“My kids don’t want to take it, but I told them, ‘These are your grades when you’re taking it, this is when you don’t,’ and they understood,” Ms. Williams said, noting that Medicaid covers almost every penny of her doctor and prescription costs.

Some experts see little harm in a responsible physician using A.D.H.D. medications to help a struggling student. Others — even among the many like Dr. Rappaport who praise the use of stimulants as treatment for classic A.D.H.D. — fear that doctors are exposing children to unwarranted physical and psychological risks. Reported side effects of the drugs have included growth suppression, increased blood pressure and, in rare cases, psychotic episodes.

The disorder, which is characterized by severe inattention and impulsivity, is an increasingly common psychiatric diagnosis among American youth: about 9.5 percent of Americans ages 4 to 17 were judged to have it in 2007, or about 5.4 million children, according to the Centers for Disease Control and Prevention.

The reported prevalence of the disorder has risen steadily for more than a decade, with some doctors gratified by its widening recognition but others fearful that the diagnosis, and the drugs to treat it, are handed out too loosely and at the exclusion of nonpharmaceutical therapies.

The Drug Enforcement Administration classifies these medications as Schedule II Controlled Substances because they are particularly addictive. Long-term effects of extended use are not well understood, said many medical experts. Some of them worry that children can become dependent on the medication well into adulthood, long after any A.D.H.D. symptoms can dissipate.

According to guidelines published last year by the American Academy of Pediatrics, physicians should use one of several behavior rating scales, some of which feature dozens of categories, to make sure that a child not only fits criteria for A.D.H.D., but also has no related condition like dyslexia or oppositional defiant disorder, in which intense anger is directed toward authority figures. However, a 2010 study in the Journal of Attention Disorders suggested that at least 20 percent of doctors said they did not follow this protocol when making their A.D.H.D. diagnoses, with many of them following personal instinct.