Teaching children to convert between number bases Richard Feynman was critical of teaching children how to convert between number bases.

I'll give you an example: They would talk about different bases of
  numbers -- five, six, and so on -- to show the possibilities.  That
  would be interesting for a kid who could understand base ten -- 
  something to entertain his mind.  But what they turned it into, in
  these books, was that every child had to learn another base! 
  And then the usual horror would come: "Translate these numbers,
  which are written in base seven, to base five."  Translating from
  one base to another is an utterly useless thing.  If you
  can do it, maybe it's entertaining; if you can't do
  it, forget it.  There's no point to it. (Surely You're Joking, Mr. Feynman!)

Are there good reasons for teaching children this skill?
 A: I am of the opinion that if you cannot add and subtract in a base other than ten, then you haven't really understood arithmetic. 
A $9 \times 9$ table of addition facts is not too hard to memorize over the course of years in elementary school. When children see the symbols "5 + 7" in that order, they can recall that the appropriate response is to write down the symbols "12" in that order. All this can be done with no understanding of the fact that the answer is meant to represent a group of ten and two "singles".
Suppose instead a child has some understanding of alternate bases and sits down to evaluate "5 + 7 (base eight)". It is unlikely she has committed all the various addition tables to memory. She is forced to visualize (maybe using manipulatives) five singles and seven singles and regroup them into a group of eight and four singles. Thus, the answer is 14 (base eight).
In summary, I do not think understanding an alternate number base is useful in itself, but is useful as a means to deeply understand base ten.
A: From my (humble) perspective:
Understanding the binary system - "base $2$" - is important, conceptually, given the prominence of computing and computer science, electrical engineering, digital "everything". Additionally, the binary system correlates with dichotic logic: $\{0, 1\}$, $\{\text{true}, \text{ false}\}$, on/off, yes/no, etc. 
So acquiring an early understanding of binary logic and the binary system can go a long way, for students in general. (Later on, understanding the hexadecimal will then be much easier, as will grasping any base that is a power of $2$).
With respect to other bases, it does seem to be more of an exercise in "mental flexibility", and its value is likely age-dependent: I suspect that consulting those who study cognitive science and developmental psychology would be helpful in answering this question, since they would likely have a better grasp of how, when, and whether students, generally speaking, are cognitively prepared for and/or will benefit from these exercises in any substantive way. 
That said:
I do know that I have acquired a much deeper understanding of the English language, both syntactically and semantically, having studied Spanish in depth -- an understanding I doubt I would have acquired without being immersed in the structure of another language. So the same may be true for helping students acquire an deeper understanding of base 10, by immersing students in tasks of converting to and from, and operating within an alternate base (a foreign numeric language, so to speak).
A: I have recently started looking closer at this topic. Let me give you a few example problems:


*

*It is currently 4:45 p.m. and I plan to go to bed at 9:20 p.m. How much time do I have before I go to bed?

*There are 432 students in the school. If 265 of them are girls, how many are boys?

*I need to make a 5 ft. 8 in. long shelf for a bookcase I'm building. How much do I need to cut off of a 6 ft. 3 in. board I found in my garage?

*I bought 5 gallons of milk at the store. If my family drank 3 quarts during dinner last night, how much milk do I have remaining?


I gave these to a sample of 60 grade 6 students. Nearly all recognized these as subtraction problems, lined them up in the traditional format, and then subtracted to find the difference. However, because they are limited to base 10 knowledge, they missed most of them by regrouping incorrectly. (time is base 60 - hour/minutes, length is base 12 - foot/inch, capacity is base 4 - gallon/quart).
Therefore, I think it is important that we teach students how to work within different bases. Obviously, though, it is important that we share other strategies (in this case, count up, convert) as well.
