I just had a most interesting conversation with a high school student who is losing marks because he isn’t showing enough working. He is really struggling to know where the line is for what working to show or not show.

The most interesting thing was his description of how he uses his own written working:
He does what he can in his head, and writes things down when he feels he doesn’t have room to hold all the information at once. The paper is for him an external drive to save information.

This is actually one of the purposes of written maths of course! But one issue is he can go a lot further than many others without the external save function that paper provides. So most of the process is not recorded.

I explained to him that the working is not just a support for his own brain in the moment. It’s also a record of the key moments when important processes were used and when decisions were made.

I helped him with some derivatives working, to notice when he used the chain or product rule and since they are key processes, to make sure there was a line when they happened

Also with solving equations. He said “I’d move this to the other side and cross multiply to get this answer”, and he was thinking of this as ONE move. I helped him hear the “and” in his description that indicates a need for two lines of working.

Another example was when he is told f(x)=thing and g(x)=thing asked for f(g(x)). “Can I just write down the answer?” he asked. I said the move where you replace all the x’s with the formula for g(x) must be somewhere, because that is the meaning of what is happening.

So all very interesting. The final comment I made to him was about the working being for future him. Even five-minutes-from-now him might need a record of the moments to be able to find when an error happened.

Overall the most useful idea for him was noticing when a named procedure was used, and when a definition was used, and when his description of what happened included an “and”. These were things he could hold onto to help decide when to write things down.