|
An inference about
learning is a conclusion about a student’s cognitive processes
that cannot be observed directly. The conclusion has to be based
instead on the student’s performance.
A valid inference
requires evidence that is adequate and relevant.
The examples "Changing
Plans in Mid-Lesson" on page 47 and "Using Evidence to
Plan Tomorrow’s Lesson" on page 49 offer illustrations
of instructional decisions based on valid inferences of students’
learning.
"Analyzing
Scores" on page 71 is an example of inferences made from large-scale
assessments.
See "A Middle-Grades
Statistics Unit" on page 30 for an example of the use of multiple
sources of evidence for assessment.
|
Assessment should promote valid inferences about mathematics learning.
Assessment is a process
of gathering evidence and of making inferences from that evidence
for various purposes. The primary technical question involves defining
procedures for making valid inferences from evidence of a student’s
learning. An inference about learning is a conclusion about a student’s
cognitive processes that cannot be observed directly. The conclusion
has to be based instead on the student’s performance. Many
potential sources of performance are available. Mathematics assessment
includes evidence from observations, interviews, open-ended tasks,
extended problem situations, and portfolios as well as more traditional
instruments such as multiple-choice and short-answer tests.
A valid inference is based
on evidence that is adequate and relevant. Valid inferences also
depend on informed judgment on the part of whoever interprets and
uses the evidence. Teachers make instructional decisions daily that
are based on inferences they have made about students’ learning.
They use their professional judgment in examining relevant evidence.
The validity of their inferences depends on their expertise and
the quality of the assessment evidence they have gathered. Similarly,
valid inferences from large-scale assessments require relevant evidence
and are based on the best professional judgment.
Using multiple sources of
evidence can improve the validity of the inferences made about students’
learning. The Curriculum and Evaluation Standards urges that
decisions concerning students’ learning be based on a convergence
of evidence from a variety of sources. The use of multiple sources
allows strengths in one source to compensate for weaknesses in others.
It also helps teachers judge the consistency of students’ mathematical
work.
|
|
"An exclusive
reliance on a single type of assessment can frustrate students,
diminish their self-confidence, and make them feel anxious about,
or antagonistic toward, mathematics."
–NCTM
(1989, p. 202)
|
A
threat to the validity of inferences comes from potential bias in
the evidence. New forms of assessment, such as portfolios or extended
projects, may create new sources of bias. Extended projects may allow
students to complete some of the work at home, with the result that
differences in home resources (including assistance from parents)
may bias the results. To ensure the equality of resources, additional
materials may have to be provided at school or in the community so
that all students can do the projects to the best of their ability.
Another source of potential bias lies in assessment activities that
rely heavily on students’ ability to use the English language
to communicate mathematical knowledge. This bias can be addressed
through additional activities that allow alternative forms of communication.
A third source of bias may derive from the forms of scoring used in
many assessment activities. Complex tasks require considerable judgment
by the scorers. Bias in that judgment is addressed through suitable
training and scoring procedures. Involving individuals with relevant
expertise not only helps guard against biased scores but also contributes
to equity and openness by offering diverse perspectives. |
|
The example "A
Balanced Assessment System" on page 60 illustrates a high-stakes
assessment that results in certification.
|
Inferences
about mathematics learning have various consequences. Some inferences
affect what students study tomorrow; others affect whether they graduate.
Regardless of the consequences, the validity of each inference must
be established. The amount and type of evidence that is needed, however,
depends on the consequences of the inference. On the one hand, an
informal interview of a student can provide a teacher with sufficient
evidence of a student’s progress to enable the teacher to determine
what mathematical task is most appropriate for the student to engage
in next. On the other hand, a large-scale, high-stakes assessment
where results are used for certification or a culminating experience
in school mathematics requires much more evidence and a more formal
analysis of that evidence.
New forms of assessment
require increased attention to the procedures for making valid inferences
about the mathematics that students know and can do. Assessments
that are based on a framework of important mathematics, draw on
multiple sources of evidence, minimize bias, and support students’
learning provide the evidence needed for such inferences. Technical
considerations relating to validity, evidence, and inferences should
be thought of not as barriers to the use of new and interesting
assessments but rather as opportunities to enhance the instructional
benefits of assessment.
To determine how well an
assessment promotes valid inferences, ask questions such as the
following:
- What evidence about learning
does the assessment provide?
- How is professional judgment
used in making inferences about learning?
- How sensitive is the
assessor to the demands the assessment makes and to unexpected
responses?
- How is bias minimized
in making inferences about learning?
- What efforts are made
to ensure that scoring is consistent across students, scorers,
and activities?
- What multiple sources
of evidence are used for making inferences, and how is the evidence
used?
- What is the value of
the evidence for each use?
|
|
