Performance standards define a student’s academic responsibilities and, by implication, the teaching responsibilities of the school. How do we determine whether students have lived up to their academic responsibilities? We assess their work—is it “good enough” by comparison with the standards.

Assessment is an integral component of the educational process. If properly designed and administered, assessments can provide important information to help guide and inform instruction. In order to perform these functions, there must be a strict alignment among standards, educational strategies and resources, and assessments. That is, what we assess must be what we teach, and both must focus on what we want students to know and be able to do—the performance standards.

Assessment takes place in a variety of formats and situations, but a convenient distinction separates informal, ongoing classroom assessment from formal, standardized assessment. The former consists of the evidence teachers collect in class on a continuous basis to track the progress of their students in mastering the skills and material that are taught. The latter are the tests and on-demand assessments administered to all students in specific grades as part of the city- and state-wide assessment programs. Both types of assessment are essential to effective instruction. Ongoing classroom assessment provides continuous feedback on student progress to students, teachers, and parents; standardized assessment measures the mastery of critical skills and concepts at key developmental milestones. Regardless of their differing perspectives, both classroom and standardized assessment must be fully aligned with the performance standards.

The state and city have begun a collaborative process to redesign their standardized assessment programs in mathematics based on the performance standards. (See the figure below.) Beginning with students who entered ninth grade in 1997, all students will be required to take the Mathematics A Regents to meet graduation requirements. The Regents examination itself will be revised to better align with the new performance standards. To ensure that all students are prepared for the state’s new commencement standards, the city and state are restructuring the standardized assessment systems in the elementary and middle school grades. The new assessment systems include a variety of components that are designed to measure attainment of the new standards at the key milestone grades 4 and 8, as well as to track student progress toward the attainment of the standards in previous and intervening grades.

Beyond standardized assessment, it is equally important to ensure that the performance standards provide the focus for ongoing classroom assessment. In the absences of standards teachers are left without a common frame of reference to determine whether the work of their students is good enough. Standards could vary widely from classroom to classroom, resulting in wide variation in instruction and achievement. The work samples that form an essential part of the performance standards provide graphic guidance to all teachers in assessing the level and quality of their students’ work.

How the assessments are connected to the performance standards
The performance standards define a domain of expected student performances. Take the Arithmetic and Number Concepts standard at the elementary level as an example. This standard begins with a definition of the kind of understanding of arithmetic and number concepts that we expect students to have at approximately the end of fourth grade. The performance descriptions go on to portray what students will accomplish in such number concepts as whole numbers, fractions, and approximations. Furthermore, students are expected to put their understanding to work and solve specific types of problems that are based on the standards.

We assess the different elements of the domain defined by a standard by using assessment methods appropriate to the expected performances. Although the assessment system that will fully align with the performance standards is currently under development, several of the components are already in place. The assessment methods are made up of a variety of on-demand, standardized, and ongoing classroom assessments.

The standardized assessments are of two types, differing in format, method of scoring, and the information they provide. One type of assessment serves the purpose of telling us how well students are performing by comparison with standards (standards-referenced assessment); the other compares student performance with that of representative samples of other students (norm-referenced assessment). Typically, the former are performance-based assessments that require students to produce work that is rated by teachers or other professionals using a rubric, or scoring criteria, based on the standards. The latter are usually multiple-choice in format and are machine scored.

In the new city- and state-assessment systems, these two different types of assessments are used to complement one another. Performance-based assessments are combined with multiple-choice tests in ways that measure both the depth and breadth of student achievement. Moreover, beginning in 1999, the state will eliminate its Pupil Evaluation Program (PEP) tests in Mathematics and replace them with assessments that align with the performance standards and articulate with the city assessments. This will allow for the elimination of duplicative testing.

The components of the new standardized assessment system in Mathematics and their relationship to the Mathematics performance standards are described below.

The norm-referenced mathematics test
The mathematics instruction of today emphasizes mathematical reasoning and problem-solving skills. This trend was supported by a set of principles recommended in the National Council of Teachers of Mathematics Curriculum and Evaluation Standards for School Mathematics (1989) and is now reinforced by the development of the New Standards™ Performance Standards for Mathematics.

The norm-referenced test in Mathematics is a multiple-choice test designed to be aligned with both of these sets of standards. Accordingly, there are no questions that test computational skills in isolation. Instead, all questions are cast in the form of word problems that focus on concepts and logic. The questions help to provide information on whether or not the student is using computational skills appropriately and understands the work.

Performance Assessment in Mathematics (PAM)
The performance standards stress the importance of a balance of conceptual understanding, skills, and problem solving. The Performance Assessment in Mathematics (PAM) measures all three. The performance tasks that make up the PAM are presented in an open-ended format that requires the students to generate their own answers, rather than choosing one from given options. Students must formulate a strategy for solving a problem and carry it through to a logical conclusion. They are asked to communicate through words, drawings, symbols, and charts the thinking underlying their solutions.

PAM tasks are representative of each of the strands of Mathematics and may be related directly to the performance standards: numeration and number theory ( , Number and Operation Concepts); geometry and measurement ( , Geometry and Measurement Concepts); data interpretation, statistics, and probability ( , Statistics and Probability Concepts); and pre-algebra, logic, and generalizations ( , Function and Algebra Concepts). Students answer three extended questions, each based on a different strand of Mathematics. Three of the remaining standards are incorporated into the work required of students by these extended questions: , Problem Solving and Mathematical Reasoning, , Skills and Tools, and , Mathematical Communication. The remaining standard, , Putting Mathematics to Work, is not assessed through on-demand examinations.

New York City teachers who have undertaken extensive scoring training score the PAM responses. Scorers use explicit rubrics that are provided by the Board of Education of the City of New York. Scorers also use sample student responses (known as anchor papers) to assist them in their objective assessment of student work and to give them a deeper insight into the goals of the performance standards. Thus, teachers are given examples of student work for scoring PAM in the same way that student work is used to exemplify the performance standards.

Aligning assessments with the performance standards
In order to ensure that New York City’s assessments are aligned with the performance standards, we will engage in an ongoing process of mapping our assessments against the standards in terms of content and level of student achievement required to meet the expectations set out in the standards. The assessment system will be modified and updated until full alignment is achieved. It is expected that this process will take place over several years resulting in successive annual improvements in the match between assessments and the performance standards.