Building directly on the National Council of Teachers of Mathematics (NCTM) Curriculum Standards, the Mathematics performance standards present a balance of conceptual understanding, skills, and problem solving.
These conceptual understanding standards delineate the important mathematical content for students to learn. To demonstrate understanding in these areas, students need to provide evidence that they have used the concepts in a variety of ways that go beyond recall. Specifically, students show progressively deeper understanding as they use a concept in a range of concrete situations and simple problems, then in conjunction with other concepts in complex problems; as they represent the concept in multiple ways (through numbers, graphs, symbols, diagrams, or words, as appropriate) and explain the concept to another person.
This is not a hard and fast progression, but the concepts included in the first four standards have been carefully selected as those for which the student should demonstrate a robust understanding. These standards make explicit that students should be able to demonstrate understanding of a mathematical concept by using it to solve problems, representing it in multiple ways (through numbers, graphs, symbols, diagrams, or words, as appropriate), and explaining it to someone else. All three ways of demonstrating understanding—use, represent, and explain—are required to meet the conceptual understanding standards.
Establishing separate standards for these areas is a mechanism for highlighting the importance of these areas, but does not imply that they are independent of conceptual understanding. As the work samples that follow illustrate, good work usually provides evidence of both.
Like conceptual understanding, the definition of problem solving is demanding and explicit. Students use mathematical concepts and skills to solve non-routine problems that do not lay out specific and detailed steps to follow, and solve problems that make demands on all three aspects of the solution process—formulation, implementation, and conclusion. These are defined in , Problem Solving and Reasoning.
The importance of skills has not diminished with the availability of calculators and computers. Rather, the need for mental computation, estimation, and interpretation has increased. The skills in , Mathematical Skills and Tools, need to be considered in light of the means of assessment. Some skills are so basic and important that students should be able to demonstrate fluency, accurately and automatically; it is reasonable to assess them in an on-demand setting. There are other skills for which students need only demonstrate familiarity rather than fluency. In using and applying such skills they might refer to notes, books, or other students, or they might need to take time out to reconstruct a method they have seen before. It is reasonable to find evidence of these skills in situations where students have ample time and access to tools, feedback from peers and the teacher, and an opportunity for revision. As a margin note by the examples that follow the performance descriptions indicates, many of the examples are performances that would be expected when students are working under these conditions. This is true for all of the standards, but especially important to recognize with respect to .
two aspects of mathematical communication—using the language of
mathematics and communicating about mathematics. Both are important. Communicating
about mathematics is about ideas and logical explanation. The travelogue
approach adopted by many students in the course of describing their problem
solving is not what is intended.
Each standard contains several parts. The examples below are cross-referenced to show a rough correspondence between the parts of the standard and the examples. These are not precise matches, and students may successfully accomplish the task using concepts and skills different from those the task designer intended, but the cross-references highlight examples for which a single activity or project may allow students to demonstrate accomplishment of several parts of one or more standards.
The purpose of the samples of student work is to help explain what the standards mean and to elaborate the meaning of a “standard-setting performance.” Few pieces of work are so all-encompassing as to qualify for the statement, “meets the standard.” Rather, each piece of work shows evidence of meeting the requirements of a selected part or parts of a standard. Further, most of these pieces of work provide evidence related to parts of more than one standard. It is essential to look at the commentary to understand just how the work sample helps to illuminate features of the standards.
The NCTM standards make explicit the need for calculators of increasing sophistication from elementary to high school and ready access to computers. Although a recent National Center for Education Statistics survey confirmed that most schools do not have the facilities to make full use of computers and video, the New Standards partners have made a commitment to create the learning environments where students can develop the knowledge and skills that are delineated here. Thus, , Mathematical Skills and Tools, assumes that students have access to computational tools at the level spelled out by NCTM. This is not because we think that all schools are currently equipped to provide the experiences that would enable students to meet these performance standards, but rather that we think that all schools should be equipped to provide these experiences. Indeed, we hope that making these requirements explicit will help those who allocate resources to understand the consequences of their actions in terms of student performance.
The elementary school performance standards are set at a level of performance that is approximately equivalent to the end of fourth grade. It is expected, however, that some students might achieve this level earlier and others later than this grade.
The middle school performance standards are set at a level of performance that is approximately equivalent to the end of eighth grade. Again, it is expected that some students might achieve this level earlier and others later than this grade. Some students will take a course in algebra before high school; their preparation, particularly in , Function and Algebra Concepts, should surpass the level of the middle school standards.
The high school standards reflect what students are expected to know and be able to do after a three-year core program in high school mathematics as defined by the NCTM standards, independent of the specific curriculum they study or its sequencing: traditional Algebra I, Geometry, Algebra II; or (Integrated) Mathematics I, II, III.