Misconceptions surrounding fractions are so well-studied that I feel a bit ridiculous sharing anything about them. Anyway…

I was chatting with this kid who was having a bunch of trouble with written fraction notation. She had been correctly solving problems that involved language such as “shade in four out of seven pieces” or “divide this shape into eighths,” but got stuck when she reached a problem that asked her to “shade in 4/6 of the shape.”

Alice:Oh, so that’s 5.

Me:Can you explain why?

Alice:Because it’s not six sixths.

Me:So, not quite.

Alice:Oh, it’s 2. Because that’s 6-4.

Me:…

Alice:Or it’s 10?

Me:See…

Alice:I’m really confused here. What’s the answer?

There’s no puzzles or misunderstandings here. Alice thought that the fraction symbol was an operation between the numbers 4 and 6. And of course she did. Every other time that she’s seen two numbers and a symbol before she’s been asked to produce a third number. This is new ground for her.

I’ve been taking the advice of Brilliant Commenters Fawn, Jenny and Avery and using the language of “out of” to bridge the gap for this kid.