Scope of mathematics This article will provide an overview of the NCTM process and content standards. Educators first studying the standards may feel overwhelmed with the amount of content addressed within each grade-level span. State frameworks that dictate standards for each grade level exacerbate this situation. However, a longitudinal view will show how the same topics are developed over several years in a spiral and interconnected pattern. For example, the concepts of multiplication and division are introduced in the PreK-2 band, but fluency with these operations isn’t expected until the 3-5 band.
Multiplication and division skills are used in grades 6-8 with problem solving and algebraic equations and in grades 9-12 with vectors, matrices, and other advanced applications. It is critical to keep in mind that deeper study of a few topics is more important for student learning than covering dozens of discrete topics at a surface level. Process Standards The process standards address ways of acquiring and using knowledge and are developed across the entire mathematics curriculum.
They also can be applied across other content areas and real-world problems. These processes are the “verbs” of math. The role of the teacher is to provide settings, models, and guidance for these processes to develop and to assess student skills in using these processes. The process standards are applied at every grade level and across all five content areas. Problem solving is a major focus of the mathematics curriculum; engaging in mathematics is problem solving. Problem solving is what one does when a solution is not immediate.
Students should build mathematical knowledge through problem solving, develop abilities in formulating and representing problems in various ways, apply a wide variety of problem-solving strategies, and monitor their mathematical thinking in solving problems. Problems become the context in which students develop mathematical under-standings, apply skills, and generalize learning. Students frequently solve problems in cooperative groups and even create their own problems. Students should learn to reason and construct proofs as essential and powerful aspects of understanding and using mathematics.
These processes involve making and investigating conjectures, developing and evaluating arguments, and applying various types of reasoning and methods of proof. Reasoning skills are critical for science, social studies, social skills, literature, and most other areas of study. Communication skills are an integral part of mathematics activities. Students must understand and use the language of mathematics-in listening, speaking, reading, and writing. Mathematics communication involves specialized vocabulary and new symbol systems, and becomes a tool for organization and thinking.
More than ever, students and teachers are “talking about math” with each other. Many new mathematics assessments require students to explain their thoughts and processes for solving problems in writing. Some mathematics teachers and mathematicians have tremendous understanding of mathematics concepts, yet have difficulty with communication skills. They can’t convey concepts on a level others will understand, or effectively use communication devices such as analogies and examples. Communication must be modeled with a full range of curriculum applications. Making connections fosters deeper mathematics understanding and assists learning.
Students are encouraged to make connections among different mathematics topics, across other content and skill areas, and into the “real” world. When introducing new concepts, it is critical that teachers assist students in making connections with previous, understood concepts. Linking prior knowledge results in more efficient and generalizable learning. Students are taught to make and apply representations across all mathematics topics. Representations assist with organization, recording, communication, modeling, predicting, and interpreting mathematical ideas and situations.
Examples of representations are graphs, diagrams, charts, three-dimensional models, computer-generated models, and symbol systems. Content Standards The five content standards are also applied across all grade levels. The process standards discussed in the previous section are critical for the development of each of the content standards. Each content standard will be “unpacked” in the following sections, tracing the development across the four grade-level spans and examining the most essential concepts. Number and Operation
Many teachers, parents, and students erroneously consider number and operation, typically called arithmetic, to be the full extent of school mathematics. Understanding number and operation is essential for progress in the other four math content areas; work with the other content areas in turn enhances number and operation understanding. Topics in this standard are the real number system, place value, and the operations and properties for the number system. Our base ten number system derives from Hindu (Indian) number notation and includes ten symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.
These numerals are often called the Arabic number system because Persian mathematicians transmitted the system to the Western world. It can be helpful to examine a concept chart of the real number system and to manipulate number charts to better understand numerical relationships and patterns. Concepts such as zero, negative numbers, primes, factors, square roots, and place value took centuries to develop; it is worth the time to explore these concepts with students to develop deeper and more connected understandings of number systems.
Numbers are ways for identifying units (things), or sets (groups of things), or their parts and relationships. Some numbers represent only names or labels of things and have no inherent meaning, such as telephone numbers or social security numbers. These are nominal (name) numbers. Other numbers designate something’s position in rank or order within a set-ordinal numbers. Examples of ordinal numbers are class rank, days of the month, and results of a foot race. Ordinals are expressed in words (tenth) or by using a suffix (10th). Numbers that are used to quantify a set are called cardinal numbers.
I have 12 library books. The children picked up 14 pennies and 3 nickels. All four arithmetic operations can be performed with cardinal numbers. Try This! Use a base-five number system (using 0, 1, 2, 3, 4 instead of inventing new symbols) to count up to 30. Hint:After the digit 4, regroup: 0, 1, 2, 3, 4, 10, 11, 12, 13, 14, 20. Now try these addition and subtraction problems: 22 + 14 = 43 – 14 = Explain why regrouping is used and how it works. The natural numbers, or the whole numbers beginning with 1, are typically called counting numbers. Including zero and negative or opposite numbers results in the set of integers.
Including all the numbers in between integers (expressed as fractions or decimals) yields all rational numbers. The term rational comes from ratio, such as 1 to 2 (1/2) or 3 to 3 (3/3 or 1). Other types of numbers are irrational (cannot be written as a terminal or repeating decimal), prime (the only factors are 1 and itself), composite (have more than two positive whole number factors), and perfect squares (4, 9, 16, 25, . . . ). There are many other special, less useful but interesting, types of numbers such as perfect, complex, Fibonacci, Lucas, random, sociable, untouchable, and weird.
Number sense is a complex concept dealing with our innate ability to individualize objects and extract numerosity of sets. It is the intuition about numbers (estimations, comparisons, simple addition and subtraction) and understanding of the meaning of different types of numbers and how they are related and represented and the effects of operating with numbers. Number sense is an ability that is further developed through experience. Students in early grades formalize early number sense through learning number symbols, working with larger numbers, and becoming fluent in basic number facts.
As they progress to higher levels, students develop abilities in estimation, representation, analyzing relationships, and working with more complex numbers. The base-ten number system allows for the manipulation of numbers of all sizes and types by using only ten number symbols. Without a place value system, we would have to memorize the name and symbol for each possible unit. The first place value systems of 20 or even 60 digits taxed memory and computational abilities – seven with special notations for places.
Different number systems are still used today, as in New Guinea, where there are 33 numbers with corresponding body parts. Of course the mostly common system in this technological age is the binary system – using only two values (often named 0 and 1) for digital transmissions. Vocabulary Lesson Algorithm refers to a set of instructions or procedures to solve a problem. Place value means that the symbol for a number, say “4,” has the value or meaning of 4 units when positioned in one place; 40, or 4 tens when positioned in the second place; and 400, or 4 hundreds if positioned in the third place.
It would mean 4/10 is placed immediately to the right of a decimal point. Place value is difficult for children for several reasons. It requires good spatial perception, new language, and multi-step cognitive manipulation. In addition, it requires an understanding of multiplicative properties of number (multiples of 10) usually before multiplication has been introduced. The previous exercise simulates the feeling of learning about number systems for the first time. How many young children make up numbers and keep on counting? “eighteen, nineteen, tenteen, eleventeen… Children who don’t understand place value have memorized numbers such as 27 and 84 in their sequence, not realizing that the digits within those numbers have special meaning because of their positions within Algorithm refers to a the numbers. These children may also be on cognitive overload: “How can I possibly set of instructions or remember more than twenty or thirty numbers? ” Further, the right-to-left order of values and algorithms dependent on place value compared with the left-to-right order of reading numbers can be confusing for some students.
The four operations – addition, subtraction, multiplication, and division – are interconnected forms of calculations with real numbers. These operations are used with integers, fractions, decimals, and within algebraic equations. In the simplest sense (with whole numbers), they are ways of counting up and back. Addition is counting on; subtraction is counting back. Multiplication is counting on by groups, and division is counting back by groups. Subtraction of integers is the same as adding the opposite. Division of fractions is multiplying the reciprocal.
There are more complicated computational algorithms to deal with larger and more difficult numbers. What is critical knowledge about operations? Students should understand the effect of each operation on different types of numbers. They should become computationally fluent, able to use efficient and accurate methods for computation (but not necessarily the teacher’s method). An important related skill is estimation – both for solving problems where exactness is not required and for determining the reasonableness of exact answers. Students hould be able to employ a variety of tools and strategies in performing computations and explain the processes used. Students who simply memorize facts and algorithms by rote, rather than understand the concepts and connections, will be less able to apply computations and adjust strategies in problem-solving situations. Numbers also have special properties that assist with operations, such as the identity, distributive, and commutative properties. Many students memorize these properties without understanding their meaning or use.
It is much more powerful to have students “discover” these magic rules and be able to depend on them. Students who don’t under-stand these rules of math may think math is a haphazard endeavor or make up their own, sometimes faulty, rules. Patterns, Functions, and Algebra In the 1989 standards document, the K-4 standards included patterns and relationships, the 5-8 standards included patterns and functions as one standard and algebra as another, and the 9-12 standards listed algebra, functions, and geometry from an algebraic perspective.
Combining these areas into one standard emphasizes the K-12 development of similar and interrelated concepts. Vocabulary Lesson A recursive sequence is a sequence of numbers in which there is a rule for getting the next number based on values of previous numbers, such as with Fibonacci and Lucas sequences. Children are encouraged to look for patterns in numbers, geometry, measurement, and data collections. Detecting patterns is critical for understanding common concepts and connections in mathematical relationships. As students gain the ability to use symbols they can manipulate more complex patterns.
One of the most important skills for problem solving is the ability to recognize patterns – of relational elements within problems and relative to similar problems. Patterns also form the foundation for understanding function and sequence. Even a seemingly simple concept as sequence becomes increasingly complex in the mathematics curriculum. Counting leads to counting by multiples which leads to concepts of exponential growth, proportional growth, recursive sequences, and related functions. Extensions of working with patterns, functions include variables that have a dynamic relationship: changes in one will ause a change in the other(s). Functions can be depicted with equations, tables, spreadsheets, graphs, and geometric representations. One of the most powerful functions is the proportion, first introduced in the study of rational numbers and operations and continued through algebra. For example, a field trip for a class of 30 students will cost $4 per student and we need to find out the cost for k students. We can easily see which numbers to treat with which operations by setting up the proportional equation: 30/k 4/c. Algebra is the study of abstract mathematical structures involving finite quantities.
It involves the symbolic representation of quantitative relationships and the subsequent manipulation of various aspects of the representation. The earliest experiences children have with algebra are with open sentences and missing numbers. For example, “If there are ten books in this stack and two are yours, how many belong to me? ” translates into: 10 – 2 = ? or 2 + ? = 10. The equals sign becomes less a symbol for an answer (doing something) and more a symbol for equivalence. While it has been taught abstractly and by rote in the past, algebra today is taught using manipulatives, technology, and other representations.
Vocabulary Lesson Theorems are assertions that can be proved true by using the rules of logic. Axioms (also termed postulates) are simple and direct statements generally accepted as true without proof. A deductive argument is a series of premises that guarantees the truth of the conclusion. An inductive argument is a series of premises leading to a conclusion that is probably, but not absolutely, true. Students’ work with number, operation, property, patterns, functions, and geometry complements algebraic understanding.
Elementary students develop fluency in working with symbols, numbers, operations, and simple graphing. Middle school students develop concepts of linear functions, geometric representations, and polynomials. And high school students explore other types of functions including rational, exponential, and trigonometric. Deeper understanding of the algebraic characteristics of our number system allows students to explore structures and patterns, pose and solve problems in a number of ways, and develop foundations for the next level of mathematics study. Geometry and Spatial Sense
Concerned with properties of space and objects in space, geometry is one of the most appealing topics for students. The world of space and objects becomes a playground for exploration. Geometry has so many real-world applications. Take, for example, building a simple backyard shed. Consider a few of the mathematical challenges: the shed should be parallel to an imaginary line drawn straight back from the house, the roof should be the same pitch as the roof on the house (rise over run), a 10 foot by 12 foot shed will require how many linear feet of siding, how many square feet of roofing, and so on.
One problem solved leads to three more questions. Geometry is fundamentally based on three undefined terms: point, line, and surface. An understanding of these terms is necessary for understanding other terms and concepts: angle, parallel, congruence, polygons, circles, and solids. Measurement, proportion, functions, and algebraic concepts are also important for the study of geometry. Geometric ideas are useful for representing and solving problems in mathematics and other fields (science, architecture, geography, engineering, sports, the arts, and social sciences).
Geometric experiences involve analyzing and manipulating the characteristics and properties of two- and three-dimensional objects and using different representational systems, methods, and tools such as transformations, symmetry, visualization, spatial reasoning, graphing, and computer animations to solve problems. Like the study of algebra and the number system, geometry involves analyzing patterns, functions, and connections and developing and using rules (theorems or axioms) within the system to solve problems or develop more complex relationships.
Students in grades K-2 study properties of two- and three-dimensional shapes and explore relative positions, directions, and distances using these shapes. Grade 3-5 students begin using coordinate systems, transformations, and other means for analyzing the properties of shapes. They use geometric models to solve problems. Middle school students create and critique inductive and deductive arguments involving geometric concepts and use coordinate geometry to examine properties of shapes.
They use geometric models to extend number and algebraic understandings. By high school, students are testing conjectures, using trigonometric relationships, and applying geometric models to solve problems in other disciplines. Measurement Imminently hands-on yet elusively abstract, measurement skills and concepts can be engaging but challenging to teach. What can be measured? Time, energy, space, and matter. Each of these physical aspects of our world has its measurable aspects, their respective measurement tools, and units of measure (see Figure 1. 7).
Some textbooks and curriculum frameworks classify money as a measurement topic; however, money is not measured but counted (unless all quantification is considered a form of measurement). Measurement is a critical topic for other mathematics applications and is related to many other topics outside mathematics. Both customary (U. S. ) and metric systems are referenced in the standards (the customary system reinforces fraction concepts and metrics reinforce the base-ten place value system and decimal concepts). Figure 1. 7 Measurement: Subject, Tools, and Units
Category| Subject| Example Tool| Example Unit| Time| long periods short periods shorter periods tempo| calendar clock stopwatch metronome| months hours seconds beats per bar*| Energy| atmospheric electric temperature earthquake hearing atomic radiation| pressure barometer electric meter thermometer seismograph audiometer Geiger-Muller tube| millibars kilowatts degrees moment magnitude decibels, Hertz* particles per minute| Space| length height of elevation capacity distance angle| meter stick altimeter tape measure odometer protractor| centimeters eters cubic feet miles degrees| Matter| volume (capacity) mass (weight) density (liquid)| flask scale hydrometer| milliliters pounds specific gravity units| When we want to measure something, there may be a standard unit (as above), more than one unit (e. g. , meters or yards), or a unit and scale can be created. *The items indicated are actually ratios of measures. Scientists tend to focus on mass, length, and time and ways they combine. For example: speed = distance (length) / time (short periods) density = volume (three-dimensional space) / mass
Vectors represent quantities with both magnitude and direction such as force, velocity, and acceleration. | Try This! Select a test item from a state or district mathematics test and analyze the content knowledge required. For example, the following item requires facility with number and operation, geometry, and possibly measurement concepts. Draw a rectangle and a triangle that have the same area. Label the dimensions. Show that the areas are the same. Young children develop the concept that objects have various attributes, some of which can be measured.
They develop the language to express measurement ideas such as longer or more. They begin to associate specific attributes with units and tools of measurement and make simple measurements fairly accurately. Elementary students gain experience with a variety of tools and measurement concepts, in both metric and customary systems. They work with formulas for perimeter, area, and volume of various shapes. By the secondary grades students gain experience with derived attributes (ratios of measurements), conversions, formulas, precision, and error concepts.
Data analysis, statistics, and probability A group of four- and five-year-olds was recently observed “doing” data analysis using probability and statistics. They were discussing their ice cream choices for the monthly parent’s night. On the wall was a chart, a pictograph, with little white, brown, and pink ice cream cones depicting their predictions the previous day of their parents’ preferences for vanilla, chocolate, or strawberry ice cream. (Chocolate was by far the winner. ) That day the children were sharing results of asking their parents about their favorite flavors.
As the new graph evolved, the children exclaimed about the prevalence of vanilla. These young children made predictions, collected data, represented data on a graph, and analyzed those data. They discussed their findings in ways that were mathematically powerful. Like the other content strands, data analysis, statistics, and probability are developed across all grade level spans. Data are quantifications of aspects of the world. It seems that everything is quantified: sports scores, weather records, income levels, test scores, stock market trends, population patterns, and political views.
Data analysis and the application of statistical methods are used across the curriculum. Students are taught how to collect and record data, to represent data in various forms, and to interpret and use data. Students are taught to describe their data collections with frequency charts, measures of central tendency, and various graphs and charts. Higher-level concepts include variability, significance, correlation, sampling, and transformations. The study of probability assists us in making more accurate estimations with problems involving uncertainty. Probability helps answer the question, “How likely is some event? Applications range from educational assessment, business, politics, and medicine to scientific phenomena. Students engaged in probability activities will use their knowledge of number and operations, variables and algebraic equations, problem-solving skills, measurement and graphing, and logical reasoning. Ultimately, skill with probability and statistics should enable students to make more informed decisions in all aspects of their lives. Further enhance your math curriculum with more Professional Development Resources for Teaching Measurement, Grades K-5.