Classroom Assessment with Mathematica
Assessing the classroom performance of students presents challenges to teachers at all
levels of education. While each grade level has its own unique and particular dilemmas,
all formal school assessments must attempt to resolve a number of common critical
problems. These include consistency--subjecting all students to the same sets of
standards; validity--measuring students on the basis of what we teach; and
reliability--the tendency of assessments to yield similar data when repeated.
Mathematica with its capabilities of processing, manipulation, and presenting a
variety of statistical constructs gives educators opportunities to examine some relevant
facets of assessment in original ways. The presentations enhance the teacher's
understanding of the most critical components of the assessment process--individual
students, groups, and the assessments themselves. For the individual student, the
teacher's assessment of achievement is typically an accumulation of measurable attributes
such as test scores and homework grades along with some less-measurable ones such as
attitude and other affective behaviors. A number of statistical measures that bring
together individual ability and an assessment's difficulty are available for use by
teachers. Using Mathematica to compute these measures can provide the teacher with
a more insightful view of the student's learning and development.
For groups of students, the teacher is interested in progress in comparison to equivalent
groups (those of similar grade level studying similar material) and to other groups such
as previous-year groups or groups taught by other teachers. By using some of
Mathematica's graphical capabilities, characteristics, trends, and other qualities of
group, data can be examined. For example,
graphed, grouped, individual scores.
Fit can generate functions that invite algebraic as well as
graphical comparisons. These graphical presentations can illuminate the subtle
similarities and differences between groups, which can more comprehensively inform the
teacher about learning, development, and achievement of the group.
Improved information about specific items that comprise an assessment can help the teacher
create better tests. Improved tests measure individual and group achievement more
precisely, at more finely calibrated levels of difficulty, and are more robust in their
measurement at wider ranges of difficulty. One of the most useful tools in item response
theory is the Rasch measurement which assesses item difficulty. In short, the Rasch
statistic transforms a nonlinear measure of distribution, the p-value (correct
responses divided by total responses), to linear measure. It does so by using the logit
function of the p-value, ln[(1-p)/p].
Computation for the Rasch statistic is a difficult and time-consuming operation, but using
Mathematica to perform it allows increased transparency and user-friendliness. The
teacher is able to make better global sense of both data and subjective information about