I was just reading http://misterbeck.weebly.com/ and in response to a recent post...
I used this (image below) to teach an application for multiplying polynomials.
Maybe showing students the two shapes separately before combining them, giving them the chance to COMPOSE before asking them to DECOMPOSE a composite shape, would help them understand the concept that a figure can be made of many shapes. This could be especially powerful with (colorful) manipulatives - even just from printer paper.
Maybe you've tried something like this before, but I'm trying to be more active in #MTBoS, and this answer occured to me as I read your blog. Please feel free to respond with any feedback !