I think I finally figured out our math problem (no pun intended). I thought that the concrete nature of mathematics was a given. Apparently I’m wrong. I just read the following from Nancy Pearcey’s Total Truth:
Today, however, most philosophers no longer even regard mathematics as a body of truths. The dominant philosophy of mathematics treats it as a social construction, like the game of baseball. “Three strikes and you’re out” is an arbitrary rule. It’s not true or false; it’s just the way we choose to play the game. By the same token, mathematical rules are just the way we play the game.
Even American schoolchildren are now taught this postmodern view of math. A popular middle school curriculum says students should learn that “mathematics is man-made, that it is arbitrary, and good solutions are arrived at by consensus among those who are considered expert.”
She then goes on to say,
Moreover, if math is arbitrary, then there are no wrong answers, just different perspectives. In Minnesota, teachers are instructed to be tolerant of “multiple mathematical worldviews.” In New Mexico, I met a young man who had recently graduated from high school, where a mathematics teacher had labeled him a “bigot” for thinking it was important to get the right answer. As long as students worked together in a group and achieved consensus, the teacher insisted, the outcome was acceptable.
Ah ha. So that’s been my problem. I’ve expected math to always lead to the same answer everytime. Now I realize that as long as Maddie and Chloe agree on what they can come up with, they are right. This should make math a LOT easier in the future.