I meant to ask my students yesterday what a number is.
I did that once with a class of College Algebra students who had appointed as spokeswoman a classmate who dared to ask me if I would please explain something “in regular English.” I told them I would try, but first I needed them to tell me in regular English not what a number does, but what a number is. They clustered up and went to work; discussing, giggling, yelling; digging in to the chore. My objective was complete when the reassembled horde reached agreement: “That’s HARD!”
Several years ago, and several years later, I griped, “It is impossible to remediate mathematical deficiencies for university students who don’t understand order or magnitude.”
More and more, university level remedial math is relegated to online drill systems which do a pretty good job of conditioning students to perform logically consistent symbolic manipulations. Any student with a tolerance for shoving symbols into formal patterns does a good job. I’ve taught many students who’ve cleared those remedial hurdles, who arrive in my room for their next course which is often “the last math they need.” These students clamor for detailed algorithms by which to arrive at the right answer: a symbol attained by careful traversal of a necessarily arduous path, in the end a trophy divorced from context.
It is students who pass through that sort of training that tell the world and their children, “I NEVER used algebra again in my life once I graduated.”