I'm thinking Grau is probably Grade (if you're in grade one, five or ten), Turma is definitely Class and Turno might just be Schedule. As for Série, I honestly don't know: could it be the student number? Hope this is some help!

Minha sugestão.

O problema com o termo "turno" é que todas as escolas dos Estados Unidos e Inglaterra, por exemplo, são em regime de tempo integral, e não existe tal conceito de turnos nas escolas, e creio que deve haver uma adaptação do termo.
Quanto à palavra "shift" é usada para turnos de trabalhadores de fábricas, ou call centers, por exemplo, que operam vinte e quatro horas por dia.

This would be a possibility, based on the reality in Brazil and the UK.
Turno is shift - In Brazil, different from other countries where education is given greater importance and children stay at school fill-time, schools have three (sometimes four) shifts, the usual names and times being Matutino (M) from 7 to 11 a.m., Intermediário (I) from 11 a.m. to 3 p.m., Vespertino (V) from 3 p.m. to 7 p.m. and finally Noturno (N) from 7 p.m. to 11 p.m.
Grau in England would probably be Key Stage, there are four Key Stages between Kindergarten and university entrance.
Série would be the year of schooling, e.g. 1a série do 1o grau (now Ensino Fundamental) which is for 7-year-olds, compared with 1a série do 2o grau (now Ensino Médio) which is for 16-year-olds.
Stream: due to the sheer number of students, often packed 50 to a class in some parts of Brazil, there are often several groups at the same level, selected at random or (rarely) by overall ability, distinguished by letters A, B, C and so forth, so 1a série A, 1a série B and so on. (Note: The practice of setting, dividing classes based on ability in different subjects, is not used in Brazil, not to the best of my knowledge anyway)