Having had a non-traditional route through learning various math topics, I'd like to advocate the teaching of calculus (the understanding of the accumulation of infinitesimally-small changes) before the teaching of trigonometry (the relationships between different geometric shapes, and the resulting association with waves).
In my (humbly biased) opinion, it's easier to learn how trigonometric concepts operate with a calculus background, than with merely an algebra-and-geometry background. Trigonometry complicates calculus education - knowing trigonometry may facilitate better calculus understanding to some, but not knowing trigonometry doesn't necessarily complicate learning calculus concepts. The trigonometry-first tradition in math education reflects a "number-phile" bias, not the most optimal route for humans to ingest and retain concepts.
Edit: The "number-phile" bias is clear - we want our teachers to be people with passion on the topic, as we're more likely to get a better understanding of the topic. Understanding sine waves as the result of an integral was much easier for me to understand than as the result of a complicated calculation.