  In this paper, we construct a derivative-free multi-step iterative scheme
based on Steffensen's method. To avoid excessively increasing the number of
functional evaluations and, at the same time, to increase the order of
convergence, we freeze the divided differences used from the second step and
use a weight function on already evaluated operators. Therefore, we define a
family of multi-step methods with convergence order 2m, where m is the number
of steps, free of derivatives, with several parameters and with dynamic
behaviour, in some cases, similar to Steffensen's method. In addition, we study
how to increase the convergence order of the defined family by introducing
memory in two different ways: using the usual divided differences and the
Kurchatov divided differences. We perform some numerical experiments to see the
behaviour of the proposed family and suggest different weight functions to
visualize with dynamical planes in some cases the dynamical behaviour.
