Chalk was a drawing and writing tool.
In Castrovalva, the Fifth Doctor used it to draw a map of the town. (TV: Castrovalva) When he visited Sair, he found chalk drawings on the floor of his room, and traced them with a piece of chalk to reveal their shape. (PROSE: Fascination)
The Twelfth Doctor found a piece of chalk in his bedroom, in the house of the Paternoster Gang, and acting irrational from post-regenerative trauma, used it to write equations on the floor. (TV: Deep Breath) Later, when the trauma had subsided and he had become collected, he kept on writing on a blackboard in the TARDIS control room. (TV: Into the Dalek onwards)
While talking alone in the TARDIS about the possibility of never truly being alone, the Twelfth Doctor wrote various points on his chalkboard. After placing the chalk on a book, he returned seconds later to find it missing and that someone or something had used it to write the word "LISTEN" in what looked like the Doctor's handwriting onto another board. When the Doctor told Clara he couldn't have just written it and forgotten, Clara asked him, "Have you met you?" (TV: Listen)