Zoe (2y. 3mo.) has recently gotten pretty good at counting up to 10 objects. Today, as we were out on a walk, I decided that it was time to start taking advantage of this and chatting math with her. Here is the conversation we had.
Me: Zoe, how many fingers do you have on your hand.
Z: one, two, three, four,five!
Me: How many fingers do you think you have on your other hand?
Z: I don’t know.
M: Ok, how about you count them.
Zoe again correctly counts the five fingers.
M: Alright, how many fingers do you think I have on my hand?
Z: Two.
M: Hmm.., lets count them.
Zoe counts five fingers on one of my hands. There is no sign of surprise whatsoever.
M: How many do you think I have on my other hand?
Z: I don’t know.
So we count once again and obtain the expected answer. But I was not ready to give up quite yet.
M: Zoe, how many fingers do you think daddy has on his hand?
Z: Three!
M: And Katie?
Z: Three!
M: So we each have five and they have three?
Z: Yep.
I am not sure what exactly I expected, but I wasn’t too surprised that she didn’t say five. The following continuation, however, left me pondering quite a bit.
M: Zoe, how many hands do you have?
Z: One, two!
M: And how many hands do I have?
Z: Two.
M: And how many hands does daddy have?
Z: Two
And it turns out that Katie has two, as do grandma and grandpa, aunts, uncles, and cousins. So what is the difference between the two questions here? She seemed to have no trouble deciding that all people have two hands – how come that same reasoning didn’t transfer to fingers?
Musings of a Mathematical Mom 
think the concept of a number does not form yet. Zoe can count, but this counting has no meaning to her. There is no relation between number five and five fingers (candies, cookies etc). The same is true for the relations greater than, less than, or equal. Give her some more time.
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But how do you explain the hand phenomena then?
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A follow-up:
I asked Zoe how many paws Alisa the cat has (one she sees fairly often). She counted “one, two, three, four!” She must have pictured Alisa in her head. Then I asked her how many paws Leopold the cat has (one she sees much less frequently). The answer: “I don’t know.” Explanation?
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She is at the point when there are one, two, and many objects. Typical Piaget 🙂
As for cats, she definitely imagined in her head one particular cat (Alisa).
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Yes, I do have an explanation. I remember watching a show about babies and toddlers and also remember reading about it in a book. Babies can “count” only up to a small number. I think that number was “3”. They had this experiment, that they show a number of dolls on a screen then one doll disappears. The baby looks surprised. However, if there are more than 3, the baby loses track and does not seem surprised (or they also tracked their eye movements).
I think here it is a similar concept. Zoe, perhaps, is more conscious and aware of the smaller numbers, conceptually. The others she can mechanically count, but cannot really “think” about them yet.
Big thanks for writing out the lessons. I will try the same on Alisa – 2 yrs and 8 months.
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This is a lovely, lovely post.
I vote with Sannichka. Babies and pigeons can subitize up to three—recognize quantity without counting. Bigger numbers require putting together a whole bunch of complicated ideas, which a two-year old is typically still putting together.
I love this conversation! Thanks for sharing it.
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Thanks everyone for the great explanations. I will try to have conversations like this with Zoe every once in a while and monitor when the transition happens. Do you think she will become comfortable with 5 before she does with 10, or will all numbers bigger than 3 click in one day? 🙂
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yes, I agree with the above commenters that she isn’t able to subatize 5 yet, but she can 2. Children have to learn how to connect the symbol for a number, such as “5” or saying “five”, to the actual quantity. As she grows, she’ll get confortable with larger and larger numbers. In Montessori, we would have a 1-10 section of work, and once that was mastered, 11-19, and then on to 20-100.
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(I’m trying responding to this via email so…it’s an experiment…not sure if it works.)
I think this is fascinating. I wouldn’t have guessed her response that others have two or three fingers per hand. As for her knowing that everyone has two hands, do you think this has to do with the fact that kids can innately subitize sets of two, but not sets of five? http://en.wikipedia.org/wiki/Subitizing
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That seems to be the consensus among the commenters 🙂
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I think the subitization is also connected with the visualization. She can imagine a person and sort of see the two hands. But each person has too many fingers to visualize, so she can’t picture and count them.
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