I was just helping a customer find some books to help her "brush up" on her Algebra before a test.

There are a lot of uses of this term, brush.

Brush up, brush off, brush your hair . . . where did it come from and how did it get so many uses?

Interesting, CW. In the OED the word 'brush' has 8 different citings. The 'brush' in 'brush up,' according to the OED, came from 'brushing up' as in 'to brush up and wash up.' In 1951 McLuhan used it to mean to 'brush up' your knowledge. I hadn't thought it was so recent.

Hon, as I look at the quotes, 'brush up as 'to brush up one's knowledge' is a good deal older. It may be that McLuhan was the first to use this as a noun ('give your knowledge a quick brush-up').

CW, you asked about word-history. In explaining the tangle in OED, I'll put verb-senses in red, with an -ing ending, and noun-senses in blue.

Brush has two basic noun-senses, which may be related, "but as their history in English shows no contact, it appears better here to treat them apart." One (call it the underbrush¹ group) relates to brush¹ as 'a thicket of trees, shrubs or underwood'. The other (hairbrush² group) is brush², 'a bunch of hairs attached to a handle', as for brushing your hair, or for painting.

Two basic verb-senses are probably from these two noun senses. First, an obsolete sense of 'to rush forcibly or speedily' (call this sense 'rushing¹'), is probably from the concept of dashing through dense underbrush¹ (but it may be onomatopia, combining hard-charging sounds hear in rush and burst or break). The other brushing verb means stroking ², rubbing², grazing² against, as a hairbrush² does. OED says some of the latter are for vigorous rubbing (as brushing² a coat, hair etc. to clean it) and others are are light going-over (brushing up² a room or brushing up³ on knowledge; brushing against² a person; brushing over² a painting or a topic.)

Now we get to the dubious parts of OED.

Brushing can also mean 'moving briskly by or against something, grazing it or sweeping it aside in passing.' I'd think this clearly relates to the brushing's second verd-sense, rubbing, grazing², but OED lists it under the first or rushing¹ sense, though 'influenced' by the rubbing sense.

OED also gives a third noun-sense of brush, including 'a hostile collision' (a brush with the enemy); at first brush (at first meeting), a brush of illness (a slight attack). It lists all of these as being from the first verb-sense of brushing¹ (that is, from the sense of brisk rushing). But again, I'd think these clearly relate to brushing's second sense of lightly grazing².

Nowhere can I find a clear statement of under which definition OED would place 'a brush with danger'.

Finally, OED has another noun-meaning: "brush: Austral. and N.Z. slang. a girl, a young woman; freq. derog.," and it says this is "Of uncertain origin." I'd think the term applies to a girl only if she is sexually mature, and that origin is obviously in the sense of brush meaning 'underbrush'. If this is too subtle, I direct you to the definition of bush found here. (OED has this definition, though it omits the word 'female'.)

Fascinating! Thank you, Shu! This is exactly what I was looking for.