Whiteboards have been very helpful for both my kids, for different reasons. DS does all his math on one (sometimes two for a long problem). When it's work for submission (such as an on-line course he took), we take photos of the whiteboard as he goes, paste them into a big PDF, and submit that electronically. No trees die! There's no reason this can't be done at school - except stigma, alas.
It worked great for DD when she was young, though - teacher just pulled out her phone and took a picture every time DD finished something.
Thank goodness for bonus challenge questions - DS always gets them easily, and they usually make up for the all the marks he lost to careless errors (at school he still does math on paper; it's the only thing he doesn't keyboard).
He too does do way too much in his head. The only solution I ever found to that was AoPS, where the complexity of the problems actually requires writing out the details to avoid errors, and where the proofs that need to be submitted in the course became an enjoyable challenge for him to produce (much to my surprise). Some of contest math we're playing with at home is getting complex enough that he's starting to make mistakes because he isn't writing enough down. That's hitting him where it hurts, and doing far more to convince him to write down more steps than anything I could ever say. Still, we work the problems side-by-side and I'm always writing at least three times what he does... sigh. Mathematician I'm not. All that to say, the willingness to "show your work" depends a great deal on being given work worth showing.
