[plinkplonk]

What helped me get a grip on learning mathematics was learning to prove theorems.

Working through Daniel Velleman's book "How to Prove It" (the only pre requisite is that you can understand boolean logic, which programmers have no problem with), and then a Set Theory book (I used Enderton) set me up to tackle (proof based) Linear Algebra, Analysis etc.

Just my personal experience. Hope this helps.

[shp0ngle]

I think theorems already presuppose some basic understanding of algebra, but I might be wrong.

It really depends on what OP actually knows and how deep he wants to learn and in what direction

[Tabular-Iceberg]

I wonder about this myself. I've always had a much easier time learning things that makes sense from first principles rather than something that I need to just take for granted, the latter being the case with the first 12 years of my own math education. The latter is much more difficult to form a mental model around.

Would it be possible to teach mathematics by theorem proving ab initio? I guess conventional algebra would be hard to digest for a first grader, but I maybe something like the Peano axioms can be thought of as rules for a game that the students can play, where subsequent arithmetic lessons will be about finding shortcuts to the tedious application of the rules in order to solve problems.