Do we need to formally teach the Greek Alphabet? This is a question that I am purely interested in because I think we never thought about this before in Mathematics education... or even so was not discussed.
When did we learn the Greek alphabets when we learnt mathematics? For example, I was pretty afraid when I saw $\text{Pressure} = h\rho g$ or even the idea where the variable for an angle was $\theta$ and that was when I was in 8th grade.
When I went to high school, I got even more confused when I saw letters like $\mu, \lambda, \sigma$ during statistics class. During calculus class, don't we remember the $\delta$ represnting small changes and our peers write it like a small d? Even only when I was in 4th year of college that I realize $\Sigma$ represents sum because Sigma and Sum start with 'S' and $\Pi$ represents produce because Pi and Product start with 'P' (it's for my memory but I'm not sure if it was taught this way.
So the question is: do we need to formally teach the Greek alphabet (not all but slowly) and tell them that researchers use these letters frequently to represent certain variables before we teach them? Of course, as student of math, we hardly (if ever) say $X$ follows a Poisson distribution with parameter $m$ or other correct, but "weird" sounding statements. I remember doing it during high school statistics... because I strongly believe that these "seemingly scary" letters "turn off" math students in the pre-college level.
 A: I think we should. The reason is that there is a correspondence between Greek letters and Roman letters, which is often kept too implicit to those that do not know the actual alphabet. $\Pi$ doesn't represent the product because "pi" begins with a "p". $\Pi$ represents the product because $\Pi$ is a "P", in Greek. 
Therefore, I think every mathematics or physics student should know the whole Greek alphabet, what sounds the letters make, and what Roman letters they correspond to (there is not an exact 1-1 mapping here: for example the vowels E and H are both roughly "E", and the letter $\Psi$ has no direct Roman equivalent). Just knowing the very basics of the alphabet is very illuminating. For example, why do we use $\alpha$, $\beta$, and $\gamma$ for angles? (Answer: because they are the first three letters of the Greek alphabet)
It's also important to know both uppercase and lowercase letters. That way, we are not so confused when $\Sigma$ and $\sigma$ correspond to roughly the same thing, and we know how to make the same correspondence with any letter, just as we should not be confused about why either is used in place of $S$ or $s$.
A: I teach physics. When a Greek letter comes up for the first time in my course, I try to make sure that it's explicitly described for them, either in lecture or in the text. They need to know that it's a Greek letter, what it's called, how to write it, and what sound it makes in Greek. They don't know these things without being told. In particular, they often see a letter like $\rho$ and just assume it's some kind of stylized letter "p." Likewise for $\gamma$ as "y," $\nu$ as "v." Teaching the whole alphabet all at once wouldn't work; they wouldn't retain it because they wouldn't be using it. It's unfortunate that the LaTeX $\gamma$ looks a lot like a "y," and someone looking at it can't tell that there's supposed to be a loop at the bottom.
You could say that students should just look these things up if they don't know them. There are two problems with this approach: (1) if they think $\rho$ is just a stylized "p," then they don't know that it's Greek and they need to look it up; (2) if you expect them to look it up, they won't, and then you'll have to grit your teeth every time you grade their paper and seem then writing $\gamma$ as "y."
A: Many successful (ex-)students seem to assume "this was no problem for me, so I assume it will be no problem for other students." (For examples of this attitude, see some of the early comments on the question, some of the now-low-voted answers, and even one of the comments on this answer.) This is not just a bad assumption; from a teaching perspective, it's possibly the worst assumption one can make. (Full disclosure: I didn't find this a problem either, but I've always been fairly good at math. However, I know a lot of people who have been turned off to math precisely for reasons like this that seem trivial to those (like myself) who don't happen to find them off-putting.)
Personally, I think there's legitimate reason for concern here, but "formally teaching" the entire alphabet probably isn't going to solve the root problem, or at least, it isn't going to be the most efficient or effective way to solve it. If you teach the alphabet all in one go as an alphabet, you're just giving students a whole bunch of random symbols to memorize all at once instead of giving them a couple of random symbols at a time with no context.
So I would suggest instead that for the first couple times each non-Latin symbol is used, the teacher should (1) mention what alphabet it's from, (2) say the symbol's name and, if there's a Latin equivalent, what the equivalent is, (3) state whether the symbol is an arbitrary choice for the example at hand (such as using epsilon rather than x as the argument of a trig function) or a standard convention that the students should memorize (such as using capital pi for a product), and (4) if applicable, whether the choice of characters has anything to do with their usage (as in the case of Pi for product, with the aforementioned "p is for product" connection).
EDIT: Even for students who (1) don't find unexplained mysterious symbols off-putting and (2) are good at looking things up outside of class, teachers can unintentionally conceal some genuine and important subtleties. For instance, it wasn't until my third or fourth year of taking classes involving calculus that I realized there's a distinction between the d symbol used for single-dimension derivation and the partial symbol used in multiple dimensions.
EDIT 2: removed reference to "most of the answers so far," which is now fortunately long-obsolete. 
A: There are a lot of tools in the mathematical toolbox that are left to incidental learning. The Greek alphabet is merely one of these, and it's a relatively unimportant one. Ideally all of these tools would be presented formally, according to a smooth learning curve, to enable future students to make faster progress than their predecessors and thus achieve greater heights, but this would be a massive undertaking and it would require an unprecedented rearrangement of mainstream educational ideas.
A: Yes you should, frequently even professors mix them up. I had a professor calling eta, "nu." It was one of those drink from the fire-hose courses. I made up a flashcard with the entire Greek alphabet on it, based on the Wikipedia table on all the letter entries. It was quite helpful in keeping me straight. 
    Greek alphabet
Αα  Alpha       Νν  Nu
Ββ  Beta        Ξξ  Xi
Γγ  Gamma       Οο  Omicron
Δδ  Delta       Ππ  Pi
Εε  Epsilon     Ρρ  Rho
Ζζ  Zeta        Σσς Sigma
Ηη  Eta         Ττ  Tau
Θθ  Theta       Υυ  Upsilon
Ιι  Iota        Φφ  Phi
Κκ  Kappa       Χχ  Chi
Λλ  Lambda      Ψψ  Psi
Μμ  Mu          Ωω  Omega

It's important to make sure students and professors are speaking the same language. For any course that seems to run out of greek letters for symbols, you all need to know all of them.
A: To me, the Greek alphabet is just another character set we use.  (For a non-Greek example, the Hebrew letter aleph, $\aleph$, is used in discussions of cardinal numbers.)
I feel that it is a waste of time to "formally" cover it in a class--a student can easily "pick up" the alphabet just by using the letters in a class.  I can recognize nearly all of the Greek letters (upper and lower case) just from casual browsing of math/science sites/textbooks.  In one introductory engineering class, I was required to memorize the lowercase Greek alphabet, but I found I didn't need to from that casual browsing...  While that was being discussed, I felt a little cheated that valuable class time was being spent discussing something so trivial.
If students didn't have access to Google or some other search engine, any good dictionary or encyclopedia should have a total list of the characters along with their names.
A: I would definitely have appreciated a brief 10-20 minute introduction to the Greek alphabet at some point during my formal education.
That being said, I think the problem isn't students' lack of experience with Greek so much as math professors incorrectly referring to and mispronouncing letters from its alphabet. This happens somewhat frequently, especially for certain letters (read: psi, xi)  and can be pretty frustrating, especially when the mathematics itself isn't super clear.
It strikes me as somewhat antiquated, but I know many PhD programs still require a sort of perfunctory foreign language exam as a prerequisite for graduation (e.g. translate a paper written in another language). At least in theory, it would be great if this included a brief quiz on the Greek alphabet and how to correctly pronounce each letter.
The other benefit to this (again, in theory) is making people aware that there are more Greek letters out there than the ones most commonly used. Running out of symbols is not an excuse for using hard-to-read subscripts. This thread reminds me of the anecdote from John Hubbard's Vector Calculus, Linear Algebra, and Differential Forms (which includes a full table of the Greek alphabet in Chapter zero) about a mathematician who refers to all Greek letters except $\omega$ as "$\xi$," while referring to $\omega$ as "w."
A: While there is a plethora of anecdotal evidence here, I think it might be useful to cite the approach of several text books I have come across.


*

*The book Vector Calculus, Linear Algebra, and Differential Forms starts off by giving an introduction to some notation and explicitly talks about the utility of learning the Greek alphabet. To paraphrase the book, learning what the symbols are and how to say them helps the reader remember the sysmbol and its importance. 

*Nearly every high school level math text book (I have seen exceptions with science) introduced the letters as you go. This is so ubiquitous, and the textbooks vary by state, so I will not link to an example.

*The majority of statistics books that I have used introduce the variable along the way, but they make an effort to tell you what it is and how to say it. The book I have the most experience with, Stats: Modeling the World, does a good job of introducing them as the text progresses. In statistics it is more common to see Greek letters early on as they are used to distinguish things such as sample vs. population.

*The majority calculus and real analysis books assume knowledge of the Greek alphabet already. Baby Rudin does not introduce anything, but considering the level of the course, it is reasonable, as the variable should have book seen in other courses. On the other hand, Apostol's calculus (an introductory text) introduces the other notations for set theory and the such, but does not touch on the Greek letters. 
In general, there seems to the trend that with the increase in the level of rigor, the less attention is paid to such matters. It is therefore unreasonable to teach the full alphabet it in a class were either the students have seen the letters before (i.e. not an introductory) or where the Greek will not be used often (however, introduction should be given through the course of the class). 
However, when it comes down to an introductory class, this is when the pedagogical methods differ. The Alphabet tends to not be taught in its entirety, but nearly all authors agree that  knowing how to pronounce the letters is helpful to the student.
A: As a below average mathematician I was grateful for the teachers who explained symbols to us.
The Greek alphabet might be a special case - but I think not.
As a student, you don't know what the "funny symbols" are, what their name and spelling is and so you have difficulty to put a name on a concept.
Suppose somebody teaches you a simple concept, draws a unknown symbol next to it and goes to the next concept - because there is no familiar name for the symbol, you will probably forget it and have trouble when you need it.
I failed quite miserably at math at one time and was quite afraid of Greek symbols until a teacher sat down and explained to me their drawing, their spelling and what concept is behind each symbol.
As an analogy: Many programming languages have dictionaries, where you can access some value by a key. But you are out of luck when you don't know the key. The Greek alphabet is similar: Sure, the concepts are more important, but it helps when the name / spelling of each Greek letter is readily available in the student's memory.
A: There is no need to formally teach the Greek alphabet; we can just look up the letters as they appear in the exposition. Having said that, it is not hard for one to look up the entire Greek alphabet online these days, and one takes only a few moments to go through them all.
A: Until the end of the last century mathematical texts written in German often used horrible Gothic letters  besides Greek letters, e.g. lowercase for vectors, uppercase for vector fields
$$\mathfrak{abcdefghijklmnopqrstuvwxyz}$$
$$\mathfrak{ABCDEFGHIJKLMNOPQRSTUVWXYZ}$$
A: Interesting question! I was wondering the same thing a couple of days ago. You didn't mention the first Greek character you encountered in your life: $\pi$. Did it scare you?
Since I am teaching Calculus in high school in Eastern Asia, I am trying to avoid culturally-oriented notations which may exclude some students. For example, $\mathscr C$ (used a lot in France to denote curves) is banned for my classes since my students are not used to script characters. When explaining that the variable $x$ in the notation $f(x)=x^2$ is just a placeholder and can be substituted by anything, I include examples in their own alphabet besides the classical $f(\square)=\square^2$. I also wish to include more local history of mathematics, to go with the usual "The Ancient Greeks did not know that...". To sum up, you have to adjust to the cultural and educational background of your students.
However, when it comes to Greek alphabet, it is so widely used, together with the Latin alphabet, that students cannot afford not to know it. But the rhythm of introduction of Greek characters in maths or physics is very slow, so formally teaching it seems to me quite useless. The time between teaching $\kappa$ or $\xi$ and using it is very large and these characters will most likely be forgotten in between. 
A: While a few Greek letters are introduced in basic algebra and geometry, pre-calculus is the first class in which they start to gain frequency.  I claim this only because I am currently enrolled in the course, and being a primary source, I can ensure that my opinion is one of the community's to which this question pertains.
I believe that I am not alone when I say that Greek letters ward me away.  I think the problem isn't that students don't know the alphabet; I think the problem is that they don't know the connotations of the letters in mathematical context (i.e. ω is angular speed).  
Compounding that issue though, is the fact that in being less familiar with a symbol, you are less likely to remember what it means.  So in that regard, I advocate the teaching of the Greek language with one caveat — you must explain the symbols' mathematical meanings.
