I assume that the 11th grade math score you mentioned is a grade equivalent (GE). This means that his achievement is the same as that of an 11th grader at the 50th percentile. This sort of score is actually fairly typical for gifted kids at your son's age, especially if they have unlimited time. A GE of 11.9 is essentially mastery of 4th or 5th grade math.
Actually, grade equivalent means that her son scored the same as an 11th grader did in his grade level maths. It does not mean that her son is at an 11th grade level in maths.
Both of you have elements of correctness. It depends on the instrument. If it was an individually-administered achievement instrument like the KTEA, WIAT, or WJ, then Kai's explanation is essentially accurate. It means that he obtained the same raw score on this particular test that the 50th percentile of students of that grade equivalent did in the standardization sample. But remember that those types of tests are designed to order students against their age-peers, not to determine if they have mastery of all topics expected at a certain grade or age.
If it was a grade-level measure, like the ITBS, ERB, or SAT-10, or a computer-adaptive test with a low ceiling, like the MAP K-2 or 2-5, then Archie is closer to correct. It's the same concept, but with an even lower cap on item complexity.
Either way, though, the principal point is correct: grade equivalents very rarely mean that the student actually is prepared for instruction at that grade level (with the possible exception of criterion-referenced tests, which are not typically administered in this situation).
