Emmy was a prospective new pupil. Her mother told me Emmy was unconvinced about the prospect; to Emmy (and I suspect her mother as well) having One-To-One sessions felt – well, not quite punishment, but certainly not a treat. School, on the other hand, believed that though Emmy was in the top group she wasn’t getting as much out of maths as she might do.
The analogy I came up with for the mother is that as children journey through mathematics they encounter lots of doors. Some are like the supermarket doors that open automatically as you approach; some you can push your way through like the villain slamming his way into the saloon in a Western. There are doors with transparent panels letting you see what’s inside, and most of the doors at school or at home are simple and easy to operate.
But some doors are locked. If you’re particularly ingenious you may be able to guess the code or pick the lock, but if somebody helps you find the key it makes things much easier.