Mathematics | |||||||||
with Core Curriculum |
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The State Education Department The University of the State of New York | |||||||||
converted for web delivery by EduTech |
Acknowledgments (back to Table of Contents)
The State Education Department acknowledges the assistance of teachers and school administrators from across New York State as well as resources from other states and countries in the development of the Mathematics Core Curriculum.
Preface (back to Table of Contents)
The Mathematics Resource Guide with Core Curriculum is one of four documents recently developed by the New York State Education Department. These documents were designed to provide guidance to districts and schools in New York State for development of local curricula, instruction, and assessment that meets local needs and resources and aids their students in achieving the mathematics standards for New York State.
The first document of the series, The Learning Standards for Mathematics, Science, and Technology (1996), introduced standards and benchmark performances in mathematics, science, and technology for grades four, eight, and high school. The Mathematics, Science, and Technology Resource Guide (1997), which is periodically updated, extended the standards into classroom practice. The Resource Guide contains lesson plans for mathematics, science, or technology classes as well as lessons that integrate them. The Mathematics Test Samplers for grades 4 and 8 were distributed in February 1998 and for Math A in May 1998. They provide a variety of assessment items similar to those which will be used in the New York State Testing Program.
Core Curriculum (back to Table of Contents)
The Mathematics Resource Guide with Core Curriculum elaborates upon the standards document and provides connections among the other three documents mentioned above. The Core Curriculum is not intended to be the mathematics curriculum for school districts but instead an outline of curriculum to be used to aid districts in developing local curriculum that is reasonable for their local resources and needs.
Although the Core Curriculum addresses only Standard 3 of the Learning Standards for Mathematics, Science, and Technology, a comprehensive mathematics curriculum would also include Standard 1 (inquiry, mathematical analysis, design), Standard 2 (information systems), Standard 6 (interconnectedness), and Standard 7 (interdisciplinary problem solving). The curriculum would also have connections with English language arts through the use of children’s literature, reading, and writing as well as with social studies, which provides many opportunities to examine and analyze data.
The Core Curriculum extends the key ideas and performance indicators of Standard 3 to additional grade-level blocks of prekindergarten to kindergarten, grades 1 to 2, grades 3 to 4, grades 4 to 5, grades 7 to 8, and Math A. A draft of Math B is also included. Suggestions for grade-level content are given for each performance indicator. Suggestions of relevant assessment items or classroom activities are provided for each performance indicator. More examples of assessment items can be found in the Test Samplers and pilot tests. More examples of classroom activities can be found in the Mathematics, Science, and Technology Resource Guide. None of the documents just described are meant to stand alone. They should be used together.
The Core Curriculum is divided into three sections: Overview, Core Curriculum (Elementary, Intermediate, High School), and Reference List.
The Overview presents a discussion on processes that students use and the roles of manipulatives, technology, and assessment. Suggestions of effective instructional strategies will be discussed and each key idea of the standards will be listed and explained.
The Core Curriculum section is divided into three parts: Elementary, Intermediate, and High School. In the beginning of each part are suggestions for manipulatives. This is followed by suggestions of content for each performance indicator. Examples of assessment items are included for grade levels in which there will be State assessments. There will be suggestions for classroom activities or problems for each performance indicator for which an appropriate assessment item is not available. The assessment items and classroom activities are found at the end of each grade-level block and are listed by performance indicator (e.g., 1A. would be the first performance indicator given for Key Idea 1, which is mathematical reasoning).
The Reference List at the end of this document includes all sources used for examples of assessment items, classroom activities, and problems. Teachers may wish to refer to them for more ideas.
I. Overview (back to Table of Contents)
Standard 3: Mathematics (back to Table of Contents)
Students will understand mathematics and become mathematically confident by communicating and reasoning mathematically, by applying mathematics in real-world settings, and by solving problems through the integrated study of number systems, geometry, algebra, data analysis, probability, and trigonometry.
Key Ideas
Mathematics curriculum and assessment of mathematical proficiency based on the mathematics standard of the Learning Standards for Mathematics, Science, and Technology revolve around the seven key ideas. The mathematics standard describes the seven key ideas listed below. Classification of mathematical content into these key ideas inevitably involves some overlap. In addition, many key ideas involve students in synthesizing knowledge across mathematical topics. A mathematics program that includes New York State’s standards and reflects the revised assessments will continue to emphasize the fundamental mathematical skills and knowledge that have been traditionally expected. Students are still expected to master basic skills of arithmetic, geometry, algebra, trigonometry, probability, and statistics. The State Education Department will continue to assess these skills and concepts with tests that will be given in secure settings, and the results of these tests will be made public each year.
The seven key ideas of the learning standards are a mixture of content and process goals. They speak to the mathematics content that a student should know and, at the same time, describe the ways in which the student ought to be able to use that content in meaningful contexts. This is possible by considering the key ideas as guides for selecting appropriate content. The following brief description of each key idea may help in that selection.
1. Mathematical Reasoning
Students use mathematical reasoning to analyze mathematical situations, make conjectures, gather evidence, and construct an argument.
Mathematical analysis is an integral part of problem solving. As a result, this key idea cuts across the content in all the other key ideas. At the elementary level, it includes the concept of pattern and at the high school level includes the concepts of logical terms such as and, or, not, if. . .then, as well as to what constitutes a valid argument.
2. Number and Numeration
Students use number sense and numeration to develop an understanding of the multiple uses of numbers in the real world, the use of numbers to communicate mathematically, and the use of numbers in the development of mathematical ideas.
Although number and numeration deal heavily with number concepts of whole numbers, fractions, decimal fractions, ratios, percents, integers, and irrational numbers, this key idea also includes procedures for ordering numbers and applying them to real-world situations.
3. Operations
Students use mathematical operations and relationships among them to understand mathematics.
Often considered to be primarily procedural, operations includes the expectation that students understand the concepts of addition, subtraction, multiplication, and division in order to be successful with problem solving. Problem solving often requires the selection of appropriate computational or operational methods. The concepts of ratio and proportion must be understood in order to recognize and solve problems that are proportional in nature.
4. Modeling/Multiple Representation
Students use mathematical modeling/multiple representation to provide a means of presenting, interpreting, communicating, and connecting mathematical information, and relationships.
Most geometry concepts are included in modeling/multiple representation but the key idea also includes procedures for geometric constructions and producing graphs and tables. Modeling/multiple representation deals with many aspects of mathematical communication. As a result, it includes the use of variables, modeling relationships both algebraically and graphically, and appropriate use of functions.
5. Measurement
Students use measurement in both metric and English measure to provide a major link between the abstractions of mathematics and the real world in order to describe and compare objects and data.
Measuring is a procedure that includes the concepts of area, volume, perimeter, and circumference. It also includes the formulas that may be applied to calculate them. Much of the content of trigonometry and statistics is included in this key idea. Measurement is a major mathematics key idea that connects with science, technology, and social studies.
6. Uncertainty
Students use ideas of uncertainty to illustrate that mathematics involves more than exactness when dealing with everyday situations.
Estimation and probability are the major topics found in the key idea of uncertainty. Most probability concepts are found in this key idea as well as procedures for calculating probabilities. Although estimation includes number sense, estimating can be used as a problem-solving strategy. This key idea includes estimation of quantity, estimation of computations, and estimation of measurements.
7. Patterns/Functions
Students use patterns and functions to develop mathematical power, appreciate the true beauty of mathematics, and construct generalizations that describe patterns simply and efficiently.
The study of patterns and functions is one of the central themes of mathematics. Patterns are a major part of mathematics study in the elementary grades. A study of patterns requires the conceptual understanding of recognizing, describing, and generalizing patterns.
The study of function helps the student to build mathematical models that can be used to predict the behavior of real-world phenomena that have observable patterns. The widespread occurrence of regular and chaotic pattern behavior can make a study of patterns and functions interesting and valuable.
Patterning and the manipulation of functions (algebraically and graphically) have procedural aspects, but pattern seeking and function finding are strategies used in problem solving. As in the key idea of mathematical reasoning, which includes looking for patterns, key idea 7 cuts across the other key ideas.
Cognitive Categories (back to Table of Contents)
Meaningful curriculum and assessment take into consideration the three categories of cognitive functioning that have served in shaping the National Assessment of Educational Progress (NAEP) and many other state frameworks and large-scale assessments of student learning in mathematics. These categories are procedural knowledge, conceptual understanding, and problem solving. Procedural knowledge involves “knowing when and how.” Conceptual understanding involves “knowing about.” They are joined together by problem solving, which involves the merging of conceptual understanding with procedural knowledge to bring students’ knowledge into use in solving problems in abstract or contextual settings.#1
Students use procedural knowledge when they:
• select and apply appropriate procedures correctly;
• verify or justify the correctness of a procedure, using concrete models or symbolic methods;
• extend or modify the procedures to deal with factors inherent in problem settings;
• use various numerical algorithms;
• read and interpret graphs and tables;
• execute geometric constructions; and
• perform non-computational skills such as rounding and ordering numbers.
Students use conceptual understanding of mathematics when they:
• recognize, label, and generate examples with a concept and without a concept;
• use and interrelate models, diagrams, manipulatives, and varied representations of concepts;
• identify and apply principles (i.e., valid statements generalizing relationships among concepts in conditional form);
• know and apply facts and definitions;
• compare, contrast, and integrate related concepts;
• recognize, interpret, and apply the signs, symbols, and terms used to represent concepts;
• interpret the assumptions and relations involving concepts in mathematical settings; and
• reason in settings involving the careful application of concept definitions, relations, or representations of definitions or relations.
In problem solving students are required to use their accumulated knowledge of mathematics in new situations when they:
• recognize and formulate problems;
• determine the sufficiency and consistency of data;
• use strategies, data, models, and relevant mathematics;
• generate, extend, and modify procedures;
• use reasoning (i.e., spatial, inductive, deductive, statistical, or proportional) in new settings; and
• judge the reasonableness and correctness of solutions.
Problem solving is the connecting thread through the seven key ideas of mathematics and the grade levels. Content which may require procedural knowledge for an older student may be a problem-solving task for a younger student. For example, a fourth grader can easily multiply two numbers by using a taught algorithm, but a first grader who may be unfamiliar with multiplication would have to devise a strategy to find the product, perhaps by making equal-sized groups of counters.
The Role of Assessment (back to Table of Contents)
Assessment is an ongoing process and not an end in itself. A combination of individual student work, student projects, and teacher observations can be used to assess student achievement.
School districts may use assessment data to adjust curriculum and instruction. Teachers may use assessment data to strengthen the teaching and learning process in the classroom, monitor students’ work and progress, and identify students’ strengths and areas in need of improvement. Students may evaluate their own learning by using assessment data. Parents may be partners in assessment and receive specific information about their child’s progress.
New York State Testing Program
The seven key ideas mentioned previously are one of three dimensions that are considered in the construction of State assessments. The three categories of cognitive functioning are another dimension. The third dimension of the model for specifying the nature of State mathematics assessments is the type of questions and tasks that students would be expected to complete when illustrating their competence in the key ideas and categories of cognitive functioning. These formats include multiple choice questions, short constructed response problems, and extended constructed response problems.
Multiple choice questions provide a highly reliable and efficient way of assessing students’ ability to select correct answers and interpretations from a listing of possible alternatives.
Short constructed response problems provide much of the same information, but the student must develop the response. The student is also expected to create the answer, or sketch a drawing, among other possible actions. Such problems can be scored with a rubric that allows for partial credit.
Extended constructed response problems require students to develop a written description of the solution to a problem or to answer a series of subquestions. This requires the student to demonstrate greater knowledge and a necessity to communicate, in some depth, about the problem and its solution. Such tasks are usually scored with the use of a scoring rubric that also allows for partial credit.
Examples of each type of question can be found in the Core Curriculum section at the grade levels for which State pilot assessments have been administered (grades 4, 8, and Math A). There will be a mixture of question types on all the mathematics assessments. Specified percentages of items will assess each of the seven key ideas and three categories of cognitive functions.
Effective Instructional Strategies (back to Table of Contents)
Students develop their own meanings of mathematical concepts and procedures when given the opportunity to become actively involved in learning. Teachers who use instructional strategies guided by this principle act as facilitators for children’s active development of mathematics. They do not act merely as dispensers of rules and algorithms for students to memorize. Some of the instructional strategies they incorporate are the use of manipulative materials, student discussion of mathematical ideas, and small group learning. Active involvement on the part of the learner has the potential to deepen understanding of mathematics.
Research concerning cooperative learning has indicated that students in classes using this learning model do at least as well and often better on standardized tests. Students from minority and low-income groups are frequently those who show the most improved scores. Other advantages of cooperative learning that have been found include development of thinking skills, improved self-esteem, improvement of attitudes toward minorities, and acceptance of mainstreamed students. Cooperative learning models have been shown to work well in heterogeneous classes. It must be pointed out that all “small group work” is not cooperative learning. The key elements of the cooperative learning model are that each student in the group is accountable for the final result and the students in the group must work together as a team in order to succeed.
Instruction that employs a wide range of representations and contextual environments enhance student growth in both affective and cognitive dimensions. The study of mathematics focuses on the representation and communication of numerical, spatial, and data-related relationships. Many classroom activities can support that focus; for example, students may translate their mental conceptions into symbolic forms and then provide a verbal description of the same situation. Other activities might include selecting the best model to physically explain a relationship; using technology in innovative ways to explore a problem; and writing paragraphs, letters, or journals to explain observations about mathematics.
Developmentally appropriate instruction takes advantage of what students are ready to learn. It provides classroom discourse that stimulates cognitive growth. It does not require students to memorize material that is beyond their current understanding. To make use of the notion of developmental stages, teachers observe their students closely and provide them with activities for which they are ready. Hands-on activities, as well as paired, group, and class discussions in which students develop and debate their ideas, contribute to the development of cognitive growth.
Mathematics environments that are embedded in real-world situations engage students in authentic problems that require creativity and demonstrate its uses in everyday life and careers, as well as play. When school mathematics builds on the mathematics that children developed on their own before they came to school, it becomes practical and relevant.
Role of Manipulatives (back to Table of Contents)
Manipulatives are physical objects that students can move around, group, sort, and use to measure as they model mathematical concepts and problems. Manipulative-rich environments may enhance understanding and achievement across a variety of mathematics topics if they are explicitly connected with the mathematical concepts and procedures they represent. Students do not automatically make connections between concrete representations of concepts or procedures and their written or symbolic forms. It is necessary for the teacher to help students make the connection.
Teachers of the early grades commonly use manipulatives but even when studying calculus, students can benefit from manipulating a physical model. Manipulatives can be elaborate and expensive, teacher-made, or simple items from home. Commercial materials often include books, games, and lessons when purchased in classroom sets. Suggestions for commonly used manipulatives will be given at the beginning of each level in the Core Curriculum section.
Role of Technology (back to Table of Contents)
Classroom technology includes calculators, computers, videos, and multimedia. Each of these can be a valuable part of the mathematics program if used when appropriate.
Appropriate grade-level calculators should be made available to students in the classroom. They may be used to assist students in their understanding of concepts and procedures. The use of calculators should not be a substitute for a student’s understanding of quantitative concepts and relationships or proficiency in basic computations. Research has shown that appropriate use of calculators in the classroom does not interfere with student knowledge of “number facts”or with their ability to perform calculations. The classroom use of calculators has been shown to contribute to improved student attitudes toward mathematics and their problem-solving ability.
Four-function calculators are appropriate for instruction with elementary students although they are not permitted on the grade 4 State assessment at the present time. They might be used at this level in the role of skip counting. Skip counting by ones, using the calculator, can help young students relate counting to the symbols. For older students skip counting by 2’s, 5’s, or other numbers can relate repeated addition and multiplication. Integers may be explored through skip counting numbers less than zero.
Calculators that compute fractions are appropriate for the intermediate grades. Fraction calculators that simplify fractions through a series of stages, for example, may be used to help students speculate on the algorithm the calculator uses to put fractions in simplest form. The discussion may include concepts of common factors that are prime numbers and may lead to prime factoring as an algorithm for determining the simplest form of a fraction. Calculators will not be permitted on the multiple choice portion of the grade 8 State assessment but will be required for the rest of the test.
Scientific and graphing calculators are appropriate for secondary mathematics students. Graphing calculators demonstrate multiple ways of representing functions: graphically, symbolically, and in tabular form. Calculators that combine geometric sketching programs and symbolic manipulator software may be appropriate for high school students. Calculators will be required on the Math A and Math B assessments. (See calculator requirements at the beginning of High School: Math A and Math B.)
Computer software that allows students to create mathematics, as well as to explore, conjecture, and investigate mathematics, provides unique opportunities for students learning mathematics. Teachers should take full advantage of this resource, if available.
Young students can benefit from computer explorations that model manipulatives. The graphics, color, and sound capabilities of the microcomputer make for a dynamic model of the manipulative on the screen. Students may explore a concept, make mistakes for concept clarification, and even pose problems. The computer program often provides the student with instant help and feedback that are not possible when the student is working only with a manipulative. Studies concerning students using geometry construction software demonstrate that students develop a deeper understanding of explored concepts and, with proper instruction, develop independent inquiry skills associated with mathematical thinking.
More recent developments in microcomputer capabilities have increased its ability to aid students in the study of real-world data. Internet capacity allows students to communicate across vast distances and provides a whole other dimension to mathematics experiments and data collection. Handheld computer-based laboratory systems which include probes facilitate the collection and analysis of real-world data and can be used to connect mathematics, science, and technology.