Apostrophe ( ' )

• The apostrophe is used when leaving out a letter or number in a contraction, e.g. can't, wouldn't.
• The apostrophe is used for omitted letters, e.g. rock 'n' roll, and for omitted numbers, e.g. the class of '72, the '20s.
• The apostrophe is used for plurals of letter abbreviations with periods and single letters, e.g. p's and q's, two A's and four B's. Plurals of multi-letter combinations and plurals of numerals end in s with no apostrophe, e.g. VIPs, 1000s.
• The possessive of singular nouns ending in s, including nouns ending in s, x, z, ch, or sh, is formed by adding 's, e.g. witness's affidavit. However, if the next word begins with s, then add only an apostrophe, e.g. witness' story.
• The possessive of singular nouns not ending in s is formed by adding 's, e.g. VIP's seat, baby's food.
• The apostrophe follows the s of a word with two sibilant sounds, e.g. Kansas', Moses'.
• The apostrophe is added for the possessive of a noun that is plural in form but singular in meaning, e.g. mathematics' formulas.
• The apostrophe follows the s for the possessive of plural nouns that end in s, e.g. girls' movies. For the possessive of a plural noun that does not end in s, add 's, e.g. women's rights.
• For singular proper nouns, add only an apostrophe for the possessive, e.g. Achilles' heel.
• No apostrophe is used for personal pronouns like hers, his, its, mine, ours, theirs, whose, your, yours. Indefinite pronouns require an apostrophe, e.g. one's lover. For other pronouns like another and others, follow the rule for singular and plural, e.g. another's and others'.
• For joint possession, the 's is added to the word nearest the object of possession, e.g. Francis and Kucera's book.
• The apostrophe is not used in names of organizations unless actually part of the legal name. The apostrophe is not used in plurals of numerals or multiple-letter combinations

Possessives

The possessive case of most nouns is formed by adding an apostrophe or an apostrophe and 's'.

• Possessive for singular and plurals nouns not ending in an S or Z sound are formed by adding 's. Examples: horse's, alumni's
• Possessive of singular nouns ending in an S or Z sound are usually formed by adding 's, e.g. hostess's, unless the next word begins with an S or Z sound.
• Possessive of plural nouns ending in an S or Z sound are formed by adding only an apostrophe, e.g. churches'.
• Possessive of plural nouns that are singular in meaning are formed by adding only an apostrophe, e.g. mathematics' rules. This is also true for a plural word in the formal name of a singular entity, e.g. General Motors' profits.
• Possessive for noun that is the same in singular and plural - is formed as if it is plural, e.g. two deer's tracks, one corps' mess hall.
• Possessive for singular proper noun ending in s - use only an apostrophe, e.g. Achilles' heel.
• Possessive for pronouns -- only for a few, such as: another's, others', someone's. There are no separate forms for the possessive for: mine, ours, your, yours, his, hers, its, theirs, whose.
• In a phrase: individual possession is shown with an 's added to each noun, e.g. 'Barbara's and Kyle's bicycles'; joint possession is shown by adding an apostrophe or 's to the last noun in the series, e.g. 'Barbara and Kyle's house'.
• Be careful of descriptive phrases. Do not add an apostrophe to a word ending in 's' when it is used primarily in a descriptive sense, e.g. citizens band radio, writers guide.
• An inanimate object can have a possessive use. It is treated in a personified sense, e.g. Time's cover.

Forbes wrote:

"Mathematics" is not possesive here and in any event functions as singular . One says "mathematics is a science". I would write "mathematics teacher" without any apostrophe, just as I write "music teacher". I just add that "mathematics(') rules" sounds a bit odd - I think I would say "the rules of mathematics".

The children's teacher was late to class.

Is your question why it shouldn't be "the childrens' teacher..."?

The rules of English are just like that -- words like children or people, which are plural, take a 's, not a s'.

Alienvoord wrote:

According to some sources, like the Oxford Companion to the English Language, the apostrophe used to mark the plural with acronyms (VIP's), decades (1970's), and family names (the Jones's) is standard.

It's highly nonstandard in my experience. Granted, different countries have different styles.

You say that "The Davis's are all coming to dinner" is the correct way to write something that says that the entire Davis family is coming?