In my years as a math teacher and tutor, I have had many occasions to help with fractions. Here’s something interesting about nearly every initial conversation I’ve ever had with someone who is having difficulty with learning about fractions for the first time. I’ll usually begin the conversation by asking something like “So you say you’re having a hard time with fractions. What do you know about them so far?” Here’s how it will often proceed from there:
“Nothing. I don’t understand them at all.”
“Okay. But you’ve probably seen them written, right? You have an idea what they look like?”
“Well, yeah, there’s a number on the top and a number on the bottom and a line in between but I don’t understand them.”
“Okay, that’s helpful for me to know. So you can recognize something called a fraction, but you’re confused about how they work or what they are for?”
“There’s one other thing I’m guessing you do already understand about fractions,” I’ll then say. [I get a skeptical look in response, but they’ll usually hear me out.] “You know what half means, right?”
The response I get tends to reflect that offense has been taken. “Well, yeah, obviously. Half is just half of something.”
Indeed. The concept of half is such old news to someone by the time “fractions” get “introduced” in school, that it seems a little insulting to be asked about it as a person of nine or ten years. One starts to hear about half and halves very early in life, long before any whisper of a thing called math, much less the more specific phenomenon of fractions. The concept of half doesn’t have a chance to feel difficult any more than any other ordinary thing that happens all the time in regular life.
But our use of fractions in everyday parlance is mostly limited to the word half. As a young child you might occasionally hear someone talk about splitting something into thirds or fourths, but you’ll hear about half very often.
Hence the snarly looks I get when I ask about half. The snarling tends to subside when I protest: “But you told me you don’t understand fractions AT ALL, so I had to ask, didn’t I!?! It’s not my fault that those pesky halves were hiding in your memory disguising themselves as Not Fractions!”
This exaggerated response from me usually gets a smile, and from there we can begin to talk about why the school fractions seem so much more torturous than the easy everyday halves, which truly are so understandable to most people I’ve talked to about them that they don’t even recognize their understanding as understanding. It’s invisible, the way sentence structure is to people who haven’t been taught grammar in their native language. They can use it, but their understanding is invisible.
Which has always made me wonder if this kind of implicit understanding could be built for other fractional parts quite simply if we just talked more about other fractions, the way we do about halves, such that they’d exist in linguistic experience for young people and thus have the chance of taking conceptual root the way the halves do.