Does having a TV on in the background - either in another room or in the same room where a child is playing - matter?

Yes, background TV can be a disruptive influence.

According to a recent study conducted by a group of scholars and published in American Behavioral Scientist, the television is on approximately six hours a day on average in American homes. Yet little is known about the impact of growing up in the near constant presence of television. They studied the prevalence and developmental impact of "heavy-television" households on very young children from birth to age 6 drawn from a nationally representative sample. Thirty-five percent of the children lived in a home where the television was on "always" or "most of the time," even if no one was watching. Regardless of their age, children from heavy-television households watched more television and read less than other children. Furthermore, children exposed to constant television were less likely to be able to read than other children. Also, other research has shown that one-, two-, and three-year-olds' play and attention spans are shorter in length in the presence of background television, and parent-child interactions are also less frequent in the presence of background television.