Saraswati begins as water. In the Rigveda she is a river, and the hymns to her are hydrological before they are anything else: she is best of mothers, best of rivers, best of goddesses, she breaks mountain ridges with her force, and she nourishes the people along her banks. Nothing about learning appears yet.
Where that river ran is one of the most contested questions in Indian studies. The hymns place it between the Yamuna and the Sutlej, and it is commonly identified with the dried bed of the Ghaggar-Hakra system in Haryana and Rajasthan. The identification is argued and unresolved, and the reason it is fought over is that a great deal follows from it: the date at which the channel carried water bears on the date of the hymns, and both bear on the long-running dispute over the relationship between the Vedic tradition and the Indus Valley civilisation. The atlas records this as an open question with political stakes and does not resolve it.
The move from river to word is traceable in outline. Vac, speech, is a goddess in her own right in the Rigveda, and a hymn is spoken in her voice. Saraswati and Vac converge, and by the Brahmanas the river goddess has become the goddess of speech, and thereafter of learning, music, poetry and all the arts. What survives of the river is the flow: the tradition reads inspiration and articulate speech as things that move and cannot be dammed.
Her classical form carries no weapon, which is uncommon. She is white, seated on a white lotus or a swan, holding a vina, a manuscript and a rosary, and one hand is often free. The swan is said to be able to separate milk from water, and stands for discrimination.