When students attempt to apply conventional algorithms without conceptually grasping why and how the algorithm works, “bugs” are sometimes introduced. For example, teachers have long wrestled with the frequent difficulties that second and third graders have with multidigit subtraction in problems such as:
A common error is:
The subtraction procedure above is a classic case: Children subtract “up” when subtracting “down”—tried first—is not possible. Here, students would try to subtract 4 from 1 and, seeing that they could not do this, would subtract 1 from 4 instead. These “buggy algorithms” are often both resilient and persistent. Consider how reasonable the above procedure is: in addition problems that look similar, children can add up or down and get a correct result either way:
Bugs often remain undetected when teachers do not see the highly regular pattern in students’ errors, responding to them more as though they were random miscalculations.
Many examples can be cited in which students attempt to plug numbers into algorithms without thinking about their meaning, a phenomenon that stretches through all grades of schooling and all mathematical subjects. Even when students are capable of solving a problem correctly informally, they are found to produce incorrect answers when they use formal algorithms. In studies by Lochhead and Mestre (1988), for example, college students who were told that there are 6 times as many students as professors, and there are 10 professors, could correctly give the number of students. But when students are asked to write the formula to represent that situation, the majority write 6S = P. The formula seems correct to students even though the solution would yield 6 times as many professors as students. The occurrence of the word 6 near the word students is sufficient to lead to a formal representation of the problem that is at odds with their informal knowledge.