Chapter 7, page 149
[the British commander] and his troops left and how fast they were walking, because that... might help
explain if they were really fatigued, and then the adrenaline started to flow in the battle, that they may have
lost control. They may have been angry—a whole range of other kinds of things. So the physical dimensions of
when they left, the fact that they had to go through a river up to the middles of their bodies meant that they
were wet, I suppose, the entire time.... (Wineburg, 1991, pp. 82-83)
This highly elaborated problem representation with extensive inference helped the historian develop a better
solution to the question of whether the British or the colonial soldiers fired the first shot.
Effective problem representation requires the problem solver to determine the relevance of
information and exclude irrelevant information (J. L. Cook, 2006; Littlefield Cook & Rieser, 2005). This
requires the problem solver to decide what information is needed to solve a problem and what can be
ignored (B. J. Barron et al., 1998). Consider this problem: “Susan and Pedro got married 5 years ago, and
now they have two children, ages 4 and 2. Assuming that the children make typical progress through
school, in what year will the younger child legally become an adult?” To answer this question, students
must recognize that only one of the three numbers provided is relevant, the age of the younger child. The
child’s normal progress through school is also irrelevant. Students must also bring in two numbers that are
not explicitly included in the problem statement: the current year and the age at which children become
adults. Students are often unskilled at determining the relevance of information, in part because they have
too few opportunities in school to grapple with such problems.
Identifying subgoals. Most real-world problems are complex rather than simple, and as problem
solvers formulate solution plans (Polya’s second strategy in problem solving), the solution plans often
include many steps. These steps mean that in order to achieve the final goal, problem solvers must work out
a series of subgoals along the way that will help them get to their final goal, step by step (Thevenot &
Oakhill, 2006). Hence, effective problem solvers identify subgoals that they can achieve on the way to the
final goal.
As an example of a problem that requires problem solvers to set subgoals in order to achieve a final
goal, consider this authentic problem that was posed to seventh graders in a recent project (Malhotra,
2006). The problem was to figure out how large a drainage trench on school grounds would need to be to
hold the runoff from an average heavy rain storm. The students’ results were used to plan the actual
digging of a drainage ditch. To solve the problem, students had to break it down into a series of steps. Each
step had a subgoal that needed to be achieved on the way to achieving the overall goal. Some of these
subgoals were:
Ɣ determine how much rain falls during a typical heavy rain storm
Ɣ determine the area of the school grounds
Ɣ determine what volume of water can be expected to run off (vs. percolate into the soil) in the direction
of the ditch
Ɣ determine the dimensions of a ditch that can hold the volume of water that would run off
Effective problem solvers learn to set subgoals when solving multi-step problems (B. Barron, 2000).
Even relatively simple multi-step problems can present a challenge to younger (and some older)
children. Consider this problem from a study by J. Taylor and Cox (1997, p. 191): “At the June Fair,
lemonade costs $0.60 for a small glass and $0.80 for a large glass. Chocolate chip cookies are $0.25 each.
How much will 8 small glasses of lemonade and 3 cookies cost?” Solving this problems (without using
algebra) involves three subgoals: Students must determine the cost of the 8 small lemonades using
multiplication (subgoal #1), the cost of the 3 cookies (subgoal #2), and the total cost by adding the results
of the previous two subgoals (subgoal #3). Without specially designed instruction, few fourth graders were
able to solve these problems.
Monitoring for sense. Effective problem solvers check to be sure that their initial problem solutions
make sense (Van Haneghan, 1990). Consider a student who is asked to solve this problem on a test: “A
cyclist is riding at 15 miles per hour. How long will it take to get to Dallas, which is 30 miles away.”
Although the student sets the problem up correctly, he accidentally hits the multiply key instead of the