The Lagrangian-history method in turbulence theory (Kraichnan 1977) is modified such that triple moments are expanded in functional powers of the Lagrangian covariance of the symmetric rate-of-strain field instead of the Lagrangian covariance of the velocity field. The simplest approximation which results corresponds to the abridged Lagrangian-history direct-interaction approximation. It is illustrated by application to the Lagrangian properties of a random velocity field whose Eulerian values are frozen in time. Then it is formulated for isotropic Navier-Stokes turbulence. The new approximation is expected to give reduced energy transfer in the dissipation range because the rate of strain along a fluid-element trajectory is statistically stationary in stationary homogeneous turbulence while the derivatives of the Lagrangian velocity with respect to initial position tend to grow and thereby have a longer correlation time. The correlation times of these two entities play corresponding roles in the new and old approximations for energy transfer, respectively.