I have a book sitting in front of me called, Introduction to Mathematical Thinking by Friedrich Waismann; the foreword is by Karl Menger, both of whom I admire greatly.

This book has opened my eyes to something very important about math education. [...]]]> Can I choke him now?

Photo by foxphotograpy (Edited by Brian)

I have a book sitting in front of me called, Introduction to Mathematical Thinking by Friedrich Waismann; the foreword is by Karl Menger, both of whom I admire greatly.

This book has opened my eyes to something very important about math education. And it’s not because the book is so good (which it is). It’s because after about my fifth attempt at getting through the book I finally realize what it has been that’s impeding my progress.

The impediment is the same impediment that has kept me from learning math and many other things that I have considered beautiful and important, but difficult in my life.

While reading the first chapter of the book again, it finally hit me. There is some really sloppy explaining at a very basic level that, if you are the kind of person that takes things seriously and wants to really understand the deeper meaning, throws a tremendous roadblock into one’s understanding.

This problem is so pervasive in the way so many things are explained in school, at work, or in the real world, that I’m sure you’ve come up against it time and time again. But nobody really calls anyone on it. Or at least not often enough. We let this problem slide by again and again, that we hardly notice it, yet it had a detrimental effect on society, probably since the first caveman tried to explain to his neighbor how to hunt the mastodon (if that’s what they hunted).

The problem is this:

The writers in question just do not have the consideration of looking at their explanation through the eyes of someone who does not understand things the same way they understand things (or doesn’t understand them at all). They’ll use a term, and they won’t explain it; they just assume you know what it means, maybe because, “everyone knows what that means.”

(Aside: Any time you hear someone say, “Everyone knows what that means,” not only is that a lie, but it pretty much shows that the person who says it hasn’t thought about it deeply enough.)

Part of this problem also consists of giving confusing (if not downright conflicting) explanations. Or giving an explanation, and then giving an example that contradicts or twists the explanation.

Have you ever opened up an instruction booklet for something you have bought, which teaches you how to assemble the product? You know that ubiquitous phenomenon in which they say, “attach widget B to the freeble?” And then they never tell you what the goddamned freeble is?!

Or when they say, “Attach widget b to the left side of the freeble,” and then they show you an illustration of widget B being attached to the right side of the freeble?

Doesn’t that make you want to just choke the moron who wrote that? Don’t you wish every idiot who wrote instructions had to put his name and telephone number on what he wrote so you could call him at 2 a.m. on Christmas morning when you were still trying to assemble the piece of crap from China that he didn’t care enough to explain well?

I do.

So back to Friedrich Waismann; no of course he’s not a moron. His Introduction to Mathematical Thinking is a classic math text, and deservedly so. But the hacks who wrote the blurb on the back cover that, “This book, presupposes no specific training in mathematics,…” need to be taken to the courtyard and caned. That wasn’t even true in 1951, when the book was first published and when some of the general population actually knew some math.

Unfortunately in the very first sentence on the very first page of the very first chapter of this otherwise fine book, comes a typical and unforgivably vague explanation and illustration concerning natural numbers.

In that sentence he says, “The numbers presented to us at the first stage of development development are then natural or cardinal numbers 1, 2, 3, 4…” In the same paragraph he goes on to talk about how numbers can be represented on the number line. At the end of the paragraph he says, “the numbers 0, 1, 2, 3… are assigned to the points thereby generated…” This is followed by an illustration of the number line from 0 through 5. He says these points are the images of the numbers.

Which numbers? One would assume he meant the numbers he’d been talking about the entire time; the only ones he had mentioned; the natural numbers. In the very beginning of the next paragraph the first sentence says, “what properties belong to the system of natural numbers?” And proceeds to tell about them, so he is obviously still only talking about the natural numbers. But he is showing an illustration of numbers from zero onward. His definition was the numbers from one onward. So which is it, wise guy?

Can I choke him now?

Am I being overly sensitive? Would you be? If someone who is considered one of the greatest experts on the subject cannot even make something clear to a beginner on the first page of his book which presupposes, “no special training in mathematics…,” how are we supposed to understand the rest of the book?

This is not an isolated case. This case doesn’t even end here. On that same first page when he talks about the system of natural numbers and their properties, he enumerates them. In point two it he talks about the concept of “betweenness” as it applies to the natural numbers. In  point three he mentions there is only one exception — “The number zero does not have a predecessor.”

Excuse me?! You originally told me that the natural numbers were from one onward. Now you really must be messing with me. Where does the freakin’ number zero enter into it? Do the natural numbers contain zero or not? There is no way a thinking person can discern that from the text. (What is a natural number will be the subject of the next post.)

No wonder so many people, myself included, grew up thinking that math class was just one big “trick question.” It was like a secret that they would test you on but they wouldn’t tell you what it was. I think this phenomenon is exactly why most of us thought that.

Can I at least choke him now?

Now I don’t want to make anybody feel that this book is not worthwhile. I’m assuming it is, although I have not gotten through all of it yet, and don’t imagine that I will in the very near future: but I am not giving up. I just resent the fact that the entire math education industry is built upon the premise that the content is so goddamned important that the people who need to learn it take second place to it.

In the last half-century or so, since the “math wars” have ravaged the educational countryside, not only has the content been more important than the people it is taught to, but now the method of teaching has become paramount. It’s so important for these pedagogical wonders to tout their méthode du jour, that not being able to see the forest for the trees is a much too gentle metaphor for their arrogant and ignorant academic-mania.

Of course not every pedagogue is an orc from the dominion of “Constructivism,” or, “No Child Left Behind.” But any teacher worth his or her salary should be very skeptical of what comes from above, and very compassionate and thoughtful towards what and whom they impart to below. That trait alone is more important than any piece of content that anyone could inculcate on the young.

Waismann is a far better expositor than the droogs that write for the “Cosmo-Demonic Scholatic Textbook Factory,” et. al. The whole industry is rife with their half-information. That’s what makes it an industry. No (sane) person could read all of them, but anyone can read enough of them to notice vaguery after vaguery. How else could an entire industry not have noticed Waismann’s error for over half a century?

There is really no hope to change the industry. It would be nice to eliminate it. There has got to be a better way to educate people than to hand that mission over to corporations and administrations.

Part of the solution is to read individual authors who write for popular consumption. I’m not talking about the kinds of new-age junk that establishes “laws” that are just some wishful-thinking pseudo-science. I’m talking about people who understand the subject well, but are brilliant at explaining.

Some of the greatest authors of mathematical explanations were neither full-time math teachers nor mathematicians. But they were full-time humans, like Martin Gardner and Isaac Asimov. Some of the great authors still writing are not just mathematicians, but philosophers and linguists.

I could be wrong, but when I read the authors that I recommended above and will recommend below, I get the feeling that they are writing because they care about their readership. They are writing to you and me. They have a mission, and the mission goes beyond “show and tell just enough to get the little tykes through the standardized hoops.”

If you want to understand the “guts” of mathematics, some of the authors you can go to are (in no particular order):

• Keith Devlin – A great place to start with him is his monthly column, “Devlin’s Angle” .l




• Jerry P. King – The Art of Mathematics




• Rosza Peter – Playing with Infinity




• Lancelot Hogben – Mathematics for the Million




• Isaac Asimov - The Realm of Numbers (out of print but you can dig up copies if you hunt hard enough)




• Bertrand Russell – Introduction to Mathematical Philosophy




• Georges Ifrah – From One to Zero




• Kaplan – The Art of the Infinite




• Apostolos Doxiadis and Christos H. Papadimitriou – Logicomix




• G.H. Hardy – A Course of Pure Mathematics




• Karl Menger – The Basic Concepts of Mathematics (This one is hard to find, but has some great information about why we need to clear up our terminology in math.)

Not all of those books will fit any one person’s style. Please don’t just read them on this list and go order them from amazon.com until you check them out at a library or bookstore first; they may not be for you at all. But they are a good starting point, especially the Asimov book, if you can get a copy

“The mathematician’s patterns, like the painter’s or the poet’s, must be beautiful; the ideas, like the colours or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics.” -G.H. Hardy

continued on the next post – More on Math Terminology Mis-Explained.

Here’s a short except:

what mathematics really is: “It was a wonderful education… Gelfand amazed me by talking of [...]]]> There was a wonderful article in the Wall Street Journal today about mathematics in the former Soviet Union. It is worth reading for anyone interested in finding out a little about the inner beauty of math.

Here’s a short except:

what mathematics really is: “It was a wonderful education… Gelfand amazed me by talking of mathematics as though it were poetry.”

In the mathematical counterculture, math “was almost a hobby,” recalls Sergei Gelfand. “So you could spend your time doing things that would not be useful to anyone for the nearest decade.” Mathematicians called it “math for math’s sake.”

How cool would that be? Can you imagine something like that in an American school? I can’t.

So check out Mathematics in the Soviet Union, at the Wall Street Journal’s website.

There was an interesting post on the Whallah! blog about an article in the Associated Press, concerning the education of math teachers in public schools.

Apparently the National Council on Teacher Quality has done a comprehensive study to come to the conclusion that everyone who is not an “expert” has known for years: [...]]]> Hey, you droogs,

There was an interesting post on the Whallah! blog about an article in the Associated Press, concerning the education of math teachers in public schools.

Apparently the National Council on Teacher Quality has done a comprehensive study to come to the conclusion that everyone who is not an “expert” has known for years: Teachers are not being taught math adequately, and generally fail to teach it well to their students. (Do tell…)

Isn’t it funny that the “establishment” will never admit that? It takes an expensive academic “study” to show what is already known, yet Universities (in general) will not do anything about the way they teach teacher how to teach math. They will try some new, expensive methods that some textbook company has lobbied for, of course. But they won’t try anything that might actually work.

That’s why homeschooling and afterschooling are becoming more and more important. Taking an interest in your own child’s education is more important than ever, as public schools tank in their ability to actually teach, thanks to the natural entropy of society, and the idiotically simple-minded ways some people like to deal with it, as with the subtly(?) sardonically named “No Child Left Behind” act.

According to the AP article:

Author Julie Greenberg said education students should be taking courses that give them a deeper understanding of arithmetic and multiplication. She said the courses should explain how math concepts build upon each other and why certain ideas need to be emphasized in the classroom.
Teacher candidates know their multiplication tables, but “they don’t come to us knowing why multiplication works the way it does,” said Denise Mewborn, who heads the University of Georgia department of math and science education.

This is the key to most of what every student needs to know – how multiplication works. Addition is almost intuitive. It is an extension of counting. Once you extend addition to multiplication, though, you need a good understanding of how the base ten system works, and the commutative, associative, and distributive laws. You don’t need to know the names of those laws, of course, but you need to understand how to use them in order to understand multiplication. You also need to know that multiplication is not just repeated addition – a misrepresentation that is prevalent in education. (I should know, I only recently “saw the light” about this.)

That’s the big issue. Just being able to recite multiplication tables is not actually being able to understand multiplication. And just going through the motions and repeating math steps that a teacher has “taught” you by show-and-tell methods, so you can prove that you can jump through the hoops for the big test at the end of the year usually does more damage to your understanding that anything.

So what is there to do about it? First, as a truly concerned parent or teacher, make sure you, yourself understand some of the nuances of multiplication. Like why when you multiply by a fraction, the product is smaller than the multiplicand. (Did I get you with that one? Leave a comment below requesting the Math Mojo take on that one, and I’ll cover it in a new post).

Second, make sure you have at least two ways of explaining to your students how multiplication works. Not just how to do it, but how it actually works. I’m working on a video series about this now. Send me a nudge (again, in a comment below) to make it a higher priority to get it done and available to you faster.

Third, make sure you have a way to assess if your child or students understand what you taught them. The assessment doesn’t have to be a test. Tests are more about beating kids over the head. Asking questions and asking to demonstrate, in a non-threatening way would be my first strategy. If you must beat someone over the head, start with someone in an administrative position.

Here’s one of the reasons why:

According to the AP article:

Since states oversee the preparation of the nation’s school teachers, the report recommends they set tougher coursework and testing standards.

Why is does the solution always involve browbeating the learners? Why are the words “tough” and “testing” so often involved? How on earth does that teach or inspire? The problem isn’t that, “those who can’t do, teach.” The people who run those studies and teach university level education courses usually can do the math they are supposed to teach quite well.

The problem is that “those that can’t teach, teach.” Then they “train” teachers, instead of teaching them. No wonder those teachers have problems teaching.

As I always say, look up when you look for where the problem lies. You can’t blame a third grader for not learning (unless there is neurological damage, of course). If it’s behavior problems, there might be an issue beyond the teacher’s scope, but most behavior problems are dealt with by good teachers.

But beyond those things, start looking up the chain for someone who needs the butt-kicking. If the teacher can’t teach, were they taught well? (Are they even allowed to teach well in that school?) If the teacher’s teacher can’t teach, were they taught well? Is their administrator constantly putting monkey-wrenches in their teaching techniques? Is something going on at the School Board mucking up the school? Is the State requiring more tests, but providing less resources for teachers and students?

Keep looking up. Here’s a hint: Besides the handicapped, who’s got the parking spot closest to the school entrance? Start with him/her.
Remember, when things are looking bad, begin to look up.

I hope to hear from some of you soon,

Brian (a.k.a. Professor Homunculus)

40 * 53 is 125. Why isn’t it 0?

On the Math.Com website, problems such as 4 to the zero power times 5 to the third power have an answer of 125 as correct. Shouldn’t the answer be zero. If not, why? Thank you!

Professor Homunculus’ response:

The [...]]]> Someone wrote in to ask:

4⁰ * 5³ is 125. Why isn’t it 0?

On the Math.Com website, problems such as 4 to the zero power times 5 to the third power have an answer of 125 as correct. Shouldn’t the answer be zero. If not, why? Thank you!

Professor Homunculus’ response:

The answer actually should not be zero, and here’s why:

Because 4 to the 0 power is 1, not 0.

So 4⁰ * 5³ would be 1 x 5³ which is 125.

Any integer raised to the zero power equals 1.

That is hard for most people to believe, so I wrote a little piece to explain why it makes sense. Here it is:

In modern mathematics, we usually use a base system to represent our numbers.

You know that we have units, tens, hundreds, etc. in our base-ten system.
Well, we also represent those columns in terms of 10 to the nth power. In other words, the thousands column is represented by 10³. So 8,000, for example is 8 *10³.

• 8,300 would be (8 * 10³) + (3 * 10²).
• 8,320 would be (8 * 10³) + (3 * 10²) + (2 * 10¹).

Now comes the problem. You see how the base stays the same, and the exponent gets smaller? To represent the units column, mathematicians have accepted the convention that 10⁰ will always equal 1.

That keeps the rule going. So 8,325 would be:
(8 * 10³) + (3 * 10²) + (2 * 10¹) +(5 * 10⁰).

Because other bases systems, (base 2, base 3, etc.) work the same way, we have further accepted the convention to be n⁰ always equals 1, no matter what the base.

Take the binary (base 2) number 1011 for example. What that means is
(1 * 2³) + (0 * 2²) + (1 * 2¹) + (1 * 2⁰).

That is the same as 8 + 0 + 2 + 1,

which is the number 11 in our normal base 10 system.

As long as we keep n⁰= 1, then the units column of any base will always mean how many ones there are in it.

Math is a network of "conventions" mankind has accepted to make it work. It is based on rules and axioms that are the most "convenient". They may seem hard to figure out at first, but when you get down to it, the rules that we use are basically the best we can come up with to get the things done that we want to accomplish.

Every once in awhile some genius comes up with a rule that makes something even simpler than how we have been doing it up until now, and that becomes the new convention. But that rule has to be solidly based on what has come before, and may not break any of the other rules.

Maybe someday you will be one of the people who comes up with something that explains, or enables something that until now was done by a convention that needed improving.

Have fun!

Professor Homunculus

Part of the math curriculum of schools is estimating, or rounding up. This is a legitimate and important concept, when it is taught by competent and interested teachers. Man, is that a big “when.”

While listening to NPR, I heard an interesting story about how political candidates affect each other. You can hear [...]]]> “Common sense is the collection of prejudices acquired by age eighteen.” – Albert Einstein

Towards the end of making math more meaningful, I’d like to discuss something in recent news that resonates with that theme.

While listening to NPR, I heard an interesting story about how political candidates affect each other. You can hear a podcast of the same story here. You can also read the Washington Post’s story (which broke first).

The story concerns what is called “The Decoy Effect” or “Asymmetrical Dominance Effect” in psychology.

In simple terms, the Decoy Effect suggests that if you are faced with two popular choices, the outcome of your choice can be subtly affected by the introduction of a third, less popular choice (the decoy). But the outcome may not be affected in the way you might expect. The introduction of the third choice would have you lean towards choosing the popular choice that is most like the decoy.

The above-mentioned article concerns itself with front-running candidates for the 2008 presidential race.

In a (very simplified) example, the Decoy Effect suggests that in the case of Gore vs. Bush in 2004, the introduction of Ralph Nader as a liberal candidate did not (as generally assumed) take votes away from Gore, but actually increased the share of votes that Gore would have gotten had Nader not run.

There are several pitfalls here. You may be tempted to think, “Hmm, so my vote for Nader didn’t help Gore lose the election, after all!” That would not be a conclusion supported by the Decoy Effect. The Decoy Effect postulates that Nader running would not negatively affect Gore. But you voting for him would, of course.

Naturally, the effect doesn’t imply that no one would vote for Nader, just that many more would vote for Gore.

(Caveat – Please don’t flood me with e-mails about your personal politics. I am merely using this to explain a concept).

Here’s a scenario that might help you visualize the effect:

You’re Lou Costello and I’m Bud Abbott (or vice-versa if that’s the way you want to be about it). We’re at a carnival with our beautiful friend, Marjory. We are standing between two rides. One is a Ferris Wheel, and the other is “The PSYCHO-TERROR-CYCLONE from HELL.”

If we choose the Ferris Wheel, we think Marjory might think we are too “chicken” to go on a real scary ride. But we both know that we are probably going to blow lunch on the “P-T-C from H”. We’re going to have to tough it out, though, and hope for the best, because we both want to impress Marjory. So it looks like we may have to opt for the “P-T-C from H.”

Now along comes our friend, Stinky, and he tells that he is going on the bumper-cars.

“Bumper-cars! Are you kidding! That’s for kids! We may not be crazy enough to go on the The Psycho-Terror-Cyclone from Hell, but we’re going on the Ferris Wheel, like sensible, real men!”

Get it?

In some sense, the decoy gives you an “out.” Sort of like when you are offered some expensive product, which you want, but you really can’t afford. You are also offered something that isn’t quite what you want, but you can afford it. You may tend to go for neither.

Then you are offered what is sometimes called a “slum prize.” It’s a piece of junk for free.

The theory is, you’ll say to yourself, “No, that’s junk. I can afford something better,” and you’ll go for the thing you can afford.

(And no, none of the free downloads at Math Mojo are not slum prizes, you cynic, you!)

In some sense, the decoy makes you base your decision on a trait that is less important that what you are really interested in. In the above example, you made your choice based on price, although you are not buying something that you really want.

You might think, “But that doesn’t make sense!”

Heck, some people get mad if you show them a better way to multiply.

I can understand your feeling that it doesn’t make sense, it’s the “but” that is the problem. Why would you think that people chose what makes sense? People do not generally process complex information well. We like to oversimplify things. That fits just perfectly with the Decoy Effect.

Understanding that people oversimplify things helps you navigate your world better. Of course, you don’t want to oversimplify things yourself (although we all do it to some extent) you just don’t want to be surprised when other people do.

This tendency to oversimplify things is what we call “intuition.”

• “Of course introducing a similar candidate will hurt the original candidate’s choice.”

• “Well somebody’s got to will the lottery, so why shouldn’t it be me?”

• “If you don’t believe in God, well, then what do you believe in?”

• “If Johnny Jones jumped off the Empire State Building, would you jump off the Empire State building?”

All of those are questions, suggestions or implications that are based on meaningless logical fallacies, but they appear to be sound to people who want simple representations of complex problems.

It is very hard for most of us to look more deeply into things that seem simple on the surface. It is difficult to sort the simple from the complex. So the simplest thing to do is assume that any complex question is simpler than it appears. That keeps us from having to put effort into dealing with it. It keeps us comfortable, up to a point. And it keeps us wrong.

Laziness is no substitute for thought.

There was a time when we may have believed in Santa Claus. When our friend told us there was no Santa Claus, we didn’t believe him, because, “everybody knows, etc.”

Eventually we matured, and understood that much of the stuff we believed when we were children was just convenient nonsense. Then we became adults, and we knew “what the truth is”.

I believe that that moment is the worst moment in most people’s development. It’s the moment when we think we know enough. It’s the moment we have effectively stopped thinking and learning.

If we are very lucky, at some time we are faced with greater truths, which are less apparent than those “truths” we “were told in school,” or “read somewhere,” – insights that not everybody is aware of. Something that most people are too hung-up to even consider. It may even frighten them.

Imagine how some people felt when they saw the Wright brothers’ fly for the first time.

• “Galileo, how dare you believe that the earth revolves around the sun! Everybody knows…”

• “What? Question the President? Of course Saddam has WMDs. Rumsfeld even said he knows were they are. Do you think they would lie?”

• “What do you mean ‘nothing can travel faster than light?’ What if you held a flashlight and walked forward with it? That beam of light would travel faster than if you held the flashlight still! See! I’ve got you there, Professor!”

All of those statements sound (or sounded, at one time) perfectly logical to many adults, in the same way that Santa seems perfectly logical to a child. But at some point everyone should get over them.

Developing counter-intuition is the next step after developing what we generally call “maturity.” Society cannot advance very far until it is expected of everyone. Unfortunately, large portions of society will do anything they can to inhibit your development of counter-intuition. They feel threatened by what they do not understand.

The funny thing is that it’s not that they can’t understand. It’s that they won’t understand. They don’t want to. It’s not as easy as being comfortably dumb.

“What you don’t know can’t hurt you.” Right. That’s like the toddler playing “hide-and-seek” who covers his eyes and says, “You can’t see me!” (Except it’s not pathetic when a child does it.)

Arrested development is the hallmark of humanity, it seems.

Sometimes you can go by your intuition, but unless you have the option of seeing the pitfalls of intuition (which you can only do if you have the benefit of counter-intuition), your intuition is going to be a very poor judge of reality.

Math Mojo is about finding alternative, better ways of understanding simple mathematical principals and arithmetical operations. It is also about critical thinking skills.

There will be more and more articles about how to develop counter-intuitive thoughts as Math Mojo develops.

The conclusion of the Washington Posts’ article is this:
“Don’t let salespeople tell you what issues to care about, and don’t let candidates define one another. More simply, think for yourself and be wary if a difficult choice suddenly feels simple.”

That is also the heart and soul of the message of Math Mojo. It’s also part of its brain.