Difference between shougaku and shougakkou?

I've been using shougakkou when referring to elementary school, but lately I've been noticing that certain sentences use shougaku instead. What's the difference?

e.g. 今年ジョンくんは小学二年生です。
In the above sentence can I substitute shougakkou instead?

Thanks.

小学　　refers to the institution of elementary schools
小学校　refers to the physical school itself

In your sentence shougakkou will not be correct.

Sorry, this is just an idea.

I think maybe it's 小学生 not 小学.
And you can count up as 小学一年生, 小学二年生...?
Because I can't remember someone say 小学 only.

ジョンくんは小学校二年生です is no problem, I had heard it.
Maybe it's understood as the abbreviation for 小学校の二年生?

"shougaku X-nen sei" to refer to "X-grade student of elementary school" is commonly used. It's like "student of Xth year in elementary school studies."

In the same way, we do say:
- chugaku X-nen sei (X-grade student in junior high school)
- koukou Y-nen sei (Y-grade student in senior high school)
- daigaku Z-nen sei (Z-grade student in college)

AK

- daigaku Z-kai sei (Z-grade student in college)

td,

That is more of a regional difference, I believe. :) I am from Kanto, and we always used "sei" for colleges as well.

For a foreigner speaking Japanese, would they use nen-sei or kai-sei? I'm just curious.

This -kai sei for colleges seems to be a usage that is limited to Osaka/Kyoto area. So if a foreign student says that, you a re likely to be asked if you stidied in Osaka or Kyoto :)