As a private contractor who traveled to a different school each weekday to teach an after school program (as 1 part of my duties), can I list mileage as an expense?

@jessestrickman unless you have a home office, the travel is "commuter miles" and is not deductible.  I assuming you are stating you go from your home to a different school each workday and then go back home. 

If you go to 2 or more job sites in a single day then the 2nd trip is deductible as a business expense. For example you leave home drive 5 miles to the 1st job then drive 10 miles to the 2nd site. At the end of the day you drive 15 miles home. Only the 10 miles between jobs is deductible. The first trip in the morning, from home and the last trip at night back home are considered commuting and are not deductible.

  To deduct mileage, from your home to the 1st job,  the home just needs to be the "Principal Place of Business"(POB).  It does not need to qualify for the Home Office deduction (the 'Regularly and Exclusively' rule does not apply).  
https://www.irs.gov/pub/irs-drop/rr-99-7.pdf
https://www.irs.gov/publications/p463/ch04.html

https://www.irs.gov/pub/irs-pdf/p587.pdf (page 3) POB defined