Tangent space, derivation defifinition

I'm reading "tensor analysis on manifolds" by Bishop and Goldberg. I have taking a course in differential geometry i R^3. The course was held on Do Carmos book. Do carmo deffined the tangent at a point on a surface as all tangents to all curves on the surface going through that point (or something like that). This diffenition is pretty clear and easy to imagine.

In the other book it is defined as all darivations on all tions from the manifold at the point to R. Then they define the tangent of a curve in the manifold to be and operator. And then they define the tangent to the ith coordinate curve of gamma, and proof a theorem about that these coordinate tangents at a point m on the manifold is a basis for all tangents.

The proof are not that hard, but my problem is that i can't see that the two definitions are the same when i choose one of the "manifolds" we worked on in do carmo, fx. the unit sphere in R^3. I now the one is a operator and the other are vectors, but there most be a clear connection?

I've seen an example where they use R^2 as there manifolds, and then state that the directional derivative corospones to the operators. And then it is clear that given and directional derivative in the direction v lets call it Dv, then i have the operator and the if i define a function f(Dv)=v then i get all tangent vectors to all points in R^2.

But somehow i can't find the connection when my surface gets more complicated, like the unit sphere. I can see that the operator is connected to the tangent of curves through m, but it is not so clear to me, in my mind there most be a function (possible a bijection) on the "tangent operators" such that i get all tanget vectors, when my manifold is so simple like the unit sphere that a "normal" tangent vector/plane makes sence. My point is that if it makes sence to define the tangents like operaors, I most have just as much information as if I had defined it the "usual" way (of cause the new defintion makes more sence when i have obscure manifolds where tangentplanes are not clear, but in R^3 there most be a clear connection in my head).

I hope someone can understand what my problem is, anbd have the commitment to read the whole post (it got a bit long, sorry). And by the way, i'm sorry for the bad englsih.