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What if it's school district property but doesn't have a "school" on it. I occasionally go for walks that take me down a side walk across the street from a school district administration building and a school district athletic field.
The statute provides:

18 USC § 921 Definitions
(a) As used in this chapter—
(25) The term “school zone” means—
(A) in, or on the grounds of, a public, parochial or
private school; or
(B) within a distance of 1,000 feet from the grounds of
a public, parochial or private school.
(26) The term “school” means a school which provides
elementary or secondary education, as determined
under State law.

So, since there's not an elementary or secondary "school" on it, it's not a "school zone," right? Or, should I change my route?

Your CHL exempts you from those statutes. Just stay out of the buildings. And/or sports fields while a school sponsored event is taking place.

In case you get more curious, previous discussions here:

FAQ: Gun Free School Zone Act

The text of the federal GFSZA is posted there as well.

Thanks.