In what age range would an infant possibly begin touching a body part as a result of pain in that area?
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There are really two questions, here, I think:
The short answer is that it depends not only on the age of the infant, but the location of the spot.
Infants gain control over their limbs slowly, with early movements being hard for them to control. Infants will be able to intentionally touch themselves in some areas earlier than others, based on the difficulty of coordinating a reach to that location. For example, the same infant may be capable of reaching to touch her mouth in an intentional, coordinated way but not her ear or her lower back.
Even as newborns, infants can coordinate an intentional reach under some circumstances (Amiel-Tison & Grenier, 1980), but they need to be held in such a way that their body is stabilized, including their other arm, and even then they can't reach across their midline (i.e. a newborn couldn't reach to something on his left side with his right arm).
If you want to learn more about this kind of thing, there are several great books that step through the details of infants' motor development. (Keep in mind that there's a tremendous amount of healthy variability infant to infant, so just because a book says, for example, that infants typically learn to grasp objects by about 5 mos doesn't mean there's anything wrong with a 6-month-old who isn't doing that yet or a 4-month-old who's doing it early. If you aren't sure whether or not your baby falls within the normal variability, your pediatrician can let you know.)
Amiel-Tison, C., & Grenier, A. (1980). Neurological evaluation of the human infant.
Infants' responses to pain (like adults') vary --- different infants will show pain different ways, and even the same infant may react differently day to day. One study of infants' responses to pain (during routine immunization) found that while there was a lot of variability, most infants showed a pattern of response something like this:
The pattern that did emerge was characterized by an initial response: a drop in heart rate, a long, high pitched cry followed by a period of apnea, rigidity of the torso and limbs, and a facial expression of pain. This was followed by a sharp increase in heart rate, lower pitched, but dysphonated cries, less body rigidity, but still facial expression was of pain. Finally, in the second half of the minute's response, heart rate remained elevated, cries were lower pitched, more rhythmic, with a rising-falling pattern, and were mostly phonated, and body posturing returned to normal.
Infants' natural, instinctive response to pain doesn't necessarily involve reaching to touch the part that hurts. For some types of discomfort, there may be a more natural reaction to touch the spot (such as an itchy kind of pain, rather than a sharp, acute pain like a needle stick), but other types of pain may not be so obvious. For example, think about your natural reaction to different kinds of pain: If you bump your knee against a desk, you'll want to rub it, but if you have a burn you'll avoid touching it at all. The last time you had a sore throat, you probably didn't walk around with your hand on your neck, you just felt the urge to swallow more often. Stomach cramps may make you want to bend over, muscle pain may make you want to stretch. Our natural response to pain is complex, and the same is true of infants.
Even though it may not be an instinct, it is possible to teach your infant to show you where it hurts by touching the spot, such as with baby sign. Infants are sometimes able to communicate a little more easily and/or a little earlier with symbolic gestures (e.g. baby sign) than with spoken language, and practice communicating with gestures might even give them a bit of a head start on spoken language later. It is unlikely that you could successfully teach an infant to show you where it hurts before about 9 months, though, since infants don't typically learn to use deictic gestures (e.g. pointing or gesturing at something to draw attention to it) until that age, although this depends on the infant's social environment and cultural practices. Most of the studies on infants' use of gesture and/or baby sign to communicate are with infants around 9-12 months old or older.