Strong Spirit
Backbone
A spirit of approximation is the approximation of spirit. My approximation of your spirit? I think you have a strong spirit. I think you have backbone.
きがい
This is a jukugo word that uses the on'yomi readings of the kanji. You should be able to read this on your own.
その教師は、教室で見て見ぬふりをされている問題に取り組む気概がなかった。
The teacher didn’t have the guts to deal with the problems in the classroom.