TITLE: Communicating mathematics
QUESTION [12 upvotes]: As a TA, my ratings have been fairly mediocre.  There's nothing to suggest I'm a problem, but I have room to improve.
What's more disturbing to me is that I even struggle to communicate math with people who know a lot more than I do.  It's no surprise that they struggle to communicate with me.  They know more, probably forgot what it's like to know less, and generally do not give the details on everything, sometimes even neglecting to mention that a particular detail needs to be filled in.  But when I communicate, I try to leave out as few details as possible and clearly enunciate when I am leaving them out, and to minimize the number of mistakes I make.  I am extremely nitpicky about this.
So there must exist other obstacles to communication other than just misrepresentations.  I suspect that what I'm lacking is a pedagogical element, which would also explain my interaction with students.  How can I go about finding out what is lacking, and then fixing it?  What are some common teaching mistakes made by people with the attitude that mathematics should be precise and airtight?  It's not that I'm such an adherent of my own philosophy that I refuse to "dumb things down."  That used to be the problem, but it appears that when I attempt to convey vague ideas, people do not understand me.

REPLY [2 votes]: A fairly random example of a great pedagogical technique for mathematics - examples first.  
For example, suppose I was trying to explain to you what a ring is in algebra.  I could start by telling you that a ring is a set $R$ together with two binary operations $+$ and $.$ such that:

$r+s=s+r$ and $(r+s)+t$=$r+(s+t)$ for all $r,s,t\in R$
There exists $0\in R$ such that $r+0=r$ for all $r\in R$
etc. ...

However, you are much more likely to understand what a ring is if I give you some concrete examples.  For example, I could say, 'Rings are like $\mathbb{Z}$.  If we consider the integers, then we have two important operations $+$ and $\times$, which have the following interesting properties...'  Then I could point out that other interesting sets like the integers $\mod n$, which also have two operations which behave in a similar way.  Only then would I give you the formal axiomatic definition of a ring.  The next step would be showing that many interesting things that are true for the integers are true for all rings, and that many other interesting things about the integers (like unique factorization) are true for rings with certain properties.  Then you would gain some understanding of what rings are and why they are worth introducing.  
Similarly, I don't think anyone would find the following definition at all meaningful if they hadn't studied topology before:
A topological space is a set $X$ together with a collection $\tau$ of subsets of $X$ (called the open sets) such that: 

$\varnothing,X\in\tau$.  
$\tau$ is closed under taking unions: for all (possibly infinite) collections of sets $(U_\alpha)_{\alpha\in A}$ with $U_\alpha\in\tau$, the union $\bigcup_{\alpha\in A}U_\alpha$ is in $\tau$.  
$\tau$ is closed under taking finite intersections: for all $A, B\in\tau$, $A\cap B\in\tau$ (and therefore all intersections of finitely many members of $\tau$ are contained in $\tau$).  

Much better is to start off by introducing the more concrete idea of metric spaces (by first showing that a lot of concepts in real analysis, like convergence and continuity, can be expressed entirely in terms of the distance between points, and showing that more abstract ideas of distance can be useful) and then showing that the definition of a continuous function between metric spaces can be expressed entirely in terms of the open sets in that space, and then introducing topology from there.  
Having given you some examples, I'll now state the general principle: if you want to explain some mathematics to someone, always start by telling them some motivating examples.  This serves two purposes.  Firstly, it shows them why they should care about the idea you're introducing, since all mathematical concepts were originally created because they were interesting in some way.  Secondly, it helps them to understand the ideas that you're telling them about, since they have some concrete examples to link them to in their mind.  
Of course, that's just one useful pedagogical technique, and it's not necessarily useful in all contexts.  But it is often very useful.  
For a much better exposition of this idea, see this blog post by Timothy Gowers.