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All students need
to know and be able to do the mathematics emphasized by the Curriculum
and Evaluation Standards.
"Assessment
should reflect the mathematics that it is most important for students
to learn."
–Mathematical
Sciences Education Board (1993, p. 32)
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Assessment should
reflect the mathematics that all students need to know and be able
to do.
The NCTM Curriculum and
Evaluation Standards for School Mathematics presents a vision
of the mathematics that all students need to know and be able to
do. Mathematics and its uses in society continue to grow and change.
Therefore, the mathematics taught in schools continues to evolve.
From time to time, mathematics
teachers attempt to formulate a statement about the school mathematics
curriculum based on current understanding of mathematics and mathematics
learning. The Curriculum and Evaluation Standards is the
most recent in a series of such statements. It represents the best
of contemporary thinking concerning not only the mathematics topics
that students need to learn but also the important ways in which
mathematical knowledge is learned and used. It reflects a shift
in the importance that the world outside the schools increasingly
places on thinking and problem solving. Procedural skills alone
do not prepare students for that world. Therefore, students deserve
a curriculum that develops their mathematical power and an assessment
system that enables them to show it.
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| See
the example "A Middle-Grades Statistics Unit" on page 30
for an example of using computers as tools in the assessment of realistic
problem solving. |
Assessments
that match the current vision of school mathematics involve activities
that are based on significant and correct mathematics. These activities
provide all students with opportunities to formulate problems, reason
mathematically, make connections among mathematical ideas, and communicate
about mathematics. Students engage in solving realistic problems using
information and the technological tools available in real life. Moreover,
skills, procedural knowledge, and factual knowledge are assessed as
part of the doing of mathematics. In fact, these skills are best assessed
in the same way they are used, as tools for performing mathematically
significant tasks. |
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See the example
"Judging Progress Equitably" on page 36 for an example
of an assessment task that encourages students to continue exploring
and learning.
The examples "Changing
Plans in Mid-Lesson" on page 47 and "Selecting Appropriate
Instructional Experiences" on page 52 illustrate how teachers
use their understanding of mathematics, their familiarity with the
curriculum, and their knowledge of how students learn in their assessments.
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Possessing mathematical
power includes being able, and predisposed, to apply mathematical
understanding in new situations, as well as having the confidence
to do so. A comprehensive program of mathematics assessment includes
opportunities for students to show what they can do with mathematics
that they may not have studied formally but that they are prepared
to investigate. Some assessments may be designed to determine how
well students, presented with an unfamiliar situation, can use what
they have learned previously. Other assessments may require that
students learn a new mathematical concept or strategy during the
assessment and use this knowledge to solve problems. Assessors need
to recognize, too, that the mathematical ideas elicited by an assessment
activity are not always those that are intended. Students respond
to open activities in creative ways, and their responses should
be judged according to the quality of the mathematics demonstrated.
Developing assessment activities
that reflect mathematics all students should know and be able to
do requires that assessors understand mathematics, be familiar with
mathematics curricula, and know how students learn. Documents such
as the Curriculum and Evaluation Standards and Professional
Teaching Standards can help, but they cannot prescribe an assessment
program for each student. Decisions about assessments should be
made in consultation with colleagues and take into account the experiences
of the students being assessed.
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For an example of how assessment tasks and scoring criteria are related
to underlying curricular and assessment frameworks, see "A Balanced
Assessment System" on page 60. |
An assessment framework
that gives appropriate weight to different facets of mathematics
presents a comprehensive view of the mathematics that is important
for students to know and be able to do. A specific assessment activity
makes sense only within such a framework. A range of assessments
that fit into the framework not only gives students multiple opportunities
to display their developing mathematical power but also increases
their opportunities to learn additional mathematics. Constructing
an assessment framework helps ensure that the mathematics assessed
over a school year, as well as throughout each student’s school
experience, forms a balanced, integrated whole.
To determine how well an
assessment reflects mathematics that students need to know and be
able to do, ask questions such as the following:
- What mathematics is reflected
in the assessment?
- What efforts are made
to ensure that the mathematics is significant and correct?
- How does the assessment
engage students in realistic and worthwhile mathematical activities?
- How does the assessment
elicit the use of mathematics that it is important to know and
be able to do?
- How does the assessment
fit within a framework of mathematics to be assessed?
- What inferences about
students’ mathematical knowledge, understanding, thinking
processes, and dispositions can be made from the assessment?
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