Let me begin by setting the stage for the uninitiated. Debates have been raging about what constitutes "knowledge" since the dawn of philosophy in the Western world, with the oft-lauded model of "justified, true belief" finding itself increasingly under fire in the modern era. Can you have a true belief count as knowledge without justification? Does it count as knowledge if the justification is somehow "flawed" by the standards of the outside observer? Can you ever really have knowledge?
A decent summary of these criticisms/fears/concerns comes from Edmund Gettier, who gives a number of counter-examples to attempts at justifying the JTB model of knowledge. I will not copy-paste them all here. Just click Mr. Gettier's name and it will take you to a nice Wiki link (I think Wiki is reliable just this once). The point of the Gettier cases is to elicit a response from the observer, specifically the response that even in cases where there is JTB, we hesitate to attribute knowledge to the subject.
Okay, I will give one example: You are driving through the countryside and see what you perceive to be a barn. That is to say that you come to judge the object as a barn, label it as a barn, and assume that you know that object is a barn. You claim knowledge! But hold on, Billy, we are not through yet. It turns out that you are driving through Fakebarn County, where there are a number of barn-like facades set up all over the place. Now, it just so happens that what you saw was, in fact, a barn. Your belief was correct and your justification was reasonable. Now knowing that it might have been a fake barn, however, do you still think you knew that it was a barn, or did you merely guess correctly? For that matter, is there a difference?
This is an issue that I take to be largely bound-up with a particular conception of knowledge that has its roots in classical Western philosophy (re: the Greeks). You see, the Greeks were not necessarily concerned with knowledge per se; they were concerned with absolute, normative truth (alethia). The interests of philosophers like Aristotle were largely concerned with getting things right; getting to the truth of the matter. Clearly, knowledge and truth are different: the former refers to a psychological state and the latter to a (the?) condition of existence. Nonetheless, by focusing on truth and using it as the target of knowledge, we come to see the beginning of the Western fascination with it.
Oh, to be sure, the Greeks did discuss knowledge on its own, a paradigm case being the Meno dialogue. In fact, that dialogue set in motion the very JTB model that modern philosophy continues to critically assess. The only real point of the Meno, however (at least in my opinion), is that having knowledge of something is better than mere belief. Even if the belief turns out to be true, you did not know that it would be so; you acknowledge the possibility of the inaccuracy of the belief. So the real prize here is certainty; nothing more and nothing less.
For example, if you have two potential guides, one who knows his way around the city and the other who does not, which would you prefer? Unless you are particularly adventurous, I am guessing you would take the former, the reason being that he is clearly capable (knowledgeable) of the city and can accurately guide you around it. Let me tell you what you might be missing out on, though: Suppose the second guide lacks knowledge but, by pure chance, is able to guide you around such that you get the exact same tour as you would with the knowledgeable guide. Is the tour itself in any way devalued, then? I can see no reason to say so, unless you are very OCD about your guide actually knowing where he is going and not just guessing. Still, you probably continue to feel more comfortable with the knowing guide. Why? My guess is that we tend to prefer a "sure thing" rather than a gamble, and knowledge means we should have certainty of whether or not we are getting that sure thing.
Now, that is my take on the Western tradition of knowledge, but there are plenty of others. A lot of people are opposed to this view of knowledge as a mere psychological state; they want it to have some greater "umph!" I have no idea what exactly they want from it, seeing as certainty is a pretty significant psychological state, but there it is. Well, that is not entirely true, I do have some idea. Linked with the Greek tradition, a lot of people seem to want knowledge to be a situation that accords with a very specific set of constraints. They want the "candidate for knower" to have a sufficient justification for his or her knowledge claim, to genuinely believe the claim, and the for claim to, of course, accord with the reality of things. Now, how you come to lay claim to this kind of ultimate knowledge is beyond me, since we are pretty much limited to our perceptions and intuitions, all of which are subject to the doubt of the aptly-named skeptic. Some people are less-demanding in the extent to which knowledge must agree with the reality of things, but they remain demanding all the same (ex: "no defeaters" models of knowledge).
The result of such an approach to knowledge, however, is extremely problematic for our approach to education. If the object of education is to generate this kind of knowledge, then we are going to be very narrow in how we think about the problems we will be addressing. If we narrowly limit the scope of what constitutes justification, we risk cutting out a number of perfectly reasonable means of coming to conclusions that are, according to the extent of human understanding, correct...but might have employed an unorthodox method of getting there. It also means limiting the goal of learning merely to propositions: I know that the solution/truth of "2+2" is "4". What more do you want, then? I mean, sure, we definitely want our students to be able to understand the general theory behind arithmetic, but is that about seeking truth or something else?
My suggestion is that what we are really looking for with knowledge is not truth itself, although I can understand why we would love to find it, but rather understanding. When we are attempting to teach students the fundamental theorems of calculus, we are not simply teaching them an accepted truth, but a skill set as well. We are teaching them how to do something as much as we are teaching them that something is the case. If you just want to teach "thats" to your students, fine. We can use rote indoctrination for that and limit our students to "knowing" a few facts...but they never know "why" this is so or "how" to do any of it. What I suggest is that coming to knowledge, as much as it is about certainty, is also about a state of concept mastery: You can claim to "know" that "2+2=4" when you accept it as true; you "know" math when you understand the "hows" and "whys" as well.
This view of knowledge is more akin to knowledge in the classical Chinese philosophical tradition, where the term "zhì " means not only knowledge, but can also be glossed as "wisdom" and, my favorite, "understanding". You see, the Chinese philosophical tradition comes to discussions of knowledge from a different understanding of the world. Oh, the Chinese were very concerned with natural patterns (look at their agriculture), and the basic psychological state they were concerned with was about drawing distinctions of whether or not something was so. This capacity, however, was not discussed or analyzed in terms of the Greek alethia, but in terms of the distinguisher's ability and simply whether or not something seemed to be the case; the obsession with absolute truth was largely avoided.
Furthermore, a lot of the philosophy was socially-oriented, and this meant that there was a lot more variability of subject. For example: How do you know the right place to live, let alone the right place? How do you know the true nature of humans? How do you know how things should be done? These are questions we continue to ask, with more and more people taking a very liberal, sometimes even pluralistic approach in their responses (Mill, Rawls, Okin, Myself). The ultimate conclusion a lot of philosophers of the era seemed to come to was that there might very well be no one set truth to all things (although maybe there were better solutions to the questions than others), and there are many points in texts where zhì can be translated as understanding just as easily as knowledge...sometimes it even makes more sense!
In no text is this more apparent than the Zhuangzi, wherein the eponymous character and possible writer discusses the distinction between "big knowledge" and "little knowledge". You can have lots of bits of little knowledge, like where you generally like to live, how to carve cups out of gourds, etc, but they do not amount to big knowledge. Little knowledge can be valuable, because it can provide a number of significant technical skills. It is, however, limited to specific instances all the same. Big knowledge comes in viewing the world more holistically and seeing the myriad possibilities for all things. It comes in realizing that we are not bound by particular customs and conventions merely because "they are so", and that there are nigh infinite possibilities for existence. In coming to peace with these realizations, one comes to the mastery that is the big knowledge.
Now, let us return to the situation of what we call knowledge in our education system. When we are teaching our students, are we trying to provide them with little knowledge or big knowledge? I think the answer is "both". We want to provide them with specific skill sets, such as how to add "1" and "1", but we also want them to understand broader theories. Then, we want them to take these broader theories and see how they fit into an even bigger picture, ultimately leading to a more holistic view of a subject. We are not just teaching them facts, which are certainly truths to the best of out knowledge, but also helping them to see that the world and all of the studies in it are broad indeed. In doing so, we create a unity of knowledge that takes on the disposition of conceptual mastery and psychological certainty. This, to me, seems to be the object of education and the genuine object of knowledge.
So what exactly does this mean for, say, a math teacher? It means that what he is attempting to articulate is not a set of facts (although we might say that such a set is part of what is being taught, even if it is merely a tool), but a way of doing something, a mathematical dào. In teaching the Pythagorean theorem, he cannot simply show the student a set of proofs and a general equation; the student, especially the new student, will not know what to do with this. The teacher must instead show the student how the theorem functions in conjunction with the theorem that what is going on is and will always be so (geometrically speaking, anyway). Here we have a coming together of theoretical and practical ability that leads to a conceptual mastery or, as I call it, genuine knowledge.
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