  A new method is presented for measuring terms in the elastoresistivity tensor
$m_{ij}$ of single crystal samples with tetragonal symmetry. The technique is
applied to a representative underdoped Fe-arsenide,
Ba(Fe$_{0.975}$Co$_{0.025}$)$_2$As$_2$, revealing an anomalously large and
anisotropic elastoresistance in comparison to simple metals. The $m_{66}$
coefficient follows a Curie-Weiss temperature dependence, providing direct
evidence that the tetragonal-to-orthorhombic structural phase transition that
occurs at $T_s$ = 97.5 K in this material is not the result of a true-proper
ferro-elastic transition. Rather, the material suffers a pseudo-proper
transition for which the lattice strain is not the primary order parameter.
