When people talk about the failure of mathematics education, we often talk about it in terms of the students inability to "think mathematically". It's impossib

When people talk about the failure of mathematics education, we often talk about it in terms of the students inability to "think mathematically". It's impossible to tell if students are capable of thinking mathematically, however, because I have not met a single (non-mathlete) student who could give me the mathematical definition of... anything. How can we evaluate student's mathematical reasoning ability if they have zero mathematical objects about which to reason?