Richard has an interesting 'problems with non-philosophers' post up. I think most people who are philosophically trained can sympathize with all the points listed; and, as he says, they really do touch on things that every schoolchild should be learning. (There's another side, of course, as Richard notes; I cannot count on my two hands the number of people I've met with philosophical background who largely use what they've learned to bully others, or to make themselves feel smarter than they are, or to nurse other despicable habits. But the existence of such people should not take away from the fact that these 'problems with non-philosophers' are serious problems.) If nothing else it should be considered a fundamental civic responsibility: a deliberative democracy depends crucially on having a large pool of people, from all represented walks of life, who can follow serious reasoning and engage in critical thought. There's no requirement that anyone be perfect at it; there's just the fundamental necessity that a deliberative democracy is, by its very nature, based on reasoning. If people are not taught to reason well, the society is not being given a solid foundation.
So the question is, what should citizens be learning? Starting with Richard's list, I think we can identify several things that are often lacking; these make natural starting points. They are:
(a) an ethos of inquiry;
(b) a familiarity with the way arguments work;
(c) good judgment about relevance;
(d) skill at avoiding fallacies.
(Of the annoyances Richard lists, the ones I'm setting aside are his #2, #5, and #7. I can see how these would be annoying in particular cases, but I don't think they can be considered general problems from the perspective that I'm taking here -- i.e., of what people should be taught as a minimum for healthy participation in civic life. Considering civic life in a deliberative democracy alone, there needs to be a healthy respect for received wisdom, a tendency to look at all the factors that are relevant, and an inclination to stick close to the factual and the likely. In particular cases these will often be problematic, but overall they are useful to the reasoning of society, as long as they are balanced by a-d.)
As Richard said over a year ago in the post he links to on the subject of philosophical education, education should primarily be a matter of skills rather than facts. Part of one's skill set should be things like inquiry into facts, evaluation of their relevance, etc., and learning most skills requires learning facts, so facts have a serious place; but they are not the point of education. They are the means. One way to say it is that good teaching will mostly be a teaching of facts but will never primarily be a teaching of facts; primarily it is a teaching of skills relevant to facts.
Naturally, however, the big question is how one would go about teaching a-d. I don't have any foolproof plans and programs, but I do have a few thoughts.
(a) An ethos of inquiry. In a sense I don't have much to say about this. Any teacher worth her salt is trying to cultivate this. I do think that we strain our schools a bit by giving them two distinct goals, which sometimes are in tension. By this I mean that we use our schools both to educate and to certify; it's possible to do both, but it's very easy for certification to overwhelm education. Everyone has dealt with this in one way or another: students who are more interested in what will be on the test than on what they can take away from their courses, teachers who expect you simply to regurgitate what they have said, administrators whose criteria of accountability have more to do with making the school look good on paper than with building thriving teacher-student-community relationships. There's nothing wrong with certification, but if too much emphasis is placed on it, it will inevitably result in people who have managed to pass a few tests but are not educated.
One of the aspects of an ethos of inquiry that I think we are completely failing to cultivate is conversibility, by which I mean an ability to treat arguments not merely as arguments-for-conclusions but elements of discussion. (In all fairness to non-philosophers it isn't hard to find people with Ph.D.'s in philosophy who are horrible at this.) After all, that is one of the primary points of formulating arguments explicitly: to be able to provide reasons to another person. To this end I think we need to do more to incorporate Socratic method into our teaching -- by which I mean not 'Socratic method' in the sense used in law, which is only loosely Socratic, but the sort of Socratic method one finds in Socrates, e.g., in the Gorgias. (I mention the Gorgias because it's a great way to see the Socratic approach in action. Not only is it, obviously, a Socratic dialogue, but Socrates also explicitly formulates elements of his approach as he goes along. It's a great text for becoming acquainted with Socratic midwivery.)
(b) Familiarity with the way arguments work. There is no way to get around this: the fact that most schools do not teach logic as a regular thing is genuinely scandalous. Every student who has reached high school, at the latest, should be able to distinguish validity and soundness. The approach should not emphasize (as I think logic courses usually do) invalidity. The fact that an argument is not valid is not actually very useful, given enthymemes, inductive arguments, analogical arguments, and the like. But there are few things more useful in all the world than being able to tell when an argument is valid, or to tell what premises could be added to an argument to make it valid. And the ability not to confuse validity with soundness -- a sound argument being a valid argument from true premises -- is crucial. I think many times the kinds of annoyances in Richard's list spring from an inability to appreciate the validity of an argument even when you have reason to think it unsound. Even more important than this, we need to teach informal logic -- and we need a better way to teach it than we currently seem to have. More on this below.
(c) Good judgment about relevance. This is the hardest thing on the list, and I think the one that teachers can do least about. However, we could still perhaps do better than we do by focusing more particularly and explicitly on what early modern Scottish philosophers called good taste. Good taste in a topic involves three things:
1) Broad familiarity with the facts and cases involved in a topic;
2) Skills useful for discerning features and making comparisons and contrasts;
3) Self-critical examination of one's own biases.
It's clear that we often try to teach, or at least encourage, these things; but we are so very piecemeal about it, despite the fact that there is probably nothing more important to a good education than the development of good taste over a large range of topics. Good taste is what good reasoners use whenever they are reasoning non-demonstratively; it is the very essence of critical thought. If you want to make good, rational judgments about (say) Shakespearean tragedy, you need a good sense of what goes into tragedies, what sort of tragedies there are, what features they have, what differences there are among them, what similarities they have, what biases might interfere with good judgment. You need experience with tragedies, skills of reasoning and perception relevant to texts and plays and the like, and the ability to know your own limitations. It is not enough to read Shakespearean tragedy, you must learn to read it well, at least to the extent that you can. Not everyone is capable of the same degree of good taste on any given topic; but everyone is capable of good taste on a lot of topics. Good teachers already go some way toward cultivating these things; but I think we would all benefit if we were more explicit about it and all in agreement on it.
(d) Skill at avoiding fallacies. This is a matter of logic -- chiefly informal logic -- but one thing that certainly does not work is giving student lists of fallacies or maxims or taxonomies or any number of other things we tend to use to introduce students to informal logic. For one thing, the maxims and taxonomies are often wrong; for another, if you give students a list of fallacies you aren't teaching them to avoid fallacies, you are teaching them how to label others -- for what students do is simply try to place (forcibly, and without good reason) arguments they don't like under the label of some fallacy.
What we need is what the medievals called dialectic: we need what they would call a topics (see also here). To put it in other terms, we need to teach students what types of inferences are based on (the seats or places -- topoi or loci -- of inferences); we need to give students not lists of fallacies, but guide them through arguments while pointing out potentially misleading features. The only way anyone ever really becomes good at avoiding fallacies is by coming to understand -- really understand -- aspects of reasoning itself. This is so both for reasoning in general (general topics) and discipline-specific inferences (specific topics). To do this we need a much more inspired logic pedagogy than we currently seem to have.
So those are a few ideas. I'm sure that other people have other ideas. How about you?