> I'm not saying that if you start out without being able to solve basic equations Well, I'd argue this is a bad place to start out. The better questions are:

> I'm not saying that if you start out without being able to solve basic equations Well, I'd argue this is a bad place to start out. The better questions are: 1. What do these equations represent? 2. Why do we care about what they represent? 3. What does "solving them" mean for the things they represent? So, maybe a more important muscle to exercise is the one that lets us go up and down the ladder of abstraction. A huge part of mathematics is seeing a class of (relatively) concrete things, finding commonalities and drawing out abstractions, reasoning about the abstractions, and then drawing conclusions about the more concrete things. You can do so much of this without the formalisms of classroom mathematics and, honestly, it's what "real" mathematicians do. The formalisms are just a high-bandwidth way of communicating these ideas between mathematicians. They're just a highly technical map of the terrain, not the terrain itself. In fact, if I was teaching someone and they invented their own bizarro syntax for "solving" equations I'd be ecstatic. Whatever it was that happened in their head that made them > but the effort to get there will be a power of that needed by a "math-inclined" person. This is fine and true, perhaps, but if so it's true of all things, not just mathematics. You can replace "math" with "writing" or "running" or "playing a kazoo." What do you think makes mathematics special?