you can, in fact, know what you know - if you know how to tell that what you think you know is what you actually know.

there was a story on the news recently about a new method of teaching math - reformed math. it's been around 15 yrs. you can google it for more details, but it's basically a loose interpretation of math that allows for estimating and getting credit for getting close. kind of like "good enough" math.

it's described as a teaching strategy that favors discovery & group interaction over memorization of math facts such as times tables. according to wikipedia: It emphasizes word problems and understanding the concepts behind mathematical operations, rather than necessarily getting the right arithmetic answers for these operations.

there's nothing wrong with group interaction, and there's certainly nothing wrong with understanding the concepts, but basic arithmetic is also one of the concepts, and the basics do not inherently preclude group interaction. what is more interpersonally interactive than using flash cards to learn times tables?

with the prevalence of calculators these days, one would perhaps imagine that arithmetic isn't all that important anymore, but without a rudimentary understanding of arithmetic, there's no way to guage an understanding of the concepts. getting the right answer demonstrates an understanding of the concepts, and getting the right answer requires knowing the arithmetic.

in reformed math, there is room for more than one "right" answer, but arithmetic is not subjective. the principles of reformed math are being applied to levels of mathematics [k-12] where they are not applicable.