Subdivision schemes are the most important techniques in Computer Aided Geometric design (CAGD) for generating curves and surfaces. The subdivision scheme is an algorithm to generate curves and surfaces as a sequence of continuously refined control polygons. At each refinement level, new points are added to the existing polygon. It repeatedly refines the initial polygonal shape. After each level, we get closer to the limit curve, which is the limit of an infinite series. We present direct methods using algebraic expressions for generating subdivision schemes. We also present the relationship of algebraic expressions in other entities of mathematics, i.e., the binomial theorem, Pascal's triangle, Fibonacci sequence, Bernstein polynomial, Bezier curve, and B-splines.