Too many students (and too many parents, and even some teachers) view the process of learning math as a one-dimensional arrow, in which the courses come in a fixed order and lead inexorably to calculus and beyond. The most successful students feel driven to move faster and faster along this arrow, the goal being to progress to calculus as fast as possible.
Of course this portrayal is an over-simplication of the popular view, but it’s not much of one. It is correct that much of mathematics is sequential and cumulative; many Algebra II topics, for example, cannot be understood without a reasonable grasp of Algebra I. And some topics are not sequential; many people realize that geometry and Algebra II can be learned in either order — after all, some schools do geometry first, some Algebra II. Many also realize that most of geometry could be left out of the sequence altogether, though that would be a pity since so many important lessons about spatial reasoning and logic in general are learned in a geometry course. But if your sole goal is to get to calculus, you could learn the geometry you need in a couple of weeks; why take a whole year?
Wherever you put geometry, a revised arrow is probably the mental model that most people believe in:
But this point of view is only slightly better than the first one. It’s still essentially a one-dimensional model, in which math has length but no depth or breadth.
Students who want to learn more math should be doing just that: learning more math, not moving faster and faster along the single-minded path to calculus. They should be exploring traditional topics in greater depth: there’s always more that’s worth learning about a topic, and there are always challenging problems that are worth attacking. And they should be exploring new topics that the majority of their classmates may never have the opportunity to learn: there are many more worthwhile topics than we can fit into the standard curriculum. So the true arrow of mathematics learning is three-dimensional. If I were more of an artist, I would represent this model effectively in a convincing picture. Maybe one of my students can draw one for me, but in the meantime this sketch will have to do: