Large chemical computations show the need for full adaptivity supporting the development of robust and highly efficient programs. For solutions possessing sharp moving spatial transitions, as travelling wavefronts or emerging boundary and internal layers, an automatic adjustment of both the space and the time step size is generally accepted to be more successful in efficient resolving of critical regions of high spatial and temporal activity. In contrast to the widespread discretization sequence, first space then time, the reversed sequence first time then space is employed. Full adaptivity of the proposed algorithm is realized by combining embedded time discretization and multilevel finite element space discretization. In this paper the algorithm is described for one-dimensional problems. The numerical results show the significantly new perspectives opened by this approach.