Recently, McDougall and Wotherspoon have developed a simple modification to the standard Newton method for finding simple roots of nonlinear equations. Their method is based on the arithmetic mean and achieves approximately 2.4 convergence order. This research presents a new iteration scheme based on the harmonic mean with the same convergence order and the comparison with the original McDougall-Wotherspoon method. Several test examples have been used and the results show good numerical behavior of the new method.