I ask Katie to do something.
Katie: This is going to take forever!
Me: How long is forever?
Katie: It’s so long that you can’t count.
Me: Can anyone count that long?
Katie: No, even if they count for 100 years.
After she finishes what I asked her to do:
Me: Did it really take forever?
Katie: No
Me: How long did it take?
Katie: Probably 5 minutes.
Indeed, it did take pretty close to 5 minutes! Recently I have been amazed by both how good and how bad her time estimates can be sometimes. A few weeks ago Katie wanted to see how long it would take me to put together a jigsaw puzzle of hers. She ‘timed’ this by counting while I was working on it. When I was done, she was up to 88.
Me: So long did it take?
Katie: 88.
Me: 88 what?
Katie: 88 Minutes.
Me: That can’t be; 88 minutes is more than an hour!
Katie: Then 88 seconds.
I told her that 88 seconds was probably closer to the truth, but also tried to impose on her that a second is a specific unit of time and that one can only estimate it by counting. What have been your experiences with kids’ sense of time?
Musings of a Mathematical Mom 
Had a great conversation in the comments section with Christopher Danielson along these lines. When traveling in the car with my 4 yr old daughter I will often answer about distance left for our trip with remarks about time. For example, I’ll say it takes about 20 more minutes -the amount of time for an episode of Powerpuff Girls. In a hotel we were playing a game where she was jumping to me from the steps of the pool. She told me to back up and when I asked her how far to go, she told me to go five minutes. She’s figured out that seconds are different from minutes and that seconds are close to counting. So if I tell her it’ll take, say, 8 minutes she will then ask what to count to.
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I will also do the same when traveling. In fact, I find that I often use time to answer questions about distance even when talking to adults. I guess that’s because we’ll ask “how far is…” when we really want to know how long it’ll take us to get there (taking into account how fast the roads are, traffic, etc.)
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It’s fun to guess how much time little household tasks take, like fetching a book from another room, or emptying the dishwasher. You can get pretty good on repetitive tasks – and then perform very close time estimate as a party trick!
For math fun: explore the finite covering theorem: http://en.wikipedia.org/wiki/Heine%E2%80%93Borel_theorem I think even the ancient Greek philosophers said something about pieces as small as you wish can cover a very large object. That is, as far as you are making progress, and your task is finite, the time’s never infinite.
I have a mini-poster about a related topic of guessing small quantities, here: http://www.moebiusnoodles.com/2014/01/math-mind-hacks-guesstimate/
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The time can be infinite if the rate at which you are making progress diminishes. Infinite series can have finite sums!
Playing time-guessing games with household tasks sounds like a fun idea.
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