Absolutely nothing, Say it again!
Well, I suppose "absolutely nothing" maybe a bit strong of a statement, but for high school science students, I stand by my statement.
I have been teaching science now (physics specifically) for 15 years. And in that time never have they been to find out what my students know and are able to do.
Any time spent telling students something you want them to know - or having them practice without the proper context is time wasted. I feel like the "teaching" of significant figures falls into that category and I'm not buying it.
I remember my chemistry classes in high school (both a college prep and an AP). Somewhere within the first month of the class at least 2 weeks were devoted to "learning" the rules for significant figures (we'll just quaintly call them sig figs). The teacher would tell us the rules an then give us all sorts of de-contextualized problems to solve. These were accompanied by all manner of cute stories, personal anecdotes, and diatribes about the importance of sig figs.
The rest of the school year was spent complaining about how none of us remembered the rules.
Then I got to college and the some section of the first chapter of the text book (don't get me started on text books) was devoted to sig figs. The professors and TAs would dutifully re-explain the rules, provide us with problems to practice and complain, for the rest of the term, that we didn't understand them.
No kidding! First of all, that is what happens when the "teaching" of any concept is done without the proper context. As we know, to really learn anything you have to figure it out for yourself or at least understand the proper context in which the ideas were developed. Simply telling us the rules for sig figs was a doomed practice from the start.
But more importantly, no matter how seriously the teachers tried to get us to take these sig figs, they just never mattered. Sure, we were supposed to put the correct number down on a test (and for those who use something evil like WebAssign) but beyond that who cared? No one!
A bunch of years ago the AP physics chemistry tests decided they no longer cared about sig figs - because knowing the correct number of sig figs does not, in any way, reveal if a student actually understands the material or the underlying concepts!
Lastly, the argument that "in industry they're going to have to know them" doesn't hold any water for me. One, very few of the students in any high school science class are going to become scientists. Not that we wouldn't like them to; but let's be realistic. Two, if any student needs a skill at any job, they will learn it there with the appropriate context anyway.
So I say - don't waste your time trying to get students to remember some arcane piece of information or useless skill set.