I am finding it worthwhile to spend a LOT more time on factoring this year than I did last year, and definitely way more than most of my colleagues do. Some of the teachers who drive by the factoring standards the fastest (to get them all checked off, of course) are also the ones with students who have the weakest and shallowest understanding when they get to the rational expressions standards. Teaching Math Support and seeing the patterns of math mistakes among the students of different teachers has been a powerful teacher to me about conceptual understanding these past two years.

One of the posts I am working on (realistically, I probably won’t finish it until early June or during state testing) is about helping students to understand the inverse relationship between the distributive property and factoring.

Some of my students exclaimed recently that they now see factoring as a process of “un-distributing” polynomials. This felt like a pretty deep conceptual leap for them to have made, and I think it can be of benefit to help more students realize this. Once they “get” the idea of factor-thinking, most of their rational expression simplification mistakes seem to resolve themselves.

I wonder if this is not because they are starting to actually “see structure in expressions.”

I think the fastest way of helping these students might be to get them to substitute a value for x at each step. Then they know that a mistake happened if the value of the expression changes. Then they can get started on uncovering what kinds of mistakes they make.

Also that way we’ll find if the problem is with the algebraic manipulation or with the understanding of fractions, by whether they can compute the value when x = 1 or 10 or whatever they picked to check.