As Richard states, there is considerable disagreement among philosophers about which philosophical questions are significant, and why. There is also considerable truth in your suggestion that studying the methods and texts of philosophy is itself a valuable way to develop one's analytical reading, writing, thinking, and communication skills.

So, as a teacher of philosophy I hope that my students will both gain useful insight by studying diverse philosophical questions and approaches and will also gain useful analytical skills. This is an extremely powerful combination, and this power is one reason why philosophy students tend to do so well in the professional job market and also tend to advance quickly within their chosen professions.

About puzzle solving: the difference that you touch on, that philosophical puzzles are rarely solved in definitive way that gain professional consensus, is the central one. I take this to mean that investigating a philosophical question is very different from solving a puzzle (and I'm very glad for this difference -- see more here: 155).

To add a word or two to Dan's great response: there is no question that mathematics deals with infinite collections, but what those are, what we mean when we make claims about them, which claims are correct — these have been hotly disputed issues for thousands of years. (In the history of mathematics, concern for these foundational questions has waxed and waned. There have been times, for instance in the early part of the twentieth century, when disputes over these issues, were very heated and split the mathematical community. There have been other times, for instance now, when mathematicians have been less interested in these issues — although of course there are always exceptions, like Dan.) The basic question — what does it mean to call a set "infinite"? — is so fundamental that it's simply astounding that we don't know how to answer it.

On one way of looking at the matter, what Dan called "platonism", to say that a set is infinite is simply to have given a measure of its size. To say that a set is infinite is much like saying that it's got 17 elements in it: if you counted up the elements in the second set you'd find there were 17 of them, and if you counted up the elements of an infinite set you'd find there were infinitely many of them.

But on another of way looking at the matter, this is insane. How can one finish counting up the elements in an infinite set? Isn't that what "infinite" means, that the process of counting never stops? On this way of looking at things, to call a set "infinite" is not to describe the size of some actual collection, but rather to mark it off from all finite collections: finite collections are ones for which the process of counting their members eventually stops, while infinite ones are collections whose elements we can keep on generating without end.

The first conception accepts the existence of the actual infinite: a collection that actually contains infinitely many objects. The second conception rejects this as unintelligible and talks instead of the potential infinite: to say that a set is infinite is not to make a claim about the size of an actually existing object but rather to say that each of its elements can potentially be brought into existence. (The two conceptions will be confused if you think that an entity that can potentially be brought into existence really exists after all — and has the property of potential existence attached to it. See here for some comments on a comparable error.)

The chief lesson I have drawn is that reaching consensus is not an important criterion for progress in philosophy. This is true in several areas of my own professional life, including in my discussions with colleagues, in my own study of philosophical texts, and in my own philosophical research.

First, there is much be learned by exploring diverse perspectives to complex philosophical issues. My best discussions with colleagues are ones where we understand and explore the differences between our philosophical views; when this occurs, I rarely end up agreeing with others but I frequently gain new insight nonetheless.

Second, this lack of agreement among philosophers—which occurs at so many levels--is itself a fascinating intellectual phenomenon. As an historian of philosophy, I am often enthralled by the ways that philosophers' thoughts about a single issue or idea have changed over time.

Finally that philosophical questions and issues are not amenable to simple answers is exciting to me as a scholar: the same intellectual characteristics that lead to a lack of consensus mean that I and other philosophers have a wealth of opportunities to contribute to the discipline in diverse, creative ways.

I agree that critical thinking skills are vital. Taking a course in critical thinking is not the only way to gain these skills, however, and so I think the most important question is whether high school students have plentiful opportunities to do this.

In part, this is a matter of curriculum. I suspect that required courses in critical thinking or introduction to philosophy are not the best ways to inspire high school students to work hard on developing critical thinking skills. Instead, I agree with the idea that critical thinking should be taught “across the curriculum,” which is to say should be taught in diverse ways in nearly every course. The trick, of course, is designing excellent curriculum that does this well.

That said, I suspect that quality of teaching matters more than curricular design: it is our relationships with individual teachers that can inspire us to work hard and learn the most, and the most important educational reforms may be those that help our teachers to learn how to inspire as many of their students as possible.