Scope of the Mathematics Curriculum

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.

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.

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.