Edit (Feb 2016): Since the OP mentioned Hacker's Algebra opinion piece in the NYTimes, perhaps this is a good place to point out his most recent follow-up in a similar direction (I exclude here my own assessment of either): The Wrong Way to Teach Math by Andrew Hacker (Retrieved: 2016 Feb 28).
Edit (July 2015): In a similar vein, here is a link to Is Math Important? David Leonhardt of the New York Times acts as host for the panel discussion; to quote directly from this blog post:
From the mathematical world there are Steven Strogatz of Cornell University and Jordan Ellenberg of the University of Wisconsin, and from mathematics education research there is Jo Boaler of Stanford University. They are joined by David Coleman, President of the College Board, education writer Elizabeth Green, author of the recent book Building a Better Teacher, Pamela Fox, a computer scientist working with Khan Academy, and financier Steve Rattner.
(The two hyperlinks were added by me: one to an MESE answer about Ellenberg's book; the other to an MESE answer about Boaler's comments on timed tests and math anxiety.)
In providing a justification for learning mathematics, I would like to split this question into two pieces (even though there are certainly more) and comment briefly about one of them (even though the other may be closer to the intended question).
A first interpretation of the italicized text above: Why does a subject, mathematics, that covers so much "abstract" material, occupy such an important place in our schools?
This is the question that I believe is being asked, and it is the sort of consideration that underlies the linked piece on the necessity of algebra, from which I quote:
The toll mathematics takes begins early. To our nation’s shame, one in four ninth graders fail to finish high school. In South Carolina, 34 percent fell away in 2008-9, according to national data released last year; for Nevada, it was 45 percent. Most of the educators I’ve talked with cite algebra as the major academic reason.
You can find other comments in this direction in an earlier opinion piece in the New York Times, Garfunkel and Mumford's (2011) How to Fix Our Math Education. Again, quoting directly:
Imagine replacing the sequence of algebra, geometry and calculus with a sequence of finance, data and basic engineering. In the finance course, students would learn the exponential function, use formulas in spreadsheets and study the budgets of people, companies and governments. In the data course, students would gather their own data sets and learn how, in fields as diverse as sports and medicine, larger samples give better estimates of averages. In the basic engineering course, students would learn the workings of engines, sound waves, TV signals and computers. Science and math were originally discovered together, and they are best learned together now.
Traditionalists will object that the standard curriculum teaches valuable abstract reasoning, even if the specific skills acquired are not immediately useful in later life. A generation ago, traditionalists were also arguing that studying Latin, though it had no practical application, helped students develop unique linguistic skills. We believe that studying applied math, like learning living languages, provides both useable knowledge and abstract skills.
In math, what we need is “quantitative literacy,” the ability to make quantitative connections whenever life requires (as when we are confronted with conflicting medical test results but need to decide whether to undergo a further procedure) and “mathematical modeling,” the ability to move practically between everyday problems and mathematical formulations (as when we decide whether it is better to buy or lease a new car).
A second interpretation of providing a justification for learning mathematics: Why should we encourage students to study school mathematics now?
This is the question that I would like to respond to, briefly.
I do not disagree with studying mathematics for its aesthetic value; I do not disagree with studying mathematics for the opportunities it provides to express ourselves and be creative; I do not disagree that pure mathematics may turn out to have important applications. But I think the strongest argument right now for studying mathematics is its role as a societal gatekeeper (google scholar).
There are normative and utilitarian meta-questions about where mathematics' place should be in school and academic endeavors, but the current reality is that "learning mathematics" is essential to moving forward (or up) in the world; such a competence seems, to me, necessary but not sufficient for working towards a "successful" life.
At present, I have been teaching mathematics to elementary school teachers. Do I try to get them excited about mathematics? Yes. Do I try to get them to think about mathematics creatively? Yes. Do they sometimes latch on to applications of their own in our discussion of pure mathematical concepts? Yes: If only you could see the dawning of epiphanies (!) as many soon-to-be-married teachers in my Spring semester course began to brainstorm, collectively, about applications of LCMs and GCFs to the construction of flower arrangements and seating charts at upcoming weddings.
But I also realize that many of them are teaching students at high-needs schools, and that their students' futures (in our current set-up - speaking specifically about the United States) can be derailed by problems that start with an inability to factor quadratic expressions - or, in many cases, even earlier.