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Have You Ever Wondered What Assessment Should Look Like in Mathematics? by the California Mathematics Council State Board
Assessment
should support the learning of important concepts and furnish useful
information. A student’s opportunity to demonstrate what he/she knows
and can do should not be hampered by time limits or facility with
language. It should be more than merely a test at the end of
instruction to see how the student performs under special conditions.
Rather, it should be an integral part of instruction that informs and
guides decisions both immediate and long term (NCTM 2000, 22). This
article will discuss why we assess, what should be assessed, what we
are currently assessing, and how we can most effectively assess.
What Is the Purpose of Assessment?
The purpose of assessment should be to support the learning of
important concepts and furnish useful information to students, parents,
teachers, and community members. What is assessed communicates what is
valued and considered important. What is assessed applies pressure to
change what is taught.
Ideally, teachers use assessment to plan instruction, to adjust
instruction as they are teaching, and to determine who learned the
planned content at the end of the unit and at the end of the year.
Students can use the assessments to demonstrate mathematical
understandings and skills, to set goals for themselves, to assume
responsibility for their own learning, and to become more independent
learners. Parents use assessments to tell them how well their children
are learning, and whether they are learning at grade level.
Stakeholders want assessments that indicate that students are mastering
grade level standards and are preparing for the workforce and future
learning. They also want assessments that indicate where resources need
to be allocated to improve the learning of all students. Stakeholders
who are concerned with equity issues want assessment information to
determine if groups within the general population are receiving an
appropriate education as well. Can a single assessment meet all these
purposes?
What Should Be Assessed in Mathematics? Schools now have
several documents that define what students should know and be able to
do in mathematics as they progress through the grades: the California Mathematics Framework , the California Mathematics Standards, and NCTM’s Principles and Standards for School Mathematics..
These documents advocate for assessments that focus on students’
understanding as well as their procedural skills. As described in these
standards, assessment should measure:
- computational skills as well as the application of these skills in familiar and unfamiliar contexts;
- the use of mathematical processes in context;
- the use of mathematics to make sense of complex situations;
- how well students formulate hypotheses, collect and organize information, and draw conclusions; and
- how well students communicate their mathematical reasoning both verbally and in writing.
Can a single assessment meet all these purposes?
What Is the Current Mathematics Assessment System in California?
At the end of the year, stakeholders want to know how well students
have learned the mathematics that is described in the standards. In
California, the mathematics portion of the STAR (State Testing and
Reporting) Program includes both a nationally-normed, standardized test
and a criterion-referenced test, which are given to students in grades
2 through 11. These tests are given toward the end of the school year.
The norm-referenced test (SAT 9) measures the basic skills of students
as compared to a nationally normed group that reflected the student
population of our nation at the time of the norming. (The normed group
may not reflect the population in a particular district). The score a
student receives is a percentile score, indicating how that student
compares to the normed group; it does not indicate how well the student
has learned the mathematics standards. Mike Schmoker (2000), author of
“Results: The Key to Continuous School Improvement,” states:
“Norm-referenced tests complicate the assessment mission by seeking to
sort and rank students. And their data are less reliable for students
near the high end, where a correct answer to even one more item can
push a student up several percentile points” (60). The
criterion-referenced test (California Standards Test) estimates how
much of the mathematics standards for that grade level each student has
learned. However, the score at this time is only a raw score, not yet
providing useful information. The content of both tests is limited.
Schmoker continues: “Standardized tests have serious limitations. They
do not fully measure students’ critical and inventive powers. Their
multiple-choice format doesn’t reveal a student’s ability to construct
a proposal, build a case, analyze an issue in writing, or originally
apply a host of mathematical processes—all things we value.”
There are two other serious drawbacks to these standardized tests.
Children who are not yet proficient in English will have difficulty
demonstrating their mathematics knowledge and understanding because
even the mathematics tests are language dependent. The tests also have
a time limit, which penalizes students who could show more of their
mathematics learning given adequate time.
The results of these end-of-year assessments provide information that
may be used to alter the next year’s program but do not inform more
immediate instructional decisions.
What Should Be in a Complete Assessment System?If the ultimate
purpose of assessment is to inform and improve the learning of
mathematics, then an end-of-the-year test is not enough. If we want to
assess how much students have deepened their understanding of
mathematics concepts during the year, we need appropriate pre- and
post-assessments. If we want to assess a student’s expertise with
mathematical reasoning, a standardized test is not enough. If we want
to know how well students can use their mathematical knowledge and
understandings in complex situations, we need to assess them in complex
situations. If assessment were used diagnostically to meet the
educational needs of students, perhaps we would not have to use the
tests punitively and create high stakes situations that erode the
fundamental purpose of education.
Just as we would not use a person’s temperature as the only measurement
to determine good health, we should not use one type of assessment to
determine quality instruction, student growth, and mastery of
standards. A complete assessment system for mathematics needs to
provide meaningful information about the strengths and weaknesses of
individual students in a variety of areas such as skills and the
application of skills in various contexts, concepts, problem solving,
and mathematical reasoning. And a complete assessment system needs to
provide this information throughout the year.
Questions to Discuss at Your School Site
 How do teachers at your school site assess where students begin in relation to the mathematics standards?
How do teachers at your school site assess progress toward
meeting the mathematics standards so they can adjust instruction to
more closely meet the needs of the students?
How does your school site assess and work to meet the needs of all populations?
Resources
California Department of Education. Mathematics Framework for California Public Schools, Kindergarten through Grade Twelve. Sacramento, CA: Author, 1999.
California Department of Education. California Mathematics Standards. Sacramento, CA: Author, 1998.
National Council of Teachers of Mathematics. Principles and Standards for School Mathematics. Reston, VA: The Council, 2000.
Schmoker, Mike. “Results: The Key to Continuous School Improvement.” Educational Leadership 57 (February 2000): 62.
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This Page was last updated: Friday, March 5, 2004 at 7:41:03 AM
This page was originally posted: 6/15/2002; 10:20:18 PM.
Copyright 2008 cmcmath
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