I was thinking an apt analogy might be making stock -- you filter out all the solid food you don't want to keep in the liquid.

And it's a doubly-good analogy, because I have occasionally gotten that confused in real-life as well. Twice in the past ten years I've had a stock boil away for three hours, and then set a colander in the sink and poured it through, only to watch my beautiful stock swirl down the drain because motor-memory made me forget that I wasn't draining pasta but should have put the colander in a bowl...

depends on the size of the sieve, but sometimes one does cook a whole stewpot of tea at once (f.e. in canteen)