In the United States, are donations to public schools like a local elementary school deductible in the same way that donations to a 501(c)(3) charity or a church are deductible?
See IRS Pub 526:
Your local school is likely nonprofit and a branch of the local government. If so, yes, the donation is tax-deductible.
See also IRS Pub 17, which has some duplication from Pub 526:
Donations to most public schools are tax deductible if it's used for public purposes; public charter schools might be an exception so I would pay attention to that. Also as mentioned above you want to make sure you are donating to the public school/government unit and not some "affiliated" organization like a booster, pto, pta, etc. which requires 501c3 tax exemption. (actually I think schools aren't considered individual entities and are part of the district - districts and schools should be clearer about how to make payment)
US Code 170 There shall be allowed as a deduction any charitable contribution (as defined in subsection (c)) payment of which is made within the taxable year. ... (c) Charitable contribution defined For purposes of this section, the term “charitable contribution” means a contribution or gift to or for the use of— (1) A State, a possession of the United States, or any political subdivision of any of the foregoing, or the United States or the District of Columbia, but only if the contribution or gift is made for exclusively public purposes.