Enhancing Understanding of Human Growth and Development
Results of the current study were discussed in terms of models of student development of academic proficiency. Spelling and reading findings support developmental theories of student growth, where students acquire important component skills relatively quickly in the early grades.
A final set of comments, about the extent to which a linear relationship adequately models student progress, also relates to our understanding of human development. In this discussion, it is important to remember that within-year analyses were conducted at the individual student level, with regressions predicting student scores from calendar weeks. By contrast, the across-year analyses were conducted at the group level, with ANOVAs assessing the relationship between grade and slope. Consequently, the analyses are not parallel. Although the current methodology does permit comparison between within- and across-year patterns of growth, future research might formulate these comparisons longitudinally, so that both the within- and across-year analyses could be run analogously, tracking the same students within and across grades.
The current set of analyses indicated that, for most students, reading and math progress made during one academic year can be characterized as increasing in linear fashion with time. For this majority of individuals, a linear regression both describes and predicts progress adequately. This finding enhances our understanding of how students develop academically, suggesting that progress occurs additively within the framework of a single academic year. These results also increase confidence in the standard CBM data-evaluation methods, which rely on regression or quarter-intersect (i.e., straight) liens to (a) describe current rates of progress, (b) predict desired rates of progress, and (c) determine when student progress is satisfactory or when a need for instructional adjustment exists (see Fuchs, Hamlett, & Fuchs, 1990 for a description of these data-evaluation methods and see Figure 1 for the application of these straight lines).
It is important to note, however, that for some students (0-31%, depending on academic area, datum, and grade), progress within one academic year was characterized by a negatively accelerating pattern. Fitting a linear function (or a straight line) to a negatively accelerating curve would result in underestimating student progress early in the year and overestimating student progress at the end of the year. Moreover, for Grade 6 oral passage reading and for Grade 6 math (as well as for many spelling grade levels), the linear relationship did not contribute significantly to the modeling of student progress for more than 60% of general education students. CBM practitioners should be mindful of the possibility that an individual student's progress may not, in fact, be characterized by a straight line -- especially for oral passage reading at Grade 6, for math at Grade 6, and for spelling. For these individuals, standard CBM methods for evaluating progress may not be adequate.
These results also bear on the long-standing question about whether equal interval or semilogarithmic graph paper is more suitable for displaying student progress over one academic year (see Marston, 1988 for related discussion). Equal interval paper assumes a linear relationship between time and score; semilogarithmic paper assumes a positively accelerating learning curve. The finding that a linear function adequately models student progress within an academic year for the great majority of individuals in reading and math supports the use of equal interval graph paper for displaying reading and math progress with CBM. It also helps