Three meanings of "to know"
I had a discussion with a physics teacher I know about knowledge you can look up and knowledge you cannot. He has been hounded by his administrators about how “the kids can look that up”, a contention that implies a serious lack of understanding about how human minds work.
Part of the blame can be laid at the feet of our verb ‘to know’, which the gramatically naive or Latinate believe is one verb and so one thing. It is not. There are two words: ‘to know how’ and ‘to know that’, and their noun forms knowhow and knowledge (which comes from the old English cnāwan, knowof in modern English). This is much clearer in Italian, which has two very different words: conoscere, to be acquainted with, and sapere, to know how to, but I will stick to knowhow and knowledge. Additionally, one of the achievements of the 20th century was the recognition of a third mode of knowing, characterized by the verb ‘to grok’.
Knowledge refers to facts that can be looked up in a book. I know that the Normans invaded England in 1066. I know my mother’s phone number. I know that in the absence of air resistance, heavy and light objects fall at the same rate in a gravitational field. I know that the value of π is approximately 3.14159.
Knowhow refers to acts I can do. I can integrate polynomials. I can swim the length of a pool. I can recognize the breed of dog by seeing it. These are not things I can look up. No amount of reference books will let me swim.
Grok refers to what I can program, that is, what I can lay out explicitly enough that I can create an automaton with knowhow.
I may not grok my knowhow. Many of the acts of distinguishing things visually that, as a human, are trivial knowhow for children, no one on earth groks. What I grok may not be part of my knowhow. The roboticists who have built robots that play violin certainly grok how it is done, but unless they have studied and learned to play themselves, they have no knowhow. Graduate math textbooks, with their sequences of lemmas, theorems, and proofs, seem inscrutable and boring to most readers because they don’t know how to take the grokked material therein and turn it into knowhow in themselves.
Given this, we can’t say, “Oh, the children can look that up when they need it.” What we have to teach may not be knowledge, and even if it is, it may be crippling to knowhow not to have it memorized. Vocabulary for foreign languages is essential for fluency. You cannot carry on a conversation or read a piece of literature stopping to look up every other word. You cannot hope to write well if you have to stop and look up the conventions of punctuation. Nor can you get through calculus unless the rules of arithmetic and algebra that govern the expressions involved are reflexive.