higher education vs teaching

higher education and teaching both are nouns.

Word	Noun	Adjective	Verb	Adverb
higher education	Yes	No	No	No
teaching	Yes	No	No	No

As nouns, teaching is a hypernym of higher education; that is, teaching is a word with a broader meaning than higher education:

higher education: education provided by a college or university
teaching: the activities of educating or instructing; activities that impart knowledge or skill

Other hypernyms of higher education include didactics, education, educational activity, instruction, pedagogy.

higher education (noun)	teaching (noun)
education provided by a college or university	the profession of a teacher
	the activities of educating or instructing; activities that impart knowledge or skill
	a doctrine that is taught