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Much depends on just how open those windows are. "All windows wide open in every classroom" during very warm or very cold weather is excessive. A window cracked open for ventilation, and either at the top of the classroom (warm weather w/ cooling active) or bottom (with heating), and a fan within the room for vertical mixing ... might help.
(Keep in mind that mixing air also mixes infectious particles.)
A slight improvement is to have a single exit point (vented with a fan) and rely on structure permiability to introduce fresh air. That's effectively a negative-pressure system.
From there, exhaust and intake with heat recovery is the next obvious step.
I mention older construction as replacing building stock is a long-term and expensive process. New construction will account for an exceedingly small percentage of all utilised buildings. New construction currently planned or in process should if at all possible incorporate lessons from the COVID-19 pandemic, but that's a drop in the bucket. Retrofits are highly likely.
If you want to read further, I'm finding numerous results for a search on school ventilation in new construction and retrofits:
https://duckduckgo.com/?q=+school+retrofit+construction+vent...
Among the first of those, from the CDC:
Reoccupying a building during the COVID-19 pandemic should not, in most cases, require new building ventilation systems. However, ventilation system upgrades or improvements can increase the delivery of clean air and dilute potential contaminants. Consult experienced heating, ventilation, and air conditioning (HVAC) professionals when considering changes to HVAC systems and equipment. Buildings that provided healthy, code-compliant indoor air quality prior to the pandemic can be improved for pandemic occupancy using less costly interventions. Below is a list of ventilation interventions that can help reduce the concentration of virus particles in the air. They represent a list of “tools in the mitigation toolbox,” each of which can contribute towards a reduction in risk. Implementing multiple tools at the same time is consistent with CDC’s layered approach and will increase overall effectiveness of ventilation interventions. These ventilation interventions can reduce the risk of exposure to the virus and reduce the spread of disease, but they will not eliminate risk completely.
...
Open windows and doors, when weather conditions allow, to increase outdoor air flow. Do not open windows and doors if doing so poses a safety or health risk (e.g., risk of falling, triggering asthma symptoms) to occupants in the building. Even a slightly open window can introduce beneficial outdoor air.
https://www.cdc.gov/coronavirus/2019-ncov/community/ventilat...
That cites an industry study by ASHRAE (professional association):
https://www.ashrae.org/file%20library/technical%20resources/...
It also recommends on exhaust fans rather than interior circulation fans to minimise strong internal air currents, with intake provide as I've indicated above.