The answer to this is to peel the ends, then chop the very base (where it turns from green/purple to white).
And the way to keep all the peelings together instead of covering your table, floor, shoes, and anything that happens to be on the work bench is a method we developed in the Claridges larder where we counted asparagus by the thousand (the most I remember preparing in one day was around 4,000 - but we had everyone on it that day).
Meanwhile, back in the asparagus factory yesterday, firstly the 'leaves' were removed on the middle 2/3 of the stem (cosmetic appearances - it looks better when cooked). Then you rest the base of the stem on a box/ bowl in the middle of a deep baking tray and peel the bottom 1/3 of the asparagus stem - this removes the fibrous/ woody part. The peelings are collected in the tray as you go along, keeping things neat, tidy and fast, and then added to your compost bin.
Grilled asparagus with tiger prawns and oyster sauce dressing
Asparagus with salmon provencal
Grilled Evesham asparagus with scallop and tiger prawn fricasse
Grilled asparagus with parmesan
Asparagus and hommous on baked new potato