The author ever had investigated teachers of 136 elementary school mathematics: After normal school graduates, whether do you often read the article of theory of foundation of maths of concerned elementary school? Whether had you reviewed knowledge of middle school maths? Whether do you often gather mathematical information in the life? Findings makes clear: Only the teacher of 31% has read the article of theory of foundation of a few maths scatteredly; Besides deal with adult the university entrance exam, had reviewed knowledge of middle school maths without the teacher; Besides attend the open class, class that judge actor, only the teacher of 14% lives to count knowledge mediumly to inscribe alertly sometimes. Most those who make a person afraid is, in investigation actually some teachers think: Knowledge of elementary school maths is so simple, the teacher should spend energy on education design, not was necessary to fluctuate in course knowledge time. Is the viewpoint of these teachers right? The case below perhaps can cause our new reflection.

[case one] education of a teacher " natural press whether the even number that be by cent of 2 divide exactly and odd " when, let a student press list to large order as a child even number and odd, form following writing on blackboard, guide dug even number and odd characteristic next.

Natural:

Even: 0, 2, 4, 6, 8...

Odd: 1, 3, 5, 7, 9...

Master: Careful observation, can you discover what characteristic even number has?

Unripe: The smallest even number is 0, without the largest even number, the even number of two photographs adjacent differs 2.

Unripe: An even number it is infinite, a natural number 2 times be even number.

Master: How you know a natural number is 2 times even.

Unripe: Because natural it is to press an even number odd, it is an even number odd such order arranges, even number and an odd number are euqally much, say 2 times of a natural several since even number so, also be 2 times odd.

Master: Your eyesight is really fierce, see a problem very comprehensive, a natural number 2 times be even number really.

[think] the true subclass that even collect and odd collect are natural market. Apparently value like even collect and odd concentration a number of the element each occupy natural concentration element the half of several, even number compares a natural number with an odd number little, actually this kind of view is wrong. Because have the element in two gather in finite gather only how much do several ability compare, the element in perhaps saying finite aggregate closes a number that number compares the element in this aggregate true subclass certainly is much, and even collect, odd collect and natural market are infinite gather, the element in cannot comparing them to assemble so how many of several. For example natural with can build between even number one to one correspondence relation: 0 → 0, 1 → 2, 2 → 4, 3 → 6... N → 2n. This shows, natural to each, an even number answers relatively to it, is how you can saying a natural number 2 times even? This is regarding two ray as point-blank like, but 2 times what cannot say linear length is ray, perhaps say to grow than ray point-blank. So we cannot the experience that rise is accumulated inside finite gather limits, blind application goes in infinite gather. If the teacher has the knowledge of this respect, with respect to the mistake that won't produce scientific sex.
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