ERIC Identifier: ED354245
Publication Date: 1990-12-00
Author: Pandey, Tej
Source: ERIC Clearinghouse on Tests
Measurement and Evaluation Washington DC.
Authentic Mathematics Assessment. ERIC/TM Digest.
There is a growing consensus among many educators that the fundamental goals
of teaching and learning mathematics are to help students solve problems of
everyday life, to help them participate intelligently in civic affairs, and to
prepare them for jobs, vocations, or professions. These goals suggest that
school mathematics should diminish the role of routine computation and focus
instead on the conceptual insights and analytical skills that are at the heart
of mathematics. New tests are needed to track progress in these areas.
This digest discusses how well authentic mathematics assessment tests can be
expected to meet these needs. It defines authentic assessment in relation to
more traditional mathematics testing methods. Aspects of the progressive
California Mathematics Program, which has been using authentic assessment for
several years, are presented.
EMERGING IDEAS IN AUTHENTIC MATHEMATICS ASSESSMENT
mathematics curriculum calls for an instructional setting which is very
different from the typical classroom settings of the past. This curriculum
combines new as well as traditional topics; mathematics is presented to students
in the form of rich situational problems that actively engage the students. The
situational lessons or real-life problems attempt to include four dimensions:
o thinking and reasoning--engaging in such activities as gathering data,
exploring, investigating, interpreting, reasoning, modeling, designing,
analyzing, formulating hypotheses, using trial and error, generalizing, and
o settings--working individually or in small groups
o mathematical tools--using symbols, tables, graphs, drawings, calculators,
computers, and manipulatives
o attitudes and dispositions--including persistence, self-regulation and
reflection, participation, and enthusiasm.
In short, students work to construct new knowledge that is integrated with
their prior knowledge. The role of the teacher is that of a facilitator. The
learning helps students acquire mathematical power to cope with ambiguity, to
perceive patterns, and to solve unconventional problems.
Traditionally, assessment has been derived from the curriculum; however,
assessment has not been part of a feedback loop linked to instruction. It is now
widely believed that assessment must be an integral part of teaching, so that it
is used as a tool not merely to collect data, but also to influence instruction.
This requires developing and implementing assessment tasks that measure
students' productivity, their performance on tasks that require mathematical
thinking in pursuit of a result that has meaning to the student. Because these
tasks have essentially the same character as instructional tasks, they also have
meaning for teachers and, therefore, are useful for improving instruction.
The new assessment tasks require students to formulate problems, devise
solutions, and interpret results. While multiple-choice test items can be used
to check students' knowledge of some concepts and some of their skills, other
modes of assessment may be better for evaluating students' products and their
choices of formulation or approach. This means that performance tasks should
increasingly become the basis for judging mathematical achievement required for
success in the technological world.
TYPES OF ASSESSMENTS
Several state assessment programs are
currently engaged in developing new modes of assessment that reflect the
emerging consensus on mathematics instruction and evaluation. In California, for
example, educators are developing the following types of assessment items:
o Open-ended questions, which are 15-minute questions that have a potential
to be adopted on a large scale.
o Short investigations, which are 60 to 90 minute tasks given to students
individually or in groups. The primary emphasis in these tasks is to assess
process skills and understanding of mathematical concepts. The students work on
these tasks independently, write answers to questions, and are interviewed by
the test administrator.
o Multiple-choice questions, which emphasize understanding of important
mathematical ideas and generally involve integrating more than one mathematical
concept. To elicit mathematical thinking, the new multiple-choice questions are
typically designed to take 2 to 3 minutes per question.
o A portfolio, which is used to assess student attainments over a period of
time and includes selections of students' work during the year. The evaluation
of portfolios is still being developed, although much has been learned from the
pilot programs in California.
In Connecticut, educators are developing curriculum-assessment modules. These
modules are made up of individual or group tasks that may take a week or more to
complete. The tasks are real-life problem-solving situations that require
students to use all available resources and tools, including computers.
THE CALIFORNIA MATHEMATICS PROGRAM
emerging ideas in mathematics assessment, California has emphasized the use of
open-ended mathematics questions. According to "A Question of Thinking," a 1989
report from the California State Department of Education, open-ended questions
o give students an opportunity to think for themselves and to express the
mathematical ideas that are consistent with their mathematical development;
o call for students to construct their own responses instead of choosing a
o allow students to demonstrate the depth of their understanding of a
problem, almost an impossibility with multiple-choice items;
o encourage students to solve problems in many ways, in turn, reminding
teachers to use a variety of methods to relate mathematical concepts; and
o model an important ingredient in good classroom instruction: openness to
diverse responses to classroom questioning and discussion.
In California, this type of assessment has been in place for the last three
years at the twelfth-grade level. Each year new questions are developed and each
twelfth-grade student answers one of the questions together with the
multiple-choice part of the test. The questions are scored holistically on a
six-point scale and reported both separately and in combination with the
multiple-choice part of the scores.
Current plans call for an annual assessment using open-ended,
performance-based items. All students take the open-ended exercises, but only a
relatively small sample of student responses are scored by the state and count
toward the total score for a given school. These exercises provide opportunities
for districts to become involved in the state assessment, to administer the
essays, and to train teachers to score them.
California Department of Education
(1985) Mathematics framework for California public schools, kindergarten through
grade twelve. Sacramento.
California Department of Education (1989) A question of thinking: A first
look at students' performance on open-ended questions in mathematics.
Mumme, J. (1990) Portfolio assessment in mathematics, California Mathematics
Project, Santa Barbara.
National Council of Teachers of Mathematics (1989) Curriculum and Evaluation
standards for school mathematics, Reston, VA.
Stenmark, J. K. (1989) Assessment alternatives in mathematics, An overview of
assessment techniques that promote learning. Berkeley: EQUALS and the California