What changes at each level
In preschool, the goals are recognition and control: naming numbers, counting small groups of objects, holding a pencil and following a line. Kindergarten turns recognition into fluency: counting to twenty and beyond, finding missing numbers, and forming letters that other people can read. First grade expects the first real arithmetic, adding and subtracting within twenty, and handwriting that keeps up with the child’s ideas.
Moving up a level
A learner is ready for the next level when the current one feels calm, not when it is merely finished. Quick, accurate answers with little counting on fingers, and the ability to explain an answer in words, are better signals than age alone. It is always fine to mix levels: Grade 1 arithmetic with kindergarten handwriting practice is a sensible combination for many children.