Chicago Public Schools’ next chapter hangs on the result of the April 4 mayoral runoff election between two candidates boasting unusually strong ties to public education — and diametrically opposed ideas for schools.

Paul Vallas plans drastic changes to CPS’ structure, bolstering principals and local leaders’ power over spending and programming — and even the ability to let a charter school take over their campus. He would prioritize standardized testing and make it easier to hold students back a grade so they don’t graduate without necessary reading and math skills...