There are some very hard problems in combinatorics that involve showing that there is a bijection (one-to-one correspondence) between two set of objects, thus showing that the two sets have the same number of elements. On the other hand, we encounter simple one-to-one correspondence problems on a daily basis without identifying them as such. The answers are so obvious to us that we don’t realize that there could be people to whom they are not self-evident. If you have 10 people coming over for dinner, of course you have to set out 10 plates! And yet there exists a fairly large group of people who would actually have to think before coming up with the answer: little children.
But enough philosophizing, and on to specific examples. A few days ago, while Katie was in the bathtub, I decided to do a little ‘mental math’ with her. I first asked her how many people lived in our house. After a brief moment she said that there were 4. I then asked her how many noses we all have (followed by how many mouths, necks, right legs, left legs, etc). Each time she would answer after a short pause, and I realized that she was counting every singe time (the answer for the right legs actually took the longest, but after that the number of left legs came immediately). We then moved on to counting the total number of legs, arms, eyes, etc. Given the previous results, it was not surprising that these she counted every time as well (for the eyes she must have messed up her counting and initially said 7, but quickly corrected herself after seeing a look of horror on my face).
Throughout, it was very tempting to start saying things like, “do you have an equal number of arms and legs? does everyone else? what does that say about the total?” but I think that would have been counterproductive. A few days after the initial encounter the subject actually came up again. Katie was saying something about knees and so I asked her how many total knees we all have. After she came up with the correct answer I asked her whether she can tell me, without counting, how many legs we have. It took her only a second to come up with 8; she got the concept after all, but if I hadn’t told her not to count I’m sure that she would have.
I think that even as adults we often redo unnecessary work without realizing that something very similar has been done before and we can reuse the results. However, if someone tell us to stop and think, then we can make the connection. I am generally against giving kids the answers when they can come up with them on their own, but what do you think about making kids aware of the fact that they already know the solution?