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 should 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 cause 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.
