(e.g., relative performance of males versus females or of older versus younger children) in no way prejudge the immutability of those differences. To continue with the example of mental rotation, the statement that Learner A has better mental rotation skills than Learner B says nothing about whether or not Learner B’s mental rotation skills might be improved (e.g., through repetition) to be just as good as Learner A’s.

Third, a statement about a difference between two learners on one specific skill need not imply something about the availability of another skill. Thus, although Learner B may be less adept than Learner A with respect to rotating a visual image mentally, Learner B may be more skilled in using a different strategy that may be just as effective (e.g., reasoning verbally about the relative locations of different sections of the geological formation).

In short, a statement about differential abilities or strategies carries no implications that differences are inherent (the first point ), fixed (the second point ), or pervasive (the third point).

Fourth, differences among learners may be considered either at the level of “group differences” or at the level of “individual differences.” The observation that there are group differences with respect to spatial learning means that—on average—groups differ in their level of or strategies for spatial thinking. Groups may be defined along a virtual infinity of dimensions: for example, biological sex (boys versus girls; men versus women), educational focus (e.g., engineering versus education majors), chronological age (e.g., elementary versus middle school students), or socioeconomic background (e.g., children from professional versus working class backgrounds). A statement that there are “significant group differences” merely means that, overall, one group performs differently than the other group, and that statistical analyses show the observed differences are unlikely to be attributed to chance, but are instead reliable differences.

Even when groups do differ significantly, it is almost never the case that every single member of one group differs from every single member of the other group on the relevant characteristic. In real life, distributions overlap. For example, consider group membership defined by biological sex with respect to two variables, first, a familiar and uncontroversial one, physical height. On average, men are taller than women. However, among the group of men, some are short; among the group of women, some are tall so that a particular man may well be shorter than a particular woman. Thus, if we want to know the relative heights of a pair comprised of one man and one woman, we would be better off actually measuring them than we would simply assuming that the man was the taller of the two.

The identical reasoning holds for a second, potentially more controversial, variable—the ability to rotate mental images of two- or three-dimensional shapes. Again, at the group level of analysis, on average, boys perform better on tests of mental rotation than do girls, on average (e.g., Linn and Petersen, 1985). Nevertheless, many individual girls perform better than many individual boys. Thus, if one were planning to teach some spatial concept that would be taught differently depending on the level of the learner’s mental rotation skills, it would be misguided to assume that all boys should get one form of instruction and all girls should get another. Rather, different instructional methods should be assigned to children based on a direct measure of mental rotation ability, not on the basis of their biological sex.

Similar reasoning holds for other differences between or among groups, including those related to chronological age. For example, on average, first-grade children may be expected to find it more difficult than fifth-grade children to employ projective spatial concepts (see Piaget and Inhelder, 1956). Yet a given first-grader may be particularly advanced and a given fifth-grader may be particularly delayed with respect to that mastery. As a result, decisions about the best ways to teach individual learners are better informed by obtaining information about specific prerequisite skills or concepts than by knowing chronological age alone. Alternatively, one may find means of presenting material at multiple levels of complexity and with multiple strategies so that the material will be appropriate for learners who have a range of strengths.



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