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.

Discussion

Students in grades 9-12 should also have some experience recognizing and formulating their own problems, an activity that is at the heart of doing mathematics. Instructional settings that encourage investigation, cooperation, and communication foster problem posing as well as problem solving. In addition, all students can profit from discussions of specific problem-posing techniques. Another important component of mathematical thinking is the process of mathematical modeling. The importance of problem solving to all education cannot be overestimated. To serve this goal effectively, the mathematics curriculum must provide many opportunities for all students to meet problems that interest and challenge them and that, with appropriate effort, they can solve.

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.

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.

Discussion

Technology is yet another avenue for mathematical communication, both in transmitting and receiving information. Calculators and computers require students to use and understand accurate, concise language. To use a calculator, students must not only understand the underlying mathematics (e.g., the order of operations or the meaning of the fraction line) but also apply the specific syntax for the type of calculator being used.

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.

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.

A third goal, also a departure from the existing curriculum for college-intending students, is to give increased attention to proof by mathematical induction, the most prominent proof technique in discrete mathematics. (The term mathematical induction refers to a formal technique used to prove statements defined for subsets of the integers. It should not be confused with inductive reasoning.)

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.

Focus

STANDARD 5: ALGEBRA

• 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;

• use matrices to solve linear systems;

• demonstrate technical facility with algebraic transformations, including techniques based on the theory of equations.

Computing technology enables schools to provide a richer set of algebra experiences for all students. Polynomial equations, which are very useful for describing relations among variables in a vast array of real-world situations, need no longer be a topic reserved for precalculus students.

• 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;

• understand operations on, and the general properties and behavior of, classes of functions.

STANDARD 9: TRIGONOMETRY

• apply trigonometry to problem situations involving triangles;

• explore periodic real-world phenomena using the sine and cosine functions;

• 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.

• 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;

• transform data to aid in data interpretation and prediction;

• test hypotheses using appropriate statistics.

Focus

Discussion

STANDARD 12: DISCRETE MATHEMATICS

• 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.

Focus

STANDARD 13: CONCEPTUAL UNDERPINNINGS OF CALCULUS

• 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;

• 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.

Focus

TOPICS TO RECEIVE INCREASED ATTENTION

ALGEBRA