I have fallen in love with the function-approach to algebra, and if I were teaching algebra 1, or even below that (as the CCSSM allows), everything would be taught from the perspective of a function.
Starting with functions from day one, you can reinforce everything about the functions, domains, ranges, families, etc with each and every lesson (just about!).
operations on expressions? Yep, start with a function that has the expressions you want. You can get so much more out of it. This can lead to so much more understanding. In class right now, students are dealing with rational functions in the form f(x)=a/(x-h) +k, finding the characteristics, then rewriting it as a quotient of two polynomials and comparing the graphs. Good stuff, and can be done with every type of function. I think it gives meaning to algebraic manipulation, why do we do it, what happens when we do it, and how does it really change the original function. These are powerful questions whose answers give real depth and meaning to the algebra classes.
I went through the EMATHS training, but the classes I taught never really let me explore the shifts that strongly (namely statistics and AP Calculus). I had a feeling that the function-based approach was, in my opinion, stronger and more flexible than the other way, but since I never had a chance to really explore it, it wasn’t cemented.
It is now. Everything that can be done in a math class can be done from a function perspective and so much more. I will never go back!
There have been questions about learning procedures, the rigor of some of the algebraic skills like “simplifying” expressions, applying properties, etc. It’s all there, all of it. It’s just a different way of getting there.
I will leave this with a story. Since we were converting a rational function from one form to another, and to get the students thinking about what is going on, I took a piece of paper and told them this was a function. We can do a lot with a function in this form, and we came up with a lot of uses for it. Then I crumpled up the paper and asked how I changed the paper. Did I add anything? Did I take anything away? Is it still that original piece of paper, just dressed up differently? Then I asked about the uses of my crumpled up piece of paper. Can I do the same things? Can I do different things? I highlighted the different uses of the wad of paper by playfully tossing it around the room. When we discuss this as a group, hopefully we will get to the point that even if a function is in a different form, it really hasn’t changed (check the graphs), but the uses for it have.
Wouldn’t it be great if that idea can be reinforced almost all the time in Algebra 1 and Geometry, and not just at the end of the 3rd unit of Algebra 2?
Thankfully, the CCSSM is all about the function in high school!