I never had any problems understanding fractions, and in a certain sense, it is a mystery to me why people have problems with them. But in a deeper sense, I think, I know exactly why they have problems with them. So when I read this, it all came to me clearly: "Multiplication makes a number bigger; division makes it smaller." Which, of course, is exactly what's not happening in multiplication or division; we aren't transforming numbers, we're just involved in elegant acts of counting. But what of fractions, then? For me, I think the key lay in the use (by my teachers) of the word 'of'. If you say, "what is one half times one quarter" it sounds deeply mysterious (especially if all you've even done is to memorize a multiplication table). But if you say "what is one half of one quarter" the meaning is transparent: one eighth. This just shows, once again, that mathematics isn't about remembering facts, it's about understanding.
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