What does Mathin show her about fire, what can he not teach her and why and what do they do after Harry's sixth week of training?

Mathin shows her three ways to make fire, but admits Corlath can show her a fourth. She realizes there is much Mathin cannot tell her, because he lacks kelar, but he does tell her much of Lady Aerin. The sixth week arrives, and the two begin heading toward the City, where the laprun trials are held.