Simple Initial Mitigation Strategies
Consideration should be given to separation walls between classrooms (and between classrooms and hallways) to provide construction details that inhibit transmission through the wall. Strategies can include multiple layers of drywall to increase mass, sound attenuation blankets (insulation) to absorb sound, as well as staggered stud construction and even separating sole and top plates to inhibit diaphragmatic action between sides of the separation wall.

STC of various 5/8-in. gypsum wall constructions. Note that this is intended as a guide and are not a substitute for reliable field and laboratory test data.
Ken Roy has suggested another table of light gauge studs here.

STC/IIC of sample ceiling/floor constructions. Note that these are intended as a guide and are not a substitute for reliable field and laboratory test data. From Cavanaugh, "Architectural Acoustics", 1999
Pay Attention to how sound travels
Attention to HVAC duct layout can have a dramatic impact on containing sound within classrooms and minimizing transfer from one room to the next. Running trunk ducts in corridors and branching off to classrooms is far more effective than running trunks lines from room to room where sound can easily be transmitted even through insulated ducts. See the Mechanical Section for more specifics on limiting the impact of HVAC design on the acoustical environment.

HVAC Layout: the main trunk is in the corridor, and branches are offset. Note VAV in the corridor and offset doors as well.

Fabric covered fiberglass panels. (source: acoustical surfaces, inc.)
Consider Absorbent Surfaces
Acoustical Ceiling tiles, Fabric-wrapped panels, Manufactured acoustical panels, Soft seating, drapes, carpeting, blinds (some blinds are designed for sound absorption), etc. all play a role in dampening sound and providing a comfortable acoustical environment.
Consider Diffusive Surfaces:
Breaking up surfaces architecturally by insetting windows, using columns inside a space and other architectural details, bookshelves (with books) and workstations with dividers at ear level will “soften” the sound in the room and provide modest absorption compared to flat hard painted walls. Diffusers are commercially available.
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