Angga ( 9 years ago )
it also means that a) I have to remember to put some on i have left the house sans pnetias before. and b) they need to be pretty!!! or hell, I could go buy clothes that fit.

Denize ( 12 years ago )
Subtracting a negative numebr is just a funny way to look at adding. This understanding comes prior to any high "concept" from the "cognitive science of mathematics," or philosophy of mathematics.Philosophy of mathematics is something mathematicians, who know their subject, dispute over after the heat of the day has passed. It's a way of elegantly organizing what they have learned. It doesn't help newbies learn, sorry.We're not really discussing maths. We're discussing how the mind learns. It isn't identical to how a subject is scientifically organized.A subject (such as maths) is scientifically organized "top-down," or deductively. Mathematicians have arrived at abstractions, and they use these to organize the field.BUT a student (any student, bright or mediocre) who is new to the subject is in a different position. He learns "bottom-up," or inductively. High abstractions cannot have any meaning to him, until they can be connected to more concrete material that he understands.Here is a simplistic example for the sake of clarity: How do you teach a child who doesn't already know it that 2 + 2 = 4?The good mathematician/good mathematics education major might answer in this fashion: "First, the child must learn the concept of sets; the sets of whole numebrs; the commutative property; and the concept of integers. I recommend that he read the chapter on sets first."But the good teacher would answer this way:"First, give the child two oranges, and hold a different pair of oranges in your hand. Next, have him count the oranges in his hand (2), and then the ones in your hand (2). Finally, give him your oranges, saying, 'My two added to your two is how many?' and have him count all the oranges (4)."The same distinction between "good science" (or a good education major paper) and "good teaching" applies, mutatis mutandis, to all subjects and all age groups. (But NOT all IQ levels - nothing can be taught to a congenital moron.)The simple is not explained by means of the complicated. The abstract is not explained by means of the more abstract.Mathematically, you are (as far as I can make out) correct about the concept of the closure property: this concept organizes known facts elegantly. The point is, students who don't know these facts will not learn them by being exposed exclusively to this piece of esoterica. A teacher will have superior results if he gets down on the ground level with his students and sees through their eyes. They have to WORK UP TO the closure property, not DOWN FROM it.