A sieve is an instrument that sand can flow through. It is like a flour-sifter, or the toy in the sandbox that has holes in the bottom. This is symbolic because Montag worries that his mind is like a sieve. Information comes in, but it filters right through his mind and goes out. This is what spurs his interest to get a mentor. That is where Faber comes in. Montag hypothesizes that if Faber is in his life, he will be able to catch more grains of sand.
Once as a child he had sat upon a yellow dune by the sea in the middle of the blue and hot summer day, trying to fill a sieve with sand, because some cruel cousin had said, "Fill this sieve and you'll get a dime!" `And the faster he poured, the faster it sifted through with a hot whispering. His hands were tired, the sand was boiling, the sieve was empty. Seated there in the midst of July, without a sound, he felt the tears move down his cheeks.
Montag has a memory that makes him remember how futile filling a sieve was for him in his childhood. He remembers that the faster he tried to fill the sieve, the harder it was to make sure it got filled up. In fact, it would never fill. This distracts and concerns Montag all at the same time.