Learn Icelandic: Kennarinn stjórnar sjálfum sér mjög vel. - The teacher controls himself very well.

The verb stjórna (to control or manage) always takes a dative object. Since the teacher is controlling himself, the reflexive pronoun must be in the dative case (sér) rather than the accusative (sig).

Why add sjálfum here? Couldn't we just say "Kennarinn stjórnar sér mjög vel"?

Yes, you absolutely could! Adding sjálfum acts as an intensifier. While stjórnar sér simply means "controls himself", adding sjálfum emphasizes that he is completely in control of his own actions and self.

How do we know to use the form sjálfum instead of another form of sjálfur?

The intensifying pronoun sjálfur has to agree with the noun it refers back to in gender, number, and case. It refers back to kennarinn (a masculine, singular noun), and it must be in the dative case to match sér. The masculine, singular, dative form is sjálfum.

If the teacher were a woman, would we change sjálfum to a feminine form?

No, it would stay exactly the same. The noun kennari (teacher) is grammatically masculine. Adjectives and pronouns like sjálfur must agree with the grammatical gender of the noun they describe, regardless of the actual person's biological gender.