Gender: boy
School:
School #13B
"Calek, Chaim" came the closest to "Chaim, Cale" and fell into the same category of birthdates for this particular school. After entering the name into jewishgen.org, it gave more detail about this student being in Lodz ghetto. Although occupation did not say "student", it said "sohn", which is german for "son." I hope this is the right one!
I googled "Chaim Calek" and an article on ushmm.org came up for a child who was in the Lodz ghetto by the name of "Chaim Cale", which is the exact name of the signature and had the same school listed. I cannot find any other information on the Name Search database or the JewishGen databases.