My daughter is in second grade also. She is just starting to figure out shifting. I highly recommend switching to a twist shifter. These seem to be way easier, more intuitive, and more fun for kids. My daughter also won't take her hands off the bars. Her idea of a turn signal is pointing with her index finger while holding the grip with the rest of her hand.
One word of warning - the kid's twist shifters require an odd size grip (small) that can be hard to find, but they are out there if you look hard enough. Mine are from SRAM. Worst case, you buy a foam grip and cut it down to size with a knife.
On a Sunday, take your son to a big open high school parking lot (or similar) that has a nice slope to it and let him go at it.