Misconceptions used to be thought of as flawed ideas that inhibit student learning and must be identified, removed, and replaced. Many common mathematical misconceptions, however, represent knowledge that is productive in some contexts but has been over-generalized. For example, “multiplication makes numbers larger” holds true for natural numbers, but when over-generalized to all rational numbers (such as fractions less than one), the idea no longer holds true. Such misconceptions are a mix of flawed and productive thinking. Since students’ advanced knowledge is built on prior understandings, misconceptions benefit most from refinement and reorganization. This means that the productive aspects of misconceptions is not dismissed as just mistakes. Better understanding of students' reasoning allows the teacher to build on students’ prior knowledge even when students’ thinking may be partially flawed.
DDE describes one way students may exhibit an incomplete understanding of data. Students consider the fundamental unit of a distribution of data as an individual case or grouping of cases with similar values. Students cannot yet conceive of a data distribution as an entity with characteristics of its own that are different from characteristics of the individual data values (e.g., center, spread).
CDD describes difficulties comparing distributions of data based on summary measures instead of individual data points in each distribution.
OMP describes students who can follow the procedure to compute the mean but have a limited statistical or mathematical understanding of the mean.