  We present a method of building operators on a Banach space $X$ that generate
distinct operator ideals in the algebra $\mathscr{B}(X)$ of bounded linear
operators on $X$. We show that there are exactly $2^\mathfrak{c}$ distinct
small closed operator ideals on the Schlumprecht space and there is a chain of
cardinality $\mathfrak{c}$ of small closed operator ideals on any Schreier
space of finite order.
