A brilliant formulation from David Pierce:
The real world consists of what *matters*. It is interesting that the verb “matter” occurs naturally here, since it is derived from the noun that is used for what the “real world”—the physical world—is supposedly made of. But if we ask, “What’s the matter with Fatma?” we may be asking about a physical ailment, or we may be asking about an *idea* that is causing Fatma to behave strangely. Ideas matter.
The whole point of education is to make things real for students, things that were not real for them before. [Things like fractions, for example.] I recall an example in a letter in the Notices of the AMS, some time in the 1990s.
Calculus students worried that on Problem X, their answer was 0.5, but the book’s answer was 1/2.
“But they are the same thing!” says the teacher.
The students then use a calculator to divide 1 by 2, getting the answer 0.5. “Oh” they say.
Fractions are not real for those students.