Abstract: The mathematical model \sum_\mathrmk=1^\mathrmn \mathrmk^2=\frac12 \mathrmn(\mathrmn+1)(2 \mathrmn+1) is widely used in the mathematics teaching and practice.The paper expounds some quite interesting foundation methods of the mathematical model with geometry,algebra,physics and differentiation method,which have a good deal of enlightment to the foundation of other mathematical models.