Posted on 12/04/2002 9:41:55 AM PST by Lizavetta
The problem with "the problem" is that it tries to get the kids to "learn" something backwards from the way it would normally be done. Specifically, the problem is designed to get the students to "infer" some of the rules (Axioms) of arithmetic from the example given. Of course, in the real world of Mathematics (as taught in decent Universities), the first thing you are given is the Axioms of the MAthematical system you are working in, and then you DEDUCE various principles (Theorems) FROM THE AXIOMS, not the other way around!
In the example, the problem reduces down to the algebraic expression:
Which when you apply the axioms for arithmetic, reduces down to
In which the variable "x" always cancels out of the equation, leaving "4" no matter what you started with.
A house is built from the foundation up, not from the roof or living room down. Mathematics works the same way; you start with the axioms (foundation) and derive the rest of the structure therefrom. You don't start from a black box and infer what the rules (axioms) are, which is what this problem is doing.
This problem would be useful ONCE the student has learned the rules of Arithmetic, but is a waste of time as a mechanism for the student to learn the rules.
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Few people here under the age of 50 have probably had real "real math" as it was once taught in this country. In 5th grade we measured and calculated the area of the schoolground in acres. In high school I had the problem, "If you drop a ball off a cliff and hear it hit 20 seconds later, how high is the cliff?" With my old math I was able to write a 1,000 line computer program to analyze horse races. I was able to derive some of the fundamenal equations of calculus such as pi Rsquared before reading that chapter in the book. I can derive a Pearson product moment around a curvilinear form.
The reason for this is that these teachers don't like the subject of mathematics, they never have (that's why they became teachers instead of, say, engineers), and consequently they find it difficult and painful to teach. They don't even understand what they are teaching well enough to do so in an interesting way, so the students lose interest.
Of course, from such a teacher's point of view, the problem must be the textbook and the "way" in which the mathematics is being taught, and the textbooks. It couldn't possibly be that the teacher is a bonehead at mathematics. Nope.
The sample question was a fine word problem for students who have already learned the basic underlying algebra concepts.
I shudder to think what goes on in regular (not the smartest) students' minds if this problem is shoved in their face before they've learned to grasp equations like "x+4=10", however.
Fuzzy math isn't about correct form but correct answers! Fuzzy math algorythms are used in the auto focus function on you camcorder. Real world correct answers are more important than form! I've seen/created great business models. I've also seen the failure of the enterprise regardless. I worked for a company which used my budget models to chart its course from a $100K/month revenue stream to $30 Million per month. We lost focus and just before a downsizing, (and eventual hostile buy out by the competitor who had forced this issue), our VP of Sales/Marketing owed up to his inattention and stated, "Well, I guess I'll have to give up my Friday Golf Games!!!"
He had mucho salary,commissions and stock options and he fled before the fall. The models were good the execution was crap!
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Amen. It's converting math to a group-think love-in with people dependent upon each other producing answers equal to the sum of their individual fears and incompetences.
Sadly this is sometimes true - especially in elementary school. Elementary teachers are often language arts people, not math. I am a math teacher. My son-in-law is an engineer. I love math for being math, he loves math to use it.
The one reason more teachers don't love/teach math is money. After 20 years, my salary as a classroom teacher is less than half of his with less experience.
My niece was having trouble with math (she was in 3rd grade, now 4th), and her mother asked me to help out. One "problem" was that she still used her fingers for math problems. Her mother wasn't too pleased when I encouraged her to continue with this - use the tools you've got at hand, so to speak... Anyway, the trick was, as I saw it, to let her use her fingers, just make the problems more difficult, so you've got to be creative.
To put this in perspective, I have a doctorate in physics, but I still use my fingers regularly when solving cross products, just to see the right-hand rule. I almost always wrap my fingers around when doing E&M problems to figure out how the magnetic field will affect things, etc. This is common in physics, we see it all the time. I see no reason a grade school student can't use the same tools.
Anyway, now she's in 4th grade, and we play a game called Nemo. Here's how it goes: Take toothpicks, as many as you want. Make an arbitrary number of piles of toothpicks, with as many as you'd like in each pile (hopefully none more than 31, or you'll need both hands for the solution). On your turn, you may remove as many toothpicks as you would like, but only from a single pile. The winner is the person to pick up the last toothpick.
Now, if you're good enough at math, adding and multiplying by 2, you can solve the problem right from the beginning, ang guarantee a win. Here's how: For each pile, figure out how many toothpicks are in the pile, and break it down into powers of 2. For instance, a pile with 19 toothpicks would be 16 + 2 + 1, or 2^4 + 2^1 + 2^0. Now, let each finger on your hand represent one of the powers of two - we usually let the thumb be 0, and the pinky 4. Put down each finger represented in that sum. Now move to the next pile, and do the same thing. Only, this time, when you're moving your fingers, put it down if it was up, and up if it was down. For instance, say our second pile has 7 toothpicks, or 4 + 2 + 1, or 2^2 + 2^1 + 2^0. Then, our thumb and index finger, which were down, go back up, and our middle finger, which was up, goes down. Continue this until you have "done" every pile this way.
If all your fingers are up, and it is your turn, you will lose unless your opponent makes a mistake. Period. If you have any fingers down, you will be able to win. To figure out your move, do the calculation again, excluding the largest group. See which fingers you have down, and use the sum above. That's how many you want to leave in the largest group. So, using the above examples, where we had 19 and 7, we would just do the 7 - leaving out first 3 fingers down. Adding them up leaves 7 (this is a simple example, we usually play with 5-20 piles), so that's how many we want to leave. We remove 12 from the larger pile, leaving 7, and will win the game. In future turns, you always want to leave your opponent with the symmetric solution (all fingers up.) At some point, he will have to leave you with all the toothpicks in 1 pile, which you then pick up and claim victory.
This makes heavy use of your fingers for doing the math, but you can't just count. My niece plays quite well, and she's also doing great in math now, since she gets so much practice. So don't knock using your fingers...
Drew Garrett
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Make that 60.
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