Calculus, which I begin teaching again on Monday, is the world of the infinitesimal. Rather than asking about changes over a decade, year, or even second, we ask about changes over lengths of time and space that are so small they approach lengths of zero. We can then talk about changes that happen in an instant, or even a point.

I thought about this as I answered the repeated question this week of “what did you do this summer?” How could I answer that? Some of what I did happened in large chunks of time; I revised a paper and, with my co-author, saw it accepted to a journal. I ade major progress on two other papers that are slated to be presented later this year, and on a book review and, when the need arose, I jumped in and taught a summer class at the last minute. Surely that is enough of an answer.

But the real accomplishments of this summer came not from the blocks of time that I put into my work, but in the slow, almost imperceptible changes I saw in my daughter. In the course of these few months I saw her grow up from an immature kindergartener to a young girl ready to take on the first grade. These changes, however, did not happen perceptibly but slowly, at an almost infinitesimal speed.

I first realized that my little girl was turning into a “big girl” when she decided she was going to read to me at night as part of our bedtime routine. Together, we read some of my favorite books from my own childhood. “Danny and The Dinosaur” was the first that she read to me, stumbling over only a few words here and there and laughing at me at the antics of the dinosaur as it played with Danny. Later, we read a collection of “Bearenstain Bears” stories, and went on to begin some “Amelia Bedelia” stories. I was most impressed that she remembered details of the books in other contexts, pointing out a yard sale sign near hour house and recalling the Bearenstain Bear story of a yard sale.

We spent as much time as possible in our local pool, with her swimming and jumping into the water and even taking a few swimming lessons. However, I didn’t realize what a true “fish” she had become until a few weeks ago, when my husband and I took her to a pool party. After swimming a few minutes, she got out of the pool and walked onto the diving board. I gasped in terror- surely I would need to get her off that board before she fell in! However, she calmly walked to the end of the board and jumped in, rising quickly to swim to the ladder on the edge and go back and do it again. WHEN did she learn to do that? I wasn’t sure, but I was amazed.

I took a day off to go with her on a field trip with her summer camp group, and was even more amazed. Somehow, the little girl whose behavior had raised concerns from her teachers last year had become one of the best behaved children in the group. I watched her stand quietly in line as the little boy in front of her laughingly punched the little girl behind her. When I complemented her on her good behavior, she thanked me and told me “it is hard, mommy”. Yes, but she had really grown up in the short time between school years.

When I think of this summer, the best symbol I have for it is the cute little gap in the front of her mouth. The adult tooth that is now coming in was there for a long time, growing in infinitesimal amounts until it finally pushed out the baby tooth, with the help of a grilled cheese sandwich about halfway through the summer. In the same way, the “big girl” was there all along, slowly growing from the little girl we all knew. As the new tooth peeks out from her gum, I am starting to see the more mature child that she is slowly becoming. And, of course, I love them both.