I found your website while looking to find out what I caught when I scooped some tadpoles for my kids today. I’m pretty sure it does not look like a stick, but rather it lives in a stick. Legs look a little crabby or crawdad like. I don’t know, here is a picture. I would really appreciate if you had any information.
Your fascinating insect is an aquatic Caddisfly Larva in the order Trichoptera. Adult Caddisflies are mothlike insects, and the larvae are aquatic shelter builders. Each species of Caddisfly, and there are many, builds a unique style of casing that it carries around with it for protection. Sone are a single twig, some many twigs cemented together, and some are even cemented pebbles and shells. You can read more about Caddisflies on BugGuide.