ANSWER:

What you have heard is absolutely wrong. 

No problem in doing other activities during athan including praying.

The best however is to reply to the athan and recite the prescribed dua after it but this is not mandatory. 

is it that we shouldn’t pray anything while the adhan is being called and just sit and listen? I have heard someone saying that even if we are praying voluntary prayers, we should stop and listen to the adhan.