In the 1930s in Scotland any child aged 8 – 16 could be sent to an approved school by a juvenile court. These schools were renowned for their harsh discipline. Corporal punishment was frequently used involving the cane, strap and sometimes the birch.
I think birching was probably used on for the most serious offences (of which absconding was considered one).
Your Grandmother would have been flogged on her bare behind with a “birch-rod”. This would consist of several long switches probably of hazel or willow rather than birch. These would be trimmed and soaked overnight in water/brine to make them more supple and whippy. The switches would then be bound together with cord in the middle and forming a grip at one end. The school gardener may have charged with the job of fabricating such an implement, although I doubt he ever saw it put to good use!