My son J, who is 3 1/2 years old, has an adorable habit of rating everything out of 100. A building in downtown Birmingham is 100 tall, and a watermelon is 100 heavy. At one point, we were playing by a creek and he said “to go from here to the other end of the creek would be 100 swimming"
Since I can’t help myself, I’ve been theorizing about how he has come up with this habit.
There are two contexts where J hears numbers used without units as a way to measure something. The first is age. I don’t typically say “I am 30 years old.” I just say “I am 30.” So J knows that he is 3 1/2, his sister is 1, and his parents are 30. Sometimes he hears us say that 30 means 30 years, but he still doesn’t have a firm idea of what a year is, so to him it’s likely just an arbitrary measure of age.
The other context where he hears number without units is driving speed. He sees speed limit signs that say 25 (which he sometimes reads as fifty two) or 60 or 70, and when I talk about driving I usually say something like “I am going 55” as opposed to “I am going 55 miles per hour.” Again, even if I used the term “miles per hour” J would have no context for understanding that unit of measure. So mostly I just say “I am going 30 now, but on the highway I will go 70.” In both cases, J hears grown-ups using unitless numbers to measure a quantity.
Moreover, in both cases adults are talking about measurements (years and mph) where 100 denotes an extreme amount. When J asks me about someone who “is 100,” I tell him that a person who is 100 would be very, very old. This connects with his existing idea that 100 is a very big number. Similarly, he often asks me to “go 100” in the car. I tell him that going 100 is very, very fast, and in fact I’ve never gone 100 because it is so fast. So again, J reasonably can conclude that 100 represents the extreme measurement of speed.
So now everything can be rated out of 100! Whether it’s height, weight, or swimmingness, J can use 100 as his benchmark.
I don’t really know what to conclude from this mental exercise I’ve taken myself on, other than to say that I think that kids come up with the ideas they do for a reason, and I think it’s fun to try to figure out what that reason might be.
I truly love teaching middle schoolers. But nothing has been as fun to me as watching a kid develop his very first mathematical ideas. I’m so excited that I have so many years of math conversations in my future, both with J and his sister.