Photo by selma90

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

http://duckcover.blogspot.com/

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

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

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

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

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

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

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

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

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

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

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

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

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

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

That’s

• x + x + 31 = 91.

That’s the same as 2x + 31 = 91.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Brian (a.k.a. Professor Homunculus)

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

What is the difference between Fractions and Rational Numbers?

Fractions and Integers

A  fraction (common, or vulgar fraction) is a number that expresses part of a whole as a quotient of integers (in the form of m/n) where the denominator (divisor) is not zero.*

OK, what’s a quotient of integers? Funny you should ask that. That term is used all over the web and in math texts, but is almost never explained.

A quotient is the result of a division problem.

An integer is all the whole numbers, including zero, or their negatives. (… -3, -2, -1, 0, 1 ,2 ,3…)

A quotient of integers is simply two integers that are being divided to obtain a quotient. In the example 6/3,  2 would be the quotient, and 6/3 is the quotient of integers.

So a quotient of integers is a number expressed like 3/5, 8/4, etc. In mathematics, a fraction must be expressed like that.

Another way to say it is, a  fraction (common fraction) is a division expression (in the form of m/n) where both dividend and divisor are integers, and the divisor is not zero.

There are different forms fractions can take. These are some basic descriptions:

• A proper fraction is a common  fraction in which the numerator (also referred to as the dividend, or the “top number,” although the term denominator is generally preferred when referring to fractions) is greater than the denominator (also referred to as the divisor, or “bottom number’” although the term denominator is generally preferred when referring to fractions.) Example: 3/4 (three fourths).

• An improper fractions is a common  fraction in which the denominator is greater than the numerator Example: 4/3 (four thirds). This can be reduced to 1 1/3 to make a mixed fraction (see below.)

• A mixed fraction is a common fraction that is combined with an integer. Example: 2  ⁵/₆ (two and five sixths).

All of the above fractions are considered simple fractions, as opposed to complex fractions (see below).

• A complex fraction (or compound fraction) is a fraction in which the numerator or denominator contains a fraction. For example, ^1/3/_3/5 (one third over three fifths)

It is important to know that the term “fraction” is not a term most mathematicians use. It is a tool used by math educators and teachers.

Rational Numbers

A rational number is a number that can be expressed as a quotient of integers (where the denominator is not zero), or as a repeating or terminating decimal. Every fraction fits the first part of that definition. Therefore, every fraction is a rational number.

But even though every fraction is a rational number, not every rational number is a fraction.

Consider this:

Every integer is a rational number, because it can be expressed as a quotient of integers, as in the case of 4 =  8/2 or 1 = 3/3 or -3 = 3/-1 and so on. So, can integers such as 4 or 1 be expressed as the quotient of integers? Yes.

But an integer is not a fraction. 4 is an integer, but it is not a fraction. Is 4 expressed as the quotient of integers? No. The difference here is in the wording.

A fraction is a number that expresses part of a whole.  An integer does not express a part. It only expresses a whole number.

A rational number is a number that can be expressed as a quotient of integers, or as part of a whole, but fraction is a number that is (must be) expressed as a quotient of integers, or as part of a whole – there is a difference. The difference is subtle, but it is real.

I’ve seen the definition of fraction described as many things, including, “A fraction is the ratio of two whole numbers, or to put it simply, one whole number divided by another whole number.” This definition also shows that an integer is not a fraction, because an integer is not a ratio. It can be expressed as a ratio, but it is not a ratio in itself; it can be divided by another whole number, but it is not being divided.

This kind of logic can be a little hard to understand, so let me make an analogy:

A person can be a chemist. But the definition of a person is not “a chemist.” A person is only a chemist when s/he is fulfilling a set of requirements that are beyond the definition of what a “person” is. So a person is not automatically a chemist. A person can be a chemist, but there is a difference between the Idea of “can be,” and the Idea of “is.”

Marie Curie was a chemist. Marie Curie was a person. Joe the plumber is a person (allegedly) but that does not make him a chemist. Therefore, the definition of a person is not necessarily “a chemist.”

The same is true of an integer not necessarily being a fraction, or a rational number not necessarily being a fraction.

So in a nutshell, the fractions are a subset of the rational numbers. The rational numbers contain the integers, and fractions don’t. Please remember that that is just the “nutshell” and not the definition, or the explanation of the difference, it’s just a rule of thumb. And keep in mind that “fraction” is not a term generally used by mathematicians. They don’t need it. They can speak in terms of rational numbers and integers.

* According to Wikipedia http://en.wikipedia.org/wiki/Fraction_(mathematics“A fraction (from the Latin fractus, broken) is a number that can represent part of a whole.” This definition is a little ambiguous. I believe the word “can” was used indiscriminantly – the word should have been “does.”*) I know this sounds like I’m splitting hairs, here, but when you use the word “can” in a definition, you are not really defining the word. A good definition implies definite  – at least as definite as you can be – (hence the name), not just possible.

The reason “can” is used in the definition of rational numbers, is because that is as definite as you can get. It is an inherent part of it’s definition, just as in the definition of “bendable” meaning “can be bent.” Something that is bendable doesn’t have to be bent, but it can be. In its unbent state, it is still bendable, just like an integer is rational number because it can be expressed as a ratio of two integers.

There are more definitions for most terms than any one person could ever know. Here’s an interesting one from Bertrand Russell

We shall define the fraction m/n as being that relation which holds between two inductive numbers x, y when xn=ym. This definition enables us to prove that m/n is a one-one relation, provided neither m or n is zero. And of course n/m is the converse relation to m/n.

- Introduction to Mathematical Philosophy  Page 64 Bertrand Russell (1919)

Remember to take all definitions with a grain of salt. The more you learn about math, the more you’ll find that some of the things you “know” have different meanings in different contexts. It can be very confusing. Understanding comes with maturity and keeping and open mind. Unfortunately those are things that schools do not value (there are no standardized tests for them).

Related posts:

Math Definitions

Explaining Math Terminology

More on Math Terminology Mis-Explained

Afterthoughts:

Some confusion can exist about such numbers as 22/7 and 1/3, as their decimals go on forever. Sometimes people misunderstand this to mean that those numbers are irrational. But irrational does not mean never-ending.

Both 22/7 and 1/3 are fractions, therefore they are both rational numbers. They can also be expressed as repeating decimals, as 22/7 = 3.142857142857142857… (notice that the 142857 repeats) and as  1/3 = .333 …

An irrational number, on the other hand, neither terminates nor repeats. If you’d like to know more about the irrationals, check out The Math Mojo Monthly Newsletter Issue #11. It will be published soon after this post is published. The only way to get it is to sign up for it. You can sign up for the Math Mojo Monthly on the top left navigation bar of this page.

(The confusion about 22/7 may come because that fraction is often used to represent the number pi. It is not the number pi, just an approximation. The number pi is a decimal that begins 3.1415… and continues on without terminating or repeating. )

Post-afterthoughts:

I just consulted John H. Conway and Richard K. Guy’s wonder-filled  The Book of Numbers. On page 25 he implies that fractions and rational numbers can be used interchangeably.  I’ve been asserting that fractions are a subset of rational numbers, because, although integers can be expressed as fractions, they themselves are not fractions.

This leads me to believe -

• a) I am utterly mistaken and presumptuous, or
• b) hell has frozen over and I may be right on some minor point concerning anything at all.

So, I may have shot myself in the butt with this whole article. I don’t think so, though. Even the big guns can miss the target occasionally. (See More on Math Terminology Mis-Explained ).

I’ve been  trying  vainly (in both senses of the word I suppose) to reach Dr. Conway to clear up this discrepancy. If any readers should have a conduit to Conway (a conduitway?) please let me know.  If my post is to be ripped apart, it should at least be by the best.

I’m still betting on Professor Homunculus, though.

Hotcha!

Brian (a.k.a. Professor Homunculus)

A challenge for you

Using the logic above, and your own reasoning and calculating skills, can you prove that every complex fraction must be a rational number? If you can, leave your proof in a comment below.

“Don’t fall into the trap of standard education – keep an open mind, and keep exploring. The more you understand, the less you know, but at least you know that.”

Click here to enlarge image

Ever obsess on something trivial when you know you have a ton of work due about an hour from yesterday? Here’s my obsession for today:

In the last Math Mojo Monthly Newsletter (look up at the top in the left menu bar of this post to make sure you get it. It’s free, of course) I posed this puzzle:

What is pizza = 307.72” ?

I just wrote the answer for this week’s edition of the Math Mojo Monthly (hurry and sign up for it now, so you’ll get the issue when it comes out, or else you’ll miss it – nudge, nudge!)

I got some interesting answers back, but the two most representative ones were:

The circumference of a circle whose diameter is 49 ” will be 307.72″, taking pi=3.14 So that is the size of the pizza — a 49″ pizza !!!

To which I wrote back:

Good try. How do you figure, though? Pi times diameter wouldn’t give you 307.72, would it?

Hint: Remember, this is a puzzle. There is a puzzle element at work here. What would make the question clever?

The reply came:

Oops. I meant to write 307.72/(2pi) = 49. So Area = 2pi*r² means a 7″ pizza. That’s your pizza.

Which was the same as the other representative answer, namely:

307.72 is the surface area in square inches of a pizza or circle with a radius of 7″

To which I replied:

Keep trying. You’re not too far off. Once you get it, you may get the “aha!” moment that the puzzle is actually after.

Both were noble attempts, but a little checking would have shown the errors. Plug those figures into the formula for a diameter of a circle, and you won’t get 302.72 . Each of those pizzas would have been pretty far off from the diameter of a standard pie, though. That could also have been a clue that they were off.

Here’s the reason I wanted to write this post,though. The first set of answers came from a mathematician. This gentleman is leaps and bounds ahead of me in his understanding of mathematics (I’m not a mathematician, just a math dilettante). He got the answer wrong. My point is: So, big deal? He was also the first one to write in. He had the interest and the effort going for him.

Too often schools surpress interest and discourage effort by their rush to judge students by testing them. It’s that false “accountability” platitude.

So I wrote back to the gentleman and we had a nice dialogue. I hope he communicates on this blog often, because his insights would certainly benefit readers.

This is what I wrote back to him, concerning his answers. I think it may be motivating and encouraging to some people, so I’m reproducing it here:

One of the things I am most concerned with about math are common mistakes, and why people make them.

Too often students feel intimidated about making mistakes. I think standardized testing is what kills this spontaneity and curiosity. I could be wrong, but I think that’s the way to bet.

One of the missions of Math Mojo is to relieve some of the fears and stress of students.

I think a great way to do that is to show them that anyone can make simple mistakes, and that there is no shame in that. The shame is simply not trying. There is no shame in trying to making honest mistakes.

I am frequently a great example of this. I make plenty of mistakes on my website. It has a great advantage for me, which I did not expect. Every time I make a mistake, I can count on more comments to that blog post than usual! I don’t have to make the mistakes on purpose — because I make enough without trying.

A problem I have when posting puzzles, is that many people, especially young people, assume that I know what I am talking about, and I am the guy who makes up the puzzles, and I am some kind of genius. I think they assumed that I could solve these puzzles each time I see them.

Nothing could be further from the truth. I found a version of this puzzle on the Internet. The answer was right there with the question, so I didn’t actually have to solve it. I just found it very clever and interesting.

To tell you the truth, I don’t think I could have solved it myself. But I certainly would have tried, and probably wasted a whole weekend doing it. Of course it wouldn’t really have been wasted, it would have been fun, and I’m sure I would have learned many things along the way.

Which gets me to the point of why I’m telling you this: I think you may have made another honest mistake.

As you know, the formula for the circumference of a circle is pi times diameter. It could also be pi times two times the radius (because twice the radius is the same as the diameter). C = pi*d, or  C = pi*2r.

The formula or the area of a circle is pi times the radius squared. A = pi * r².

I think the confusion stems from the fact that 2r looks like r². But they are not the same. One is two times the radius, and the other is the square of the radius.

In second answer, you  said, “ Area = 2pi*r²“.  There is an extra, unneccesary 2 in that equation. I think the extra 2 might be due to the above explanation.

It is a totally common mistake, that even a mathematician can occasionally make. I would like to let my readership know that there is no reason to feel that someone must be “bad at math” to make such mistakes. Clearly, if a mathematician can have a foggy day, then so can others. It doesn’t mean you’re “bad at math.”

When I explain the solution to the puzzle in the next Monthly, I think you will find it very entertaining. I’ll bet you will try it on some of your mathematician friends in the future.

I really appreciate our dialogue, and I thank you for playing the puzzle in the Math Mojo Monthly. I hope you keep working on it, because I really appreciate talking with an actual mathematician occasionally. It keeps me on my toes.

This all brings me back, once again, to the value of puzzles. Puzzles are a great place to be able to make mistakes. You don’t get graded on them (unless you have a teacher who’s making a great pedagogical mistake). Puzzles are a safe place to practice and  improve your intellectual ability.

Another benefit of puzzles is that they are not always straightforward. They test and hone your ability to think laterally, creatively. This wasn’t just a simple equation. It was a challenge to think differently. It could have been solved without doing any equations at all if you thought like a puzzler instead of a mathematician.

The mathematician might only be looking at the numbers, where a puzzler may have been looking from a wider perspective, and seen the letters, and recognized right away that the “pi” in “pizza” might stand for π

Maybe I’m giving away too much here, but you might think about what the double z could stand for. And don’t forget the a.

I won’t be giving away prizes for anyone who solves this (I offered one in the last Chronicles, but nobody won). But if you have an complete answer, send it in before the end of the weekend and I’ll give you an “honorable mention” in the upcoming issue of  The Math Mojo Monthly newsletter. (Wow, it just doesn’t get any better than that!)

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 .

You may also be aware that the Friday and Saturday NYT Crossword puzzles are harder than the Sunday puzzle. Sometimes much harder.

I don’t know if I could solve many of them alone, but together, we are a pretty good team. Mimi can brainstorm and come up with things I could never get, and I can help filter out some of the wild Ideas she comes up with that would jam the puzzle.

By “jam,” I mean, if you put in an answer that turns out to be wrong, it will mislead you from getting the answers that cross that answer in the puzzle. It can send you down wrong paths, and keep you from noticing the right ones.

One thing that we’ve found on many puzzles, at least over the last few years since we started doing them, is that there will often be a mention of something that appears in the NYT that day. Often it’s from the magazine section.

Unfortunately, we can’t get delivery of the NYT where we live (too rural). I used to get the paper from the truck stop each Sunday, but I stopped a few years ago. So we don’t get a lot of the references anymore. (Stay with me – this will be relevant soon.)

We usually do the puzzle during meals, and we get Sunday in one meal, on average, and Friday or Saturday usually take two or more meals. That leaves lots of meals during the rest of the week without puzzles, so we go back into the NYT online puzzle archives when we run out of puzzles. We subscribe to the puzzle section of the NYT online. It’s about $39 a year, and totally worth it.

This week we were doing the puzzle for Friday, Nov. 29, 1996 (from the archives). We almost finished it by the third meal. But there was one particular spot that was getting us. 24 down was: “Heraldic bands.” We have no Idea what this could be.

We had all the letters for it except for two. The first was the second from the top, which crosses 28 across: “U.S.D.A power agcy.” We originally thought it must be the F.D.A, but the second letter in it is definitely “e.”

The other missing letter was the second from the last letter, which crosses 39 across: “Els with tees.” We had all of the other letters in 39 across. “E ,R, N, I, ___. Yes, I know some of you know this right away, out of context. But we were totally lost. We know nothing about the context. Of course we could guess at “Ernie” and we imagined it would be right. But we had no corroboration.

Back to the point I made above, about how sometimes you get hints from what’s in the newspapers that day. This is a very cool thing the NYT does. It reinforces what you’ve read that day. And this is one of the things that I believe puzzles help your mind with. Any time you cross-reference information in your mind, you are building connections. These are sometimes referred to as “neural pathways.” You can read more about such things here:

• neural pathways: http://www.neuralpathways.org.uk/articles/repetition.htm

• neuroplasticity: http://en.wikipedia.org/wiki/Neuroplasticity

(Please keep in mind that I am not an expert in this field, and you should take these thoughts and websites as a “jumping-off point” and not as Gospel.)

As we did the Friday, Nov. 29, 1996 puzzle, we figured we could get no help from the current newspaper. We had started the puzzle two days ago.

So how surprised was I when my wife brought home the paper from the nearest “city” (population: about 15,000, Motto: “City of Pizza and Beer – We Used to be Somebody, but then they took the Trains Out”) and there was an article in the sports section which gave us a hint? We normally don’t read the sports section – but Mimi scanned the caption of the photo on the cover. The picture was from this week’s U.S. Open at  Pebble Beach. There were two golfers kneeling and checking out a shot.

Amazingly, the caption started, “Tiger woods and Ernie Els Line up putts on the 12th hole…”

“Els with tees.” Ernie Els, the golfer! Get it?

Here’s the thing – If we’d have simply gone with our “instincts” and put in Ernie to begin with, we would have forgotten about it later, because we would have given ourselves the feeling that we had “solved” it. We may have gotten the right answer, but it would have been a dumb guess, and we would have learned nothing.

By not taking the easy way out, and staying curious, we learned something. This was a shining example of how that works.

Some people may think, “Hey, big deal – so you learned something about a sport you don’t care about.” But they’d be missing the point. We are developing a healthy habit. By not being mentally lazy, and keeping our curiosity open instead of giving ourselves credit for “solving” something without understanding it, we are creating more neural pathways. Not just this time – it is a habit.

Doing puzzles is a perfect way to develop such habits. Think of it as like playing a sport. There really is no sense in hitting a little dimpled ball around with a crooked stick. But you’d be developing patience, hand-eye coordination, and the ability to handle hours and hours of boring nonsense. (Whoops… sorry golf fans.)

As much as I like puzzles, I’ve come to think of math as about the ultimate puzzle. Not in the sense of it being confusing (because by its nature it aims at reducing confusion and creating clarity), but in the sense that it has logical rules but requires creativity to understand and use in any meaningful sense .

If you like to sharpen your mind, make yourself more mentally resilient, learn something useful, and have a good time doing it, I don’t think you can do better than learn math and mathematical philosophy.

P.S. -  After getting Ernie there was one space to go. We were stumped. We’d have to guess. So we did. In other words, we didn’t fully solve this puzzle. We had to “cheat” and check the answers on the puzzle we’d downloaded.

What’s an orle? Check out this picture. The blue border is the orle. I told  you Friday puzzles could be hard!

Did I say hard? How about diabolical? Although REA is the answer to “USDA power agcy.”  I still had to do some serious Googling to find out what that meant. it turns out that it’s the Rural Electrification Administration one of the New Deal agencies created under President Franklin Delano Roosevelt, which was the foreunner to the Rural Utilities Service.

I also checked out Ernie Els. They guy was competing fourteen years ago when this puzzle was made. Now he is tied for second place in the U.S. Open. Pretty impressive. Maybe more impressive is that he has established the charitable organization, Els for Autism Foundation.

P.P.S. - There is another Math Mojo Chronicles post along these lines at Crossword Puzzle Digression. It mentions a good resource to help you understand the logic of the answers of the NYT crossword puzzle. Unfortunately that resource did not exist in 1996, so I was reduced to googling this time .