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This volume is organized into three main sections: Elementary School, Middle School, and High School. Each section follows the same format.

Each standard is identified by a symbol.
There are eight standards for Mathematics at each of the three levels, each identified by a symbol. The symbol for the Arithmetic and Number Concepts standard is . This symbol appears wherever there is a reference to this standard.

  1. Most standards are made up of several parts.
    All of the standards are made up of several parts, for example, the Arithmetic and Number Concepts standard has seven parts at the elementary school level. Each part is identified by a lower case letter; for example, the part of the Arithmetic and Number Concepts standard that refers to kinds and forms of rational numbers is c . These symbols are used wherever there is a reference to the relevant part of a standard.

    Performance descriptions tell what students are expected to know and be able to do.
    Each part of a standard has a performance description. The performance description is a narrative description of what students are expected to know and be able to do. It is shown in bold type.
  2. Examples are the kinds of work students might do to demonstrate their achievement of the standards.
    Immediately following the bold-typed performance descriptions for the standard are examples of the kinds of work students might do to demonstrate their achievement. The examples also indicate the nature and complexity of activities that are appropriate to expect of students at the grade level. However, we use the word “example” deliberately. The examples are intended only to show the kinds of work that students might do and to stimulate ideas for further kinds of work. None of the activities shown in the examples is necessarily required to meet the standard.
  3. Cross-references highlight the links between the examples and the performance descriptions.
    The symbols that follow each example show the part or parts of the standard to which the example relates.
  4. Cross-references also highlight links among the standards.
    Often the examples that go with the performance descriptions include cross-references to other parts of the standard and parts of other standards.
  5. Cross-references also highlight opportunities for connecting activities across subject matters.
    Some cross-references shown following the examples identify parts of standards in other subject matters. The cross-references highlight examples for which the same activity may enable students to demonstrate their achievement in more than one subject matter.
  6. Margin notes draw attention to particular aspects of the standards.
    The notes in the margin draw attention to particular aspects of the standards, such as the resources to which students need access in order to meet the requirements of the standards.

    Comparing the grade levels.
    Each page showing performance descriptions for the standards has a note in the margin that directs attention to the Appendix which shows the performance descriptions at each of the three grade levels: Elementary, Middle, and High School.

    Work samples and commentaries.
    Work samples and commentaries appear on the pages immediately following the performance descriptions.
  7. Standards are highlighted in the bar at the side of the page.
    The bar along the side of the pages showing student work highlights the standards that are illustrated by each work sample. (N/A for online version)
  8. The box at the bottom of the page shows what is illustrated in the work sample.
    The shaded box at the bottom of the page lists the parts of the standards that are illustrated in the work sample.
  9. Work samples illustrate standard-setting performances.
    Each work sample is a genuine piece of student work. We have selected it because it illustrates a standard-setting performance for one or more parts of the standards.
  10. The commentary explains why the work illustrates a standard-setting performance.
    The commentary that goes with each work sample identifies the features of the work sample that illustrate the relevant parts of the standards. The commentary explains the task on which the student worked and the circumstances under which the work was completed. It draws attention to the qualities of the work with direct reference to the performance descriptions for the relevant standards.

The commentary also notes our reservations about the work.
The commentary also draws attention to any reservations we have about the student work.

Performance Standards = performance descriptions + work samples + commentaries on the work samples.
Performance standards are, thus, made up of a combination of performance descriptions, work samples, and commentaries on the work samples:

  • The performance descriptions tell what students should know and the ways they should demonstrate the knowledge and skills they have acquired.
  • The work samples show work that illustrates standard-setting performances in relation to parts of the standards.
  • The commentaries explain why the work is standard-setting with reference to the relevant performance description or descriptions.

Each of these is an essential component of a performance standard.


Most work samples illustrate a standard-setting performance for parts of more than one standard.
Most work samples illustrate the quality of work expected for parts of more than one standard. For example, some of the work samples selected to illustrate parts of , Arithmetic and Number Concepts, also illustrate a standard-setting performance for part of , Mathematical Skills and Tools, or for part of , Mathematical Communication, or, possibly, all of these.

Feverish Freddy” is an example of a work sample that illustrates parts of more than one standard.