Learn Turkish: Öğretmen çocukları saydı. - The teacher counted the children.

The -ı is the accusative case suffix, which is used to mark a specific direct object. Because the sentence says the teacher counted 'the children' (meaning specific children rather than children in general), we must add -ı to the end of çocuklar.

In çocukları, which suffix attaches first: the plural or the case ending?

The plural suffix always attaches first. You start with the root çocuk (child), add the plural -lar (children), and finally add the accusative case suffix -ı to get çocukları.

How is the verb saydı put together?

It comes from the verb root say- (to count). We add the simple past tense suffix -dı. Because the subject is the third person singular (öğretmen), there is no extra personal ending needed after -dı.

Why isn't there a word for 'the' before öğretmen?

Turkish doesn't have a separate word for 'the'. A subject like öğretmen at the beginning of the sentence is naturally understood as 'the teacher' based on the context.