I still remember struggling with fractions when I was young. The funny thing is that I ended up majoring in math in college (at a highly ranked one at that), an

I still remember struggling with fractions when I was young. The funny thing is that I ended up majoring in math in college (at a highly ranked one at that), and I did very well, even publishing research in differential topology as an undergrad. How can someone who struggled to even grok fractions end up being successful in pure mathematics? I think it's because all my teachers operated off the assumption that kids can't possibly understand abstract definitions, so everything must be explained via analogies. So nobody could tell me that fractions are equivalence classes. Everything was instead explained in terms of pies, which certainly seems like it would be intuitive, and there's really nothing technically wrong with it, but it encouraged me to think of, say, 1/2 and 2/4 as different entities instead of two representations for the same entity (since, after all, cutting a pizza into two slices instead of four is just not the same thing). I didn't "get" fractions until on my own I came to the conception that these are actually kind of abstract concepts that go by many different names. For me, I was taught too much to identify numbers with concrete objects in the real world. This sort of works for integers (as long as you ignore negative integers and even zero to some extent), but it basically breaks down for all other types of numbers, including rational numbers. So it's a really bad expectation to set in my opinion. But you can hardly find an elementary school teacher who will dare approach this topic. I honestly think that if someone had just explained that "1/2" is a symbol we've devised to represent the solution to the equation 2x = 1 -- an equation which otherwise has no solutions at all in the integers -- I would've grokked it a lot faster. It would've been clear to me that we're introducing an entire new set of numbers distinct from the integers. These new numbers do not have a direct correspondence with physical objects in reality. And it would've been easier to see that the same solution must go by many other names as well. For instance, multiplying that equation by 2 on both sides gives us a new equation which must have the same solution: 4x = 2. So "2/4" must represent the same entity as "1/2". And so on. But because US educational culture has decreed that basic algebra is more advanced than fractions, the very idea of discussing solutions to an equation will never be allowed in an elementary school classroom. It is assumed as a basic fact of human life that kids cannot understand the concept of solving an equation before learning about fractions. We'll teach kids to perform mechanical manipulations of decimal expansions before we ever begin explaining what any of those digits actually mean.