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.

Be warned that the video below is a George Carlin video, and is filled with some of “The Seven Dirty Words you can’t Say on TV”

If you are offended by  raw language, please don’t watch it.

Okay, now I’m mad. I’m reading an article in my local paper about how the school board may lower the passing grade.

At the moment the passing grade is 75%. Most schools in this area have already succumbed to the stupidity of lowering the passing grade to 65%. This school is one of the hold–outs.

The bullshit argument is basically that the local schools are handicapped when sending their kids to college because it appears that they have lower grades than kids from other schools. Also that it will help when the local schools are compared to other schools in this state reporting.

I think the latter part is what’s really at work here. It’s about how the administrators look. It’s not anything remotely about students.

Instead of sticking with their guns, and doing what is right, they are like schools across the country, succumbing to the dumbing down of The education most public schools subscribe to.

The article said that the principal said that the change should not be considered a lessening of standards. I don’t know the principal, who might be a fine person otherwise, but that statement is just a load of crap. Lowering the same grade is exactly a lessening of standards. It is practically the definition of a lessening of standards, and any statement to the contrary is just mealymouthed horsefeathers.

The lessons that student get from this is that you should do what’s expedient, and who cares about the consequences of your decisions. Nothing matters, just do what makes you look good – you don’t actually be any good. it’s just another example of how our society values appearances above substance.  And it is a pitiful commentary on us.

So how do they think this is going to play out? They know how it will look to the colleges, they know how it will affect their funding, they know how it will effect their graduation rate, and they know how it will affect how their school’s rating looks compared to other  schools that have dumbed-down standards.

What they don’t know, haven’t mentioned, and obviously don’t care about is how it will effect student’ s learning. Think about it – the vast majority of students exist in the B–C–D  realm. Those slackers  (and I know them well, because I was one of them) who were sliding along at 75%, will now feel free to slide along at 65%. Same grade, less work, and all happily sanctioned by the school. You’re basically giving them a license to learn less. Great freakin’ lesson!

Those students who actually excel will get the message that it’s okay not to do as well in general, because there are no apparent consequences. The consequences are not apparent, because the damned school board members never consider them.

You know, the more I’m exposed to public schools, the more I feel for the parents  who have to send their kids to them. There used to be a lot of things about public schools that you could like. At one time they were even a jewel in society’s crown. But those days have pretty much gone the way of  the stegosaurus, bell-bottom pants, and eight–track tapes.

And one time you could be proud to send your kid to the local public school. Now you probably worry about your kid all day when you send him or her to one. I don’t know the answer to this. But I do know that there is no reason at all to leave your child’s education up to a public school.

It’s a sad fact, but nowadays parents  have less time than ever to spend with their children,  but more reasons than ever to spend time with them. So they have to make each moment count.

It’s a shame that it’s so hard for parents to help with their children’s education, because curriculum and standards get changed more often than partners at a square dance. One of the things Math Mojo would like to help you with is your child’s math education. I try to put out no-nonsense, encouraging material to help you teach your child, and possibly learn something as you do it, so you share in the learning experience, have some fun, and have some meaningful time together. (Imagine that!)

I’m working on a new, nuts-and-bolt e-booklet about how to get great at addition. If you’ve signed up for the Math Mojo Monthly newsletter  (the sign-up  form is at the top left this page) you’ll be notified as soon as it is available  It will be inexpensive, and have a lot of material that you have probably never seen.  It will help you turbocharge your addition skills, and give you insights into speed math and mental math that you would use for the rest of your life.

I wish you all the best with your child’s education, whatever form or shape it takes.

Hotcha!

Brian (a.k.a. Professor Homunculus at MathMojo.com )

Afterthought: If schools would get a little bit creative  with their public-relations, and maybe create a social networking campaign that would extol the legitimate virtues of their schools, maybe institutions would give the individual score a little more weight assessing their student admissions.

If a school legitimately was known for its high standards, it could set the bar for other schools, instead of the other way around.

Of course this is naïve of me. As soon as it was known that a school could get a higher rating because of its  higher standards, others schools would start faking it, and rely on public relations more than actual facts. Same old same old – it’s about appearances, and not substance. What have we become?

If you’ve read much of Math Mojo, you’ll know I am an advocate for practicing basic skills until you know them “in your bones,” or “cold,” or “until you can do them in your sleep.”

While listening to to the story “The Writer Who Couldn’t Read” by Robert Krulwich  on the radio on NPR’s Morning Edition this morning, I realized that they were talking about that very thing.

The story is about, how:

“In January of 2002,” writes the neuroscientist Oliver Sacks, “I received a letter from Howard Engel, a Canadian novelist describing a strange problem.”

Howard is an author of detective novels.

One morning he work up and got the morning paper, the Toronto Globe and Mail, an English-language journal — but he found that it was written in Serbo-Croatian or Korean, or some other language he didn’t recognize.

Now he has a problem. Howard has lost the ability to read.

Howard lost a part of his brain from a stroke in the night. He now suffers what is called word blindness.

He thought he was done as a writer. But what he discovered was something that I talk about often in Math Mojo – knowing something in your bones. It turns out that Howard can read what he writes with his own hand. His mind recognizes the words from his muscle memory, or “motor memory.”

I remember from my own childhood times when I couldn’t remember, say, 7 x 8, but if I traced the numbers in the air with my fingers, or even in my mind, I could come up with “56.” Now I understand that it was because I had written 7 x 8 so often, as I did written exercises, that it had gotten “into my bones” – into my motor memory.

I knew that I wasn’t learning the “tables” as well as some of the other kids in the class by doing “worksheets.” I also found them so boring and passive that it caused me to resent them. So on my own I just wrote the charts and tables myself, over and over.

Does this mean that I take back my constant complaining that we use too many charts and mind-deadening worksheets when we teach multiplication skills or other basic math skills? Not at all. Both of those things have most of the written material given to you, and you just either look at them, or simply write the answer.

You may write “56″ over and over again, but your muscle memory won’t necessarily associate that with 7 x 8. That’s what you have to write the problem and the answer over and over. Of course not everyone has to do this. Some people learn by staring at charts. But I’ll bet every single person would learn better if they used some “muscle.”

And I’m sure there is a very high percentage of the population that doesn’t learn well at all from staring at charts of seemingly random material. I’m not saying that you shouldn’t use charts at all. Sometimes it’s nice to supplement real learning with stuff like that. But it should not be the main source. I think dependency on passive learning is a pathology of our modern society, and is a lot more counterproductive than most people realize.

If you Google multiplication you will find thousands upon thousands of “free worksheets.” There is a reason that they are less than a dime a dozen. It’s because they are worth less than a dime a dozen. Sometimes I believe they should pay you to use them.

Howard says, “I also started writing the words with my tongue on the roof of my mouth…” This is a good, creative way to practice. It also shows thought and creativity. Thinking like this about writing, math, or anything else gives more meaning to what you are doing. It creates more neural pathways (See the post about How Puzzles may Improve your Mind.

A great way to practice is with playing cards. You are active with them in both the visual and kinesthetic sense. I have had a lot of success teaching people who have otherwise had very limited success was learning math skills, by having them use playing cards. Not only is your brain more active when you use them, but they are more interesting to use as well. Face it, playing cards are fun. (Maybe that is why they don’t usually use them in schools.)

P.S. you can listen to the audio of Howard’s story for free here.

P.P.S. You can find dozens of videos of how to use playing cards to practice multiplication skills at in the “Numbers Juggling–Times without the Tables” “ course, at learn2multiply.com .

It was sometime before noon, when my wife and I heard a boink, boink somewhere in the house. What the…?

A short investigation revealed that a female cardinal and her mate had perched upon a branch just beyond our second floor porch *. We think they may have had a baby or babies with them.

The female appeared to be attacking her reflection in one of the large windows on the south side of our house. Every few seconds she would furiously fly into the window with a thud. This went on for quite some time.

Mimi, my wife, told me that ,down the hill at her sister’s house, they had a cardinal that did that for a long time and they couldn’t figure out a way to stop it.

After a while we decided to take action. I got a long stick and tried to shoo the bird from the branch. To no avail. Every time I approached the tree from the porch the bird would see me and fly away. As soon as I would leave, the bird would come back.

Then I went and got a bucket of water. I thought I might douse the bird. The bird was having none of that, though. Just as before, as soon as I’d show up with the bucket, the bird would hightail it, and as soon as I left, she’d return.

Mimi suggested we hang something from a branch that would make noise when the cardinal landed. It sounded like a good idea to me, so we went and got a set of measuring spoons and tied them to the end of a string. Then, from the porch, I tried to throw them over the branch that the bird had perched on. After several attempts, it seemed like it worked. So Mimi went back to her reading, and I went back to my writing.

Minutes later, the little birdbrain was back at it. By this time she had flown into the window at least 100 times in the last half hour or so. No wonder they call them birdbrains. What is it in her little noggin that kept her at this unproductive activity?

We then tried another tactic. We have a broom with a telescopic handle that extends to about 20 feet. We stretched it to reach meanfrom the inside up to the top window, which the bird had been attacking. We thought that maybe the sight of something on the other side of the window would scare the bird.

I must admit that by now I had started feeling like Wiley Coyote versus the Road Runner.

Have you ever heard the expression, “The definition of insanity is doing the same thing over and over again and expecting different results?” At first I was wondering why this bird would keep repeating her actions and expecting different results. Now I was wondering the same thing about me!

It’s been a week since I first started this post. Who do you think has won? Yes, that little feathered perpetual motion machine is still at it. Furthermore, she must have phoned her family and told them about her success, because as of today we’ve had another cardinal smashing herself into one of the windows on the western side of our house. I don’t know what they get out of this besides blunted beaks. But my wife and I have learned acceptance and patience with our fluttering friends. At least we had for a while, until the groundhog showed up in our bedroom. But more about that in the next post.

* (This sentence just reminded me of a line from Edgar Allan Poe’s “The Raven”… “perched upon a bust of Pallas, just above my chamber door…”)

Regarding the the expression, “The definition of insanity is doing the same thing over and over again and expecting different results” …

I’ve seen it attributed to Albert Einstein, but I don’t believe that one bit. It seems like too much of a trite truism to be from Einstein. Whenever someone considers something slightly pithy, they tend to want to attribute it to either Einstein, Mark Twain, Churchill, or George Carlin.

The quote has some merit, but not the depth that is generally ascribed to it. It may concern an attribute of insanity, but it certainly isn’t the definition.

Definitions are funny things. So often, people are content to settle for generalizations that have some truth to them as a full definition. If you are following the idiotic rhetoric from some American “news personalities,” then you’ve had lots of chances to see how minor, occasional attributes can be twisted into seeming like full-blown definitions.

People like simple solutions. And the simpler the person, the simpler they like their solutions, even if it concerns complex questions.

Somehow, in benighted corners of society, “simple” has become a badge of honor. “I’m just a simple man, and I just want simple answers…”

Yeah, and I want a flying pony.

Simple answers for complex questions are for simpletons, or “simps.” One thing that Einstein did say, is something like, “Everything should be made as simple as possible, but no simpler.”

It can scarcely be denied that the supreme goal of all theory is to make the irreducible basic elements as simple and as few as possible without having to surrender the adequate representation of a single datum of experience.

wikiquote.org: “On the Method of Theoretical Physics” The Herbert Spencer Lecture, delivered at Oxford (10 June 1933); also published in Philosophy of Science, Vol. 1, No. 2 (April 1934), pp. 163-169. [thanks to Dr. Techie @ www.wordorigins.org and JSTOR]

(That quotation is very similar to Occam’s Razor. You can read more about Occam’s Razor at a previous Math Mojo Chronicles post.)

Mathematical Definitions – Why are they so elusive?

Usually, as you learn math in school, you are given certain definitions for concepts. For example, you are told what a digit is. You are told what a whole number is, what a fraction is, etc. Then you are told what addition is, what subtraction is, etc.

From there, you are taught how to manipulate numbers, based on those concepts.

But are those definitions correct, complete, useful, or perhaps even misleading and harmful? Many problems people have with math stem from the poor explanations they were given. This leads to frustration, and often to the wrong conclusion that, “Oh, I’m just not good at math.”

Maybe it’s not that the person is not good at math – maybe it’s that what they were taught was not good math. Had they been properly introduced to math in a better way, they would surely have had less problems with it, and maybe even come to appreciate the joy and romance of it.

Let me give you an example. Often, we are taught that “multiplication is just repeated addition.”

Oh, really? How does that work for fractions? It doesn’t. The definition of multiplication as repeated addition is what drove you crazy when you first learned to multiply fractions.

There are plenty of definitions in math that aren’t definitions at all, just descriptions of some attributes. How about this one, “A fraction is a number that can be put into the form of m/n.”

That is a description of an attribute. It is not incorrect, it’s just not a good definition. It’s sometimes useful, and sometimes not.

Calling things “definitions” is one of the problems in math education. We are a society that is obsessed with easy answers for complex questions. Ah, if only life worked that way! But of course, it doesn’t. But we like to pretend it does, so we give expedient and incomplete definitions, to get the matter out of the way.

The problem is that later, it comes back to bite us in the butt. A child is faced with multiplying fractions, and is confused that you cannot just repeatedly add to get the answer. Now he’s mad that you told him one thing, then, but it doesn’t work, now. You pulled the rug out from under him. That’s probably what makes math so frustrating for most people.

Or in another case, we may teach children that the “natural numbers” are “1,2,3…” Then some day they read that the natural numbers are “0,1,2,3…” (This time including zero). What’s up with that?

Both are correct. The former is used for number theory, and the latter is used for set theory and other branches of math. The former is probably what you want to be using when you are talking about arithmetic.

Do we even hint that there are distinctions? Of course not. Most teachers don’t even know that there are distinctions. No matter – the kids are to be tested on it anyway, and if they don’t give the answer we want, why, then we fail the little suckers.

Maybe you, like I, remember sitting in math classes throughout your school years, listening to teachers drone on, and wondering “What the hell are they talking about?” Among the many reasons that many children have this experience, is because we expect them to accept definitions that only work in certain cases, as black-and-white all-encompassing final definitions. Then we reinforce this by testing them on it, and hammering in the notion that simple explanations are sufficient to explore complex subjects.

What’s the solution? At some point we, as a society, have to mature to the point that we understand that not everything is simple. There is always more to be learned. Take everything with a grain of salt. It’s best to teach that we cannot pin everything down at the level of the learner. There are things about math that you have to try now, and try later, when you have more experience with the subject, and that later you will grasp it better.

We also have to stop the obsessive testing. Things take time to sink in. I dare you to define the word “number” right now, or even “math.” Of course your answer will be more or less incomplete. So if you and I can’t define these things adequately, how fair is it to be testing children on concepts we cannot adequately define?

It turns out that what we are testing for is not what is important. Even the way we are testing makes it harder to teach and learn. We’re doing exactly the wrong things, and we’re doing them systematically.

Don’t take my word for it…

One of the greatest mathematician/philosophers of all time, Bertrand Russell, in Introduction to Mathematical Philosophy (1919) concerning definitions of basic concepts, like “number”, wrote on pages 3-5,

But though familiar, they are not understood. Very few people are prepared with a definition of what is meant by ” number,” or ” 0,” or “1.” It is not very difficult to see that, starting from 0, any other of the natural numbers can be reached by repeated additions of 1, but we shall have to define what we mean by “adding 1,” and what we mean by “repeated.” These questions are by no means easy. It was believed until recently that some, at least, of these first notions of arithmetic must be accepted as too simple and primitive to be defined. Since all terms that are defined are defined by means of other terms, it is clear that human knowledge must always be content to accept some terms as intelligible without definition, in order to have a starting-point for its definitions.

It is not clear that there must be terms which are incapable of definition: it is possible that, however far back we go in defining, we always might go further still. On the other hand, it is also possible that, when analysis has been pushed far enough, we can reach terms that really are simple, and therefore logically incapable of the sort of definition that consists in analysing. This is a question which it is not necessary for us to decide; for our purposes it is sufficient to observe that, since human powers are finite, the definitions known to us must always begin somewhere, with terms undefined for the moment, though perhaps not permanently.

In Introduction to Mathematical Thinking, by Friedrich Waismann, 1951 (Originally Einfuehrung in das mathematische Denken ) Karl Menger wrote in the foreword on page VII, concerning a particular definition:

In regard to the definition of numerical equivalence, which goes back to Cantor and Frege, one will willingly grant that it represents only one of many possible attempts at precise formulation of the vague and ambiguous use of this word in colloquial language-a one-sidedness that, to be sure, it shares with every attempt at precise definition of any part of colloquial speech. Furthermore, one will perhaps agree that the criteria given by this definition of numerical equivalence for its application to experience, are none the more precise if compared with those of the usual definitions of fundamental physical concepts, such as equality of lengths, simultaneity, etc. Neither does it seem to me, however, that they are less precise. Indeed, the great mathematical significance of this definition rests preponderantly on its fruitfulness, i.e., on the fact that so many conclusions can be deducted from it. However, just because this definition has proved to be so very fruitful that no other definition up to now has been able to compete, it certainly is of value to point out other similar possibilities (especially to prevent the erroneous view that it is the only conceivable definition).
(Emphasis mine)

Paul Lockhart, in his absolutely must-read, A Mathematician’s Lament, puts it like this,

The point is you don’t start with definitions, you start with problems. Nobody ever had an idea of a number being “irrational” until Pythagoras attempted to measure the diagonal of a square and discovered that it could not be represented as a fraction. Definitions make sense when a point is reached in your argument which makes the distinction necessary. To make definitions without motivation is more likely to cause confusion.

So, in essence, you have to start somewhere, but you should not get locked into the Idea that your working definition is complete, nor the only possible definition. There is always more to learn and refine. Keeping an open mind is essential to understanding.

By the way, I can see no reason why these three resources should not be read and digested by everyone who teaches mathematics, especially elementary school teachers. They contain the seeds of the answers to almost all the questions that curious students will ask when taught about numbers and math, as does Karl Menger’s The Basic Concepts of Mathematics (1957) (Out of print – Okay, I can see why this one might not be read by all teachers).

All too often teachers are expected to teach “a mile wide and an inch deep.” Fortunately, many pupils’ curiousity extends a lot deeper than that. It’s books like these, which delve deep into the “why” of math, that help teachers, students and the general public get a grip on what is a lot more important than simply teaching the mechanics of passing less-than-meaningful tests.

She said she’d send her friend and her friend’s daughter to my site. Then she called wanted to know if she could give them my number for them to get in touch with me.

They never did.

The next week I spoke my acquaintance again, and she said, “Wait, my colleague is in the next room. I’d like to tell her you’re here.” I told her, “OK, but I don’t really tutor. There are a lot of good resources at my website for a ninth grader, though.”

She came back a few minutes later and said that they couldn’t use Math Mojo because her friend’s daughter’s math teacher said, “She has to do it her teacher’s way.”

What a$%#ing &tard! That teacher should be cited for criminal education neglect. Enforcing the benighted notion that math has to be seen from only one angle is the reason most kids don’t get math in the first place.

I explained the Parallax to my acquaintance, and she completely agreed. I also talked about the reason that we have two eyes and not one is for depth perception and a parallax view. If you only use one eye you are handicapped.

In a nutshell, a parallax is the use of more than one point of view to get an overview of something. That, of course, is not a complete, or entirely accurate explanation, if you want a more complete scoop, check out parallax on Wikipedia.

Only having one method to accomplish anything handicaps you. Having a second method does not degrade the first. It enhances it. It makes each part greater, and it makes the whole greater than the sum of the parts. (No, that is not a logical contradiction. Ever hear of nuclear fusion?)

Unfortunately, the members of the school system who should know this most (math teachers, who should be versed in basic logic) often don’t, and are the greatest enemies to the mathematical reasoning skills of their students.

Why do people who should know better insist that everything must be a zero-sum game, and that their way must be defended at all costs, even though everyone suffers in the long run when they only use that one way?

And of all the logical farts in that teacher’s argument – if her way was so damned good, why does the kid need tutoring? If that way didn’t work for her before, why does she assume more of it, and nothing else, is going to be what helps her most?

Look, I know I should be producing more “nuts and bolts” lessons for people to use. To tell you the truth, the more experience I have with people, the more I start thinking, “What’s the use?” I know that is wrong, and I’m trying to fight it. OK, it’s not wrong – it’s absolutely right. But it’s not helpful. I’m trying to reconcile the two. Anyone got any suggestions before I give the whole “trust your brain” thing up, and become a televangelist or a politician?

By the way, “The Parallax View” was a great Warren Beatty movies from the 70′s. Check it out.

An insightful reader wrote in:

Susan Grigor wrote:

Good morning, Brian,

I want to consult your wife as one who works with little people.

I have been looking at the Grade 3 and Grade 4 curricula. There is a
lot about looking for alternate strategies such as 26 + 35 = 30 – 4 +
30 + 5 = 61. Note, however, that positive and negative numbers are
not taught until Grade 7.

Kids even up to Grade 7 are literal/concrete thinkers, not abstract
thinkers. How well do they use these abstract ideas? They are
mental gymnastics.

Susan,

I don’t see anything in Math Mojo that is less concrete than the pap they feed kids in “standard curriculums.”

Math Mojo doesn’t use the 26 + 35 = 30 – 4 + 30 + 5 = 61 kind of strategy. I use what I call the See-Say-Write strategy. It is subtly different, but it does not use any subtraction. It is also extensible. I use it for adding gigantic columns and rows in my head, easier than most people can use a calculator. I also use it for advanced multiplication. (More on that in a forthcoming course.)

The only problems I’ve ever had teaching second-graders or above any Math Mojo stuff has been because of what you mention in the next paragraph – some of them apparently “like” to do it the tired, old way.

I see that as one of the most important things I can do in this world – break them of that miserable, school-learned, brain-deadening habit.Naturally, kids like to learn. They like to learn cool and new things. But when you get “partial credit” for showing work that doesn’t need to be done, and you are bent-over and forced through artificial hoops long enough, it just beats the soul out of you.

Here’s my dirty secret – Math Mojo isn’t about any math techniques. It’s about re-humanizing the learning experience. My goal is to get people to realize, “Hey, that’s amazing! I can really learn meaningful stuff if I want to! And it’s always more fun and rewarding than that stuff the drones do!”

Susan continued -

My experience is that kids in elementary school (and high school) do
a lot of math by ritual and rote, rather than real understanding–
and they like it that way. They want to be told the right way to do
things, and then to practise until they get good at it. Only one to
three per class are interested in mental mathematics and alternate
views.

I am working on presenting these ideas. But I wonder about those
others for whom this is not interesting, but frightening, those who
want numbers to be sure and solid, not slippery and subject to
interpretation like story- and essay-writing.

Okay, I’ll admit that some kids get scared of this in the beginning. But those kids are so severely damaged that they need to learn this stuff. It’s so important that we teach kids that math really is a free-ranging, adventurous, imagination-filled world. If they don’t learn to appreciate that in the early grades, they will be the kind of people who grow up to say “I’m just not a math person.” That is like saying, “I’m just not a reading person.”

Kids that insist their numbers be “non-slippery” are kids who are going to have a very, very difficult time with irrational numbers someday. They are also the kinds of kids who get the heebie-jeebies when they are faced with operations with negative numbers.

All of that stress can be avoided by not reinforcing the bad pedagogy that standard curriculums present

So one of the ways I present cool stuff is not to tell them that they are learning the same stuff a new, alternative, better way (which of course it is), but instead to present this stuff to them as an extra bit of “math magic.” Not a trick, but real magic.

For example, when teaching multiplication by 5, I write a long number, with all even digits, on the board, like 68,462.

Then I ask one of the kids who may not be the swiftest in the group to come up for a magic experiment. I them simply ask him or her what half of 6 is, and to write that below the six. Then what’s half of 8, and to write that below the 8, etc. down to the final 2, and then imagine there’s a zero at the end, and to write half of that (which will be zero, of course) below that.

Then I tell the class that little Spatula (or whatever the kid’s name is) has done an amazing magic trick. It’s one that even David Blaine probably can’t do (true). Then I have the kid sit down, and ask for applause.

Everyone thinks I’m nuts.

Then I ask the best math student in the class to come up and be part of an experiment. (It’s still the same “trick”, but I don’t tell them that).

Then I ask the “smart” kid to multiply the original number on the board by 5. Normally the little genius re-writes the number, writes a 5 below it, writes an “x” for multiplication, draws a line, then goes through the ritual of doing times-tables in his or her head, complete with writing the carries, until finally little Pippin (or whatever the kid’s name is) arrives at the very same number that Spatula did, without carries, re-writing or other machinations. (And Spatula even did it from left to right!)

I do so enjoy the little tykes’ “Oh, my God!“ses when they realize what just happened.

They have seen, and convinced themselves, that this new thing “rocks.”

Being a magician, I’ve spent most of my life helping people reach a “suspension of disbelief.” This is one great way to do that with math.

In general, I think this is the way to go. It makes you appear to be less of a “teacher.” It also avoids trivializing magic as a “trick.” The actual magic happens when the kid goes “Oh, my God!” – when the light bulb goes off. It’s that light bulb, not the method, that’s important. As a true teacher you know that already. As a magician, I’m just giving you another way to light that sucker up.

It is questioning like this that interfered with my career, you know

Yeah, I know. That’s why I can’t really even consider working for anyone but myself. But it speaks wondrous volumes towards your human-ness.

By the way, the little people my wife works with are usually more at the stage of being able to tell their nose from their arm, or simply being able to count, than doing simple arithmetic. I’d love to help her with this kind of stuff, but there’s not much call for it with the kids she works with. Her magic with them is waaaay beyond any help I could offer her, anyway.

I think teachers in general (the good ones) are magicians in a sense I never could be – you deal with administrations. That is some heavy mojo (voodoo?)

All the best,

Brian

A commenter wrote in what I consider a classic example of misrepresenting what he or she is commenting on. The argument is so full of holes that are all too often repeated in schools and other institutions, that I thought I’d better stand up for students everywhere in order to protect them from the kind of rhetoric that is so often used to make them  obey “rules” that either aren’t rules, or if they are, are not adequately  explained at to  why they are necessary.

The commenter wrote in:

The use of the word “and” to indicate the decimal point is an agreed-upon “rule” in math. Math has many of these “rules,” which try to clarify mathematics. This is a standard rule taught in most elementary schools. PLEASE don’t confuse the students by saying they can do as they please. Then can 2 + 3 x 5 be equal to both 25 and 17? NOT. The order of operations “rule” says multiply first, so 2 + 3 x 5 = 17.

Professor Homunculus sez:

I’m afraid you’re going to have to show me where that “rule” is. I think what you are calling “an agreed-upon “‘rule’ in math” is just a convention made up for the convenience of elementary schools and is not a mathematical law.

Click here to view the embedded video.

Great logic, Dick!

Regardless of how much you want to believe you are right, your argument is full of logical holes. It is a good lesson to students about one of the things that is so wrong about some schools and teachers, though. Using bad logic to bully students into accepting anything just because you say it’s “agreed-upon” is a crime against education. Unfortunately it is the  ”agreed-upon” crime .

Here are a few things readers can keep in mind when making up their minds concerning what is valid and what isn’t. When someone uses the words “agreed upon” in an argument, you should ask yourself the question, “agreed upon by whom?” The above argument does not address that. That fact doesn’t negate the above argument, but it certainly weakens it.

Also, keep in mind that no matter how strongly someone feels about an argument, it does not make their argument any more or less valid. What does make it less valid, though, is when someone unfairly and untruly twists the other side’s argument to try to make it sound wrong. This is exactly what the commenter here has done. His or her insistence that I , “PLEASE don’t confuse the students by saying they can do as they please,” is a disingenuous ruse. Go back and read the article if you want. You won’t find that I said “students can do as they please.” I didn’t even imply it. I even say the “and” rule is a good Idea, but it’s just a convention, and not the final law. It’s a good Idea not to fart in class, but there’s no law against it (no matter how strongly we both may feel about that).

Another disingenuous tactic is, “Then can 2 + 3 x 5 be equal to both 25 and 17? NOT. The order of operations “rule” says multiply first, so 2 + 3 x 5 = 17.”

To put it bluntly, that is a crap argument. I don’t say or imply that you ignore mathematical laws. I LOVE mathematical laws. The order of operations is necessary, doesn’t contradict anything else. It is a convention that grew into a rule. It’s not an axiom. It’s not even an actual “law”. But it is a valid rule. So, no, I don’t say or imply that 2 + 3 x 5 can be equal to both 25 and 17, and it would be lying imply that I am.

To put a fine point on it, that ruse is a very pernicious one, and is often used by people to promote ideas that are just idiotic. “Oh, so you think it’s good to help poor people? So you’re saying you’re a communist?” See how dumb that would sound? Of course the commenter is not the same point, but he or she is using the same form of argument. It’s not valid in either case, and to visit this kind of BS argument onto children, and say they should abide by it, is an injustice.

To be fair, maybe the commenter isn’t intentionally twisting the argument and using logical fallacies just to prove a point that he/she believes in. It may be ignorance, not intent. Either way, regardless of how strongly one feels about a subject, they don’t get to make the rules for you to have to obey, unless they make sense, are necessary, and don’t contradict existing laws.

Unless the commenter can prove that it is a mathematical law, you can dismiss his/her argument. It doesn’t mean it’s wrong – there might actually be a proof – it just means his/her argument is meaningless without the proof.

It’s hard to believe that a person in the twenty-first century can still come out with an argument like, “This is a standard rule taught in most elementary schools.” That is an argument? It’s still standardly taught that “multiplication is repeated addition.” That doesn’t make that right. (It isn’t. See this article by Keith Devlin, and this article by Keith Devlin)  Does any self-respecting human think that something is right just because it’s a standard rule taught in most elementary schools? For pete’s sake, segregation was a standard rule in most elementary schools up until the point when someone figured out how bogus it was. Sheesh.

 I recently read an article about Srinivasa Ramanujan. He was one of the greatest mathematicians who ever lived. He died in 1920, aged 32.  Mathematicians still study his notebooks today, trying to understand the brilliant insights he came up with. According to the chapter “The Formula Man,” in Ivars Peterson’s wonderful book, A Mathematical Mystery Cruise, “Ramanujan’s work reveals a genius for finding numerical patterns, hidden laws, and relationships in the wilderness of numbers. No one really knows what led him to his astonishing array of mathematical discoveries or how he proved his results. He spent most of his life far from centers of mathematica activity, and working in his own way, drawing formulas and theorems from mental landscape unconnected with the frontiers of contemporary mathematics.”

Until Ramanujan was 17, he was mostly self-educated. He had no teachers to tell him he was wrong for thinking the way he did. Can you imagine if he had a teacher who was fixated on an imaginary law like “and?” Yikes!

I’m not saying that everyone is a Ramanujan. What I am saying that O.D.-ing on what people tell you “are the rules” is a sure-fire way to make sure you will never be a Ramanujan.

I think this whole issue is a clash between mathematics and school rules. You are not doing mathematics unless you are thinking. You need to understand why certain things are accepted or not, and it is part of the duty of those who “make the rules” to prove that the rules make sense, are necessary, and don’t contradict existing laws.

The biggest reason for people “hating school” is that the rule makers tend to be ill-advised school boards, the administrators who have to kneel before them, some uninformed teachers, and text-book lobbyists. Real teachers are usually too busy trying to teach, or buying materials that the schools won’t supply, or trying to fill out lots of unnecessary paperwork to be making the rules. And real mathematicians are seldom understood or consulted by elementary schools. (Does your school have a mathematician – and I don’t mean a math teacher , or math-ed consultant – on it’s staff?)

Mathematics is an invitation to think. Elementary math education is more and more becoming an order to obey. There are exceptions, but I think one reason that a lot of people seek help from Math Mojo and other alternative sources is because the exceptions are too few and far between.

What do you think? Please leave a comment.