Sigh....this isn't an example of "new math", this is an example of misunderstood new math. Yes, a child should be able to explain how they got an answer--but only when given work that is at level. Asking a child who is several years above to explain how they "got" an answer to a problem like 10-7 is like asking a proficient reader to explain how they know that w-h-a-t spells what (I choose this example because it isn't phonetically regular: once you know it...you know it. You can't sound it out to "show" understanding).
I'm sure the teacher is well intentioned. There is a lot to be said for being able to explain rather than just memorize routines, but for teachers who are transitioning in their own understanding of what this means, there is too often a misapplication of the ideas. I cringe to think that I was that teacher a few years back, but.....
Two possible thoughts:
1) Ask the teacher to let your child do the same work with different numbers (since it appears that the type of work must not be negotiable). Given numbers that can't be calculated mentally by your child, there will actually be something that she can show/explain.
2) If appropriate numbers are not an option, ask your child to show/tell how she would explain 10-7 to a child who didn't know how to subtract. Often that makes more sense to a student than trying to explain to the teacher (who obviously already knows that 10-7 is three). It may help to ask your child how they would explain 10-7 if they weren't able to use mathematical language (minus/subtract/equal). Sometimes just the act of translating the equation into different language will demonstrate the understanding for the teacher, and for your child it can become a language activity--kind of like the game "taboo" where you have to get someone to say something but can't use any of a particular group of words as clues.
