On Wednesday we hiked to Corona Arch, just outside of Moab, Utah. The trail cuts across red rocks, too sturdy for human feet to cut a clear path, so much of the trail is marked with faded blue paint strokes every few yards. As we followed along, my 6-year-old said, “I’m counting the blue marks, but without numbers.”
“That’s impossible,” her older brother retorted. “You can’t count without numbers.”
“Yes I can,” she insisted.
“Are you noticing each mark?” I asked.
She agreed that this was what she was doing.
“But that’s not counting,” my son pressed.
“It’s like the ‘fish, fish, fish’ sketch from Sesame Street,” my husband said, and he was exactly right. In the “123 Count With Me” Sesame Street sketch (brilliantly commented on by Steven Strogatz in the New York Times), a furry monster tries to communicate a certain number of fish to Ernie as “fish, fish, fish, fish, fish, fish,” whereupon Ernie explains that a better way to communicate the quantity is with the number “six.”
So when my daughter said she was “counting without numbers’, she was fish-fish-fishing the blue marks – noting each one, without assigning a number word. I knew what she was doing. But the dispute was about what to call what she was doing. Certainly she was engaging in a component of the counting process. I teach my college students that successful counting involves a one-to-one correspondence between words and counted objects, a consistent ordering of counting words, and a recognition that the last number said is how many there are. On the surface, she was just doing the first of these.
But I want to go deeper here, because the next thing that happened was that my son started wondering about how to say a number in binary, something that had come up the other day. In base ten, we group numbers by tens. As we count, we begin with “1, 2, 3, 4, 5, 6, 7, 8, 9”, but then instead of inventing a new number for one more than 9, we write 10, which means 1 ten and 0 ones. This means we can count as high as we want without continuing to invent new numbers to place in the sequence.
We don’t have to use ten, though. Various languages and writing systems have used 5, or 20, or even 60 for grouping. Binary uses 2. 1 is followed by 10, which doesn’t mean ten but rather 1 two and 0 ones. This is followed by 11 (1 two and 1 one), then 100 (1 group of two twos, or 1 four), and so on. If you count in binary, it’s still counting.
But juxtaposing the counting without numbers discussion with the binary discussion made me wonder: what if we go still further and just count by ones? What if we called 1 thing “one”, 2 things “one one”, 3 things “one one one”, 4 things “one one one one”, and so on? Is this base one? And wouldn’t this still be counting in a sense? It’s like a tally. If I keep track of points in a game with a series of hash marks, I’m still counting points, aren’t I? If Ernie can ask, “How many fish?” and the monster can respond, “fish, fish, fish, fish, fish, fish”, and Ernie can provide him with the correct number of fish, haven’t the fish been counted?
Granted, if you asked my daughter how many blue marks there were, she would not have been able to respond “mark, mark, mark…” with the correct number of marks. I don’t think what she was doing could actually be defined as counting. Her “counting without numbers” was not actually counting. And once I started asking my family if what she was doing was counting in base one, it became clear that I was the only one still interested in pursuing this line of thought.
Still, I did find it intriguing that my daughter called what she was doing “counting without numbers.” Counting doesn’t stop being interesting once kids move out of preschool.