看 is aiming your eyes; 见 is what lands in them

Where the picture shows up in 见

1. 1

   reaching the ear

   你听见了吗？有人敲门。

   nǐ tīng jiàn le ma? yǒu rén qiāo mén

   Did you hear that? Somebody's knocking.

2. 2

   not reaching the ear

   说大声点，我听不见。

   shuō dàshēng diǎn, wǒ tīng bu jiàn

   Speak up a bit, I can't hear you.

3. 3

   reaching the eye

   我刚才看见他进电梯了。

   wǒ gāngcái kàn jiàn tā jìn diàntī le

   I saw him getting into the lift just now.

   看 is aiming your eyes; 看见 is the thing landing in them. That is why 看了半天也没看见 makes sense.

4. 4

   reaching the nose

   我闻见煤气味儿了。

   wǒ wén jiàn méiqì wèir le

   I can smell gas.

5. 5

   reaching each other's eyes

   那明天见！

   nà míngtiān jiàn

   See you tomorrow, then!