## Circular Reasoning

Have you ever been really impressed by something your child did, just to realize a few moments later that it was probably unintentional?

Today Katie was doing some exercises from a fun workbook; there are three circles with arrows between every two of them as well as three numbers.  The goal is to place the numbers in the circles in such a way that the arrows always point from larger numbers to smaller ones.  I explained this to Katie and left her to it.  When I came back, she had done three of them, but everything was in reverse; the arrows were pointing from smaller to larger numbers.  Ok, so she understood the concept but mixed up the directions.  I point this out to her and tell her to fix them.  She looks at the first one, erases just the smallest and the largest numbers, and switches them, leaving the middle number intact.  I was impressed!  But then, she goes on to the next exercise and erases all three of the numbers, proceeding to replace them correctly.

My guess is that for the first example she had just forgotten to erase the middle number without realizing it, since ultimately the answer was correct.  As she was about to redo the third one, I had pointed out to her that in the first two examples one number ended up staying in place, while the other two were switched.  I asked her whether she could correct the problem by erasing just two numbers.  After staring at it for a bit, she switched the correct two.  I guess I could still be a little impressed :-).