Ms. MacIver asked her senior math students to solve the following problem:
Make three rows of three dots on a piece of paper. Without lifting your pencil, and using four straight lines, connect all of the dots.
The students tried to solve the problem, but could not. Ms. MacIver then said to her students, "I'll show you how to connect the dots." She then drew the following solution on the overhead.
"You have to go outside the boundaries of the dots in order to solve the problem," explained Ms. MacIver.
How did the students initially represent the problem? In other words, state how the students described the problem to themselves—how they pictured it, interpreted it or defined it?
Explain how the students' representation of the problem interfered with finding the solution. What could students learn from this experience that might help in solving problems in the future?
What will be an ideal response?
Ms. MacIver's students represented the problem as limited, in space, to the boundaries of the dots. This confining representation required them to change direction at the dot boundaries, making too many lines. A representation such as "a large parking lot where there is plenty of room for three-point turns," may have allowed them to find the solution. Through experience, students may learn to be more flexible in looking at a problem from different perspectives, i.e., exploring different ways of representing a problem.
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