From my point of view, it is not a good idea to award children with money for their accomplishments at school because it can affect children's attitude towards school and cause unrealistic expectations.
To start with, award system for children can negatively affect children's attitude to education. Education is an important part of a child and requires effort to learn more. Parents have a crucial role in a child's success at school, and many parents use different ways to support them. When it comes to awarding a child for every high grade, even though it seems reasonable for a child, it can change child's attitude towards success. What I mean is that if a child get awarded by money for a good grade, his purpose will evolve to get money instead of getting a good score. One of my friend in high school, for instance, got awarded with remarkable amount of money when he gets the best score in the class. However, after a while his aim has changed to get best score, not learning new things, and he started to cheat to reach his goals. A recent study conducting a university in the US has revealed that using an award system in a student's education cause psychological problems later on.
Another point to consider is that children can expect an award for everything they achieved later on life. Life is full of challenging situations and one may face many different choices. Children who are used to get awarded by their parents have an approach to get awarded even in the work life. For example, one of my coworker in the company I worked couple of years ago had a tendency to see some satisfaction from others. To be more specific, she was expecting others to thank her every time she did her work. She was not able to see the fact that she is her responsibility to make work done.
To sum up, it is better not use an award system since it has a negative impact on children such as wrong evaluation of the importance of education and expanding expectations.