ERIC Identifier: ED359069
Publication Date: 1993-07-00
Author: Brosnan, Patricia A. - Hartog, Martin D.
ERIC Clearinghouse for Science Mathematics and Environmental Education
Approaching Standards for Mathematics Assessment. ERIC/CSMEE
It is acknowledged that the typical mathematics curriculum of a generation
ago emphasized teaching facts, standard procedures, and skills to groups of
passive recipients. In the last decade, a more integrated, child-centered
curriculum presented to more active, participating students has emerged in
response to deteriorating public confidence in the quality of American
education. However, curriculum, designed on the finest principles with the very
best intentions, effects no change in classroom practice if assessment
procedures remain the same [National Council of Teachers of Mathematics (NCTM),
1989; Mathematical Sciences Education Board & National Research Council
(NRC), 1993]. Realizing this, the mathematical community is addressing multiple
challenges to articulate and implement effective standards in the key areas of
testing, assessment, and accountability. American students are subjected to a
variety of tests, usually standardized, multiple-choice tests, from kindergarten
to graduate school. Such tests are, according to widely held beliefs, inhibitors
to change and improvement in education--especially in mathematics (MSEB &
NRC, 1993). To make assessment instruments agents of change rather than
preservers of the status quo, there are several major issues that must be
Are current assessment practices and reporting methods problematic?
What guiding principles should the mathematics community uphold when revamping
Will alternative assessment methods really make a difference?
Should use of alternative assessments be mandated?
Should technology be available at all times? @With these issues emerge problems
How can assessment be designed equitably?
Do all students have equal access to technology?
How can standards be implemented nationally?
Who will assess the assessment process? Who will enforce the standards?
What needs to be done to convince teachers to change?
WHAT MAKES CURRENT ASSESSMENT PRACTICES AND REPORTING METHODS PROBLEMATIC?
States and districts providing leadership in American
education have identified the lack of shared standards for student achievement
and good methods for assessment as the greatest obstacles to creating high
performing schools. It is all too clear that current tests used for assessment
of educational performance fail to measure adequately progress toward national
standards (MSEB, 1993). This is especially true in mathematics, where curriculum
and teaching standards recommended by the NCTM reflect broad nationwide
The new curriculum Standards call for an instructional setting that is very
different from the classroom settings of the past. The curriculum combines new
as well as traditional mathematical topics. Mathematics is presented to students
in the form of rich situational problems that actively involve the students.
Nonetheless, commonly used tests continue to stress routine, repetitive, rote
tasks instead of offering children opportunities to demonstrate the full range
of their mathematical power, including such important facets as communication,
problem solving, inventiveness, persistence, and curiosity (MSEB, 1993).
Traditionally, instruction has been driven by the curriculum, but assessment
has not been an effective part of a feedback loop linked to instruction.
Assessment is most valuable when it is an integral part of teaching, not merely
a tool for ranking students, but a mechanism for influencing instruction. To
realize the full potential of the assessment process requires that the
profession develop and implement assessment tasks to measure student
productivity and performance on tests that require mathematical thinking
For too long teachers have been "teaching the test," students have been most
interested in learning what will be on the test, and administrators have gauged
progress from the results of these tests. If administrative personnel continue
to demand and value static scores, teachers will continue to gather the type of
data needed to make such a limited report. So, to change assessment practices,
administrators must change what is required on reporting mechanisms.
WHAT GUIDING ASSESSMENT PRINCIPLES SHOULD THE MATHEMATICS
EDUCATION COMMUNITY UPHOLD?
A National Summit on Mathematical Assessment was held
23-24 April 1991 (MSEB, 1991). From the summit emerged a consensus that the
evaluation standards of the NCTM Curriculum and Evaluation Standards for School
Mathematics (1989) must be expanded. The following list of principles and goals
for mathematical assessment was developed to guide the proposed expansion.
The primary purpose of assessment is to improve learning and teaching.
The primary use of results of assessment is to promote the development of the
talents of all people.
The content of assessments is derived from the consensus of the discipline.
Assessments will be aligned with the mathematical knowledge, skills, and
processes that the nation needs all of its students to know and be able to do.
Assessment practices will promote the development of mathematical power for all
A variety of effective assessment methods will be used to evaluate outcomes of
Adequate accountability systems will be used to assess mathematics.
Guidelines will be developed for judging the quality of all forms of mathematics
Mathematics teachers and school administrators will be proficient in using a
wide variety of assessment methods for improving the learning and teaching of
The public will become better informed about assessment and assessment practices
(Mathematical Sciences Education Board, 1991, pp. 16-19).
As a result of this summit, the National Center on Education and the Economy
(NCEE) based in Rochester, NY and the University of Pittsburgh Learning Research
and Development Center (LRDC) were charged with the task of developing The New
Standards Project. This Project team would produce standards for assessment that
would align with previous standards in curriculum and teaching in mathematics,
as well as in other academic disciplines. These standards will emphasize the
ability to think well, to demonstrate a real understanding of subjects studied
and to apply what one knows to the kind of complex problems encountered in life.
The New Standards Project system will employ advanced forms of performance
assessment, including portfolios, exhibitions, projects and timed performance
examinations, all based on the use of real-life tasks that students are asked to
do alone and in groups. Some tasks can be completed in minutes, but others will
take weeks or even months. The first valid, reliable, and fair exams will be
available for use in mathematics by 1994.
The goal is not simply to measure student performance, but to improve it
significantly. The following principles, developed by the New Standards team,
provide the framework from which the new assessment standards will emerge.
It is possible to have an educational system that is both equitable and
Student performance standards and assessment systems will be designed to help
bring all groups of students to high levels of performance.
Children should not be denied the right to a first-class education because they
go to school in one state rather than another or in one school district rather
The standards set will emphasize thinking and students' capacities to use what
they know outside a school setting.
Although the standards will be uniform, schools will be free to use very
different methods to help students achieve the new standards through a variety
of means that will enable them to take advantage of the strengths of their
backgrounds and experiences.
Standards will be set through a public process that engages all sectors of
society throughout the nation.
Assessments will be deliberately designed as targets for instruction and
Teachers and content specialists will be actively involved in developing,
grading, and interpreting the assessments.
Students, parents, and teachers must be convinced that competence as
demonstrated on the assessments will lead to real opportunity for further
education and jobs with advancement prospects (The National Center on Education
and the Economy, 1991, pp. 9-10).
Standards for Teacher Competence in Educational Assessment of Students were
concurrently developed by the American Federation of Teachers (AFT), the
National Council on Measurement in Education (NCME), and the National Education
Association (NEA) (1990) requiring that:
Teachers should be skilled in CHOOSING assessment methods appropriate for
Teachers should be skilled in DEVELOPING assessment methods appropriate for
Teachers should be skilled in ADMINISTERING, SCORING, and INTERPRETING the
results of both externally-produced and teacher-produced assessment methods.
Teachers should be skilled in USING assessment results when making decisions
about individual students, planning teaching, developing curriculum, and school
Teachers should be skilled in DEVELOPING valid pupil grading procedures which
use pupil assessments.
Teachers should be skilled in COMMUNICATING assessment results to students,
parents, other lay audiences, and other educators.
Teachers should be skilled in RECOGNIZING unethical, illegal, and otherwise
inappropriate assessment methods and use of assessment information. (pp. 3-5)
With these teacher competence standards and the proposed assessment standards
on the horizon, the question must be raised, "Will alternative assessment
methods really make a difference?"
WHAT DO EARLY RESEARCH RESULTS SUGGEST ABOUT ALTERNATIVE
Large scale assessment programs are beginning to assign
assessment tasks in which groups of students work collaboratively to solve
problems or complete projects. Research results show that, in the group setting,
performance is much better than in the individual setting. While the data
provide important insights into students' mathematics skills and their behavior
in cooperative groups, little is known about the validity of data from group
assessment for making inferences about the competence of individual students
The effects of portfolio assessment on mathematics students' attitudes about
grades, their awareness of their mathematical strengths and weaknesses, and
their ability to set reasonable goals were studied . Results suggest that
portfolio assessment might help students see their strengths and weaknesses so
that they are more able to link successes and failures to performance. Portfolio
assessment might also facilitate goal setting (Owings & Follo, 1992).
Results of innovative assessment indicate that some students performed
surprisingly well and were able to explain concepts in detail. If a question was
rephrased, students were able to do very well explaining a problem in contrast
to drawing a blank on traditional tests. Others did not do as well as expected,
demonstrating an inability to explain concepts, and relying on memorized facts
and simple computation (Lehman, 1992).
In response to the call for assessment reform, many states such as
California, Connecticut, Vermont, Kentucky, Virginia, and Ohio have been
experimenting with alternative forms of assessment on a statewide basis and
early reports include similarly promising results. Likewise, on a national
level, the Educational Testing Service, the National Assessment of Educational
Progress, and the College Board are all working towards implementation of
open-ended assessment items as well as incorporating the use of technology in
their testing programs. Scientific calculators will be allowed on the 1994 SAT
and graphing calculators will be permitted in 1995--a clear movement in the
direction of mathematical empowerment.
As the United States moves resolutely towards
standards-based education, we must learn how to measure the quality of students'
learning through performance. Traditional testing in the United States measures
only the individual, valuing competition over cooperation. Just as our nation
needs standards in curriculum and teaching, so too we need standards for
assessment. Without such standards, we will continue, unwisely, to measure what
is convenient rather than what is most important. Our nation's drive towards
standards-based education will not be complete until joined with appropriate
assessment (MSEB, 1993). Operating from the point of view that instruction and
assessment are closely linked, that good teachers constantly assess students
informally, that student self-evaluation is a vital part of learning, that
formal assessments are stronger if they relate closely to the content and form
of classroom instruction, and that documentation of assessment is important in
connecting classroom work to external evaluation, mathematics educators may
indeed be able to formulate assessment practices that will elicit improved
mathematical achievement (Stenmark, 1991).
American Federation of Teachers, National
Council on Measurement in Education, & National Education Association.
(1990). Standards for teacher competence in educational assessment of students.
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Lehman, M. (1992, February). Assessing assessment: Investigating a
mathematics performance assessment. East Lansing, MI: National Center for
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National Council of Teachers of Mathematics. (1989). Curriculum and
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Owings, C. A., & Follo, E. (1992, August). Effects of portfolio
assessment on students' attitudes and goal setting abilities in mathematics.
Pandey, T. (1990, December). Authentic mathematics assessment. Washington DC:
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questions, and practical suggestions. Reston, VA: National Council of Teachers
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The National Center on Education and the Economy. (1991). The new standards
project. Rochester, NY: Learning Research and Development Center, University of
Webb, N. (1992, November). Collaborative group versus individual assessment
in mathematics: Group processes and outcomes--Review of group assessment issues.
Los Angeles, CA: National Center for Research on Evaluation, Standards, and