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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.
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.
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 fractionin 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.
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:
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.
The cartoon below is similar to a small part of the method I teach in “Numbers Juggling – Times without the Tables” e-book and course (see http://learn2multiply.com)
You’ll need a password for it. You can get the password by requesting it below
(After you fill out the form, hit the “back” button on your browser to return to this page)
What’s interesting about the comic is that the method works the same way, (using slightly different fingers for each number) and for the same reasons, but it comes to a very discouraging and misleading conclusion at the end. The “Numbers Juggling – Times without the Tables” e-book and course clear up the problem the cartoon describes.
Although the comic is funny and makes an ironic point at the end, I find disturbing and not entirely true. Read it for yourself, and then check out my notes at the end. Continue reading Geek-Speak →
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.
As many readers of the Math Mojo Chronicles know, my wife and I do the New York Times crossword puzzle together every Friday, Saturday and Sunday.
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. Continue reading How Puzzles may Improve your Mind →
As many readers of the Math Mojo Chronicles know, my wife and I do the New York Times crossword puzzle together every Friday, Saturday and Sunday.
