Hintikka’s Paradox comes from Deontic Logic, a form of Modal Logic. I first read about it in Raymond Smullyan’s “Alice in Puzzleland’ (a brilliant book about logic, and Alice in Wonderland, that is worth looking into.)

In the introduction to “… Puzzleland,” Smullyan describes Hintikaka’s Paradox this way:

“Is it proper to call morally wrong something a person cannot do? Hintikka has a notorious arguent designed to show it is wrong to try to do something impossible. There is now a large literature on this strange question…”

I’ve yet to encounter much of that literature, and boy, I have looked. I probably wouldn’t have understood most of it, anyway. 

But it boils down to this, Hintikka’s Paradox implies that, “What is not possible is positively forbidden.” 

It’s important not to approach this from a “common sense” frame of mind. Common sense is usually neither, and is often a disadvantage when approaching counter-intuitive material. So try to keep an open mind. 

As far as I can tell, the logic of the argument goes something like this:

What cannot be done without something wrong being done, would itself be wrong to do.

(1) To do something that cannot be done without something wrong being done would itself be wrong. But (2) what cannot be done at all cannot be done either with or without something wrong being done.

 So, if x is impossible and y is wrong, I can neither do both x and y, nor do x but not y.

But, (by 1) if y is wrong and doing x but not y is impossible, it is wrong to do x. “

Hence (3) if it is impossible to do x, it is wrong to do it.” 

From - Encyclopedia of Philosophy, Vol. 4, Ppg 509-514

OK, I can follow the argument. 

Here’s my question - Why do they have to addend ‘y’ to the argument to make the point? I mean, to me, in a “common sense”  way (always wrong to do), I would think* that I could just as equally say, “If x is impossible and y is not wrong, I can neither do both x and y, nor do x but not y. The “But, (by 1) part is not not applicable, thus the conclusion of above (“if it is impossible to do x, it is wrong to do it” ) cannot be drawn.

In my simple mind, it seems as though they are throwing in a red-herring (something that is not logical, but appears so, in order to muddle the argument). Sort of like saying, “If Joe is a guy, and Max is a communist spy, then Joe and Max cannot both be “good guys.” But if Max is a communist spy, and Joe existing without Max is impossible, then Joe must be a communist spy.”

Of course I am aware that my comparison is wrong. I just don’t understand (yet) why it is wrong. My not understanding is due to my incompetence, I know, and not to any flaw in Hintikka’s logic. I’d just love to be able to understand the logic. 

The reason for my interest in this paradox, is that if I can understand it better, I think I can use it for a philosophical quandary I am in. It relates to politics. (See kids, I am always trying to use math or logic to get more meaning out of life. Feel free to play along at home.)

I am not a logician. But if someone out there reading this is a logician, or understands the paradox, could you please explain it to me in a comment? Remember, neither I nor most of my readers are logicians, so take it easy on the jargon, K?

*By the way, whenever most people, including myself, use the phrase, “I would think,” or “You would think,” you can be fairly sure that they will follow it by saying something that is wrong to think. I’m probably guilty of it in thinking “y” is a red-herring.

If you have some mature, fermented thoughts on this, please let me know. Remember, I’m looking for clarity, not more common-sense muddling. If your not a logician, or something like it, please ask your logician uncle Raymond, or someone, and have him/her take a look at it. 

All the best, 

Brian

The same person wrote a follow up comment:

“you are not pretending i’m stupid!!!!! Okay is a base the number you can multiply by?????
“example: base two is 2,4,6,8,10,12,14,16,18 ?????????i don’t know what you mean!”

Yeah, keep trying to convince me that you’re stupid. From your grammar and your tone, you’re starting to make some headway. 

But I generally don’t believe a child can be stupid. Misguided, full of anxiety about themselves and the world, OK, but stupid is reserved for adults (where a lot of people make up for lost time). 

Maybe I wasn’t clear enough, so let me try again.  A base is a way to write a number using place value (columns). The amount of digits you decide to use in the columns determines the number of the base. If you use ten digits per column, the number will be in base 10. If you use three digits per column (the digits 0, 1 and 2), the number will be in base 3. You will understand this better as you read on.

You know what place value is. We use it in our daily number system. It’s the ones, tens, hundreds, thousands, etc. columns. Because the amount of digits we use in each column is 10 (those are the digits from zero to nine), we call our system the base 10 system. It’s also called the decimal system. The word decimal comes from the Greek word for ten - “deka.” 

Once you’ve used up the ten digits in a column, you must start filling up the next column. If you still don’t understand it, there is a good lesson about why we regroup and carry when we add at:

http://mathmojo.com/interestinglessons/regroupingandcarrying/regroupingandcarrying.html

That lesson explains what happens when we “go over” in a column, and why we use the next one. 

In the  base ten, as the numbers grow, each higher column is ten times the amount of the previous column. So in base 10, the columns go, ones, tens (because one times 10 is ten), hundreds (because ten times 10 is a hundred), thousands (because a hundred times 10 is a thousand), ten thousands (because a thousand times 10 is ten thousand), etc. 

In base 2, it would work the same way, except each time we would multiply the column by 2  (instead of 10) to get the next column. So in base 2, the columns go, ones, twos (because two times 1 is two), fours (because two times 2 is four), eights (because four times 2 is eight ), sixteens  (because eight times 2 is sixteen), etc.

We would need more columns to represent a number in a smaller base than in a larger one. Can you see why? 

It’s because If you had five columns in base two, the most you could represent with the fifth column would be 1 group of sixteen. 

If it was in base ten, by the time you got to the fifth column you could represent 9 groups of ten thousand! That is a lot more than sixteen!

In base 2, the columns go like this:   

By the way, the base 2 system is also called the “binary” system. “Bi” for “two,” as in bicycle, bisect, etc.

One thing you might be asking yourself, is, “What the heck is this stuff good for?” Fair enough question. 

Personally, I think the best thing this stuff is good for is to improve your mind, and learn how things work. Real, true, honest-to-goodness curiosity. Imagine that!

On a more everyday level, we use base 12 when we talk about dozens and grosses. When you pack things in dozens, you are giving the second place value a value of 12. And a gross it twelve twelves (or 122), which is 144. Base 12 is called the “duodecimal” system, from the Greek “dodeka.” (Can you se how that word looks like “dou deka?” or “two-ten?”  That’s where the English word “twelve” comes from.)

Base 2 happens to be the base which almost all computers function on. Computers don’t recognize anything but the digits 0 and 1. 0 means there is no flow of electricity, and 1 means there is flow. Like an on and off switch. 0 means off, and 1 means on. 

So why are there more numbers than that on a keyboard? Simple – every time you type a symbol in to a computer, the computer translates it to a base 2 number automatically, and that decides which switches are turned on and off, which makes certain things happen. 

(That is about the simplest explanation I can think of. Of course there is much, much more to it, but I hope it will do for now.) 

Since computers seem to be here to for the long-run, it pays to learn a little about programming, and learning base 2 is a good place to start. 

There are other reasons for learning other bases, and some of them have to do with how computers work, as well. For example computers use a modified base 16 (the hexadecimal system) to represent colors. As you know, there are millions and millions of colors in the world, and computers represent them with numbers. For example, the color bright red is #FF0000. (You’ll learn why we use letters with bases beyond 10 in a future post.) Using the hexadecimal system you can represent many millions using only six place values (columns). 

Cryptography (the science of codes) often makes use of bases other than 10. 

Many other  uses of other bases are too complex to explain, here, so I hope you will be patient and stick with math till you get to them. Some of them are about logic and decision-making, some are about better ways to do simple math (like using the abacus – base 5 has a lot to do with that), some are about geography and geometry (base 60, base 360), some are about history (base 20 and base 60 were commonly used), and some are about measurement. If you stay curious about the way things work, as you get more mature you will find lots of ways that bases are used. 

Believe it or not, I use this stuff for magic tricks. Some of the subtlest and best tricks that even fool other magicians are based on subtle math principles. Not the dumb kind like “take a number, multiply it by something, divide it by the number of coins in your pocket, add the number of teeth in your grandmother’s head,” etc.. but some great ones that are usually only seen and performed by elite magicians (and I don’t mean Chriss Whatshisface.)

Another good use for binary is in game theory. One good example is the game of Nimm. It’s sometimes used as a betting game, and if you are quick with using the binary system, you can win almost every time. I’ll teach it to you when we’re done with all the lessons about bases. That may take a few weeks, so make sure you learn all this stuff, and you’ll be ready to kick some butt at Nimm. 

I’ll also teach you how to count on your fingers in binary. You can use that skill to help you with huge additions in base 10. I know that sounds impossible, but if you learn it you can definitely become quicker than a calculator at large additions. 

Enough philosophizing for now! Let’s try converting one more number from base ten to base 2: 

Let’s try the number 7 (base 10):

A good rule of thumb, is to go through all the columns of the base you are converting to, until you reach a number that’s higher then the number you are converting. 

In other words, in this case we’, go through the powers of 2 until you reach a number higher than 7.

• The first column in base 2 is the ones column. 1 isn’t higher than seven, so keep going.

• The second column in base 2 is the twos column.  2 isn’t higher than seven, so keep going. 

• The third column in base 2 is the fours column.  4 isn’t higher than seven, so keep going. 

• The fourth column in base 2 is the eights column.  8 is higher than seven, so it is two much. You have to go back a column. 

This tells us that we’ll start writing the number 7 in base 2, starting with the fours column. We ask ourselves, “is there a 4 in 7?” There is, so we write a 1 in the fours column. Now we subtract the four from the 7 (because we’ve already used it in the fours column). That leaves us with 3.

Now look at the next column (the twos column). As yourself, “Is there a 2 in 3?” Yes, there is. Now we subtract the 2 from the 3 (because we’ve already used it in the twos column). That leaves us with 1.

And that goes in the ones column. 

So we have a 1 in the fours column, a 1 in the twos column, and a 1 in the ones column. That gives us the number 111 in base 2. 

That means that 7(base 10) is written as 111 (base 2). 

In the next post, we’ll learn how to read any base 2 number as a base 10 number. Let’s see if you can guess it before then. Think you can change 10101(base 2) into base 10? Leave the answer in a comment if you think you’ve got it. 

Also, for practice, turn 37 into base 2. Leave that in a comment as well, if you like. 

photo by lsiegert

A curious reader asked this question:

What is a base?? I’m sorry but I’m in the sixth grade and never heard of a base and then all of the sudden it’s in my homework. Will you please explain to me in easy fifth or fourth grade words what a base is? Pretend I’m stupid or something!

 Professor Homunculus replies:

Well, that’s going to be hard to pretend, because you are obviously smart enough to ask for help. You also did a good job expressing your question, so here goes:

Bases are different ways to express numbers. Like languages are different ways to express thoughts. You could say, “butterfly” in English, or “mariposa” in Spanish, “papillion,” in  French, or “schmetterling,” in German, but they would all mean the same thing, just different names for it.

You can write the number 11 in base ten, or as 21 in base five, or as A in base eleven, and they all stand for the same amount.

Just as in different languages, there are specific times you need to use different bases. That is a little hard to understand, right now, I know, but first you must learn how to translate into different bases, before you can understand anything about them.

Fortunately, it is much, much easier to learn how to translate from base to base than from language to language.

A base is the amount of digits we use to represent our numbers with.

We normally use what it called the base ten system. As you know, we normally use only ten digits - 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 to make up all of our numbers. After 9, we have to start a new column (called the “tens” column, because it tell us how many tens we have). 

Then we go through the numbers again, all the way to 99, when we have to start another column, called the hundreds.

The hundreds column actually is the same as the “ten tens” (or “ten groups of tens”) column  because there are ten tens in a hundred.

Then, of course we continue, till we get to 999, when we have to represent a thousand (ten hundreds).

You see, because we only have 10 digits, and one of them is zero, we have to make a new column to carry another digit every time we get to more than 9 of anything in any column. So there is no single symbol for 10, we have to use two symbols in each of two columns.

So far, I’ll bet this all sounds simple - probably because it really is.

But what would happen, if no one had ever invented the symbols we use? Or what if they had only invented, say, two of them?

We always start with zero, because we always need a 0 to represent any empty column, no matter what base are using.

So, if you have a base 2 system, there would be only two digits, starting with 0. That means the only digits or symbols we could use, would be 0 and 1.

But how can you represent all numbers using just two digits?

It is actually simple. You will just need more columns. It usually works like this: The more digits you have, the less columns you need, and vice-versa.

In the  base 10, as the numbers grow, each higher column is 10 times the amount of the previous column. So in base ten, the columns go, 1s, 10s (because 1 times 10 is 10), 100s (because 10 times ten is 100), 1,000s (because 100 times ten is 1,000), 10,000s (because 1,000 times ten is 10,00), etc. 

In base two, it would work the same way, except each time we would multiply the column by two  (instead of ten) to get the next column. So in base two, the columns go, 1s, 2s (because 2 times one is two), fours (because two times 2 is four), 8s (because 4 times two is eight ), 16s  (because 8 times two is 16), etc.

So, in base 2,  the furthest column to the right would be the ones column (just like in our normal base 10 system). The column which is to the left of that is usually the tens column in the base ten system. But in the base 2 system it would be the twos column.

Let’s see how that would work. If we wanted to turn the number 3 (base 10) in to base two, this is how we would do it:

We would think, “Well, there is no digit for 2 or 3, so I can’t just write the 3. So I have to see if the number is high enough for there to be anything that would have to go into the twos column.”

And of course, there is. You figure out how many 2s would fit in the twos column. There is only one 2 in the number three, with one 1 left over.

So you write a 1 in the twos column (because you only have one group of 2). How many ones do you have left over? Just 1, so you write a 1 in the ones column, too. That means 3(base 10) = 11 (base two).

If that doesn’t make sense to you, read it again. If it still doesn’t, read tomorrow’s post, where we’ll continue about bases.

You may wonder why the furthest column to the right would be the ones column in all bases. There is a pretty interesting twist to the answer to that. 

You may also wonder how we represent base 1, if all we can use is one digit, and since we always start with zero, our only digit would have to be zero. If anyone knows the answer to this, please feel free to post it in a comment today. Otherwise, I’ll answer it in a future post as part of this series. 

Professor Homunculus intends to continue this series on bases until you know more about them than most college students (although, to tell you the truth, that’s not much! OK, lets say most advanced high-school or homeschool students.)

So for the next week or so, I intend to post articles about how to change from base 10 to other bases and back, how to change from non-base-ten bases to each other and back, what bases are good for, bases and their relationship to exponents, bases higher than base ten, adding, subtracting, multiplying and dividing in different bases, game theory and bases (I’ll teach you the winning strategy for a cool game), magic tricks and bases, and an awesome way to mentally add huge rows of numbers in base 10, using base 2 (it’s much easier and faster than the way we normally do it with paper, and it’s quicker than using a calculator!) 

So read on to the next post for more about bases!

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.

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. And no wonder the

As I always say, look up when you look for where the problem lies. You can’t blame a third grader for not learning. 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. Also, I’m sure you understand that I am not talking about children with neurological damage. That is an issue outside the realm of my expertise.

But beyond those things, start looking up the chain for someone who needs the butt-kicking. If the teacher can’t teach, was  s/he taught well? (Are they even allowed to teach well in that school?) If the teacher’s teacher can’t teach, was s/he taught well? Is the administrator constantly putting monkey-wrenches in the teacher’s 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? Did some idiot in the White House set everyone else up to fail so he can push some hidden agenda?

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)

“How do you know when you are writing in standard form, expanded form? For example, is the expanded for of 30,048
30000 + 40 + 8 ?
Or for 29,486, the expanded form = 20000 + 9000 + 400 + 80 + 6 ?”

Professor Homunculus replies:

Precisely! Oddly enough, there is no “standard” for “standard.” What I mean is, for 30,048 the standard form could also be considered:

3(10,000) + 4(10) + 8(1)
as well as
3(10⁴) + 4(10¹) + 8(10⁰).

To make it even more confusing, some teachers might require you to include the empty columns. That would mean they might require you to write:

30000 + 0 + 0 + 40 + 8
or
3(10,000) + 0(1000) + 0(100) +4 (10) + 8(1)
or
3(10⁴) + 0(10³) + 0(10²) + 4(10¹) + 8(10⁰)

Even though seems silly to be that “complete” (I can’t imagine having to include the empty columns in “the real world”) it isn’t bad to teach it like that in order for students to get the feeling for the whole concept. It helps solidify it in one’s mind. It’s just when kids get tested on it and writing out the whole thing is required, that it gets a little ridiculous.

They may be other ways to express “expanded notation.” Of course, it depends on what one mean’s be “expanded.”

One of the problems with the way schools teach is that they usually teach you one way and imply that is the “official” way. Most often, the teachers are not even aware that they could be confusing students by not at least making them aware that there are other ways to do “standard” things.

Schools do that with many things in math, and in other subjects as well. I think that is an educational crime, because it encourages students to stop thinking and accept the narrow information they are given as the “absolute truth,” which, of course, it never is.

One thing you may want to keep in mind, is that although school lessons on “standard and expanded notation” tend to be boring and leave you with the feeling that those things are not important, and just basically fodder for tests, that is not necessarily so. To find out more about expanded notation, and why you should understand it, check out:

http://mathmojo.com/interestinglessons/expanded_notation/expanded_notation.html

There is also something called scientific notation which is often confused with expanded notation. They are not the same. For a good explanation of scientific notation, refer to:

http://www.nyu.edu/pages/mathmol/textbook/scinot.html

“Mom” also asked:

“3205; is this number writen in standard form?”

Prof. Hunc sez:

Yes. You could also have written it with the comma, as in 3,205. (You don’t have to write the comma in the real world, but sometimes schools require it.)

Thanks for writing in. Reader input is what drives Math Mojo. Keep it coming!

I haven’t been posting much. Sorry. Been in a kind of existential funk.

But today I had to include this. It is a link to an article that was in our local paper today, about a heinous phenomenon in a local school. Apparently, some second-graders have been actively plotting to kill a little girl in the school. You can read about it in this digg.com post.

OK, no big deal right? After all, this is America, where everyone is “entitled” to their lunacy, no matter how depraved.

But this is in a rural, upstate New York school. No inner-city, no whacked out Waco, no out-in-the-hills survivalist community.

I have done afterschool Math Mojo programs in this school. It is a nice place with (generally) nice kids. I’m not amazed, though, because our society has become all about abuse of power, from the highest, to the lowest, levels.

I know that you can’t make a sweeping judgement about public schools in general from a local anecdote. But the anectodes are getting to be pretty thick in our public schools.

It is depressing as hell. I don’t mean to depress you. There has got to be a solution, and I believe that readers of this blog are generally part of it. Homeschooling, unschooling and afterschooling are good, positive movements.

The big difference is the amount of parental involvement. If your child knows that you truly take an interest in them by spending time with them, your conscience becomes part of their conscience, without having to lecture them or make them feel “watched.”

But you know that. I just want to say that from the comments this blog gets, and the e-mails I receive, it’s people like you that give me hope. I hope I give you some to, at least as far as encouraging you to play around with math.

To that end, I went out and bought a cool little piece of software at the Apple Store yesterday. It’s a Wacom writing tablet, and I hope to make some really easy-to-follow math tutorials for you with it in the next few days. I’ll have one up here by tomorrow probably.

See you then (if I didn’t bum you out too much.)

There is an alternative argument that argues for the opposite. Many pedagogues try to plead the case that first you must teach the “basics” (meaning the basic skills, like the “multiplication facts”) before you can expect a child to acquire any meaning about it.

First, a short digression:
I HATE it when they call them the this-or-that “facts.” Firstly because it implies that anything outside the stupid, limited chart they are pushing are not “facts.” Is 84.5 * 63 = 5,323.5 somehow not a fact? And secondly because the way they use the word “facts” somehow implies that the fact is “outside driven.” By that I mean that it is a random kind of thing that you just have to memorize, instead of something that you can experience and learn from conceptualizing. Okay, back to our theme…

There is some validity to those pedagogues argument. Or at least there would be if some conditions were fulfilled first. But they almost never are in any public school system, which renders them pitifully invalid. Before I rip their into little girlie-man shreds, let me present the kind of case where their arguments would be valid:

“… wax on - wax off …”

Maybe you remember the cool seventies flick, “The Karate Kid.” (If you haven’t seen it, you should. It will illustrate several important points about teaching - and it’s a fun flick.) In it, the kid wants to learn Karate, and somehow convinces the somewhat reluctant Mr. Miyagi to teach him martial arts.

Mr. Miyagi ends up having the kid paint his house, wax his car, etc. Even worse, Miyagi is apparently a control-freak, and insists the kid do those chores with certain arm movements.

After a few weeks of this, the kid gets impatient, and asks about when Mr. Miyagi finally will teach him some self-defense. Mr. Miyagi pretends to try to slap the kid, and the kid blocks the slap perfectly. The kid is totally astounded that he could even do that.

It turns out that the arm movements that Miyagi Sensei was so obsessive-compulsive about are exactly the arm movements you need to learn for those basic blocking moves.

There was a perfect example of learning the skill mindlessly, by rote first, and then grasping the concept later.

This may actually be the best way for many things, including basic math facts. But certain conditions must be fulfilled in order for them to work.

What condition our condition is in

The first condition, is that the teacher must not only be a master practitioner, but a master teacher of his art. And it must be an art, in a deep sense.

I have only met one teacher in my lifetime in a public school (university) who met these conditions. He is a decorated and brilliant retired theoretical physicist and professor emeritus, and the founder of “Eduction” (or “Edux“). (More on Edux at some later date.)

How many teachers or parents out there want to try to fulfill those conditions? When you do, you’ll have nothing but my admiration. But until any of us do, we’d be well advised to teaching the concepts first, I think.

So if you’re short of being a Zen Master of your art, the next best thing seems to be being a thoughtful teacher, who doesn’t insist on teaching with a style that you can’t really fulfill, but will do the best with what is within your capacity.

There is a tendency to demand certain things from students without providing the means for them to attain those things. We “raise standards” and enact a cruel and, well, basically retarded “No Child Left Behind” act. Then we don’t fund it. We demand “accountability” (one of those words from this hypocritical era in American history that will go down in infamy) with “standardized testing,” yet we don’t give teachers the kind of training, the time, and the resources that they need to do their jobs. Then we blame them for it. And we blame kids for failing. You can’t load thirty kids into a class, undermine the teacher’s ability with some random “standards,” and then expect any meaningful outcome.

Those pedagogues who have the “lofty” Ideas, but not the means to make them work, are only going to confuse more and more children, and make sure no one wants to remain or become a teacher. It’s one thing to be an academic, with your head stuck firmly up your butt, and it’s another thing to try to stick innocent young kid’s heads up there with it.

“You vill do it first, und zen you vill undershtant it!”

If you are interested in reading some interesting thoughts about how to learn something to a degree of meaning that most of us might otherwise never dream of, consider getting a copy of “Zen in the Art of Archery” by Eugen Herrigal. It is a classic, and I don’t believe I can think of any other book that has a greater right to be on everyone’s bookshelf (and read).

If you’ll allow me another digression:

The archer in the animated movie below was scanned from the cover of my original copy of “Zen in the Art of Archery.”

This was one of the first “flash” animations I ever made. Please excuse the soundtrack if you speak Japanese. I don’t speak much Japanese (Suimasen watashi wa Nihongo hanesamasen), so when I chose the soundtrack, I chose it for the actual sounds, not the meaning. The sentences came from the soundtracks of old Japanese movies, and I have no Idea what they mean. If you are Japanese, you are either rolling on the floor laughing your ass off, or are pissed at me. Please forgive me. I meant no disrespect. Much to the contrary.

A lesson from contemporary China

I have been corresponding with a student at Beijing University who put up a video on YouTube. It is of him rolling a coin over his fingers and doing other magician’s “flourishes” with coins. The kid is amazing. The coin-roll is one of my specialties in magic (I am a semi-retired professional magician), and this kid’s coin-rolling skills just blow mine away. So I wrote to him to express my admiration, and we began corresponding to exchange Ideas about magic, performance, skill, philosophy, etc.

In one of his e-mails he mentioned this wonderful thought: “… and as in a old saying in China ‘Interest is the best teacher.’”

Tao, I couldn’t agree more.

Here is Tao jamming with coins:

BTW, as you watch this one, crank your speakers. The soundtrack is by Jeong-Hyun Lim, a Korean kid who recorded this at home. His youtube video was featured on these Chronicles on this post.

If you’d like to see an old (1991) video of yours truly rolling two coins at once, check out the video below.

Afterthoughts:

As you watch those young Korean and Chinese kids perform those intricate, creative and absolutely astounding feats, you might ask yourself why many of us in the US set such wussified standards for our own kids. What will our kids accomplish when they are their age? More important, what will turn them on?

So if you can have the items related to each other, it’s easier to remember them. It’s certainly easier to make sense out of them.

Now, when you consider the following numbers from the point of view of a child learning them:

9, 18, 27, 36, 45, 54, 63

they seem to have nothing in common, except that they are on a chart of “the multiplication tables” that the child is told to “shut up and memorize!”

On the other hand, if you show the child groups of objects, which have nine in each group, and you put them next to each other, they can readily see how nine grows into 18 if you add two groups, then into 27 if you add another group, and so on.

This increases the child’s appreciation for what he is learning, and decreases resistance to that learning. So showing the concept definitely helps learn the skill.

The school of thought that says, “Teach the skill and the concept will be easier for the child to understand later” is based on what we want the child to learn, not how it’s best for the child to learn it. This is utterly misguided.

It’s the “authoritarian” way. It’s the same school of thought that says, “You will do as I say whether you like it or not, because I know what’s best for you.” That is the most counterproductive way that was ever thought of. It is the default, brain-stem, neanderthal way. And it doesn’t work to produce intelligent children.

It may produce obedient drones if that’s your goal. But I’m pretty sure that if you read this blog, that’s not exactly what you’re after.

Actually, the “authoritarian” way doesn’t even create obedience. But it’s a great way to produce anti-social rebels who will revolt against the thing you are trying to teach them. I don’t mean that they’ll become the creative-rebellious types who produce great things for society, I mean they’ll become the destructive little shits who only complain, and then grow up to be exactly the kind of disappointed authoritarian types as the dunces who tried to inculcate them with “facts” and “skills” that became neither to them.

A Theory of Relativity…

Imagine giving your child a list of names, like: Mark, Simone, Leonora, Derrick, Romulus and Matthilde. Now tell them what each looks like and acts like, and that Mark and Simone are married, and Mark is their father’s brother, and Leonora and Derrick are their children, and Romulus and Matthilde are Derricks friends from school. You can even make a chart of all their relationships and characteristics.

Or, you an invite them all over for a barbeque and letting the child play with them and get to know them. They gradually learn how each is related to whom, and they get a personal relationship with each. Those people become meaningful to the child (imagine that!)

Which do you think will stick with the child more? Which would he rather do, sit and stare at a list and a chart while you grade him on his progress, or play and interact?

I hope this little analogy, imperfect as it is, nailed the difference between understanding and drilling for you.

Respect your Mind

When we teach concepts first, and skills later, we are reinforcing the Idea of respecting the minds of the children we teach. It may take a while, but when you do that, the child slowly gets the Idea that it’s okay to trust your brain. They’ll see that when you don’t expect them to memorize something and then use it quick (because, say, you have to “cover material”, and it’s in this “module” of the “curriculum”, and that “it will be on the test soon”), that they can actually grasp it, and understand it more deeply. That way, everyone gets more satisfaction from the teaching/learning endeavor.

Tomorrow I’d like to mention a kind of caveat to all of this, concerning a bit of Zen philosophy.

Apropos of nothing:

I’ve recently gotten into the “Twitter” thing. Not sure if I like it, or what it will bring, but it is new, interesting, and a great distraction from work. I’m supposed to be using it as an organizational and motivational tool. We’ll see.

Recently I read a “tweet” from http://twitter.com/bcubbison. He asked, “What kind of people would name their planet after dirt?” It made me wonder what the people who named their planet “Uranus” are like.

BTW, do you use Twitter? If you do, check me out at twitter.com/Prof_homunculus, and follow me.

- Georg Cantor

Today is our little dog’s birthday. Our “little” dog is an eighty-pound Golden Retriever named Maia . (Our hundred-ten pounder is a Golden Retriever named Galileo.)

I went down to the local slaughterhouse (yes, we have one in our little rural village, and it’s right along the railroad tracks) and got a bag of bones for Maia. I came home with them and she was ecstatic. She even shared one with Galileo.

I couldn’t resist putting a video of her up today. I hope you like it.

Happy Birthday, Maia!

• A graduate with a Science degree asks, “Why does it work?”
• A graduate with an Engineering degree asks, “How does it work?”
• A graduate with an Accounting degree asks, “How much will it cost?”
• A graduate with a Liberal Arts degree asks, “Do you want fries with that?”

The reason for this post is that Angela (Mother Crone) left a very interesting comment on yesterday’s post concerning how mental math has helped her daughter, who is dyslexic.

How many screwbulbs does it take to light in a dyslexic?

(Yes, that was unbelievably cheap.) Although I have no insights into clinical dyslexia, I have fought my whole life against certain dyslexic-like symptoms. I also suspect that any person who is at least mildly aware of his or her thought-processes struggles with similar symptoms.

For example: when I do a crossword puzzle - when I write in an answer that had been partially complete, I often transpose some letters. Say the answer to 32 across is four letters long, and I’ve already filled in the first two from “down” clues. They are O-D. Now when I come to the clue for 32 across, it’s, “What are the … ?” So I know that the answer is O-D-D-S. When I go to fill in the D-S, I will often write S-D. I won’t even notice it unless my wife is there to point it out to me (which is usually the case). Or, if I’m doing the puzzle alone, I won’t figure it out until I have beaten my head against the wall trying to get the right answer to one of the “down” clues that crosses the D or the S. Because they are transposed, they will “block” any correct answer I come up with until I finally see what I have done.

It’s interesting that I almost never transpose letters when I am writing in a whole word or phrase. If none of the letters of the answer for O-D-D-S was already present, I’d simply write the word “odds,” but if any letters are missing from a complete word, I have a good chance of transposing them.

It happens so often, that over the years I’ve had to train myself to occasionally step back during the puzzle and stop searching for clues, and just “sift” through the answers I have so far and look for discrepancies. (Actually, I’m glad I’ve had to do this. It’s a great habit to get into when trying to solve almost any type of problem.)

The reason I don’t play Bingo

Also, I often transpose the digits of the number of a clue when I go to write it into puzzle. For example, I see clue 47 down is “Plant of rock and metal.” (For the answer, see below) I immediately think, “Got it! Now where is 74 down?” And I’ll search for 74 down, and either notice that there is no 74 down, or it has the wrong number of letters for my answer. After a few seconds of confusion, I’ll have to look back at the clue and notice the correct number.

My wife isn’t as aware of my problem with this as she is of the “transposed letters” problem. There is almost no chance for an observer to notice me doing this, as I don’t fill in anything incorrectly. The whole phenomenon takes place in my head, and I don’t write anything down before I notice what I’ve done.

And this is exactly one of the points I want to address. Many people (not dyslexic themselves) see dyslexia as simply a social problem. “Oh, how embarrassing that he made such a goof in front of people! It must be hell to live with the shame.” (Or something like that.)

Public goof-ups are just the tip of the iceberg. When you have a cognitive problem (Dyslexia, ADD, depression, etc.) it’s not really about your public ego. It’s something you deal with when your on your own, as well. The frustration of not being able to solve a problem that you understand is sometimes unbearable. I can speak from personal experience with ADD (still struggle with it) and depression (have pretty much overcome it).

People just don’t see that you are struggling with the issue using very different thought patterns than they are. Sometimes while I know they are thinking, “Why doesn’t he just…” I’ll be thinking, “Why doesn’t that ignorant bastard solve his own problems before giving misguided advice to me?”

While some people are embarrassed for the dyslexic person, some of us are more embarrassed that the others don’t have the depth and breadth of thoughts that some dyslexics, or others have.

I am fortunate enough to see that there are some people who can appreciate the “different” thought patterns of dyslexics, ADD “sufferers”, depressives, etc. My wife is not only a special education teacher (which explains her patience with me!) but she has a non-judgmental persona. She can notice things and remark on them without jumping to a conclusion.

Lots of people can do that, but a lot of people can’t. Unfortunately, many of them that can’t hold positions of “power” in the education system. They are the people who make the “standards” and “curriculum.”

…and those who can’t teach, administer.

They are the people who have successfully ruined what once gave me hope that the American school system could someday be excellent. They are the people who are trying to take curiosity and learning, and “otherness-thinking” away from the younger generation.

They are the ones that make life hell for talented, dedicated teachers. They’re the ones that want to put your thinking right back in the box, and hermetically seal that box.

So are you just going to rant, or are you going to give us something we can use?

Hmmm, maybe. Angela mentioned that her dyslexic daughter is “stoked” about doing math mentally. I know how she feels. When I write answers I tend to transpose digits, but I almost never do what when I do math mentally. Other people have written to me that other Math Mojo methods have helped their dyslexic children or students. Although I can make no claims for it’s clinical efficacy, if you struggle with cognitive problems, you might want to give mental math a try.

One thing that has gotten a lot of positive comments from people helping dyslexic children is the Abax. I’ve even gotten testimonials from special education teachers about it.

If you would like to know more about this fascinating manipulative, and how to use Math Mojo methods with it, check out Introducing the Abax, at MathMojo.com

Take a number…

I’d like to address the use of mnemonic devices for memorizing numbers. When I use mnemonic devices, once a number has been “translated” into a device, I never transpose any digits. The system helps prevent it.

I would like to know if anyone out that has any experience with full-blown dyslexia and high-powered mnemonic systems. If you do, please leave a comment about them. I don’t mean the “Most Very Educated MathMojoers Judiciously Serve Under No Public-school-system” stuff (hey, I just made that up! Someone stop me before I pun again…) I mean the phonetic system for memorizing gigantic numbers.

Anyone else, please feel free share your experiences and thoughts about dyslexia and math in a comment, below.

By the way, there is a word for “dyslexia with numbers.” It is called discalculia. Please don’t self-diagnose yourself as discalculic just because you have problems with math. Discalculia is rare, and needs an in-depth diagnosis. It is a lot more severe than the symptoms I have mentioned.

Which leads me to my final point:
I am very grateful to Angela for letting us know about her daughter’s progress. People with dyslexia, ADD, etc. can be the luckiest people in the world. When you see a problem as a challenge, not as a handicap, you have an advantage over people who will go along in life never imagining what it is like to make a great improvement over terrible odds.

It is the “motivational dissatisfaction” with your situation that can lead you to take steps that the hoi-polloi only dream of. It is a kind of “sublimation.” Sublimation is a kind of coping mechanism (or defense mechanism). I recently read that it is considered the only defense mechanism that positive.

An example of sublimation would be the famous Charles Atlas, who was the original “ninety-eight pound weakling” who used to get sand kicked in his face at the beach regularly. Instead of not going to the beach anymore, he started lifting weights, became one of the fittest men in the world, and marketed the “Charles Atlas Fitness Program,” which made him rich and famous to boot.

So here’s to all you mental Charles Atlases out there!

Hey, Hey, Mama…

Answer to 47 down (damn, I really almost wrote 74 - true!): Plant of rock and metal = Robert. (Why, what did you think?) For you young ‘uns, Robert Plant is the lead singer for Led Zeppelin (By the way, I just made that clue up. I better make a note of it…)