Mathematics Curriculum Framework - November 2000
"Can you do addition?" the White Queen asked. "What's one and one and one and one and one and one and one and one and one and one?"
"I don't know," said Alice. "I lost count."
- Lewis Carroll, Through the Looking Glass
The study of numbers and operations is the cornerstone of the mathematics curriculum. Learning what numbers mean, how they may be represented, relationships among them, and computations with them is central to developing number sense.
Research in developmental psychology and in mathematics education has shown that young children have a great deal of informal knowledge of mathematics. As early as age three, children begin counting and quantifying, and demonstrate an eagerness to do so. Capitalizing on this informal knowledge and interest, education in the early years focuses on developing children's facility with oral counting and recognition of numerals and word names for numbers. Experience with counting naturally extends to quantification. Children count objects and learn that the sizes, shapes, positions, or purposes of objects do not affect the total number of objects in a group. One-to-one correspondence, with its matching of elements between two sets, provides the foundation for the comparison of groups. Combining and partitioning groups of objects set the stage for operations with whole numbers and the identification of equal parts of groups.
In the early elementary grades, students count and compute with whole numbers, learn different meanings of the operations and relationships among them, and apply the operations to the solutions of problems. "Knowing basic number combinations—the single-digit addition and multiplication pairs and their counterparts for subtraction and division—is essential. Equally essential is computational fluency—having and using efficient and accurate methods for computing."8 Once teachers are confident that students understand the underlying structure of a particular operation, they should teach students the conventional algorithm for the operation. While students will not be asked to demonstrate use of standard algorithms on the grade four MCAS mathematics tests, they are expected to be introduced to them as theoretically and practically significant methods of computing.9 After students have learned how to use the conventional algorithm for an operation, whatever they then choose to use on a routine basis should be judged on the basis of efficiency and accuracy. No matter what method students use, they should be able to explain their method, understand that many methods exist, and see the usefulness of methods that are efficient, accurate, and general.10
As they progress through the elementary grades, students compute with multi-digit numbers, estimate in order to verify results of computations with larger numbers, and use concrete objects to model operations with fractions, mixed numbers, and decimals. By the end of their elementary school years, students choose operations appropriately, estimate to solve problems mentally, and compute with whole numbers.
Mathematics in the middle school centers on understanding and computing with rational numbers, and on the study of ratio and proportion (what they are and how they are used to solve problems). Students achieve competence with rational number computations and the application of the order of operations rule to prepare for high school.
At the high school level, understanding systems of numbers is enhanced through exploration of real numbers and computations with them. Thereafter, students investigate complex numbers and relationships between the real and complex numbers. Students expand their knowledge of counting techniques, permutations, and combinations, and apply those techniques to the solution of problems.
As students develop competence with numbers and computation, they construct the scaffolding necessary to build an understanding of number systems. Students not only compute and solve problems with different types of numbers, but also explore the properties of operations on these numbers. Through investigation of relationships among whole numbers, integers, rational numbers, real numbers, and complex numbers, students gain a robust understanding of the structure of our number system.
Technology in the Number Sense and Operations strand is used to facilitate investigation of mathematical concepts, skills, and strategies. Calculators and computers enhance students' abilities to explore relationships among different sets of numbers (e.g., the relationship between fractions and decimals, fractions and percents, and decimals and percents), investigate alternative computational methods (e.g., generating the product of a pair of multi-digit numbers on a calculator when the multiplication key cannot be used), verify results of computations done with other tools, compute with very large and very small numbers using numbers in scientific notation form, and learn the rule for the order of operations.
Number Sense and Operations Learning Standards for:
I continued to do arithmetic with my father, passing proudly through fractions to decimals. I eventually arrived at the point where so many cows ate so much grass, and tanks filled with water in so many hours. I found it quite enthralling.
- Agatha Christie, An Autobiography
Algebra emerged through the analysis of solutions to equations, while the concept of a function developed as the insights and techniques of Calculus began to spread. Patterns, relations, and algebra are integral elements in the study of mathematics. All students should understand how patterns, relations, and functions are interrelated; be able to represent and analyze mathematical situations and structures using algebraic symbols; use mathematical models to understand quantitative relationships; and analyze change in various contexts.
The foundation for the study of patterns, relations, and algebra can be constructed in the PreK-K years and expanded gradually throughout the other years. All students should be aware of the mathematics in patterns and use mathematical representations to describe patterns. Young students can identify, translate, and extend repeating rhythmic, verbal, and visual patterns. They can recognize patterns and relationships among objects, and sort and classify them, observing similarities and differences. They can then probe more deeply into the study of patterns as they explore the properties of the operations of addition and multiplication.
Through numerous explorations, elementary grade students deepen their understanding of pattern and work informally with the concept of function. It is important that the concept of a variable is developed for them through practical situations, for example, as they engage in such basic activities as listing the cost of one pencil at 50¢, two pencils at ?, three pencils at ? , ... n pencils at ?.
Investigating patterns helps older students understand the concept of constant growth as they analyze sequences like 1, 3, 5, 7, .... These students should contrast this type of change with other relationships as evidenced in sequences such as 1, 2, 4, 8, ... ; 1, 3, 6, 10, 15, ... (the sum of the first n positive integers); and 1, 1/10, 1/100, 1/1000, .... In middle and high school, students build on prior experiences as they compare sequences and functions represented in recursive and explicit forms.
As students advance through the grades, their work with patterns, functions, and algebra progresses in mathematical sophistication. They learn that change is a central idea in the study of mathematics and that multiple representations are needed to express change. They identify, represent, and analyze numerical relationships in tables, charts, and graphs. They learn about the importance and strength of proportional reasoning as a means of solving a variety of problems. While understanding linear functions and their graphs is a realistic goal for the middle school student, students deepen their study of functions in the secondary years. They engage in problems that feature additional types of polynomial functions, and rational, exponential, logarithmic, trigonometric, and other families of functions.
Graphing calculators and computer software with spreadsheet and graphics capabilities are ideal resources that help students make connections among different representations of the functions. The meaning and importance of domain, range, roots, optimum values, periodicity, and other terms come alive when experienced through technology. With appropriate instruction, students move readily among symbolic, numeric, and graphic representations of functions. Through insightful examples, secondary students learn that functions are a key concept with connections not only to Calculus but also to transformational geometry and topics in discrete mathematics.
Patterns, Relations, and Algebra Learning Standards for:
[The universe] cannot be read until we have learnt the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word.
- Galileo Galilei, Opere Il Saggiatore
Geometry, spatial sense, and measurement were among the sources of the earliest mathematical endeavors. Ancient people, in their efforts to manage their lands, conduct commerce, and describe natural forms and patterns, began to rely on abstract shapes and standard units of measurement to communicate with each other. Euclidean geometry, a landmark in the development of mathematics and other academic disciplines, is the study of points, lines, planes, and other geometric figures, using a modified version of the assumptions of Euclid, who is thought to have lived about 300 BC. The geometry in Euclid's Elements of Geometry was a logical system based on ten assumptions. Five of the assumptions were called common notions (axioms, or self-evident truths), and the other five were postulates (required conditions). The resulting logical system was taken as a model for deductive reasoning and profoundly influenced all branches of knowledge. Indeed, the development of the axiomatic approach to geometry extends to present.11
Today, students have many of the same needs that their ancestors had thousands of years ago. They need to understand the structure of space and the spatial relations around them, measure many aspects of their environment, and communicate this structure, these relations, and their measurements to others. Instruction in geometry and measurement is designed to address this need.
Before students begin school, they have developed some knowledge of the physical and spatial world. They have explored size, shape, position, and orientation of objects in everyday activities at home, on the neighborhood playground, or at a supermarket. They have become familiar with two- and three-dimensional shapes. The language describing location and orientation of objects-words such as right, left, above, below, top, bottom, and between-are part of children's daily talk. However, everyday use of the language of measurement and geometry often differs from formal use in mathematics. These differences must be addressed through instruction. Students should be expected to use clear and increasingly precise language in their mathematical talk.
In the early grades, children explore shapes and the relationships among them that build on their natural understanding. As they progress through elementary school, students identify the components, attributes, and properties of different shapes, including sides, corners or vertices, edges, interiors, and exteriors. With time, they develop procedures to identify and categorize shapes by referring to their components, attributes, and properties. Students investigate these features dynamically by using mirrors, paper folding, hand drawing, and computer drawing. Still operating on concrete objects, students develop the idea of transformations by recognizing changes effected by slides, flips, and turns, not only on individual objects but also on combinations of objects. Investigations of simple transformations lead to the concept of congruence.
In middle school and high school, students solve problems in other areas of mathematics using geometric concepts, including coordinate geometry, perspective drawings, and projections of three-dimensional objects. They use mechanical and electronic tools to construct common geometric shapes and patterns, and to develop the idea of geometric similarity, which can be integrated with the ideas of ratio and proportion. Students can also apply methods developed in the geometric context to make sense of fractions and variables, construct graphs and other representations of data, and make and interpret maps, blueprints, and schematic drawings.
In high school, students use formal reasoning to justify conclusions about geometry and its relationship to other areas of mathematics. They recognize the logical structure of the system of geometric axioms, become increasingly proficient in proving theorems within the axiomatic system, and use axioms and theorems to verify conjectures generated through their own work or by their peers. Students apply coordinate geometry to the solution of problems and extend transformational geometry to a variety of congruence and similarity transformations and their composition.
Geometry Learning Standards for:
Measure what is measurable, and make measurable what is not so.
- Galileo Galilei
Measurement is best learned through direct applications or as part of other mathematical topics. A measurable attribute of an object is a characteristic that is most readily quantified and compared. Many attributes, such as length, perimeter, area, volume, and angle measure, come from the geometric realm. Other attributes are physical, such as temperature and mass. Still other attributes, such as density, are not readily measurable by direct means.
In PreK - K, students begin to make qualitative comparisons between physical objects (e.g., which object is longer or shorter, which is lighter or heavier, which is warmer or colder), and begin to use nonstandard units of measurement for quantitative comparisons. Building on existing measurement ideas, students in grades 1 and 2 become competent with standard units of measurement. Students gain understanding of ratio and proportion in the middle grades, and apply their new found knowledge to making scale drawings and maps that accurately reflect the dimensions of the landscape or the objects they represent. Greater familiarity with ratios enhances students' understanding of the derived attributes (speed, density, and trigonometric ratios), their applications, and the use of conversion factors to change a base unit in a measure.
At all levels, students develop respect for precision and accuracy by learning to select the tools and units of measurement appropriate to the situation. They also learn to analyze possible and real errors in their measurements and how those errors may be compounded in computations.
Measurement Learning Standards for:
Life is a school of probability.
- Walter Bagehot
Education in a free society must prepare citizens to make informed choices in all areas of their lives. They must be able to grasp the information being presented, analyze it, and make reasoned decisions. To accomplish these goals, students learn to collect, organize, and display relevant data to answer questions that can be addressed with data; use appropriate statistical methods and predictions that are based on data; develop and evaluate inferences and predictions that are based on data; and apply basic concepts of probability.
In the early grades students learn how to collect data, observe patterns in the data, and organize and analyze the data to draw conclusions. To organize and display their data, they begin by using concrete and pictorial representations, and gradually learn to use tables, bar and line graphs, and data line plots. As students advance through the grades, they explore more complex forms of representation, including multiple-line graphs, circle graphs, and frequency tables.
In their study of data and statistics, students shift their perspective from viewing data as a set of individual pieces of information to an understanding of data as a coherent set with its own collective properties. This shift is emphasized in the middle grades when students study characteristics of sets of data, including measures of central tendency and techniques for displaying these characteristics, e.g., stem-and-leaf plots and scatterplots. Students learn how to select and construct representations most appropriate for the data and how to avoid misleading and inappropriate representations.
In high school, students gain insight into the use of trend lines and measures of spread for analyzing data. Students use technology to estimate and find lines of best fit for scatterplots. They categorize data by the type of model that best represents them, design surveys to generate data, and learn to choose representative samples and identify biases in the samples and survey questions.
Probability may be called the study of the laws of chance. In the elementary grades, students begin the study of probability by conducting experiments with spinners, counters, number cubes, and other concrete objects. They learn to record outcomes of individual experiments, and to organize and analyze results. They identify certain, possible, and impossible events.
In the middle grades, students enumerate all possible outcomes of simple experiments and determine probabilities to solve problems. Through the exploration of various problem situations, students learn to distinguish between independent and dependent events. By representing problem situations both numerically and geometrically, students can begin to develop an understanding of probabilities for simple compound events.
In high school, as they compare results of experiments with their theoretical predictions, students gain an understanding of the difference between predicted and actual outcomes. They apply counting techniques, use multiple representations to solve complex probability problems, and investigate probability distributions.
Data Analysis, Statistics, and Probability Learning Standards for:
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Number Sense and Operations
Learning Standards for Grades PreK-K