Hi lifeman9:
Yes, It is normal, but If you dont want him to continue growing, pause it by pressing A button(the one on the left) and B button(te one on the center)and thats it!
but it can mesup his growth, and also, when it grows older(6, 7, 8 yrs, etc) it will appear another tamagotchi with a book, its thye match maker, so press B or A button, it will appear another tamagotchi, and then it will appear the question: LOVE?
YES
NO
and if you choose yes, they will give a kiss, and your tama will have a baby!
Hope it helped!
Muffin_Tama