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NCTM Math Standards 9-12

 

 

enc4.gif (4101 bytes)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--

Focus

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.

 

 

enc4.gif (4101 bytes)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-

Focus

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.

 

 

 

enc4.gif (4101 bytes)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--

and so that, in addition, college-intending students can--

Focus

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.

 

 

 

enc4.gif (4101 bytes)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--

Focus

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

 

 

 

enc4.gif (4101 bytes)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--

and so that, in addition, college-intending students can--

Focus

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.

 

 

 

enc4.gif (4101 bytes)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--

and so that, in addition, college-intending students can--

Focus

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.

 

 

enc4.gif (4101 bytes)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--

and so that, in addition, college-intending students can--

Focus

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).

 

enc4.gif (4101 bytes)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-

and so that, in addition, college-intending students can-

Focus

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.

 

 

enc4.gif (4101 bytes)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--

and so that, in addition, college-intending students can--

Focus

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.

 

 

enc4.gif (4101 bytes)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--

and so that, in addition, college-intending students can--

Focus

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.

 

enc4.gif (4101 bytes)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--

and so that, in addition, college-intending students can--

Focus

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.

 

 

enc4.gif (4101 bytes)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--

and so that, in addition, college-intending students can--

Focus

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.

 

 

enc4.gif (4101 bytes)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--

and so that, in addition, college-intending students can--

Focus

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.

 

 

enc4.gif (4101 bytes)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--

and so that, in addition, college-intending students can--

Focus

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