BOX 2.3 Understanding and Problem Solving In mathematics, experts are more likely than novices to first try to understand problems, rather than simply attempt to plug numbers into formulas. Experts and students in one study (Paige and Simon, 1966) were asked to solve algebra word problems, such as: A board was sawed into two pieces. One piece was two-thirds as long as the whole board and was exceeded in length by the second piece by four feet. How long was the board before it was cut? The experts quickly realize that the problem as stated is logically impossible. Although some students also come to this realization, others simply apply equations, which results in the answer of a negative length.      A similar example comes from a study of adults and children (Reusser, 1993), who were asked: There are 26 sheep and 10 goats on a ship. How old is the captain?      Most adults have enough expertise to realize that this problem is unsolvable, but many school children didn't realize this at all. More than three-quarters of the children in one study attempted to provide a numerical answer to the problems. They asked themselves whether to add, subtract, multiply, or divide, rather than whether the problem made sense. As one fifth-grade child explained, after giving the answer of 36: "Well, you need to add or subtract or multiply in problems like this, and this one seemed to work best if I add" (Bransford and Stein, 1993:196).