periodically, perhaps at monthly intervals, to ensure that students' word spelling is improving gradually, along with LS.
The following statistics are presented, by grade level, for digits and problem scores, in Table 5 for Year 1 and in Table 6 for Year 2: (a) average slopes and standard deviations, (b) percentages of individual student regressions for which the quadratic term significantly contributed to the modeling of student progress, (c) information about the distribution of the slopes across students, and (d) percentages of students with negative slopes. See footnotes to these tables for results of ANOVAs, testing for the effect of grade level on slope.
How well does a linear relationship model student progress within 1 academic year? For the digits score, analyses indicated that the quadratic term significantly contributed to the modeling of student growth for 0-15% of the individuals in Year 2. For the problems datum, the quadratic term significantly contributed to the modeling of progress for 412% of the students in Year 2 (see Table 6). Again, for almost all cases, these curvilinear relationships revealed a negatively accelerating pattern across one academic year. The linear term significantly contributed to the modeling of progress for the following percentages of students at Grades 1-6, respectively: 84, 60, 98, 75, 60, and 38 for digits, and 92, 64, 89, 84, 61, and 45 for problems.
How well does the normal distribution characterize the distribution of slopes? Distributions of the Year 1 slope (see Table 5) were normal for approximately half the grades (i.e., kurtosis and skewness fell within 2 standard errors). As shown in Table 6, with the larger sample sizes of Year 2, most distributions appeared to conform to normality. In all cases, when kurtosis exceeded 2 standard errors, the distribution was lepokurtic (i.e., more peaked than a normal distribution). With respect to skewness, distributions that exceeded 2 standard errors were, in all but one case, positively skewed with a few extreme positive slopes.
What is the weekly rate of student progress and, to what extent does it vary with grade level? CBM math slopes for the digits correct scores (i.e., the primary CBM math score) ranged from .20 at Grade 2 to .77 at Grade 4 during Year 1; from .28 at Grade 2 to .74 at Grade 5 during Year 2. As described in the footnotes to Tables 6 and 6, in both years, ANOVAs revealed statistically significant differences in slope as a function of grade.
For Year 1 slopes, a linear relationship existed between the digits slope and grade level; Grade 2 slopes were reliably lower than Grade 3, 6, and 1 slopes, which in turn were reliably lower than Grade 4 and 5 slopes. For the Year 2 slopes, both the linear and quadratic terms were statistically significant; Grade 1, 2, and 3 slopes were reliably lower than Grade 6 slopes, which in turn were reliably lower than Grade 4 and 5 slopes.
Consequently, across the samples in both data collection years, patterns in the slopes for digits were similar. Two notable exceptions were (a) the significant quadratic relationship revealed in Year 2, but not Year 1, and (b) the apparently higher Year 1 first-grade slopes compared to those in Year 2. Given the larger sample size for the Year 2 data, greater confidence should be placed in the figures shown in Table 6.