As a couple of other posters have said you build muscle after your workout not during it.
Your workout is the time during which your muscles are damaged and made weaker. This muscle damage then switches on various biochemical processes both in the muscle cells themselves (where various genes are switched on to express themselves) and also in the various glands which release hormones that signal your muscles to repair themselves to grow bigger and stronger. (See the latest article on the site for an example of the hormonal response to exercise).
Point is its the rest period between workouts which is when your muscles grow and the purpose of the workouts is to stimulate this growth during your rest periods.
Your muscles take some amount of time to recover it doesnt happen instantaneously. After enough rest not only will your muscles have fully recovered but with any luck theyll be even stronger than when you damaged them with the workout. Again reaching this super compensation phase doesnt happen simultaneously but takes time (I have seen at least a couple of studies showing peak strength 4 days i.e. 96 hours after exercise).
The point of the above is that it is possible to exercise your muscles again and hence damage them again before they have fully recovered from the last work out (or after they have recovered to previous strength levels but not yet supercompensated i.e. grown even stronger). I would argue that giving your muscles the signal to grow stronger (by working them out) when they havent even recovered to previous strength levels would be pointless. This would be over training and would result in your muscles not getting stronger but instead will potentially result in them getting successively weaker between workouts (at least weaker to the point that the amount of damage you can do per workout matches how long you give them to repair i.e. they wont keep getting indefinitely weaker to the point you can no longer use them but they will be no where near as strong as they could be).
It is of course also possible to wait too long before exercising your muscle again (i.e. after it has past its supercompensation peak), this would be undertraining.
Point is there is some optimum rest period between workouts for a muscle group. Working out too often is pointless (as is working out too infrequently) so that more is not necessarily better for getting body building results. In other words although it may seem unfair, sometimes working harder can make you weaker not stronger.
You have to find out what this optimum rest period for you is and in the case of arms it may not be three times a week (in fact it very likely isnt three times a week unless you are a real beginner or on steroids. A beginner doesn't do as much damage per workout so can recover quicker while someone on roids can also recover quicker).
The exact length of that optimum rest period is hotly debatable (some people will say 48hrs, other 96hrs, others like myself a week). In my case I find once a week (direct) workout per muscle group optimum, otherwise I personally just get weaker between workouts (though when I train I train very intensely).
It is debatable as to whether it is better to train very intensely and hence have to put up with the longer rest periods (like a week) that that entails or to train less intensely but more frequently (I favor the former but I have seen articles by the experts article writers here at T-nation favoring the latter).
Youve got all the time in the world to experiment though (I mean youve got another 20, 30 or 80 whatever years body building in front of you) so Id recommend experimenting with different workout frequencies for months at a time to find out what works for you (i.e. note whether you are getting stronger or weaker or staying the same between workouts). If you follow a bad routine for a month or two and so dont make progress for a month or two no big deal like I say you have decades of working out in front of you to make up for it and at least then you will now what works for you.
I would strongly recommend trying out the once a week direct tricep and once a week direct bicep routine though. You can also experiment with doing direct and indirect tricep and bicep routines on the same day or on different days; which means doing tricep or bicep workouts effectively once a week or twice a week.
My own theory is that as you get closer to your genetic potential/limit and as you also get older you are better off doing bicep/tricep direct work on same day as you do indirect work so that they are effectively only exercised once per week but if you are on steroids or younger or for what ever reason have better recovery potential you may be better doing biceps on same day as chest/shoulders etc and triceps on same day as back or legs so as that they are effectively exercised twice or even more per week. (In an ideal world youd have such great recovery ability that you could train each body part again after a mere 24hrs rest and maybe some genetic freaks or those on ridiculous doses of hormones can do so but the reality for the rest of us is that we may need a number of days recovery for optimal results).
You can read all the articles you like on this site but youve really got to put it to the test to find out what works you (I am willing to bet though that youll do better on less than three direct arm workouts per week, if your advanced enough one direct arm workout per week may be fine. I know in my case biceps are by far the easiest muscle to over train and I really have to back off how much bicep training I do compared to what I am capable of to prevent over training them).