Sophistication of Methods

So I've been thinking about what constitutes a sophisticated method and when. Until recently, I've always considered that solving equations "algebraically" using inverse operations is obviously the most sophisticated method, and it's the one that we should be training our students to do. I'm usually a teacher who wants students to develop different methods of reasoning, and to make sense of the mathematics (and I encourage that during our first unit, when we're working with linear relationships), but I've always considered the method of inverse operations for solving equations a necessity for higher levels of math because students will struggle to solve a quadratic or a system if they don't know how to use inverse operations.

I was recently pushed to reconsider this by a few people (teacher educators who I admire/respect) that perhaps this is not the ultimate way for students to BEGIN learning how to solve such abstract relationships, and there might be some stepping stones they need on the way to developing that abstract method. But then, I was pushed even further in my thinking about this.

I've recently been exploring the Contexts for Learning Math book about the California Frog Jump, and using it to explore equivalence and solving equations. And I'm now wondering whether this method might be more powerful than algebraic expressions, at helping students to understand the relationship between quantities.

I think one major problem with our math curriculum is that we force students who are 13 and less into formal reasoning about math before their minds are ready for it. Most people don't develop formal operations (a la Piaget's 4 stages) until later in life, yet we expect 12 year olds to reason formally about abstract algebra concepts...