Poll: What level of math is programming roughly analogous to?

Disclosure: I am a mathematics teacher. (I teach prealgebra to elementary-age pupils.) I am NOT a programmer. But I think the correct answer to the question is "it depends." In other words, even though you have specified a level of programming, "bog standard, journeymen, CRUD programming in Objective-C," and even though you have laid out the usual progression through the usual school mathematics topics in the English-speaking world, you haven't given enough information to match the levels.

Some people program insightfully, or so I am told by people who program, and other people program nothing but kludges. Among mathematics learners, there are great variations in the levels of "profound understanding of fundamental mathematics" (PUFM) among people who are studying the "same" level of mathematics. It interested me that you mentioned neither number theory nor combinatorics among your list of secondary mathematics courses, because in some other countries, those courses would come well before calculus. In general, any reader of Hacker News could learn a lot about what is missing in the United States (or, more generally, in the English-speaking world) in elementary mathematics instruction by reading Knowing and Teaching Elementary Mathematics: Teachers' Understanding of Fundamental Mathematics in China and the United States by Liping Ma.[1] There is a lot to basic mathematics that is far from easy, if deep understanding is what you are looking for. In the countries that excel over the United States in mathematics instruction, proof is part of every mathematics course, not just part of geometry, for example.

[1] http://www.amazon.com/Knowing-Teaching-Elementary-Mathematic...

Ah, apparently I do not fully understand the hardware implementation.

Consistent, in software at least, is the concept of memory being linear; references using a start and end point, and points between.

It's not.

There are certain levels of math or logic needed for various programming tasks, but it is not 'analogous' to any level.

For example, function composition is very useful in programming and thus Algebra (or was it Algebra 2???) comes in handy for understanding that at a basic level.

However, if you're doing graphics programming, then you often need experience with geometry (composition of basic shapes to form more complicated shapes, linear algebra (matrix operations), and modern algebra (quaternions).

> In the countries that excel over the United States in mathematics instruction, proof is part of every mathematics course, not just part of geometry, for example.

There's room to wonder about how much of a causative role this has, and how much it reflects consumer demand. When my sister started taking algebra, my parents asked me to help her, and I was careful to provide proofs of all the algebraic properties of exponents (e.g. x^m * x^n = x^{m+n}; x^0 = 1; like that). That's not usual in the US, and didn't feature in her textbooks or supplements, but I did it basically just because it was important to me.