A new method is presented for measuring terms in the elastoresistivity tensor mij of single crystal samples with tetragonal symmetry. The technique is applied to a representative underdoped Fe-arsenide, Ba(Fe0.975Co0.025)2As2, revealing an anomalously large and anisotropic elastoresistance in comparison to simple metals. The m66 coefficient follows a Curie-Weiss temperature dependence, providing direct evidence that the tetragonal-to-orthorhombic structural phase transition that occurs at Ts=97.5 K in this material is not the result of a true-proper ferroelastic transition. Rather, the material suffers a pseudoproper transition for which the lattice strain is not the primary order parameter. [ABSTRACT FROM AUTHOR]