First, we need to stop insisting that "math is all around us!" (usually said with Carl Sagan-like starry-eyed wonder ;-) (jk)
As a STEM major, I had a natural interest in things that math talked about. However, I was not quite so enthralled with Chaucer, The Stamp Act, Art, or most of the humanities (I've gradually come to enjoy them too).
So, I sympathize with students who are not interested in a STEM career -- unlike, say, history, writing/literature, or even science, mathematics is quite removed from daily life. And it's not just because most people will never have to solve the quadratic equation. It's that its supposed "knock-on" benefits appear oversold to most non-STEM folks. Here's a possible view from their perspective:
Take history -- I'll never write a history article or scour the archives, yet, knowing history helps put everyday life into perspective. History directly affects our lives and we constantly talk about it in politics and our own lives. Relevant!
Now, take English/writing -- I'm not going to be an author or journalist, but I need not tell anyone here that being able to write clearly and coherently is probably one of the biggest determinants of success in your career. From cover letters, to reports, to presentations, if you cannot express yourself, you're limiting yourself. Relevant!
No, onto our sister discipline -- science. I'm not going to be a scientist or engineer (well, OK...I did become one, but that's beyond the point!) but science is about tangible things and is responsible for our technical wealth. Not to mention global warming, stem cells, cancer treatment, space exploration, and others are things many people are interested in talking about or having an opinion on. Relevant!
Now, on to our collective favorite -- math. While the above subjects give knowledge that is quite useful outside the academic discipline, math gives us points, lines, planes, quadratic formulas, and calculus. I've found that "everyday examples" of these ideas are quite contrived, as most people will never need to write a formula down or solve anything but the simplest equation.
The problem is the other fields are geared towards being useful to the general citizen while math classes appear to be attempting to groom the next generation of STEM majors. I'd say, leave that to college. To people who aren't interested in solving equations or proving theorems, math class is just one puzzle after another, with no relevance (and, guess what, they're right!).
Now, I'm not being down on math. I think there is hope, but it's not by making non-STEM-focused students love the quadratic equation, trigonometry, or integrals by making up examples of cannonballs falling, bridge building, or leaky buckets. It's by copying from the playbook of History, English, and, to a lesser degree, science. We need to focus on areas where complex mathematical concepts creep up in actual daily life. In my experience, this area is statistics.
Statistics pops up everywhere in life, and even if you're not going to be a statistician, the quantitative reasoning skills (and knowing the logical traps) required to understand statistical arguments is extremely useful in everyday life. Politics, sports, theology (Bayesian arguments for/against god .. Ok, hot button issue, but still ;-), medicine, gambling...these are things ordinary people actually do and actually could be helped by some numeracy.
Until the curriculum is changed, I think we'll be stuck with contrived examples from algebra, trig, and calculus. Also, I agree that we need some basic arithmetical numeracy like finding areas/volumes, and understanding arithmetic operations. However, I think that Algebra/Trig/Calculus sequence is really best left for students who will be doing STEM.
I'd really, really, like the general ed math to focus more on asking quantitative questions and being skeptical of numbers. I think that number-phobia is manifested by people either trusting or mis-trusting an argument simply because it has numbers. They've never been given the confidence to "challenge the teacher or the numbers", which is what you need for correct quantitative reasoning.
Imagine students leaving after spending four years honing their ability to deconstruct a quantitative argument in a newspaper, to question a politicians statements, to be able to appreciate clinical research, to incorporate uncertainty and risk properly in their lives...that would be something!