LEARNING ENVIRONMENT STANDARD 18
All students will be evaluated using a diversity of assessment tools and
strategies to provide multiple indicators of the quality of every student's
mathematical learning and of overall program effectiveness.
(This "learning environment standard" was developed and approved by the task force that prepared the
Mathematics Standards and appears in the Introduction to the Mathematics Standards chapter of the New
Jersey State Department of Education's Core Curriculum Content Standards; however, since it was not
considered a "content standard," it was not presented to the New Jersey Board of Education for adoption.)
A variety of assessment instruments should be used to enable the teacher to monitor students' progress in
understanding mathematical concepts and in developing mathematical skills. Assessment of mathematical
learning should not be confined to intermittent standardized tests. The learning environment should embody
the perspective that the primary function of assessment is to improve learning.
An important goal of this chapter is to broaden understanding of both the purposes and the tools of
assessment. The popular conception of assessment is restricted to evaluating individual student performance
by tests designed to determine, at the end of a unit of time or instruction, what the student has already learned.
But assessment should also be used during the learning process to enable teachers to monitor students'
understanding and to modify curriculum and instruction, as well as to assess the effectiveness of school
programs. Assessment of individual student performance should be a continuous process that involves many
types of assessment activity. Students should play active roles in assessment so that each assessment
experience is also an educational experience.
This chapter therefore has three main sections, dealing with, Alternative Assessment Strategies, The
Student's Role in Assessment, and Educational Purposes of Assessment.
The Assessment Standards for School Mathematics of the National Council of Teachers of Mathematics
describes assessment as "the process of gathering evidence about a student's knowledge of, ability to use, and
disposition toward mathematics, and of making inferences from that evidence for a variety of purposes." The
kinds of inferences that can be drawn from that evidence are discussed in the section on Educational
Purposes of Assessment.
However, it is important to establish at the outset the perspective that the major purpose of assessment is to
promote learning. The assessment is not the goal, but a means to achieve a goal. Measuring What Counts, a
1993 Policy Brief of the Mathematical Science Education Board, begins as follows:
New Jersey Mathematics Curriculum Framework -- Standard 18 -- Assessment -- 593
"You can't fatten a hog by weighing it." So said a farmer to a governor at a public hearing in order
to explain in plain language the dilemma of educational assessment. To be useful to society,
assessment must advance education, not merely record its status.
Measuring What Counts lists "three fundamental educational principles which form the foundation of all
assessment that supports effective education":
The Content Principle -- Assessment should reflect the mathematics that is most important for students
The Learning Principle -- Assessment should enhance mathematics learning and support good
The Equity Principle -- Assessment should support every student's opportunity to learn important
These three principles are reflected in the first three cumulative progress indicators for this standard.
Experiences will be such that all students:
1. Are engaged in assessment activities that function primarily to improve learning.
2. Are engaged in assessment activities based upon rich, challenging problems from mathematics
and other disciplines.
3. Are engaged in assessments activities that address the content described in all New Jersey's
The Content Principle, the Learning Principle, and the Equity Principle were incorporated into the first three
of the six assessment standards in the NCTM Assessment Standards for School Mathematics:
C Assessment should reflect the mathematics that all students need to know and be able to do.
New Jersey's Mathematics Standards provide a vision of the mathematics that all students
should know and be able to do. Assessment should match this vision.
C Assessment should enhance mathematics learning.
Assessments should be learning opportunities as well as opportunities for students to
demonstrate what they know and can do. Although assessment is done for a variety of reasons,
its main goal is to improve students' learning and inform teachers as they make instructional
decisions. As such, it should be a routine part of ongoing classroom activity rather than an
C Assessment should promote equity.
Assessment should be a means of fostering growth toward high expectations rather than a filter
used to deny students the opportunity to learn important mathematics. In an equitable
assessment, each student has an opportunity to demonstrate her or his mathematical power; this
can only be accomplished by providing multiple approaches to assessment, adaptations for
bilingual and special education students, and other adaptations for students with special needs.
Assessment is equitable when students have access to the same accommodations and
modifications that they receive in instruction.
594 -- New Jersey Mathematics Curriculum Framework -- Standard 18 -- Assessment
C Assessment should be an open process.
Three aspects of assessment are involved here. First, information about the assessment process
should be available to those affected by it, the students. Second, teachers should be active
participants in all phases of the assessment process. Finally, the assessment process should be
open to scrutiny and modification.
C Assessment should promote valid inferences about mathematics learning.
A valid inference is based on evidence that is adequate and relevant. The amount and type of
evidence that is needed depends upon the consequences of the inference. For example, a teacher
may judge students' progress in understanding place value through informal interviews and use
this information to plan future classroom activities. However, a large-scale, high-stakes
assessment such as the HSPT11 requires much more evidence and a more formal analysis of that
C Assessment should be a coherent process.
Three types of coherence are involved in assessment. First, the phases of assessment must fit
together. Second, the assessment must match the purpose for which it is being conducted.
Finally, the assessment must be aligned with the curriculum and with instruction.
These principles should be kept in mind as changes in assessment strategies are contemplated, developed,
tested, and implemented. They should be kept in mind by classroom teachers and all others involved in
assessment -- for example, district committees selecting a standardized norm-referenced test, district
supervisors or department chairs analyzing data from a collection of student portfolios, and state mathematics
content development committees reviewing proposed test items for the statewide tests.
Alternative Assessment Strategies
The next cumulative progress indicator for this standard refers to a wide variety of assessment techniques that
are now available to help make informed judgments and to assure continued progress.
Activities will be such that all students:
4. Demonstrate competency through varied assessment methods including, but not limited to,
individual and group tests, authentic performance tasks, portfolios, journals, interviews,
seminars, and extended projects.
Making use of a variety of assessment methods provides a more complete picture of students' learning. Some
types of assessment tasks provide information about students' abilities to perform mathematical procedures.
Others involve higher-level thinking and problem-solving skills, represent meaningful instructional activities,
and/or invoke real-world applications. Stenmark (1991) describes some of the changes in mathematics
learning that result from using these alternative assessment strategies:
C think more deeply about problems;
C feel free to do their best thinking because their ideas are valued;
C ask deeper and more frequent questions of themselves, their classmates, and their teachers;
New Jersey Mathematics Curriculum Framework -- Standard 18 -- Assessment -- 595
C improve their listening skills and gain an appreciation for the role of listening in cooperative work;
C feel responsibility for their thoughts and ownership of their methods;
C observe that there are many right ways to solve a problem;
C experience the value of verbalization as a means of clarifying one's thinking;
C form new insights into mathematical concepts;
C learn ways to identify the places they need help;
C increase their self-confidence and self-esteem as a result of genuine interest shown by a teacher or
C feel more tolerance and respect for other people's ideas;
C focus their energy on exploring and communicating ideas about mathematical relationships rather
than simply finding answers;
C develop strategies for conducting self-interviews while solving problems in other settings;
C find satisfaction and confidence in their ability to solve problems; and
C look less to the teacher for clues about the correctness of their methods and focus less on imitating
the "right" way.
C gain access to student thinking;
C enhance their ability to use non-threatening questions that elicit explanations and reveal
C strengthen their listening skills;
C show respect for students by being non-judgmental;
C use interview results as a source of questions to pose on written assignments for the whole class;
C encourage respect for diversity by modeling appreciation of varied approaches;
C pose questions that encourage students to construct and share their own understandings;
C feel reinforcement for letting go of "teaching as telling."
A good source of samples of different types of assessment tasks is Mathematics Assessment: Myths, Models,
Good Questions, and Practical Suggestions (Stenmark, 1991). Many of the examples and definitions in this
section come from that source.
Individual and Group Tests
Traditionally, the dominant mode of assessment has been paper-and-pencil testing of individual students.
This testing often includes both selected-response items -- such as matching, multiple-choice, and true/false
questions -- and constructed-response items -- such as problems to solve, or short-answer, fill in the blank,
or "show your work" questions. Large-scale testing often uses primarily selected-response items, since these
are easy to score. However, constructing good selected-response test items is quite difficult, so many teachers
rely more on constructed-response items for classroom assessments. Some teachers check only answers,
while others ask students to show their work and provide partial credit to varying degrees. More recently,
individual tests have also begun to include open-ended questions ("solve and explain your solution"), such as
those found on New Jersey's Eighth-Grade Early Warning Test (EWT) and Eleventh-Grade High School
Proficiency Test (HSPT), since these provide more insight into student thinking. The following are some
596 -- New Jersey Mathematics Curriculum Framework -- Standard 18 -- Assessment
suggestions for creating and/or selecting open-ended questions, as well as for lightening the burden created by
having students write.
T The questions should be constructed so that they cannot be answered by simple multiple-choice
responses. They should address the ability of the student to form and communicate mathematical ideas
and arguments, see and make connections between the various content strands of mathematics, make
conjectures, justify results, organize and analyze data, and make estimates and predictions based on
incomplete data or patterns of events.
T When students have been using an arithmetic operation or algebraic procedure, ask them to explain,
in writing, or with a diagram, what that operation or procedure means and how it works.
T The New Jersey Department of Education's Mathematics Instructional Guide: Linking Classroom
Experiences to Current Statewide Assessments provides many examples of open-ended questions for the
7-12 grade levels. Problems used in other states' assessment programs can be found in the National
Council of Supervisors of Mathematics' Great Tasks and More!!, A Source Book of Camera-Ready
Resources on Mathematics Assessment.
T Ask students to explain how they got their answers. This works quite well not only for textbook
word problems but also for mental math and estimation problems.
T Sample student papers randomly. Select a few papers or a few questions on each paper each day to
review. Be selective about what is commented on; choose one or two aspects for evaluation or scoring
and detailed feedback.
T Teach students to assess each others' work. Review sample student answers with the class, asking
students to suggest improvements. Share scoring rubrics with the students.
T Evaluation of open-ended questions can be done in several ways. For classroom tasks, the "piles"
method is often appropriate and efficient; put each paper as it is read in one of three (or more) piles so
that comparable papers are in the same pile. Labels or points can be assigned to the piles after all of the
papers are sorted. For assessments with more substantial consequences, it is important to ensure the
reliability of the scoring (two different graders will give the same paper the same score); thus, scoring
rubrics have been developed. Some of these are quite general, such as the one on the following page that
is used in New Jersey's statewide assessments; this scoring guide can be found in the New Jersey State
Department of Education's Grade 11 High School Proficiency Test: Directory of Test Specification
and Items. Other scoring rubrics can be found in the National Council of Supervisors of Mathematics'
Great Tasks and More!!, A Source Book of Camera-Ready Resources on Mathematics Assessment.
New Jersey Mathematics Curriculum Framework -- Standard 18 -- Assessment -- 597
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