The 1 1 1-2 system is called unit counting and, as others have said, it's very useful for younger students. I myself start young students with nominal counting: "quarter, quarter, half-note" so there's no possibility of confusing counting numbers with finger numbers. The Fabers actually offer a variation with "walk" for quarter notes and, at Level 2A, "run-ning" for two eighths.

However, everybody needs to do metric counting (1 2 3 4) eventually. So if you're working with older students, I'd say start them out with metric.

By the way, there is another system: syllabic. That's the "ta" system often used in vocal music programs. (Started by Kodally, I believe.) In the long-ago days when I taught elementary band, I used that system in addition to metric counting. The ta's and ti-ti's seemed well suited for wind players. (Similar to tonguing.)

Susan Bagot