutoring and specialized schools are also available.
Submitting a good college essay is another aspect that has to be considered by the high school student who wishes to attend a competitive school. This essay, together with the SAT (see my article on SAT Test Preparation), are two key aspects that must be satisfactorily completed by the college-bound student. Sometimes the college essay is what makes or breaks a student’s chances of getting into the school of his or her choice.
When people think of additional school help for their child who is struggling in math, they usually think of one-on-one private tutoring. Private tutoring is a great way to get help for your struggling child. As mentioned in my article on effective math tutoring, the knowledge that a tutor brings to the table is not so important as his ability to relate well to your child and to identify with their source of difficulty. Even though private tutoring is the way parents think they need to go, there are alternatives such as email tutoring that can be a welcome help to your child.
People may think that the key to effective math tutoring, or tutoring in any area for that matter, is contingent upon the knowledge that the tutor brings to the table. While knowledge of one’s field is certainly important if a tutor is going to be effective, more important is the tutor’s ability to relate to your child and give him or her the necessary confidence to succeed in the math subject at hand. During my many years of math tutoring, students would come to know that I knew my field well, but more importantly they came to know that I could relate to them on a deeply personal level. Once this aspect of the tutor-student relationship is established, tutoring becomes markedly more effective.
Fractions are the pits. You know you can’t just add or subtract them even though multiplying and dividing them is not too bad. But since addition is the most popular arithmetic operation, that’s where the darn problem is. I mean those pesky denominators always get in the way. Yet fractions appear everywhere you look: look at the price of gas, which is hovering about $3.00 per gallon and you see something like “Unleaded Regular - $2.79 9/10"; or take a look at the unit prices in supermarkets and you might see something like 33 ½ cents per pound, or 16 1/3 cents per ounce. Let’s face it, you’re not escaping these little monsters so you better just get used to them.
You know it troubles me more and more each day as I watch what is going on around us in this world. It seems today more than ever that individuals are so focused on themselves, exhibiting a level of self-centered, egocentric, and selfish tendencies that are unprecedented. Caring about one another or showing empathy toward others does not even factor into any equation whatsoever. In the New Testament book of Hebrews, Paul talks about letting brotherly love continue; yet today more and more people are concerned about letting their selfish ways perpetuate. This is most evident in the youth of today. This condition is sad, as we prosper or stagnate in proportion to the support and help we lend to each other as human beings.
Having just returned from the wake of a friend whose sister was killed in an apparent murder-suicide, I found myself wondering about the fragility of life and how close we all are to the portal of death. This too was the second time in a month that I was in the same funeral parlor room, the first time to pay my respects to a beautiful young woman of forty-four years who succumbed to Lou Gherig’s disease, whose middle son was good friends with my younger son; and now to pay my respects to the two brothers of this thirty-two year old beautiful woman who was murdered by her husband. What made these two wakes particularly tragic was that each woman left behind young children, each was both beautiful and young, and each had great families and very successful lives from a financial perspective.
As a follow up to my article “The Magic of One Numbers Part I” I now continue with Part II in this fascinating series. For those who have not read the first article, please do so now so that you can better understand this one. Here I will show you a method to perform the multiplication of any two “one” numbers regardless of size. The result of such multiplication—once this method is studied and learned—can be obtained effortlessly and usually within seconds. So let’s get started.
Do you know that numbers are the key to life? Indeed numbers form the foundations of mathematics from number theory all the way up to partial differential equations. Without these curious creatures, we could not calculate, estimate, or compute (as in computer); nor could we transact any of the business that occurs daily throughout the known world. Yet most people find these most interesting entities just a bothersome part of life. Why should this be so? Well maybe a person’s ambivalence towards numbers derives from childhood frustrations experienced when struggling to learn the basics of arithmetic and the rote calculations associated with this discipline.
You must excel in math for many reasons. Math by its very nature is a rigorous subject which conditions the mind to work more efficiently than it otherwise would. One of the reasons I teach my short-cut arithmetic techniques is so that people who learn them can exercise their brains regularly. Indeed such exercise not only stimulates the mind, but also helps keep your brain younger and less prone to degenerative diseases and decreased cognitive functioning. And there is no better brain stimulation than that obtained while you navigate within the mathematical disciplines, trying to work and solve problems of myriad levels of difficulty.
“Aim High! There is little virtue in easy victory.”
—Sir Edmund Hillary, who with Sherpa guide Tenzing Norgay, was the first man to successfully summit Mount Everest.
Think of the effort that Sir Edmund Hillary must have put in to mount the 29,035 feet of the Himalayan giant Mount Everest. And we think that internet marketing is hard! Yet in spite of the difficulty that is inherent in mountain climbing expeditions, there are challenges in climbing the oft reputed internet peaks that dot the cyperspace landscape. These are obstacles that any person new to this arena faces. Web site optimization, traffic regulation, search engine submission—where does it end? Each of these mountain peaks has sub-peaks, and every time we think we have reached the top, we see that—as the famous poet Alexander Pope said—“Alps on Alps arise.”
So you think you can’t do math in your head, huh? Well, it all depends on how you do such math. After all, if given the right tools to do a job, then the job comes out right often enough. So why fret over the small stuff? Too often mathematics is made out to be some mysterious subject, only to be mastered by an esoteric group of like-minded nerds who co-habitate in some forsaken land. Well, that’s not the case and one of the things I’m fond of doing is debunking this oft-held yet misguided notion.
If there is one thing I cannot stand more than anything else, it has to be the internet shark-hypocrite. What do I mean by this? An internet shark-hypocrite is one who goes around parading like your best friend in cyberspace, one who can help you achieve whatever your heart desires, when in fact he is really a lying shark ready to pounce on you at the first scent of blood. In the days since the internet boom, shark-hypocrites have multiplied to the point at which a person in search of some help on the internet, particularly as concerns making some extra money, cannot avoid being attacked by these omnipresent and conniving beasts. Despite their growing numbers, I will not cease to expose them at every turn, trying to educate readers of my articles so that they do not become a sumptuous meal for these lurid creatures.
As a former teacher of high school mathematics, I understand the day-to-day frustrations that any teacher might experience, particularly when trying to teach a subject like mathematics. The first day of class was always interesting. As teacher, I felt like the enemy who was bringing messages of death and despair to the students. I could see in many of their faces how dreaded a subject this truly was. But I would win them over. Yes, one by one I would quench their fear and instill new hope.
You know it’s true. If you want your children to master mathematics, you need to give them certain tools so that they can whip the heck out of this difficult subject. One of those tools is a mastery of arithmetic. Another is a mastery of fractions. These two together will give your children a formidable arsenal against this notorious subject.
You know it’s funny. I’m making so much money online that I can’t believe it’s true. These internet gurus out there who sell all those money making programs are right. I’m about to quit my job because of them and spend most of my time at the gym and on the beach this summer. I might even cruise down to some of the islands—maybe even take a stroll through Hawaii. My Paypal account is about ready to explode. What a great life!
All right. I’m sick and tired of all the hype on the internet. And I can’t believe people actually fall for this stuff. You know, the promises of six-figure incomes—not yearly—but monthly; the promises to help you leave your day job within three months; the promises to give you the knowledge to work three hours per day or less and spend the rest of the time on the golf course or at the gym. Come on. As John Stossel says, “Give me a break!”
How important on a scale of 1 to 10 would you say knowing how to do basic percents was? Personally, I cannot think of a more important basic math function than this. Yet most people struggle with the very basics of percent calculations. This should not be the case. My aim in writing Arithmetic Magic was to show people that they can do basic arithmetic, and this once realized would empower them to study math at a higher level and become more interested in this most precious discipline.
Why is it that math geeks are considered squares? That is, why is the straight-A math student considered Mr. Un-popular in school? It’s funny how twisted things are in life. The cool kids in school are the ones smoking pot and cigarettes, perhaps taking other illegal substances, having sex with other students, and getting into at least one fight per quarter. Meanwhile, the straight-A kids are considered geeks, even though they are the ones that will build the country, write the future laws, and find the cures for the most vicious diseases that affect us all today.
Math rules, and everything else drools. Well maybe not everything, but the truth is that math really is the king of subjects to study. For ultimately every subject in school—indeed everything—revolves around math. Even subjects that you would never think. Like English, for example. How so? you say.
As we continue the “Why Study Math” series of articles, here we look at the conic section called the hyperbola. The hyperbola is obtained by intersecting the double-napped cone (see the other articles in this series on this point) with a plane so that both parts of the cone are cut. Those familiar with the parabola might note that this curve almost looks like two parabolas pasted back to back with a space in between them. Mathematically, the hyperbola is not a parabola, although these two conic sections have a similar outward appearance.
As we continue the “Why Study Math” series of articles, here we look at the conic section called the parabola. The parabola is obtained by slicing a nap of the cone (see the other articles in this series on this point) with a plane parallel to one of the generators of the cone. In plain English, this means the following: take a cone made out of styrofoam; draw a line from the apex, or point, straight down to the base; on the opposite side of this line, slice the cone with a knife, starting midway down from the top, and such that the cut is made parallel to the line on the opposite side. The resulting cut produces a shape called the parabola.
Admittedly fractions are trouble for most students. In my previous article I talked about why this is so. Percents and decimals too present their share of problems to young students-adults as well. There is an interesting connection between these three mathematical entities and here it is: fractions, percents, and decimals are variations of one and the same thing.
So you think you’re not good with math. Hmmm. Well that’s something I hear all the time so you shouldn’t feel too bad. The truth is, many, many people, at one time or another, have felt the same way, and have experienced something which has made them feel bad about math and numbers in general. So the common complaint, "I was never good at math," or "I hate numbers and math," is as common as the common cold and sometimes just as annoying. Interestingly, what I have discovered from working with students through the years is that this feeling of inadequacy often stems from some negative experiences with the subject. This might very well be the case with you. Perhaps you had a hard time in one of your math classes and felt very frustrated by it. Perhaps you struggled year after year because you could’nt understand the material being taught, or because the pace was too fast. Perhaps you didn’t understand because the subject matter was presented in a way that did not get across to you. Sound familiar?
In continuation of the "Why Study Math" series of articles, here we look at another conic section: the ellipse. The four conic sections, in order from most popularly known to least, are the circle, the ellipse, the parabola, and the hyperbola. Remember that these shapes can all be obtained by slicing a right circular double-napped cone with a plane. As a visual exercise, picture an ice cream cone-without the ice cream-upside down standing on a table. This is a single-napped right circular cone. (To get a double-napped cone, add another cone on top, right-side up, balanced at the point.) To get the circle, take an imaginary plane (picture a piece of paper) and intersect the cone parallel to the base. The plane has just cut out a circle on the cone. Similarly, to get the ellipse tilt the plane slightly up or down and intersect it with the cone. What you have then is an elongated circle, or ellipse.
Let’s face it. We all want to be popular. Popular with friends, relatives (well, maybe not relatives—just kidding), coworkers, bosses, and so on. Popularity is associated with fun times, success, prosperity, and a whole array of other positive thoughts and emotions. After all, when you think of the most popular boy or girl in high school, what pictures come to mind? In the case of the girl, you might envision the prettiest girl with the best smile and personality, perhaps captain of the cheerleaders. For the boy, you might think football quarterback, best athlete, or just plain best looking kid with the highest academics. Granted, these descriptions might not match exactly what came into your mind, but I think you get the point: when people conjure up images of the “popular ones,” they tend to think “the best looking,” “the best dresser” or something along this line.
Fractions. Ugh! I could just hear the squeals coming from my students any time we entered the realm of these nasty little demons. Anytime we embarked on an area of mathematics that would require heavy fraction work, students would act as though we were entering Hades after an arduous crossing of the river Acheron, led by the fearless ferry-man Charon and his three-headed dog Cerberus. Ouch! It was that bad.
Analytic Geometry is a branch of mathematics that treats the relation of algebraic functions and their respective graphs, or pictures that can be drawn from these functions. Students are first introduced to analytic geometry in Algebra II courses, and delve further into its study in both pre-calculus and calculus courses. Essentially, this branch of mathematics combines geometry and algebra to show what certain mathematical relationships, called functions, look like in the real world.
This article will launch my new series in the Why Study Math? category. This will be the start of a fascinating series which show that mathematics is more than just a pain in the neck; indeed these essays will permit a fascinating peek into the world of this curiously strange discipline. We start in the land of geometry, where shapes and figures and their many properties drive students to the breaking point of insanity. Each article will feature a specific aspect of a particular branch of mathematics, such as algebra, trigonometry, and calculus; and each will endeavor to shed some light on how these fields are used in the real world. So come on board and enjoy the ride.
In Amazon’s Significant Seven interview with Jim Frey, the controversial author of the Oprah Book Club selection, “A Million Little Pieces,” he was asked: “If you could have one superpower what would it be?” Since Mr. Frey has been the focus of media attention lately, could anyone familiar with this issue guess as to what his answer might be? Given the circumstances surrounding this whole matter, one might make a good stab at the answer. No, still can’t get it? Well, Mr. Frey’s answer was...immortality.
If you want to write well, one of the stylistic elements that you need to pay heed are those nasty dangling modifiers, whether in participial, gerund, or phrasal form. Although sometimes insidious to spot, these grammatical faux-pas will tinge your writing and confuse your readers’ train of thought. Much like a poorly segued musical transition, dangling modifiers impinge somewhat strangely on the ear and often lend unintended yet grotesque humor to your writing in general.
The other night I was watching an episode of the show 20/20, in which John Stossell was exposing the dreadful situation of education in the American schools. In this episode, the host made a comparison between our schools and those in Belgium. For this comparison a test was administered to both American and Belgian students of the same age. To wit, the Belgian students clocked the Americans on this test. What stung even more for the American cause was that the American kids were from a rather decent school. Wow! How dreadful. Yet this state within American schools is something I witnessed firsthand.
In continuation of Part I, we now plunge more deeply into the Quick-Add Method and show how this makes doing addition quite easy. This procedure hinges on two key ideas: 1) the method of complements; and 2) the Quick-Add Conversion. To refresh your memory (also see “Teach Your Kids Arithmetic - The Quick-Add - Part I), complements of a number are those numbers, which when added to the given number, yield a sum of 10, or some multiple of 10. For example, the 10-complement of 8 is 2, since 8 + 2 = 10. The 10-complement of 4 is 6, since 4 + 6 = 10. The Quick-Add conversion is simply the way in which we convert our given addition problem into a “quick-add;” for once done, the problem becomes—well, what the method says: a quick-add. That is, the addition can be done quickly and easily. As mentioned previously, the Quick-Add works as follows: in analyzing 10 + 7, we rewrite this example as 10 + 07. We insert a 0 in front of the 7 as a placeholder for the empty “tens column,” and to bring the numbers into parallel structure. The brain performs 1 + 0 in the “tens column” and 0 + 7 in the “ones column,” thus capitalizing on the “Additive Identity Property” of 0.
Are attacks on the Bible really justified. Children believe in Santa Claus, the Tooth Fairy, and the Easter Bunny up till a certain age. Do we dare attack these cherished beliefs because we know otherwise? We uphold these out of love for our children, our grandchildren, our nephews and nieces. So dare men risk attacking something that they cannot prove false? This is the epitome of both arrogance and foolishness.
So the cliche rings out: “Nice Guys Finish Last.” When this expression was coined by the famous baseball coach Leo Durocher, what he meant was that winning requires toughness, ruthlessness, and singleness of mind in order that one aspire to champion status. We all want to win and be champions—at least on a conscious level. However, losing is also part of life and we need to deal with this aspect at times. What has happened with this statement is that it has been accepted by the populace as a truism. Yet is it really true? If you are nice, does this imply that you must necessarily finish last? Most people give a knee-jerk response to this statement without really contemplating its true import or implications. This statement, along with bromides like “It’s a dog-eat-dog world,” serve only both to disempower and force us to believe in false doctrines established by society.
In continuation of my series on arithmetic, I present here a topic that was one of the cornerstones in my book “Arithmetic Magic.” To fully understand how this concept aids one in arithmetic operations, we need to lay some foundational ideas first. The “Quick-Add” is an enormously valuable tool to help children master quick arithmetic, particularly applied to summing numbers. Today the calculator has crippled even the ablest students. Hardly a one knows his fundamental multiplication facts, as the omnipresent calculator does this operation for him. This situation is understandable, and a comparison I can make is one regarding remembering telephone numbers. Since the ubiquitous cell phones store numbers, I no longer memorize people’s numbers as I have no need to. Analogously, students no longer can add or multiply because the calculator does it for them. This is a problem for the following reasons: 1) not mastering arithmetic leads to problems in mathematics down the road; 2) not being able to add or multiply engenders frustration when doing basic homework assignments; and 3) lack of doing = future lack of doing, which further increases the chance of mathematical illiteracy.
“Infinity is a floorless room with no walls or ceiling.” —Anonymous
For those who have followed my series on infinity, this article gets to the crux of the matter concerning the existence of different types of infinity. Having read the two previous articles, you get the sense that the topic of infinity is a strange one indeed. Throughout the ages both philosophers and mathematicians have been debating this topic and puzzling over its implications. From the fifth century Eleatics (the Greek philosophers from Elea: Zeno, Parmenides, and Melissus) all the way through to the famous German mathematician Georg Cantor, who is known as the father of modern set theory, great thinkers as these have pondered and labored feverishly in trying to nail down a precise formulation for this seemingly surreal idea.
Boy this math stuff can be hard to understand. I mean concepts like infinity, differential calculus, set theory. Good grief. John von Neumann, the famous mathematician, said “Young man, in mathematics, you don’t understand things, you just get used to them.” That’s exactly the way I feel sometimes when I myself read and study about this most curious subject. I guess the reason why I excel at explaining concepts in mathematics is that I have experienced so much frustration in trying to understand this subject from another’s perspective that I go out of my way to break things down so that even a baby could understand. Well, not a baby, but you know what I mean. Often I go and do research on the internet and read articles there. It is sad that well written, intelligible articles are hard to find. And this is coming from someone who has studied mathematics at the graduate level!
The third millennium has brought with it a new paradigm in society morals, one which, quite frankly, I can live without. Is it just me that sees the steady decay in society morals, particularly with our youth? My principal concern is where we are heading as a nation, as a country, as a people. As a former teacher of high school mathematics, I got an inside peek into the world of adolescence and the attitudes carried by such breed. It pained me on a daily basis to see what motivated and drove students.
In continuation of my discussion on infinity and its implications with the divine, I should mention that the concept of there existing infinities beyond imagination is quite difficult to comprehend. If you read my poem, “How Can this Be? (See the link below to this collection of poems) you read in verse the proof that shows clearly that there is no such thing as one kind of infinity. The extension of this most curious fact is that there are actually an infinite number of infinities!
Just recently, I was reading a very good article titled “When Your Back is Against the Wall” by another ezine writer named Anne Stewart. As stated in the title, this article dealt with the uncomfortable position we are in when the difficulties of life hit us hard. Inevitably, life will hit us. Hard indeed. Of this one thing we can be sure. However, some seem to get the onerous privilege of being pounded by life the way Rocky was in his movies. The good thing for Rocky was that he ended up the winner. Time would heal his bruises from the fight.
As a continuation in my series of articles on the merits of mathematics, I submit this topic for consideration by parents who want to give their children a serious edge in mathematics. You see folks, it all begins with the basics. Give your children the basics and then the rest is assured. You cannot build a house on a weak foundation, nor can you expect to build your children’s education on a weak substructure. One of the problems with education today is that we have moved away from stressing the importance of the basics. Although this pedagogical approach is still adhered to in many school districts, too much tampering with basic approaches to the learning of mathematics has resulted in a perversion of this basic tenet. Because mathematics is inherently difficult to learn, the common frustration which results from its study has led educators and parents to seek alternative learning approaches.
Christmas is a week away, temperatures have been hovering in the teens, and I’m writing about fun in the sun and something about using mathematics to tell time on the beach! Good gracious. You must be thinking how weird I am. Well maybe, but for those who have been following some of my other articles, you know then that I seek to show how mathematics—yes even basic mathematics—functions so universally throughout our everyday lives. Even the layman can use this beautiful tool do so many common and ordinary things!
Life is hard, yes even difficult beyond measure sometimes. As adults we scramble ferociously trying to eke out a living and make do with all of our family, personal, and other obligations. Of course, I am referring to the experiences of the average citizen, not the ones I read about on the AOL headlines that flash on my computer screen; you know, the ones that are super rich and live in eight million dollar mansions, or the celebrities and music artists that made it big in their twenties and never really knew what is was like to struggle to pay bills.
People hate math, at least that's what they're always telling me. Of course, being the way that I am, I have to make converts of everyone. I don't know what my passion is exactly, that drives me to convert the "math haters," but I simply cannot overcome the compulsion to do so. To me, there is not just beauty in mathematics but something else which is spiritual. It's as though God speaks to us through this strange and mysterious language. For this reason, I always approach the subject with a certain degree of humility and reverence, knowing that I can be both abased and exalted by its magnificence.
If you read the previous article on this topic, then I imagine you were quite piqued by the nature of its contents. How we use mathematics to find a mall parking spot is not a typical thing you would hear people discussing at their Christmas parties. Yet I think anyone with a modicum of human interest would find this a most curious topic of conversation. The reaction I usually get is one of “Wow. How do you do that?”, or “You can really use mathematics to find a parking spot?”
Just how does an abstract discipline like mathematics find itself mixed up with a notion as difficult to pin down as that of faith. What is this thing called faith anyway? As far as I can see, I never saw faith walking around, nor was I ever able to touch it. As much as I might have wanted a heavy dose of faith as a Christmas present some years, I do not ever remember anyone telling me that they just picked me up a nice piece of faith in the local mall and got a great deal on it.
I am always amused when people become fascinated with simple demonstrations of the power of even basic mathematics. We have become mostly a jaded society depending on the technological progress of corporations to give us more and more electronic gadgetry and the like. We have forgotten how all this technological stuff originates: via the harnessing of the knowledge of mathematics and science and their applications in practical ways.
