The numbers $f_λ$ of standard tableaux of shape $λ\vdash n$ satisfy 2 fundamental recursions: $f_λ= \sum f_{λ^-}$ and $(n + 1)f_λ=\sum f_{λ^+}$, where $λ^-$ and $λ^+$ run over all shapes obtained from $λ$ by adding or removing a square respectively. The first of these recursions is trivial; the second can be proven algebraically from the first. These recursions together imply algebraically the dimension formula $n! =\sum f_λ^2$ for the irreducible representations of $S_n$. We show that a combinatorial analysis of this classical algebraic argument produces an infinite family of algorithms, among which are the classical Robinson-Schensted row and column insertion algorithms. Each of our algorithms yields a bijective proof of the dimension formula.