The answer can be found by turning the fractions into blocks. We need a big supply of blocks from which we take 6.
A third of 6 blocks is 2 blocks. We need two thirds=4 blocks.
Half of 6 blocks is 3 blocks. We need three times this many blocks=9 blocks. So we need take extra blocks from the supply.
Which of the two sets of blocks is larger? The set of 9. So since this came from 3/2, 3/2 must be bigger than 2/3.