If mathematics is taught well and the students learn it, then mathematics can help train the mind. Other subjects can as well. But the key is that teachers encourage critical thinking and not just mere recitation of facts and mere regurgitation of solutions to drill and standard problems (for instance, the kinds of problems we often see as worked- out examples in textbooks). But it is best that we are exposed to a variety of subjects if we are to learn general critical thinking skills and not just critical thinking for a specific subject.
Much of mathematical reasoning is inductive for trying to discover patterns and discover theorems, but then only deductive reasoning is used to prove theorems. The catch is that deductive reasoning is rarely used outside of mathematics. But I would think that adding critical thinking to any subject--whether mathematics or not--can help students learn to think. But most courses in school today focus on memorizing a bunch of facts rather than on learning to think. Reducing any subject to rote-- whether math or not--destroys the higher purposes of education. Teaching students to think should be our main goal as teachers. Perhaps much of the thinking behind mathematics does not apply directly to real life, but I do wonder if that thinking behind mathematics can still complement these goals.
In fact, reducing education to all job training also destroys the higher purposes of education. That does not mean that it is necessarily a bad idea to try to motivate students about the uses of various subjects in careers and in everyday life, but I think we get too carried away about this. As Underwood Dudley says--and I think he is right--those drawn to mathematics are drawn to the subject for reasons that go beyond getting a good job. Of course, such people are most likely thankful for the good jobs they did get with their mathematical knowledge but also find pleasure in mathematics for additional reasons as well. Furthermore, that does not mean that I oppose career-oriented schools such as Kaplan University or Argosy University or other similar schools; we still need them. Employers do want potential employees who can think but also want them to have certain career-specific skills as well. And we must face reality: Many students do want to go to college to train for a specific career. Some of them do want to learn to think, and others can be convinced that this is a good goal to acheive, but they also want to learn career-specific skills as well: Most of them are not going to college to become pure scholars.
I do question Dudley's claim that the public wants more mathematics taught since most people in our culture fear and hate mathematics. But it is possible that many of these same people wish they understood mathematics better and might support more mathematics being taught in schools if mathematics were taught well in schools, which is often not the case right now. I don't know since I don't recall reading anything that tells us what the general public thinks about whether math should be taught in schools and how much should be taught (this article is the only exception I can recall).
I do not think it is reasonable to conclude that kids turned off by the traditional curricula of math cannot be interested in any kind of mathematics. Mathematics is often taught as a boring, uncreative, uninspiring subject, so it should be clear why so many kids do not like math. If we were to fix these problems with math teaching and work harder at helping students find something enjoyable about math, then I believe we would see far more students liking math or not seeing math as such a burdensome or tortorous subject. We should shed the notion of the "one-size-fits-all" approach to teaching because students are not clones of each other: What works well for one student may not work well for another student. Maybe some of those turned off by traditional curricula might like math better because they have more options that now appeal to them or simply because the traditional curricula was taught to them in these bad ways. In short, our definition of "school mathematics" is too narrow, so I think it is a good idea to consider expanding students' choices of which math courses to take in middle and high school and college. Kirby Urner on the math-teach list has plenty of good ideas worth considering for expanding these options: He proposes including more discrete and digital and computer mathematics in school. Courses on mathematical modeling are worth considering. Berea College in Berea, KY, has a freshman mathematics course on mathematical modeling using computers (called Math 101). Case Western Reserve University has an interesting freshman mathematics course (Math 150) called "Mathematics from a Mathematician's Perspective."
Does Dudley prove in this article that mathematics is not useful to most people or that mathematics applies only to a few careers? No. First, he focuses just on algebra, not on mathematics as a whole. Second, his article does not say that mathematics is not important to these various professions but instead argues that the math can be done without going through all the algebra because formulas and other rote rules and tables have been developed to help professionals get the necessary information. For example, problems requiring a system of linear equations are done by using formulas that give the solution to the system of equations; all we need to do is plug in the given data and crank out the solution. But mathematics lies behind these rules and procedures and other principles, and I find it at least a bit distressing that many people apply these rules and formulas and use these tables without having at least some idea of what justifies what they are doing.