Counting

前置知识

1 项必需的前置知识（一直追溯到底）

你将学会

• Count a collection of up to about 100 objects, saying exactly one number word for each object.
• Explain that the last number word said tells how many objects there are, whatever order they were counted in.
• Compare two collections by pairing their members one-to-one and by counting.

定义

• Counting matches each member of a collection with exactly one number word, in order, without skipping or repeating.
• Two collections have the same number of members when their members can be paired up exactly, with none left over.
• Zero is the number of members of the empty collection.

定理

• The number of members does not depend on the order of counting or on how the objects are arranged.
• If the members of two collections can be paired with none left over, counting them gives the same number.
• Every counting number has a next one, so there is no largest counting number.

常见误解

• Thinking that spreading objects out makes more of them.
• Counting an object twice, or skipping one, when the objects are not in a line.
• Believing the last number word names only the last object rather than the size of the whole collection.

通向

可选发现

• Countability and Cantor's diagonal argument Pairing members one-to-one, rather than counting, is how mathematicians compare the sizes of infinite collections.

学习路径

• Conventional K–12 progression

延伸阅读

• Hung-Hsi Wu, Understanding Numbers in Elementary School Mathematics (2011)