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NCTM Math Standards 9-12
STANDARD 1: MATHEMATICS AS PROBLEM SOLVING | Examples of Problem Solving |
In grades 9-12, the mathematics curriculum should include the
refinement and extension
of methods of mathematical problem solving so that all students can--
- use, with increasing confidence, problem-solving approaches to investigate and understand mathematical content;
- apply integrated mathematical problem-solving strategies to solve problems from within and outside mathematics;
- recognize and formulate problems from situations within and outside mathematics;
- apply the process of mathematical modeling to real-world problem situations.
Mathematical problem solving, in its broadest sense, is nearly synonymous with doing mathematics. Thus, whereas it is useful to differentiate among conceptual, procedural, and problem-solving goals for students in the early stages of mathematical learning, these distinctions should begin to blur as students mature mathematically. In grades 9-12, the problem-solving strategies learned in earlier grades should have become increasingly internalized and integrated to form a broad basis for the student's approach to doing mathematics, regardless of the topic at hand. From this perspective, problem solving is much more than applying specific techniques to the solution of classes of word problems. It is a process by which the fabric of mathematics as identified in later standards is both constructed and reinforced.
NCTM Math Standards 9-12
STANDARD 2: MATHEMATICS AS COMMUNICATION | Examples of Communication |
In grades 9-12, the mathematics curriculum should include the continued development of language and symbolism to communicate mathematical ideas so that all students can-
- reflect upon and clarify their thinking about mathematical ideas and relationships;
- formulate mathematical definitions and express generalizations discovered through investigations;
- express mathematical ideas orally and in writing;
- read written presentations of mathematics with understanding;
- ask clarifying and extending questions related to mathematics they have read or heard about;
- appreciate the economy, power, and elegance of mathematical notation and its role in the development of mathematical ideas.
All students need extensive experience listening to, reading about, writing about, speaking about, reflecting on, and demonstrating mathematical ideas. Active student participation in learning through individual and small-group explorations provides multiple opportunities for discussion, questioning, listening, and summarizing. Using such techniques, teachers can direct instruction away from a focus on the recall of terminology and routine manipulation of symbols and procedures toward a deeper conceptual understanding of mathematics. It is not enough for students to write the answer to an exercise or even to "show all their steps." It is equally important that students be able to describe how they reached an answer or the difficulties they encountered while trying to solve a problem. Continually encouraging students to clarify, paraphrase, or elaborate is one means by which teachers can acknowledge the merit of students' ideas and the importance of their own language in explaining their thinking. Providing opportunities for discussions about issues, people, and the cultural implications of mathematics reinforces student understanding of the connection between mathematics and our society.
NCTM Math Standards 9-12
STANDARD 3: MATHEMATICS AS REASONING | Examples of Reasoning |
In grades 9-12, the mathematics curriculum should include numerous and varied experiences that reinforce and extend logical reasoning skills so that all students can--
- make and test conjectures;
- formulate counterexamples;
- follow logical arguments;
- judge the validity of arguments;
- construct simple valid arguments;
and so that, in addition, college-intending students can--
- construct proofs for mathematical assertions, including indirect proofs and proofs by mathematical induction.
Inductive and deductive reasoning are required individually and in concert in all areas of mathematics. A mathematician or a student who is doing mathematics often makes a conjecture by generalizing from a pattern of observations made in particular cases (inductive reasoning) and then tests the conjecture by constructing either a logical verification or a counterexample (deductive reasoning). It is a goal of this standard that all students experience these activities so that they come to appreciate the role of both forms of reasoning in mathematics and in situations outside mathematics. Furthermore, all students, especially the college intending, should learn that deductive reasoning is the method by which the validity of a mathematical assertion is finally established.
A second goal of this standard is to expand the role of reasoning, now addressed primarily in geometry, so that it is emphasized in all mathematics courses for all students. In addition, this standard proposes that college-intending students should learn the more formal methods of proof required for college-level mathematics.
NCTM Math Standards 9-12
STANDARD 4: MATHEMATICAL CONNECTIONS | Example of Math Connections |
In grades 9-12, the mathematics curriculum should include investigation of the connections and interplay among various mathematical topics and their applications so that all students can--
- recognize equivalent representations of the same concept;
- relate procedures in one representation to procedures in an equivalent representation;
- use and value the connections among mathematical topics;
- use and value the connections between mathematics and other disciplines.
This standard emphasizes the importance of the connections among mathematical topics and those between mathematics and other disciplines, connections that are alluded to in many of the other standards. Two general types of connections are important: (1) modeling connections between problem situations that may arise in the real world or in disciplines other than mathematics and their mathematical representation(s); and (2) mathematical connections between two equivalent representations and between corresponding processes in each
NCTM Math Standards 9-12
STANDARD 5: ALGEBRA | Examples of Algebra |
In grades 9-12, the mathematics curriculum should include the continued study of algebraic concepts and methods so that all students can--
- represent situations that involve variable quantities with expressions, equations, inequalities, and matrices;
- use tables and graphs as tools to interpret expressions, equations, and inequalities;
- operate on expressions and matrices, and solve equations and inequalities;
- appreciate the power of mathematical abstraction and symbolism;
and so that, in addition, college-intending students can--
- use matrices to solve linear systems;
- demonstrate technical facility with algebraic transformations, including techniques based on the theory of equations.
Algebra is the language through which most of mathematics is communicated. It also provides a means of operating with concepts at an abstract level and then applying them, a process that often fosters generalizations and insights beyond the original context.
NCTM Math Standards 9-12
STANDARD 6: FUNCTIONS | Examples of Functions |
In grades 9-12, the mathematics curriculum should include the continued study of functions so that all students can--
- model real-world phenomena with a variety of functions;
- represent and analyze relationships using tables, verbal rules, equations, and graphs;
- translate among tabular, symbolic, and graphical representations of functions;
- recognize that a variety of problem situations can be modeled by the same type of function;
- analyze the effects of parameter changes on the graphs of functions;
and so that, in addition, college-intending students can--
- understand operations on, and the general properties and behavior of, classes of functions.
The concept of function is an important unifying idea in mathematics. Functions, which are special correspondences between the elements of two sets, are common throughout the curriculum. In arithmetic, functions appear as the usual operations on numbers, where a pair of numbers corresponds to a single number, such as the sum of the pair; in algebra, functions are relationships between variables that represent numbers; in geometry, functions relate sets of points to their images under motions such as flips, slides, and turns; and in probability, they relate events to their likelihoods. The function concept also is important because it is a mathematical representation of many input-output situations found in the real world, including those that recently have arisen as
a result of technological advances.
NCTM Math Standards 9-12
STANDARD 7: GEOMETRY FROM A SYNTHETIC PERSPECTIVE | Examples of Synthetic Perspective |
In grades 9-12, the mathematics curriculum should include the continued study of the geometry of two and three dimensions so that all students can--
- interpret and draw three-dimensional objects;
- represent problem situations with geometric models and apply properties of figures;
- classify figures in terms of congruence and similarity and apply these relationships;
- deduce properties of, and relationships between, figures from given assumptions;
and so that, in addition, college-intending students can--
- develop an understanding of an axiomatic system through investigating and comparing various geometries.
This component of the 9-12 geometry strand should provide experiences that deepen students' understanding of shapes and their properties, with an emphasis on their wide applicability in human activity. The curriculum should be infused with examples of how geometry is used in recreations (as in billiards or sailing); in practical tasks (as in purchasing paint for a room); in the sciences (as in the description and analysis of mineral crystals); and in the arts (as in perspective drawing).
NCTM Math Standards 9-12
STANDARD 8: GEOMETRY FROM AN ALGEBRAIC PERSPECTIVE | Examples of Geometry from an Algebraic Perspective |
In grades 9-12, the mathematics curriculum should include the study of the geometry of two and three dimensions from an algebraic point of view so that all students can-
- translate between synthetic and coordinate representations;
- deduce properties of figures using transformations and using coordinates;
- identify congruent and similar figures using transformations;
- analyze properties of Euclidean transformations and relate translations to vectors;
and so that, in addition, college-intending students can-
- deduce properties of figures using vectors;
- apply transformations, coordinates, and vectors in problem solving.
One of the most important connections in all of mathematics is that between geometry and algebra. Historically, mathematics took a great stride forward in the seventeenth century when the geometric ideas of the ancients were expressed in the language of coordinate geometry, thus providing new tools for the solution of a wide range of problems.
NCTM Math Standards 9-12
STANDARD 9: TRIGONOMETRY | Examples of Trigonometry |
In grades 9-12, the mathematics curriculum should include the study of trigonometry so that all students can--
- apply trigonometry to problem situations involving triangles;
- explore periodic real-world phenomena using the sine and cosine functions;
and so that, in addition, college-intending students can--
- understand the connection between trigonometric and circular functions;
- use circular functions to model periodic real-world phenomena;
- apply general graphing techniques to trigonometric functions;
- solve trigonometric equations and verify trigonometric identities;
- understand the connections between trigonometric functions and polar coordinates, complex numbers, and series.
Trigonometry has its origins in the study of triangle measurement. Many real-world problems, including those from the fields of navigation and surveying, require the solution of triangles. In addition, important mathematical topics, such as matrix representations of rotations, direction angles of vectors, polar coordinates, and trigonometric representations of complex numbers, require trigonometric ratios, further underscoring the connections between geometry and algebra.
NCTM Math Standards 9-12
STANDARD 10: STATISTICS |
Examples of Statistics |
In grades 9-12, the mathematics curriculum should include the continued study of data analysis and statistics so that all students can--
- construct and draw inferences from charts, tables, and graphs that summarize data from real-world situations;
- use curve fitting to predict from data;
- understand and apply measures of central tendency, variability, and correlation;
- understand sampling and recognize its role in statistical claims;
- design a statistical experiment to study a problem, conduct the experiment, and interpret and communicate the outcomes;
- analyze the effects of data transformations on measures of central tendency and variability;
and so that, in addition, college-intending students can--
- transform data to aid in data interpretation and prediction;
- test hypotheses using appropriate statistics.
Collecting, representing, and processing data are activities of major importance to contemporary society. In the natural and social sciences, data are also summarized, analyzed, and transformed. These activities involve simulations and/or sampling, fitting curves, testing hypotheses, and drawing inferences. To enhance their social awareness and career opportunities, students should learn to apply these techniques in solving problems and in evaluating the myriad statistical claims they encounter in their daily lives.
NCTM Math Standards 9-12
STANDARD 11: PROBABILITY | Examples of Probability |
In grades 9-12, the mathematics curriculum should include the continued study of probability so that all students can--
- use experimental or theoretical probability, as appropriate, to represent and solve problems involving uncertainty;
- use simulations to estimate probabilities;
- understand the concept of a random variable;
- create and interpret discrete probability distributions;
- describe, in general terms, the normal curve and use its properties to answer questions about sets of data that are assumed to be normally distributed;
and so that, in addition, college-intending students can--
- apply the concept of a random variable to generate and interpret probability distributions including binomial, uniform, normal, and chi square.
Probability provides concepts and methods for dealing with uncertainty and for interpreting predictions based on uncertainty. Probabilistic measures are used to make marketing, research, business, entertainment, and defense decisions, and the language of probability is used to communicate these results to others. In grades 9-12, students should extend their K-8 experiences with simulations and experimental probability to continue to improve their intuition. These experiences provide students with a basis of understanding from which to make informed observations about the likelihood of events, to interpret and judge the validity of statistical claims in view of the underlying probabilistic assumptions, and to build more formal concepts of theoretical probability.
NCTM Math Standards 9-12
STANDARD 12: DISCRETE MATHEMATICS | Examples of Discrete Mathematics |
In grades 9-12, the mathematics curriculum should include topics from discrete mathematics so that all students can--
- represent problem situations using discrete structures such as finite graphs, matrices, sequences, and recurrence relations;
- represent and analyze finite graphs using matrices;
- develop and analyze algorithms;
- solve enumeration and finite probability problems;
and so that, in addition, college-intending students can--
- represent and solve problems using linear programming and difference equations;
- investigate problem situations that arise in connection with computer validation and the application of algorithms.
As we move toward the twenty-first century, information and its communication have become at least as important as the production of material goods. Whereas the physical or material world is most often modeled by continuous mathematics, that is, the calculus and prerequisite ideas from algebra, geometry, and trigonometry, the nonmaterial world of information processing requires the use of discrete (discontinuous) mathematics. Computer technology, too, wields an ever-increasing influence on how mathematics is created and used. Computers are essentially finite, discrete machines, and thus topics from discrete mathematics are essential to solving problems using computer methods. In light of these facts, it is crucial that all students have experiences with the concepts and methods of discrete mathematics.
NCTM Math Standards 9-12
STANDARD 13: CONCEPTUAL UNDERPINNINGS OF CALCULUS | Examples of Conceptual Underpinnings of Calculus |
In grades 9-12, the mathematics curriculum should include the informal exploration of calculus concepts from both a graphical and a numerical perspective so that all students can--
- determine maximum and minimum points of a graph and interpret the results in problem situations;
- investigate limiting processes by examining infinite sequences and series and areas under curves;
and so that, in addition, college-intending students can--
- understand the conceptual foundations of limit, the area under a curve, the rate of change, and the slope of a tangent line, and their applications in other disciplines;
- analyze the graphs of polynomial, rational, radical, and transcendental functions.
This standard does not advocate the formal study of calculus in high school for all students or even for college-intending students. Rather, it calls for opportunities for students to systematically, but informally, investigate the central ideas of calculus--limit, the area under a curve, the rate of change, and the slope of a tangent line--that contribute to a deepening of their understanding of function and its utility in representing and answering questions about real-world phenomena.
Most of the mathematics described in the other 9-12 standards involve finite processes, such as determining a sequence of transformations that maps a figure onto a congruent figure or approximating a zero of a polynominal function using an iterative technique. In contrast, the concept of limit and its connection with the other mathematical topics in this standard is based on infinite processes. Thus, explorations of the topics proposed here not only extend students' knowledge of function characteristics but also introduce them to another mode of mathematical thinking.
NCTM Math Standards 9-12
STANDARD 14: MATHEMATICAL STRUCTURE | Examples of Structure |
In grades 9-12, the mathematics curriculum should include the study of mathematical structure so that all students can--
- compare and contrast the real number system and its various subsystems with regard to their structural characteristics;
- understand the logic of algebraic procedures;
- appreciate that seemingly different mathematical systems may be essentially the same;
and so that, in addition, college-intending students can--
- develop the complex number system and demonstrate facility with its operations;
- prove elementary theorems within various mathematical structures, such as groups and fields (e.g., vector to scalar operations);
- develop an understanding of the nature and purpose of axiomatic systems (e.g., Distributive Law).
The structure of mathematics is like the steel framework of a modern building. Students should become aware of this structure, how it provides a strong foundation on which a variety of content strands are built, and how it simultaneously holds these different strands together. For example, one of the girders in this building is the associative property to which are attached objects and operations in such wide-ranging mathematical subjects as arithmetic, algebra, functions, and geometric transformations. An awareness of these broad structuring principles frees students to take a more constructive approach to new mathematical topics and provides them with a conceptual framework that facilitates long-term retention.
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Updated: March 12, 2004