Gaining familiarity with mathematical ideas
There are ideas from advanced fields of study that kids are capable of comprehending. I think it's much easier to work with a particular idea if you have a warm familiarity with it. Kids can't understand most ideas within knot theory, but we can give kids certain problems involving knots. Even if their conclusions aren't very deep, when they hopefully study knot theory later in life, they'll have a familiarity that may give them the confidence they need to approach the more difficult ideas. These "first brush with X" topics should be peppered into the curriculum. Here's a lovely article with knot theory questions for kids:
As another example, consider the concept of isomorphisms. A student's first exposure to isomorphisms is likely during graph theory, where they learn about graph isomorphisms. But the Game of Nine, otherwise known as the Pie Game, is a game that kids can play that is isomorphic to Tic-Tac-Toe. The correspondence between these two games can be shown without formally proving them isomorphic. I believe that by seeing that one game is isomorphic to another, students will be more accepting of the idea of isomorphisms when seeing them in graph theory. I would love to find research that confirms or denies this conjecture.