Attitudes towards Adoption and the need for Privacy
Society in general and children in particular tend to feel awkward when confronted with an adopted child and reactions vary from being overly (and unnecessarily) sympathetic to even hostility. Unless your child has encountered such situations before it can be a very hurtful experience for them. What is worse, it can fuel their own insecurities if they have any and thus cause much more harm.
Varying Approaches to Privacy
As a result of this, some parents instruct their children not to mention their adoption status at all. This effectively protects them from an adverse reaction from their schoolmates, old and new. However, it does not prepare them for awkward situations that may arise when the topic of adoption is broached for one reason or the other. For example, if another child in the class is also adopted, and the question of adoption arises in general, they may not be prepared to face the difficulties of how to react.
The other approach to adoption and privacy is to maintain none. When adoption is plainly admitted, there is no need to inwardly cringe and hide facts from one's friends. The disadvantage of this is that once the fact is out, there's no putting it back in and such children might find themselves a target of various adverse reactions by their colleagues.
In the end, it's for each family to decide their level of comfort with others knowing about an adoption. It will be tied to the maturity of the child as well as his or her colleagues. Also a factor is how the parents feel about the situation and decide to handle it based on their own wisdom and knowledge.