Math Relearning/Progressions/High School Algebra

These are my comments on the Common Core Math Progressions document Algebra: High School (PDF).

Overview
"This insight allows for the method" - Occasionally I pause and have to find the point again--why are we learning these things? What's interesting or useful about them? Right now I'm thinking of the set of all math facts as an infinite, dense, undifferentiated field of necessary possibilities. That is, they're concepts that could become instantiated by real events and would then dictate certain features and outcomes of the situation. When we use or study a particular math concept, we're letting it stand out from the field for some reason. Maybe it helps us accomplish certain real-world goals; maybe it's a general concept that leads us to others we care about; maybe it gives us an organized way to think about the larger concepts they're a part of.

Seeing Structure in Expressions
"try possible manipulations mentally" - Keep this in mind as a skill.

"simplest form ... [vs] equivalent forms that are suitable" - Important distinction.

Arithmetic with Polynomials and Rational Expressions
"equivalent expressions ... naming some underlying thing" - It would be nice to explore both approaches, (1) identifying the function the equivalent expressions define and (2) using the properties of operations to transform polynomials, treating polynomials as elements of a formal number system. Apparently a particular curriculum will or should use only one of the approaches, I assume to minimize confusion.

"Polynomials form a rich ground" - Sounds great. I wonder what features will transform these math terms from vague concepts into familiar ones. Will I imagine a paradigmatic graph? Will I have in mind paradigmatic uses for each kind of formula?

All of this reminds me of the progressive, building nature of math. I wonder what ways you could gamify math education. I'm sure people have at least started to do that somewhere.

"Binomial Theorem" - Long equations like this look a little terrifying. I wonder if it would help me learn them with less intimidation if I knew how they were (or could be) developed. It looks like that's one of the exercises for learning this theorem ("why this rule follows algebraically from ...").

"construct polynomial functions with specified zeros" - I imagine the thought processes are similar to writing a program to achieve a particular result.

"a computer algebra system" - This reminds me of reading SICP, which so far is almost as much about math as it is about programming. This reminds me that knowing math might make the difference in my programming classes between sitting in a puddle of insecurity and participating energetically. This makes me wonder if I could approach my math education partly by thinking about what I'd like to know comfortably when I have math discussions with other people. What would I like to be able to reason about with them?

Creating Equations
"much more strategic in formulating" - It's like a game. This happens a lot in the games I play.

"solution to an equation might involve more" - At what point does math get more complicated than is useful to an average person? I'm sure we've passed that point in these progressions. But how much math does a programmer generally need to know? This makes me think of the GCF discussion in SICP, which is used in later sections of the book. The main mathematical issue in programming is probably deciding on an algorithm. Sometimes you know of several that will achieve a solution. The way math algorithms are presented often makes it sound like mathematicians are only guessing with a shrug about good or best ways to calculate a formula. Well, how did we come up with the formulas in the first place, and does that give us any clues about good/best algorithms? It's kind of disturbing that algorithms don't seem as perfect or logically necessary as math generally is.