Officials across the country, including Mayor Bill de Blasio of New York, are looking to efforts like those in New Jersey as they seek to broaden access to free, full-day prekindergarten. President Obama embraced the policy last year, and politicians in several states, including Maryland, Texas and Washington, are considering ambitious expansions. Though experts differ on the long-term benefits of preschool, the programs in 31 low-income districts in New Jersey are widely acknowledged for strong results. In New Jersey, students attend prekindergarten for two years, not one. Class sizes are limited to 15. Teachers generally have more training than in New York. And it took more than a decade to perfect the model: The districts faced a host of challenges, including persuading families to enroll and rolling out a robust curriculum.