It's fine if this wouldn't work for you. Nothing works for everyone, and that's sort of the point. Kids are funneled through an extremely rigid sequence of topics focusing entirely on rote application of manipulative techniques. Anyone who doesn't conform to this sequence is just left to painfully deal with it on their own.
I remember a bit later in school I discovered algebra on my own and became really interested in it. I asked if I could take a class on algebra ahead of schedule and was virulently refused by the school counselor, who scoffed at the very notion and declared it to be impossible. Her response was so indignant that it offended me very deeply and caused me more or less to rebel against school in general. I ended up turning away from math completely for many years and didn't come back to it with any real interest or passion until I was about to graduate high school.
> If I explain 1/2 and 2/4 using pie or pizza slices and ask you how "much" pizza did you get, I think the equivalence idea clears up quite nicely.
Sort of. But childhood me would have just told you that, no, in one case I got one big piece and in the other case I got two smaller pieces. Combined they might have the same mass or the same number of calories or the same area, but the numbers you just gave me don't have units and can't be measures of area since they're independent of the radius of the pizza.
> > If I explain 1/2 and 2/4 using pie or pizza slices and ask you how "much" pizza did you get, I think the equivalence idea clears up quite nicely.
> Sort of. But childhood me would have just told you that, no, in one case I got one big piece and in the other case I got two smaller pieces.
Why are we using pizza pies in the first place? I vaguely remember being taught something about pie as well. Instead how about this:
You have a piece of string 1 m in length. You cut it into 2 equal pieces. We can denote the length of one piece as 1/2 m.
You have a piece of string 2 m in length. You cut it into 4 equal pieces. We can denote the length of one of these pieces as 2/4 m.
These pieces of string are the same length, therefore 1/2 m is the same as 2/4 m.
Although in my head this feels more like a nice way of demonstrating they are the same, I'm not so sure if it helps a lot with reasoning about fractions. But that's fine, like the article said it's good to be able to explain something in multiple ways.
Personally that's one of the things that absolutely fascinated me about arithmetic when I was young; That you can play around and have multiple ways of arriving at the same answer, and that if you follow the rules of math right, they always end up as the same answer, sometimes even unexpectedly so (for young me).
I like your analogy a lot. It's hard to me to say definitively but I think myself as a toddler would have found it more compelling than pizza and pie analogies. Of course, the same sort of manipulation can be done with pizza/pie slices, but it's a much more natural and straightforward manipulation with strings.
[I would edit this onto my comment above, but it's too late so I'm just adding it as a reply.]
I realize this is going to go far beyond the thoughts of a toddler learning about fractions for the first time via the pizza analogy, but I think it demonstrates that even the seemingly obvious pizza analogy is more nuanced than it seems.
I've never bothered thinking about this before, but it seems obvious to me now. Using just a straight-edge and compass, it is not always possible to cut a pizza into N standard-shaped slices for all positive integers N. Doing so is equivalent to constructing a regular N-gon, and this is only possible when N takes on the special form
N = 2^k * p_1 * ... * p_M
for some nonnegative integer k and some sequence of distinct Fermat primes (that is, primes of the form 2^(2^j) + 1).
Thus, you cannot slice a pizza into 7 slices using a straight-edge and compass. So in the pizza analogy the number 1/7 corresponds to something that is not physically achievable with standard implements.
I remember a bit later in school I discovered algebra on my own and became really interested in it. I asked if I could take a class on algebra ahead of schedule and was virulently refused by the school counselor, who scoffed at the very notion and declared it to be impossible. Her response was so indignant that it offended me very deeply and caused me more or less to rebel against school in general. I ended up turning away from math completely for many years and didn't come back to it with any real interest or passion until I was about to graduate high school.
> If I explain 1/2 and 2/4 using pie or pizza slices and ask you how "much" pizza did you get, I think the equivalence idea clears up quite nicely.
Sort of. But childhood me would have just told you that, no, in one case I got one big piece and in the other case I got two smaller pieces. Combined they might have the same mass or the same number of calories or the same area, but the numbers you just gave me don't have units and can't be measures of area since they're independent of the radius of the pizza.