GERUNDS AND GERUNDIVES
The gerund is a verbal noun, equivalent to an -ing word in English. A good way to identify it in English is to give it a possessive pronoun. If that feels natural, you’re probably dealing with a gerund. “My leaving home was sad for all of us.” “You should catch up on your gardening.” “Our clapping made them play louder.”
The Latin gerund is identical in form to the gerundive, but, unlike the gerundive, it only has four forms. It has no nominative and no plural. And it is always active, never passive. So for the gerund singing in Latin there is cantandum (accusative singular), cantandi (genitive singular), and cantando (dative or ablative singular). The genitive and the ablative work as you would expect, and the dative is used where a particular construction demands the dative.
The accusative has a special role, used with ad to express purpose. amor cantandi, “a love of singing,” is in the genitive. gaudemus cantando, “we rejoice in (or by) singing,” is in the ablative. But venimus ad cantandum, “we came to sing” is in the accusative with ad.