Assessment/Accountability
Massachusetts Comprehensive Assessment System

III. Mathematics Content and Skills to be Assessed by MCAS

Overview of Mathematics Assessment

Background

The Mathematics section of the Massachusetts Comprehensive Assessment System (MCAS) is based exclusively on the learning standards described in the Massachusetts Mathematics Curriculum Framework (1996). These learning standards were developed in collaboration with teachers, school and district administrators, mathematicians, college faculty, parents, and representatives of business and community organizations across the state. They were developed to address concerns of clarity, accessibility, consistency, and mathematical accuracy, and to be aligned with the National Council of Teachers of Mathematics Curriculum and Evaluation Standards for School Mathematics.

The Mathematics Curriculum Framework identifies expectations for student learning, organized by content strands and substrands for grade groupings K-4, 5-8, 9-10, and 11-12. Since MCAS is administered at grades 4, 8, and 10, Mathematics questions for each of the tested levels focus on the learning standards specified for that grade level, plus the standards identified at all preceding grade levels. Consequently, students will be required to demonstrate cumulative content knowledge and mathematical thinking skills, e.g., grade 8 students will be tested on all learning standards identified in the Framework from kindergarten through grade 8.

Content Knowledge and Mathematical Thinking Skills to be Assessed

The MCAS Mathematics Assessment is designed to assess two fundamental dimensions of learning: content knowledge, and thinking skills in using and applying mathematics.

Content Knowledge

Four major content strands identified by the Mathematics Curriculum Framework serve as the foundation for the MCAS Mathematics Assessment and its reporting categories:

• Number Sense
• Patterns, Relations, and Functions
• Geometry and Measurement
• Statistics and Probability

The table below shows the approximate distribution of MCAS questions by content strand for each grade level.

For further definition of these content strands, refer to the Mathematics Curriculum Framework.

Mathematical Thinking

In addition to content knowledge, students will be expected to demonstrate problem-solving and mathematical communication and reasoning skills, as well as skill at making connections between math content and its real-world application.2For the purposes of the MCAS Assessment, these skills are grouped into three major areas: conceptual understanding, procedural knowledge, and problem solving.

Conceptual Understanding. Questions in this area assess student skills in labeling, verbalizing, and defining concepts; recognizing and generating examples and counter-examples; using models, diagrams, charts, and symbols to represent concepts; translating from one mode of representation to another; and comparing, contrasting, and integrating concepts.

Procedural Knowledge. Questions in this area assess student skills related to executing procedures and verifying results; explaining reasons for steps in procedures; recognizing correct and incorrect procedures; developing new procedures, or extending or modifying familiar ones; and recognizing situations in which a procedure is appropriate, necessary, or correctly applied.

Problem Solving. Questions in this area assess student skills in selecting appropriate mathematical concepts and procedures for both real-life and mathematical problem situations and appropriately applying these concepts and procedures; selecting and using appropriate problem-solving strategies; and verifying and generalizing solutions.

___________

2 The core concept of the Massachusetts Mathematics Curriculum Framework "is that students develop mathematical power through problem solving, communication, reasoning and [making] connections" (p.1).

The table below shows the approximate distribution of MCAS questions by mathematical thinking skill for each grade level.

All questions on the Mathematics Assessment test

• knowledge of learning standards from one or more Mathematics Curriculum Framework content strands, and
• one or more mathematical thinking skills.

The table below illustrates the overlap of content knowledge and mathematical thinking skills as tested by MCAS questions.

For reporting purposes, MCAS questions are linked with the reporting category that most closely represents the standard(s) assessed. See pages 6 and 14 for more information on reporting categories.

Types of Mathematics Questions on MCAS

Three types of questions will be used at each grade level tested:

• multiple-choice;
• short-answer; and
• open-response.

Multiple-choice questions on the MCAS Mathematics Assessment require students to select the correct answer from a list of four options. Both short-answer and open-response questions require students to generate, rather than recognize, a response. Short-answer questions require a brief response, usually a short statement or numeric solution to a computation or simple problem. Open-response questions require students to show their work in solving a problem and require responses in writing or in the form of a chart, table, diagram, or graph, as appropriate.

Students will be required to answer questions that assess the content knowledge and mathematical thinking skills described above as is developmentally appropriate for each grade level. The sample questions contained in this document often measure more than one mathematical thinking skill, and some measure knowledge of learning standards across multiple content strands/substrands. The set of sample questions provided in this document is included for illustration purposes only, and does not address all content strands or mathematical thinking skills that will be included on any actual MCAS Assessment.

The approximate distribution of test questions by type for each grade level is shown below. For scoring purposes, the weighting of questions by type will be approximately 50% multiple-choice/short-answer and 50% open-response.

Guidelines for the Scoring of Open-Response Questions

Open-response questions require students to provide evidence of content knowledge; understanding of mathematical concepts, principles, and procedures; and problem-solving and mathematical communication skills. Answers will be individually read and evaluated against a scoring guide, or "rubric," that is developed for each individual test question.

Student answers to open-response questions will be judged on the following, where relevant:

• accuracy of solutions;
• knowledge of mathematical facts and procedures;
• understanding of mathematical concepts;
• quality of mathematical reasoning;
• efficiency of application of mathematical procedures; and/or
• correct use of mathematical communication terms, diagrams, and symbols.

Answers will not be judged directly for quality of written communication as it relates to grammar, punctuation, or other conventions of writing. However, it is important to note that students' written communication skills are important insofar as such skills may determine the clarity and effectiveness with which mathematical reasoning, concepts, and solutions are expressed.

Use of Mathematical Tools during Test Administration

Calculator use

The importance of using a calculator as a tool in solving multistep mathematical problems has been well documented by mathematicians, business, and industry alike. In order to address the need for students to develop necessary skills in calculator use, the first administration of MCAS in the spring of 1998 will include two different Mathematics Assessment sessions for all students at grades 8 and 10. One session will allow the use of calculators. The other session will require students to compute "by hand" without using calculators. The use of calculators will not be allowed for the grade 4 Mathematics Assessment.

For the purpose of the MCAS administration, each student at grades 8 and 10 will be expected to have sole access to a calculator. Optimally, all students will provide their own personal calculators for the Assessment since it is recommended that students use a calculator with which they are already familiar. However, schools should be prepared to provide calculators for use by students unable to provide their own. A basic four-function calculator with a square root key, which costs only a few dollars, will be adequate for the Assessment. However, in accordance with the Massachusetts Mathematics Curriculum Framework, more sophisticated calculators, such as scientific and graphing calculators, are strongly encouraged and will be allowed on the MCAS Assessment. It should be noted that, because of the assignment of questions to calculator and non-calculator sessions, these types of calculators will not offer a significant advantage on the Assessment over a four-function calculator with a square root key. Sample questions within this document allow the use of calculators, unless the symbol is shown.

Tool Kits and Reference Sheets

Tool kits and/or reference sheets will be provided to students at the time of the test administration. Samples of the tool kits and reference sheets to be provided at the spring 1998 administration of MCAS are attached as Appendix A.

Categories for Reporting Results

As described on page 6, students, schools, and districts will receive reports of test results based on established levels of performance for each content area. Each report will contain

• a score that designates, overall, a particular level of content-specific performance (for example, "Proficient" in Mathematics), and
• details of performance results for each reporting category within the content area.

Reporting categories for the MCAS Mathematics Assessment are based on groups of learning standards that have been logically clustered within (and occasionally across) Mathematics Curriculum Framework strands and their subdivisions (substrands). The categories that will be used for reporting Mathematics MCAS results are shown in Table 5 on the following page.

Please note that the names used for reporting categories are sometimes identical to the strand or substrand from the Framework. In other cases, the learning standards contained in a particular substrand have been divided into two reporting categories, and occasionally, learning standards from two substrands have been combined into a single reporting category. (See page 1 for information on the use of reporting categories in the organization of this document.)

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