For quite a long time now I'm having some trouble to bridge the gap between two different approaches to Special Relativity. The first approach is the traditional one. It is the approach that Einstein presented in his paper and that is taught in most of the basic textbooks. In this approach, Special Relativity is introduced by two means of two postulates. In that setting the theory is based on the two postulates (the relativity principle and the constancy of the speed of light) and the Lorentz transformations arise as the transformations between inertial reference frames which obeys the two postulates. In this approach, for example, it is the postulates that forces us to consider that time may be reference frame dependent, because the relativity of simultaneity is a consequence of the postulates alone. On the other hand there is the modern approach. In the modern approach Special Relativity is introduced directly as a theory of the structure of spacetime. The spacetime metric is introduced right away and the Lorentz transformations are defined as the transformations of spacetime which keep the spacetime metric invariant. On this approach, nothing leads to a new spacetime structure, nor to the metric. It all appears out of thin air, inasmuch as the postulates in the original approach. Now I want to make something clear here. I do know that the only important thing is that both approaches work. One can look into the second approach, and just use it. The point is that IMHO, there's a huge gap between the two approaches. The original approach by Einstein is well motivated in his paper. Talking about Electrodynamics, Einstein is able to motivate quite well the need for his two postulates. It all flows quite smoothly. The second approach is not that nice to grasp. It doesn't seem to flow smoothly from anything else. It is usually not motivated, it is just presented as: "the mathematical framework is this because it works" and nothing more is said. Furthremore, it is not at all obvious that the Einstein's postulates are totally equivalent to invariance of the spacetime metric. In that sense, how can we bridge the gap between the two approaches? How can we present the second approach in a quite smooth and natural way as the first approach was presented by Einstein?