Supose that X is a homogenous variety and $T_0 ,...,T_m$ be a basis for $A^*(X).$ Consider $T_i \cup T_j\cup T_k.$ This means that we consider varieties $\gamma_i,\gamma_j,\gamma_k$ correspond to $T_i,T_j,T_k$.Now we can intersect these varities and we have a class in $A^*X$ again.
Now does $\int_{X}T_i \cup T_j \cup T_k$ counts number of points that the varieties $\gamma_i,\gamma_j$ and $\gamma_k$ intersect?