Yes. The rules will depend on the rules of the pension fund, though. In Illinois,
If you return to state employment on a contractual basis, […] or for the private sector, your SERS benefit will not be affected.
So to continue employment, the superintendent would need to go to a different state, or work in the private sector. In general, this is the same for most states and true of the private sector (there's nothing stopping you from taking out of a retirement fund given you meet all the requirements to draw from it).
In fact, the state of Alabama had a program specifically designed to prevent experienced employees from going to work for the benefit of other states in their early retirement years called DROP that basically mimicked the effect of pension+salary to keep them in Alabama.
Back to your example. In order to retire receiving 75% of one’s salary (based on the formula on the page I linked to), the employee would need to have worked a little over 34 years continuously. If retiring at age 59, there is a reduction of benefits by up to 6% as opposed to age 60 (0.5% per month before age 60), so the full benefits would not be realized.
Note, however, that educators do not pay into (and thus do not receive, barring other employment) Social Security in the state of Illinois. If teachers did (and I'm guessing there are some who did way back when and the law may have changed, else it wouldn't be in the calculation formula), they receive only 1.67% instead of 2.2% per year, which requires 45 years of employment for maximum benefits — impossible for a 59-year-old employee.