In many practical applications the underlying graph must be as equitable colored as possible. A coloring is called equitable if the number of vertices colored with each color differs by at most one, and the least number of colors for which a graph has such a coloring is called its equitable chromatic number. We introduce a new integer linear programming approach for studying the equitable coloring number of a graph and show how to use it for improving lower bounds for this number. The two stage method is based on finding or upper bounding the maximum cardinality of an equitable color class in a valid equitable coloring and, then, sequentially improving the lower bound for the equitable coloring number. The computational experiments were carried out on DIMACS graphs and other graphs from the literature.