**Riemann sums and my mom.**

I just defended my dissertation in mathematics last week. This article is devoted to my mom, who gives me everything.

I grew up in a small village in Vietnam in the early 90s. There were no bookstores in the town, and learning aids were not always available.

Occasionally, in my history lessons, we had to give presentations, which often required big geographical maps to show, for example, how one army moved from one location to another. Some of my friends overcame the shortage of study aids by sketching figures/images from the textbook. For me, it was not an option because I was always a terrible drawer! Luckily, I had my mom. Here is how my mom drew a map.

(1) She first inscribed the image from the textbook inside an m x n rectangular. She then divided this rectangle into m x n smaller square grids.

(2) On a larger paper, she drew another m x n rectangle using a bigger scale. As in step one, she divided this rectangle into m x n smaller grids.

(3) For each small rectangle in the textbook, she chose a representative point. So, in total, there were m*n mark points. Typically, these points were selected at places where the map’s boundary changed its direction (in other words, breaking points.)

(4) She then chose a similar set of points for the larger rectangle.

(5) Connecting the chosen points in step (4) based on their original connections.

My mom knew that the more grids she created, the more precise the final result. However, this came at the cost of time-consuming. Furthermore, as the number of grids increases, choosing breaking points in step (4) would not matter much.

With these simple procedures, my mom helped me create countless precise maps for my history lessons. Very later on, when I entered college and learned calculus, I realized that what my mom did was very close to the formation of a partition and its cousin, the Riemann sum. Her observation on the relation between the number of grids and the maps’ precision is precisely the observation that a finer partition gives a more precise estimate of the Riemann integral.

My mom never finished secondary high school: her family was too poor to support her with proper education. So, to come up with the above method is not trivial. I always admire her for that (and other things, such as her mental calculations capacity.) I firmly believe that if she had a chance, she would become a good mathematician!

Thank you and love you, Mom!