* In some parts of the US (mostly the Northeast), "grammar school" refers to elementary school (grades K-5 or K-6).
* In other parts of the world, "grammar school" may refer to a secondary school (roughly equivalent to middle or high school in the US).
Middle school, in the US, typically encompasses grades 6-8 (though this can vary slightly). It's a distinct level of schooling between elementary and high school.