Using overload to implement the equivalent of a switch

Hello!
I've been trying to use polymorphism -via- overloading+overriding to
simulate a dynamic switch

class AbstractBase{
 public abstract void doSomething(Object anObject);
 public void activate(Object anObject){
   doSomething(anObject);
 }
}

class ResponsibleChild{
 public void doSomething(Object anObject){
   System.out.println("Type not supported");
 }
 public void doSomething(Integer anInt){
  System.out.println("Integered");
 }
 public void doSomething(String aString){
  System.out.println("Strung");
 }

}

class Main{
 public static void main(String args[]){
   AbstractBase reference = new ResponsibleChild();
   reference.activate(Integer.valueOf(10));
   reference.activate("Strung!!");
 }
}

The output I expected from this was ;

Integered
Strung

However the actual output is ;
Type not supported
Type not supported

Shouldn't the call
a. be made in the context of the object, and
b. narrow to the best possible match for the method signature

?

Regards,
Abhishek