Bash from scratch
The script has a correct syntax, and works for me. If I enter anything from 0-10, it does not echo any text.
For future reference, I recommend you install shellcheck and make a habit of checking your script syntax with it.
Alternatively (since you mention that you are learning bash) you could use an arithmetic expression:
if (( VAR > 10 ))
then
echo "The value is greater than 10."
fi
Or the more concise one-liner:
(( VAR <= 10 )) || echo "The value is greater than 10."