All searches I did for Schoof's algorithm to count points on elliptic curves over finite fields show the same as in Wikipedia https://en.wikipedia.org/wiki/Schoof%27s_algorithm This algorithm says under section "Computation modulo primes" that "The scalar multiplication of $\bar{q}$(x,y) can be done either by double-and-add method or by using the $\bar{q}$th division polynomial". Invariably, all examples I found on the internet use "division polynomials."
What will Schoof's algorithm be if double-and-add method is used instead ?
I would appreciate if someone could formulate it with explanations so that non-math background people like me can understand.