Hi,
Finite Element Procedures by K.J.Bathe is originated also for educational purpose at MIT. It doesn't teach you "how to invert big matrices by hand", no, absolutely not!, but rather the methode to invert those big matrices and some consideration about numerical difficulty. Futhermore it explain the mathematical derivation of some elements and also about dynamic aspect.
May be you have just read the first paragraph about matrice formulation and solution of Bathe, don't you? I think, Bathe is not so stupid to teach you how to invert matrices by hand in the age of computational matrice softwares like MATLAB & co. Beside that, i solve simple problems not with a pencil and paper, but conveniently with mathematical software like MATHCAD.
Your propose to practice, practice, practice… on simple problems with known solutions was also proposed by zuardy before : "learn FEM by using delivered manuals of any FEM software". FEM manuals, e.g. NASTRAN, have always problems examples called verification problems, right?
cheers