We show that the chemotactic movements of colonies of the starving amoeba Dictyostelium discoideum are driven by a force that depends on both the direction of propagation (directional sensing) of reaction-diffusion chemotactic waves and on the gradient of the concentration of the chemoattractant, solving the chemotactic wave paradox. It is shown that the directional sensing of amoebae is due to the sensitivity of the cells to the time variation of the concentration of the chemoattractant combined with its spatial gradient. It is also shown that chemotaxis exclusively driven by local concentration gradient leads to unstable local motion, preventing cells from aggregation. These findings show that the formation of mounds, which initiate multicellularity in Dictyostelium discoideum, is caused by the sensitivity of the amoebae due to three factors, namely, to the direction of propagation of the chemoattractant, to its spatial gradient, and to the emergence of cAMP “emitting centres”, responsible for the local accumulation of the amoebae.