This post is for all you math gurus out there who really love your field of study but are having trouble finding a practical place for that love in the modern job market (which, granted, shouldn’t be that hard in the first place in less you, say, strictly studied chaos theory and refuse to work on anything else … even then). The most marketable and out-branching aspect of your mathematical expertise is probably so embedded into your brain that it’s really just an extension of who you are: your logic.

Essentially, mathematics is a formal codification of logic. Using high levels of problem solving for mathematics involves a great deal of abstraction. In topology, you will be abstracting shape from numbers, and in algebra, you are dealing with structure. Many people don’t see math in this way, but this has to do more with their own shallow grasp of the study, understanding primarily arithmetic and not much else.

This is how many of the concepts that you learn in mathematics can apply to computer science and even philosophy and semantics. Now granted, if you actually write a computer program, you probably won’t be using much if any applied mathematics. In fact, you probably won’t be using much computer science to simply write a program (particularly a simple one at that).

It’s only once you begin attempting to solve especially difficult programming problems that your mathematical mindset really begins to bloom. As you go further into the specifics of a program and try to flesh it out, you will be using lower levels of abstraction. These lower levels will begin to reveal the values behind the programming and the significance of what their variation represents.

I’m not definitively calling computer science a subset of math, but the two are definitely related. And having a flexible mind that can give shape to dynamic abstract concepts and problems is a must for both fields of study. In this same vein of thought, you can also reason that having that mental perspicacity could easily lend itself to the sciences and even language. After all, what is language but a series of abstractions arranged to represent a bigger “problem,” or meaning. The only difference is that, while computer science and mathematics are interpreted by a strict and robust system, the interpretation of language is left to the whims of any nitwit with ears or eyes.

So, in the spirit of this post, please make ever attempt you can to branch out your mathematical mind to new levels of abstraction. Seek out relationships between the field of study you love and other topics you might find interesting. Who knows, you may create a whole new career from it.

This guest post is contributed by Barbara Jolie, who writes for online classes.  She welcomes your comments at her email Id: barbara.jolie876@gmail.com.

A Look at 9 Awesome Careers that Use Math

Many students have the same question when they have to study math in high school: “When am I going to use this?” The answer to that question is multifaceted. First of all, math is one way that students can learn logic, problem solving and critical thinking skills, which are all extremely important for furthering one’s education, especially if a student is considering getting an advanced degree like an online Ph.D. Second, basic principles of math are important in everyday life, even if it is just as simple as keeping financial records, calculating costs and estimating simple probabilities. Third, and finally, more advanced mathematics can be directly useful in many different careers involving mathematics and science. As such, when determining how their math degree will best serve them, students might want to consider the following nine possibilities.

Computer Scientist

Computer science is a practical field that requires some knowledge of math. In fact, computer programming in general requires the same step-by-step logical thinking involved in mathematics. Some areas of computer science require knowledge of advanced mathematics; and math and computer science are often so integrated that universities combine their math and computer science departments. Computer scientists solve problems, program computers, develop new technology and may work in fields such as robotics and artificial intelligence.

Physicist

Of the “hard sciences,” physics is the most math-heavy. It is often the case that abstract, or “pure” mathematicians will work on an entirely theoretical problem that they think has no real-world application, and it will turn out to have an application later in quantum mechanics or some other field in physics. Physics is a difficult subject, but for those who are interested in the fundamental principles of the universe, being a physicist can be an extremely rewarding career. Physics has both theoretical and applied divisions.

Chemist

Chemists require knowledge of mathematics both for theoretical and research calculations. In chemistry research, different types of mathematics such as statistics, algebra and calculus may be utilized. Students in basic chemistry classes are required to know algebra and basic mathematical operations, but in some areas of chemistry, the mathematical calculations can get much more advanced. Like physicists, chemists can work in the chemical industry as applied chemists or do research as theoretical chemists.

Biologist

The biological sciences are also great career option for those that like math. This is due to the fact that all scientific researchers need to be familiar with statistics, as well as basic math to perform tasks in the laboratory and for biochemical calculations. Likewise, some occupations in biology such as ecology and population genetics have a heavy theoretical foundation in math. Not all fields in biology require the knowledge of advanced math, but this knowledge is beneficial and useful.

Social Scientist

Social scientists and anthropologists study populations of people, their traits and how they interact. All researchers in the social sciences need a solid background in statistics. Statistics help researchers determine whether a trend they are observing is significant or if it is due to chance alone. Believe it or not, this is not easy to determine without using math to account for observer biases. People who do research in psychology also need to know statistics for the same reason.

Medical Professional

Pharmacists, doctors, medical technicians and others in the health care industry are required to understand math. This is incredibly important when calculating dosages for medications and the parameters for the treatment of certain patients. Anesthesiologists, for example, are required to calculate the correct dosages of anesthesia that a patient receives before surgery. Pharmacists are also required to calculate and measure correct dosages of medications. Similarly, people who work in a medical laboratory need to use math during certain lab tests.

Financial Analyst/Business Person

Mathematics is extremely important in the financial world. People with a background in math can work in any field of financial mathematics, from banking to becoming a tax professional on Wall Street. Likewise, people with a background in mathematics are very qualified to run a business or work for a large or small company. Having a degree in math shows that you are capable of logical and critical thinking.

Epidemiologist

People with degrees in math may choose to specialize in statistics or epidemiology. Statisticians are hired to analyze data, and statistics is an important part of any scientific research. Epidemiology is basically statistics as applied to health care. As such, epidemiologists may study outbreaks of infectious diseases, risk factors for certain health conditions or transmission of genetic diseases.

Bioinformatician

A relatively new field that has formed in the past few decades is called bioinformatics . Bioinformatics is basically a combination of the biological sciences, computer science and mathematics. This type of discipline may appeal to someone with a wide variety of scientific interests. People who work in bioinformatics help to develop databases to store biological information and create algorithms to analyze this data. Bioinformatics is, at its heart, a way to make scientific research more efficient and informative. It is useful in many fields, such as genome mapping, protein folding and evolutionary biology.

People who get a degree in math may want to stay within the field of mathematics, as opposed to one of the careers described above. These people may teach in at a high school or college, or they may do mathematics research of their own.

Contrary to popular belief, mathematics is actually a very useful subject. All scientists and businesspeople need to have some degree of mathematical knowledge. Plus, the critical thinking skills learned in solving math problems are applicable in all careers and in everyday life.

Jeremy Fordham, a contributing writer for onlinephdprograms.com, is an engineer who addresses issues at the boundary of many fields with the hope of inspiring dialogue in unique niches. He is an advocate of process optimization and renewable energy.

“Why is it that when you multiply a negative number by another negative number, you get a positive number? Is there any way to show this; any examples?”

Professor Homunculus answers:

I love this question because it makes so much sense. It makes sense to question the premise.

But keep an open mind, because the answer, too, will make sense to you in a moment.

OK, let’s assume Lenny the Fish owes Vinny the Shark $5. So Lenny is down five bucks. That means Lenny has, in effect, negative 5 dollars.

Now, along comes Lenny’s friend, Sammy the Clam. Sammy is loaded, so he gives Lenny the five buck to pay off Vinny. In other words, he has NEGATED Lenny’s negative sum. Lenny is now even. So the negation of a negative is it’s opposite positive.

Sammy has negated that negative five bucks, one time, by giving Lenny the five. Had he negated it twice, by giving him another five, he would have given him ten. If he had negated it another time, he would have given him fifteen dollars, four times would have been twenty, and had Sammy negated Lenny’s negative five dollars five times, he would have given him a cool twenty-five slimmolians.

In other words, multiplying something by a negative is the same thing as negating something that many times.

Got it? Does this make any sense to you, because it is starting to make sense to me, and I hate to be alone with my delusions.

Here is another way to look at multiplying by negatives, (in case the above analogy is too weird).

Multiplication of integers can be seen as repeated addition. (That is not a good definition of multiplication – I repeat, that is NOT the definition of multiplication, for many reasons. To understand that, and get a great lesson in mathematical thinking, see Keith Devlin’s article.  )

Although it’s not the definition, sometimes it’s useful to see a simple multiplication problem as being able to be represented as repeated addition. (I may catch hell for saying this by some math puritans. I think sometimes they miss the bigger point, though.)

So multiplying by -n can be accomplished by adding that thing -n times. (And even the puritans will have a hard time arguing with that.)

That needs and example to make it more clear.

Let’s take -6 x -3. That can be accomplished by adding -6 negative three times. So you take the negative of -6 three times.

The negative of -6 means positive six, right? (Because you should know that the negative of a negative is a positive). So, if you add positive six 3 times, you get what? Positive 18, of course!

That, in a nutshell, is why a negative times a negative gives you a positive.

I think you probably get it by now, a bit.

Remember, though, having something explained to you is not the same as really understanding it. If you just say, “I get it,” that means you will probably forget it by the time you need it again.

Go over what you have just read, until you have it “in your bones.” By that, I mean until you can really explain it to someone else who doesn’t get it, and make them understand it.

That is one of the secrets to learning math. Teachers only care if you can answer quizzes and pass some tests.

The real test is if you can have a discussion with someone and use your knowledge when you need it.

So take my advice, and re-read all the above, then go out and try to make someone else understand it. If you have no family member, friend or teacher who gives a hoot, you can write your explanation for why a negative equals a negative in comment below.

Today a thoughtful reader left a comment on a previous post (Why You Suck at math Pt. I) Dan Marks commented:

Kids suck at math because their teachers only share the secret with the smart kids. Everybody know the ones who are successful wait until a blue moon in an odd-numbered month and drink pickle juice at precisely 3 a.m. while facing east. Hard work has absolutely nothing to do with success.

Seriously, students with bad grades in any class are generally (there’s that word) lazy and irresponsible.

I love it! OK, generally, I feel the same way, but there is a lot more going on deep down. I run into a lot of kids who are, yes, lazy and irresponsible.

Now, for me, it boils down to this: Does that mean that there is something bad or wrong with them? I think not.

Unfortunately, part of our country (at least my country) was founded on some ill-conceived Protestant work ethic. There is a pernicious myth about that ethic being some kind of moral thing.

Yeah, it’s good to work. Whoopie. It’s also good not to beat children, undermine their confidence (or, on the other end of the spectrum, overpraise them for no good reason); it’s not good to feed them coke and oreo cookies, chips and McDiabetes, send them to school without teaching them any work ethic, yet chastise them for what we have never helped them avoid.

I find it hard as hell to morally justify judging a fourteen year-old for his or her attitude, unless

that attitude is ruining something for someone else. When a kid is being mean to a less-fortunate person, it’s time to get the cane out. But if a kid isn’t reading “Moby Dick” because the teacher who introduced him/her to it was a raving authoritarian, well, how can I blame the kid? How can that kid know better if s/he hasn’t been exposed to anything better.

You must excuse me for defending the lazy little rats, because I was one of them. I constantly got that vacuous chant, “You’re not living up to your potential!” shouted at me,  And that from some stuffed-shirt functionary, or some elder who made me feel bad about myself  for my “own good.”  It made me resent the hell out of math, and and those who “know better”.

Then one day I found out, through some wonderful introduction into what math could really be, that, hey, it’s those judgmental, ignorant authoritarians who had ruined it for me. As soon as I realized that I wasn’t lazy (how could I be lazy? I practiced magic rudiments for 8 hours a day for year after year, when other kids were watching moronic sit-coms or doing homework in subjects that they would forget about entirely as soon as senior year ended). I had simply had the inspiration crushed out of me by people who derided my lack of “Work Ethic.”

Luckily, teachers are generally not as un-empathetic as they were in the 60′s. And it wasn’t really the teachers then, anyway (except for the math teachers and the coaches – they were the psycho-nazis from hell). It’s always been the creeps who reside in the offices. The ones who never spend time in classrooms, yet make all the policies for them, with the blessed assurance that only small-minded flunkies can ascend to.

I can tell you that most of the kids who were the “smart ones” from my school years are the boring ones of today. Many of them make a good living at providing nothing of value, and sucking everything they can from the rest of society, whom they consider to be slackers.

Many of the greats of the past and the present were not exactly industrious in school. We know what Einstein thought of school – 
One had to cram all this stuff into one’s mind for the examinations, whether one liked it or not. This coercion had such a deterring effect on me that, after I had passed the final examination, I found the consideration of any scientific problems distasteful to me for an entire year,” and ““It is a miracle that curiosity survives formal education.”

Just a thought – Jesus was supposed to have been a carpenter. What exactly did he build as a carpenter? I know he spent a lot of time not doing carpentry. They wrote a whole book about it and never even mentioned him once picking up a hammer or a saw. (Please, no religious comments – no disrespect is intended.)

“I’m lazy. But it’s the lazy people who invented the wheel and the bicycle because they didn’t like walking or carrying things.”

- Lech Walesa

Haim G. Ginott’s quote from “Teacher and Child: A Book for Parents and Teachers”:

“Dear Teachers:

I am a survivor of a concentration camp. My eyes saw what no person should witness. Gas chambers built by learned engineers. Children poisoned by educated physicians. Infants killed by trained nurses. Women and babies shot and burned by high school and college graduates.

So I am suspicious of education. My request is: help your students become more human. Your efforts must never produce learned monsters, skilled psychopaths, or educated Eichmanns. Reading, writing, and arithmetic are important only if they serve to make our children more human.

Here’s a list of some people you might have heard of, who either dropped out of, or never went to, college:

• Hans Christian Andersen
• Edward Albee
• Mortimer Adler
• Ansel Adams
• Paul Allen
• Woody Allen
• Jane Austen
• Steve Ballmer
• Warren Beatty
• Irving Berlin
• Carl Bernstein
• Michael Saul Dell
• Charles Dickens
• George Eastman
• Millard Fillmore
• Robert Frost
• R. Buckminster Fuller
• Bill Gates
• George Gershwin
• John Glenn
• Barry Goldwater
• Patrick Henry
• Peter Jennings
• Steve Jobs
• Andrew Johnson (Of the 43 people who served as president of the United States, 8 never went to college.)
• Dean Kamen
• Stanley Kubrick
• Stan Lee
• Rush Limbaugh (big surprise, there)
• Charles Lindbergh
• Jack London
• Steve Martin
• Me
• Karl Menninger
• Jacqueline Kennedy Onassis
• Yoko Ono
• George Orwell
• Richard Pryor
• George Romney
• Karl Rove
• J.K. Rowling
• J.D. Salinger
• George Bernard Shaw
• Quentin Tarantino
• Nina Totenberg
• Harry Truman
• Steve Wozniak
• Jerry Yang
• Frank Zappa As he noted in liner notes for his Freak Out album, “Drop out of school before your mind rots from our mediocre educational system.”
• Emile Zola
• Mark Zuckerberg

Here are some more who either dropped out of, or never attended high school

OK, let’s start out with four “gimmes”:

• Paris Hilton
• Avril Lavigne
• Jessica Simpson
• Britney Spears

Now on to ones you might find surprising:

• William Blake
• Marlon Brando
• Richard Branson
• Humphrey Bogart
• Andrew Carnegie
• George Carlin
• Neal Cassady
• Tom Carvel
• Scott Carpenter
• Charles Chaplin
• Winston Churchill
• Winston Churchill
• James Fenimore Cooper
• Charles Culpeper
• Robert De Niro
• Walt Disney
• Thomas Edison
• Albert Einstein – but later attended and graduated from Federal Polytechnic in Zurich
• William Faulkner
• Henry Ford
• Benjamin Franklin
• Eric Hoffer
• Harry Houdini
• J. Paul Getty
• Andrew Jackson
• Jack Kerouac
• Frederick “Freddy” Laker
• Abraham Lincoln
• Herman Melville (Dropped out because of the way they taught “Moby Dick.)
• Rod McKuen
• Claude Monet
• Florence Nightingale
• John D. Rockefeller Sr
• Carl Sandburg (Had little formal education but later attended Lombard College and graduated.)
• William Shakespeare (Only a few years of formal schooling.)
• Alfred E. Smith
• W. Clement Stone
• Mark Twain
• Leon Uris
• Martin Van Buren
• Anton van Leeuwenhoek
• Alfred Russel Wallace
• George Washington
• Walt Whitman
• Frank Lloyd Wright
• Orville Wright
• Wilbur Wright (It can’t be wrong to drop of of high school –  because three Wrights don’t make a wrong.)

(Most of the above are from http://www.collegedropoutshalloffame.com/ ) (The snide comments are my own.)

Interesting note:

Sirs John, Paul, George and Ringo did not graduate from college. George and Ringo did not graduate the English equivalent of high school. Yet Yanni graduated from the University of Minnesota with a Bachelor of Arts degree in Psychology. Go figure.

On the Other Hand…

Some people who got college degrees or beyond:

• Osama bin Laden - studied economics and business administration[10] at King Abdulaziz University. Some reports suggest bin Laden earned a degree in civil engineering in 1979,[11] or a degree in public administration in 1981.[12] Other sources describe him as having left university during his third year,[13] never completing a college degree, though “hard working.”  (From http://en.wikipedia.org/wiki/Osama_bin_laden#Childhood.2C_education_and_personal_life)
• George W. Bush – Yale University, Bachelor of Arts
• Dick Cheney - University of Wyoming – Bachelor of Arts and a Master of Arts in political science
• Walter Jackson Freeman II, M.D – graduated from Yale University and the University of Pennsylvania Medical School.
• Jim Jones – Butler University, earning a degree in secondary education
• Pastor Terry Jones honorary degree from the unaccredited California Graduate School of Theology
• Ted Kaczynski – PhD in mathematics from the University of Michigan
• Joseph McCarthy – law degree at Marquette University
• Josef Mengele -Ph.D in Anthropology from the University of Munich
• Maximilien Robespierre – Lycée Louis-le-Grand in Paris

Some might say that you can’t draw any conclusions from this. All this “evidence” is anecdotal. But you can draw the conclusion that college is not necessary or even beneficial for everyone. Nor is high school. That doesn’t make it always bad, but it makes it undebatable that it isn’t always good. And it shows that you don’t have to be smart or good to get into college. You can be evil as hell and/or a downright idiot. Sometimes it’s the luck of the draw, or daddy’s influence.

So any time people want to get moralistic about education and hard work, you can chalk that up to maybe good intentions, but certainly bad logic and bad advice. It is not a moral issue. Don’t let it discourage you from learning your own way, as long as you learn.

Recently, an interested reader wrote in this question:

Hi ,

I am doing a quiz and one of the questions asks what is the lowest number which divides evenly into 1,111,111.

What is the way to find the answer? Is there some sort of formula?

Thanks (hopefully!)

- A Reader

Professor Homunculus replied:

Hi, Reader,

Well, of course that number would be 1 (because 1 is the lowest whole number that can divide evenly into any positive whole number. But 1 isn’t normally considered a “prime factor.” So, if the question was phrased as you wrote it below, the answer is “1.”

If it asked for the lowest prime factor, the answer would be “God, that is so mean!” because it would take you all day to figure that out in your head, or on paper. I have no Idea how long it would take with a calculator, because I don’t use calculators.

Can I ask what grade, or where this quiz was used? And what answer did they finally expect?

Good luck with your quiz,

Yours truly,

Brian Foley (a.k.a.”Professor Homunculus”)

Then , Reader wrote:

Hi, Prof. Homunculus,

The question comes from The Great Brain Quiz (196) and is actually worded:

“During a plague of rats in Hamelin the local dignitaries set the town cats on them. Each cat caught the same number of rats and a total of 1,111,111 were killed. What is the fewest number of cats who could have accomplished this task?”

So they must be looking for the smallest number that can be divided evenly into 1,111,111.

Good question!

To which Professor Homunculus replied:

Dear Reader,

Not exactly. As I mentioned, the smallest number that can be divided evenly into 1,111,111 is 1. But the answer to the question cannot be 1, because it asks for number of cats (plural). What they are asking for is the lowest prime factor for 1,111,111.

This is a big part of what math is about – I don’t mean prime factors – I mean looking to see what the problem is and how to find the answer using reasoning.

If you understand what prime factors are and how to find them, that’s how to get your answer. If you don’t, head out to “The Pretty Good Guide To Prime Factorization.”

The first (lowest) prime factor you can find for 1,111,111 is going to be your answer.

If you write to me and let me know how far you got with your prime factoring, and show me how you eliminated the first dozen or numbers as possibilities, so that I know you really understand prime factorization, I’ll send you the answer and the sneaky way I got it, OK? But no deal if I think you are fudging on your abilities with prime factorization.

All the best,

- Professor Homunculus

By the way, the number 1,111,111 is very interesting. It is the kind of number that is called a “repunit.” You can find out some interesting things about repunits at “How to Learn and Teach Multiplication.”

Photo by selma90

If  you’re a teacher, my condolences go out to you at the beginning of this semester.

With the nation polarized concerning just about any issue, the hate- and fear-mongers have pounced on teachers with a sick, perverted glee.

You may have the misfortune to have your local newspaper publish an alleged “comic” strip (an odd name for a propaganda-strip devoid of any comic relief at all) that will go unnamed here. No need to publicize the hate-filled, malevolent ravings of a malicious maladroit. If you don’t know the one I mean, I wouldn’t dream of sending you to his reactionary website, but you can see a copy of an offensive cartoon at http://cartoonistsgroup.com/store/add.php?iid=51181

For years, that malcontent has been spewing venom about how teachers are the bane of modern society because of constructionist education reforms.

A typical case of the debate around education reform is the “Math Wars.” The Math Mojo Chronicles have tackled this issue a bit in some of my posts about the math wars.  In those writings I’ve tried to be open minded, and not lay blame at any one group’s doors. We’re all culpable to some degree. But to blame teachers for the things they have no control over is more than a little wrong, yet that is exactly what that foul creature does.

Alas, many people, like the questionable cartoonist, prefer simple answers to complex questions. They apparently never learned to think beyond scapegoating the easiest target, even if it’s the wrong one. It is apparent that they had bad teachers. Something has to explain their lack of logical thought. It’s a typical logical fallacy to assume that even though you are poorly educated and suffer from a lack of critical thinking, that everyone shares your ignorance because they had bad teachers, as well, but that is where this odd duck is at.

Let’s put it bluntly – teachers are up against some terrible odds today. They are expected to be babysitters, lawyers, nurses, bean-counters, crisis-negotiators – anything but teachers. They are often stuck with class-sizes that are over the legal limit, and can’t do anything about it. They deal with some parents who spend no time on their own children’s education, yet expect every teacher to be Annie Sullivan (“The Miracle Worker”)

Then along come some lobbyists for different curriculums and textbooks, both traditionalist and reform-minded, that fight for the administration’s budget (your tax dollars) and want to inculcate your kids with their ideology.

It is a canard that teachers are at fault for this. Teachers are more at the mercy of a treacherous industry that is bound by a bottom line that the student’s welfare plays no part in.

Administrators are held hostage to a high-stakes testing strategy that makes no sense, and a sardonically misnamed “No Child Left Behind” act that handcuffs them to standards that are haphazard and unenforceable, and will change as often as the wind. In turn, they hold the teachers responsible to “make it work.”

Ignorant people like to blame teachers and/or their unions. I think these people are under the impression that teachers unions are behemoths that wield impressive political power, and are responsible for everything from the fact that many schools need weapon-detectors at the door to the the 9/11 bombings, to, what the heck, the fall of the Roman Empire.

If you’re a teacher and need protection from asinine school-board decisions or administrative abuse, you’ll soon find out that teacher’s unions have about as much clout as a crepe paper shillelagh club. Teachers are routinely overworked and undervalued, and forced to fill out useless form after form about “student achievement” and IEPs that will be ignored by the people who have to make decisions about how to help that student along in the future, because of “budget restraints”. If schools don’t meet the “academic standard du jour,” the administrators will look to pass off any culpability on the teachers, first. The buck stops “down there, with them.”

The money may go to put air-conditioning in the superintendent’s office, or go to a new football scoreboard more than it will to actually effective teaching supplies. Then the teacher will have to shell out his or her own money to equip the class. Or the PTA will hold a bake sale. How would you like to have to bake cookies to have your employer be able to give you the supplies you need to do your job?

How would that propaganda-pushing caricature of a journalist like to have to educate some kids who come from homes that have never had a real book in them? Or maybe kids who only know abuse at home and cannot behave in public?  Would it be the teacher’s fault to have a class with kids who have an intelligence span wider than that of the cartoonists’ core audience, yet have to have all of them meet a one-size-fits-all standard?

Families move more often than in the years of “Leave it to Beaver.”  A class may have pupils from school districts with entirely different standards than that class. The teacher has to make up for all the discrepancies that he/she had nothing to do with, yet that teacher will be responsible for the grades of children that may have come into the class woefully ill-prepared, because the children may have come from one or the other doofus-districts.

Teachers are hampered by exactly the type of ignoramuses who insist on “standards” that the ignoramuses themselves do not understand, and could not enforce. It’s easy to set up well-meaning (if naive and ineffectual) criteria for other people, and through wishful thinking and some notion about “tradition” expect others to live up to your unreasonable expectations even as you pull the rug out from them and disrespect them.

Scapegoating is the oldest, meanest, and dumbest propaganda tool in the simple-minded person’s playbook. Scapegoating the lowest man on the totem pole is the cheapest shot one can take.

If you really want to find who is responsible, look for the person who’s assigned the closest parking spot to the building (besides the handicapped spot – the disabled are picked on enough) and keep looking up from there. Anyone who’s saying, “… but we’re doing the best we can…” and who’s not speaking out for the rights of students and teachers first, is, in the words of a true cartoonist and journalist who actually stands for something meaningful (Doonesbury), “Guilty! Guilty! Guilty!”

It needs to be pointed out that I am not re-assigning blame to all administrators. There are those who are fighting the good fight. I actually met one recently. It was refreshing and sobering. Those ladies and gentlemen need encouragement, just as the many good and great teachers out there.

If you have an administrator who is in trouble with the school-board for standing up for his or her teachers, support that person. If you have school-board members who are trying to shake up the establishment by standing up for administrators who support teachers, shake their hands. Buy their cookies.

As for the status-quo, unimaginative, bean-counting functionaries who take up office space and never deal with students or parents directly, let’s have the courage to stand up against them. Speak up at meetings, inform yourself as to who is running for school board member, and then go vote. Write letters to the editor. Don’t jump to conclusions.

And as for the propagandists who only put false words into speech bubbles and attribute them to people they attack, and draw unflattering pictures of people they don’t like or understand (just like when they scribbled nasty graffiti and malicious stick figures of the teacher on their school desks when they were children), well, there’s no need to fill more space concerning them.

Here’s to all the teachers who love their mission but are hampered by their jobs. You’re the best! (Along with the school librarians!)

Check out these other sites that have notices the same thing about that mindless strip:

http://duckcover.blogspot.com/

This one is great: http://www.huffingtonpost.com/chris-kelly/mallard-fillmore-makes-me_b_38124.html Here’s a quote from it:

“Mallard Fillmore” is an actual disgrace. Reading it is like watching the loneliest creep at the gun show try to pick up a waitress by quoting George Will, throw up on himself, and cry.

It goes on to point out that the strip repeatedly attacked Ted Kennedy for driving drunk in the Chappaquiddick incident, then shows the mugshot of the cartoonist when he was arrested for driving drunk, himself. If it walks like a hypocrite, and quacks like a hypocrite…

The sum of two numbers is 91. And the difference is 31. What are the two numbers?

The first  answer to it simply  gave the answer as,  “The two numbers are 66 and 25.”

I think the whole thing is a waste of time for all concerned, so far. The asker has learned nothing, and the answerer has taught nothing.

Just giving an answer is “show–and–tell” teaching. It serves no purpose except to show off that you have an answer. It doesn’t teach anything. It’s the old phenomenon of,”Give a man a fish and you make him dependent on you.”

Let’s see if we can make this meaningful with math, by figuring out how we could come up with this answer.

You could try this by guess-and-check, But you wouldn’t really learn a system that would help you solve these kind of things in the future. So let’s use what we know to discover what we don’t.

We know that the two numbers have a difference of 31. So both numbers are unknown at the moment, but we do know that there is a relationship between the two.

So we can call one number x, and describe the other one as its relationship to x. In this case, we know that if one number is x, the other one is x+31. Alternately we could call them x and x -31.

I will randomly choose the first pair:  x and x + 31.

So now we can write it as an equation, using only one variable to express what we don’t know.

x (one number) + x + 31 (plus some other number that is 31 more than the first number) = 91.

That’s

• x + x + 31 = 91.

That’s the same as 2x + 31 = 91.

• Subtract 31 from both sides (because according to the Addition Principle, if two sides of an equation are equal, you can add or subtract equal amounts to each side, and the solution set will remain the same) and you’ll get 2x = 60.

Can you take it from there? I’m sure you can, but for the sake of completion, let’s do it together.

• Divide both sides by 2 (because according to the Multiplication Property of Equality, if two sides of an equation are equal, you can multiply or divide equal both sides by the same non-zero number, and the solution set will remain the same) and you will get x = 30.

• You know the answer to one of the numbers  (x = 30). You also know its relationship to the other number, which is x + 31.  Add 31 to 30, and you will get 61.

Therefore,  the numbers are 30 and 61. Check it.  Do 30 and 61 add up to 91? Yes. Is the difference between 30 and 61 31? It is.

Do we want to leave it at that? I don’t think so. It would be good to check if other numbers could fit in the equation, too, so we know if our answer is unique.

If you make either number higher, you would have to make the other one lower in order for them to still equal 91. But if you make one lower and the other higher, you will increase the difference between them, so the difference could not remain at 31.

In other words there is no way that you could change either one of the addends and still get the sum of 91.

So we have gotten the answer, and proven that it can be the only answer.

Why do we care if it is the only answer? Ah, here is where we try to make this meaningful in real life…

There are many questions, decisions, etc. that we are faced with in life that can have more then one answer. Simple people  (by this I mean “simpletons,” not “regular people”) like simple answers to complex questions. Say, for example, there are many ways to solve an argument. One is by violence. It might work. But logic might work, flattery might work, mutual benefit might work, as well as many other things. But a simpleton would grasp for the first thing he or she could think of, and tell themselves, “I feel threatened by this situation – If I club the other person, it will end the threat.”

Yes, it might. And it might not. And other things might work better. If the simpleton stopped with the first answer that might work, they wouldn’t realize that there are other solutions that are better for everyone, the simpleton included.

If you filter out all the answers that don’t work, you are left with a much clearer choice. Math is a great way to learn effective decision-making strategy.

Thinking through things, and working them out together, and making sure we understand each step of the way together is a lot more productive than having someone hand you the answer, isn’t it? If we teach this way to our children, it encourages them, enables them to explore their own minds, and demonstrates how to use critical thinking to evaluate solutions to problems. It also encourages them to check their answers, and not just blindly accept “whatever works.”

This is what Math Mojo is about – making math meaningful.

Even if you knew all of this before you read this, I hope you got something out of it, even if it was just to reaffirm your healthy appreciation of critical thinking.

All the best to you in your learning/teaching endeavors,

Brian (a.k.a. Professor Homunculus)

Afterthoughts: Can any reader here see how understanding why we don’t divide by 0 in arithmetic also illustrates the lesson from this post? Leave a comment if you do.