An elementary proof or use of elementary technique in mathematics means use of only real numbers rather than complex numbers, which relies on manipulation of the imaginary square root of (-1). Elementary proofs are preferred because they are do not require additional assumptions inherent in complex analysis, such as that there is a unique square root of (-1) that will yield consistent results.
Mathematicians also consider elementary techniques to include objects, operations, and relations - thus orientational metaphors are also elementary in this sense. Sets, sequences and geometry, e.g. chord factors and tensegrity, are not included, so most structural metaphors would invoke assumptions analogous to complex numbers.
The prime number theorem has long been proven using complex analysis (Riemann's zeta function), but in 1949 and 1950 an elementary proof by Paul Erdos and Atle Selberg earned Selberg the highest prize in math, the Fields medal.