Mathematics Grades 9-12

Philosophy

In its July 31, 1992, "Mathematics Policy," the Tennessee State Board of Education states: "All students must have access to a rich curriculum emphasizing mathematical thinking and problem solving in order to ensure a mathematically literate work force and to promote equal opportunity for all citizens." The document sets forth the following goals for all students: "that they (1) learn to value mathematics, (2) become confident in their ability to do mathematics, (3) become mathematical problems solvers, (4) learn to communicate mathematically, and (5) learn to reason mathematically."

The following year the Tennessee State Board of Education articulated the "High School Policy: A New Vision for Tennessee High Schools." (September 17, 1993) Based upon current educational research, these recommendations for the mathematics curriculum were made:

Students will:

1. Read, write, and orally communicate mathematical concepts.
2. Use various methods, including mental math, estimating,
3. Organize, analyze, depict, and interpret data to make decisions and predictions related to real-world situations.
4. Use appropriate tools, such as measuring instruments, calculators, and computers, to solve problems.
5. Solve theoretical and practical problems using essential concepts of algebra, geometry, probability, and statistics.
6. Understand the relationship between mathematics, the sciences, technology, and society ("High School Policy: A New Vision for Tennessee High Schools"; 1993).

This committee has strived to outline a Secondary Mathematics Curriculum Framework that is in accordance with these. This Framework outlines several course sequences which set high, but not unreasonable, expectations that are designed to help prepare all students for citizenship, for work, and for higher education.

In the past, pencil-and-paper algorithms have been the primary tools of mathematics that were taught in school. Such a focus is inadequate for today's world. Because calculators and computers are widely accessible, the nature of the problems of mathematics and the methods used to investigate them have changed tremendously. Indeed, technology is pervasive in government, industry, and business. Students now need to know how to use a variety of tools, including technology and mental calculations as well as pencil and paper. The proposed Framework strongly supports the implementation of technology throughout the mathematics curriculum.

The organization of this Framework is a two-level hierarchy of expectations, all of which are intended to advance the goals previously stated. The first level of expectations consists of the four Process Standards: (1) problem solving, (2) reasoning, (3) communication, and (4) connections. The second level is made up of the five Content Standards. The four Process Standards permeate the Content Standards; that is, we acknowledge that teaching any particular content well involves creating an environment in which students are engaged in significant problem solving, in sense-making, in mathematical discourse, and in recognizing inter-relationships.

The issue of equity, particularly in mathematics education, is crucial. Mathematics is a filter for employment. That is, lack of mathematical background limits job opportunities. Therefore, it is vital that all students, especially females and minorities who have traditionally been underrepresented in mathematics-intensive fields, be strongly supported in mathematics education. In order to promote equity, this Framework is designed so that all students who meet the three mathematics credit graduation requirement will have studied these five content standards: Number Sense and Number Theory; Estimation, Measurement, and Computation; Patterns, Functions, and Algebraic Thinking; Statistics and Probability; and Spatial Sense and Geometric Concepts.

As suggested by the Tennessee State Board of Education "Mathematics Policy" (1992), the Framework includes the option of a three-year integrated curriculum. An integrated curriculum would allow opportunities for learning that reflect the connections among the various strands of mathematics. For instance, students could experience the usefulness of taking a geometric approach to a probability problem. It also provides for students to visit the same mathematical concepts from several perspectives. However, moving to such a curriculum will require time for professional development, for exchange of information, and for students to complete currently offered courses of study they have already begun. Courses in the traditional sequence also include an increased emphasis in the four process standards. School districts should carefully weigh the merits of each option and offer that which they think is most effective for their students.

Tennessee joins other states in basing the State Curriculum Framework on systemic reform. In fact, our task was made less daunting by the prior work of committees in other states, such as Delaware, Georgia, and Virginia, has made. It is the belief of this committee that implementation of the recommendations stated in this document will enable Tennessee to improve the mathematics education of its students. This committee understands that the changes delineated here are not merely adjustments of the former curriculum. Instead, this document represents systemic change in mathematics education that will require extensive staff development and necessitate new forms of assessment that reflect the emphasis on higher order thinking.

History

The Basic Skills First curriculum initiatives that began in the late 1970's involved elementary and middle school teachers and mathematics curriculum specialists in identifying by grade level, K-8, the skills and concepts that should be taught. This effort resulted in the first statewide mathematics curriculum document which, with later revisions, became the Blue Book. Key skills at the 6th-8th grade level were identified as minimum proficiency skills and policy was adopted which required students to pass a test based on these skills in order to receive a high school diploma. The list of proficiency skills has been revised several times since the late 1970's. In 1993, another revision occurred which resulted in the TCAP Competency Test. The Competency Test is based on 8th grade skills as identified in the state framework. Passing it remains a graduation requirement in order for students to receive a high school diploma.

In 1984 the Comprehensive Education Reform Act (CERA) was passed by the Tennessee Legislature with the stipulation that within five years instructional programs should be improved in a measurable way. Mathematics, as described in Academic Preparation for College: What Students Need to Know and Be Able to Do (the Green Book), was included as a subject to show measurable improvement. In 1986 the Tennessee State-Wide School-College Collaborative for Education Excellence (the Collaborative) was organized and included several Task Forces, including one for Mathematics. The Mathematics Task Force looked at the existing State Mathematics curriculum documents and produced the two documents, Mathematics Framework (K-8) and Mathematics Curriculum Guide (9-12). These documents were correlated to the Green Book as required by CERA.

In 1989 the NCTM Curriculum and Evaluation Standards for School Mathematics was published. A cadre of mathematics educators from across the state, including K-16 representation, was assembled to receive training for leaders offered by NCTM and to then conduct awareness workshops across the state regarding the Standards. In addition, this cadre reviewed the K-8 and 9-12 frameworks in light of the Standards and made some revisions of those documents in 1991. Also in 1991 Math for Technology I was added to the state curriculum for 9-12 and the Professional Standards for Teaching Mathematics was published by NCTM.

In 1992 the Tennessee Legislature passed the Education Improvement Act based on the 21st Century Challenge Plan of 1990. As a result, several initiatives began. The Tennessee Comprehensive Curriculum Guide, Grades K-8 was assembled. It included skills and concepts for Language Arts, Mathematics, Science, and Social Studies for these grades in one document that became known as the Blue Book. Also, the State Board of Education adopted a Mathematics Policy endorsing the NCTM Curriculum and Evaluation Standards for School Mathematics and several mathematics professional groups and individuals across the state began efforts to review the existing frameworks and guides in light of the Standards. One of these groups was formed by the Systemic Initiative Steering Committee and received funding from the Eisenhower Math/Science Consortium at Appalachia Educational Laboratory to prepare a companion document to the frameworks that would connect these two frameworks with the NCTM standards documents. The document, Mathematics for All Tennessee Students, was written in 1994 with the goal of making the frameworks more useful to teachers and more compatible with the Standards.

In 1993 Math for Technology II was approved as a high school course and was certified by governing bodies as meeting the Algebra I admission requirement for state universities.

In 1994 the high school policy requiring the graduation requirement of 3 years of mathematics including Algebra I or the equivalent, Math for Technology II, was implemented.

In 1995 preparation for instituting End of Course Assessments in high school mathematics courses began. Task forces of teachers from across the state were convened to write a bank of items in all the existing mathematics courses included in the existing curriculum framework. The items were submitted to CTB/McGraw Hill to be used to construct subject area tests for Pre-Algebra, Math for Technology I, Algebra I, Geometry, and Algebra II. Preparation for tests for other mathematics courses was delayed.

In the spring of 1996 a team of mathematics teachers and curriculum specialists representing K-16 and the entire state was convened to rewrite the K-8 state frameworks based on the Standards and current practice. The revision was then reviewed by mathematics educators statewide. This effort resulted in the production of the K-8 Mathematics Framework which identifies the Process Standards, Content Standards, and Learning Expectations that should guide state school systems in making the changes needed to improve the learning of mathematics by students across the state.

Building upon the expectations of the K-8 Mathematics Framework, a team representing high school and university mathematics educators began revising the Secondary Mathematics Framework in February of 1997.

Process Standards

Mathematics as Problem Solving

The study of mathematics must emphasize Problem Solving opportunities which require various approaches to investigate, understand, and apply mathematical concepts.

The development of each learner's ability to solve problems is essential if he or she is to be a productive citizen. We strongly endorse the first recommendation of An Agenda for Action (NCTM, 1980): "Problem solving must be the focus of school mathematics." To develop such abilities, students need to work on problems that may take hours, days, and even weeks to solve. Some may be relatively simple exercises to be accomplished independently; some should involve small groups or an entire class working cooperatively; and some problems should also be open-ended with no single right answer.

"Mathematics as Problem Solving" emphasizes the learners' use of a broad base of strategies to:

• Investigate and understand mathematical content
• Recognize and formulate problems from within and outside of mathematics
• Use mathematical modeling and appropriate technology to solve a wide variety of problems, including real-world problems.
• Generalize solutions and strategies, applying them to new problems.
• Increase confidence in their ability to use mathematics meaningfully and to become independent problem solvers.

Mathematics as Communication

The study of mathematics must emphasize Communication by requiring opportunities to explain, conjecture, summarize, and defend one's ideas orally, in writing, and through the use of technology.

The development of a learner's power to think mathematically involves learning the signs, symbols, and terms of mathematics. This is best accomplished in problem situations in which students have an opportunity to read, write, and discuss ideas in which the use of the language of mathematics becomes natural. As students communicate their ideas, they learn to clarify, refine, and consolidate their answers.

"Mathematics as Communication" focuses on the learners' development of using language and symbols to:

• Reflect and clarify thinking about mathematical ideas and situations.
• Express mathematical ideas and relationships, orally, in writing, and with physical material, pictures, and diagrams
• Understand and value the role of mathematical notation.
• Realize that representing, discussing, listening, writing, and reading mathematics are vital aspect of mathematics study and use.
• Use mathematical notation to formulate generalizations.

Mathematics as Reasoning

The study of mathematics must emphasize Reasoning which requires critical thinking, logical argument, and justification of solutions, of thought processes, and of conjectures.

Making conjectures, gathering evidence, and building an argument to support such notions are fundamental to doing mathematics. In fact, a demonstration of good reasoning should be rewarded even more than the learner's ability to find correct answers.

"Mathematics as Reasoning" concentrates on leading the learners to:

• Make and test mathematical conjectures.
• Make, follow, and judge the value of mathematical arguments
• Draw logical conclusions.
• Justify solution-finding processes and answers.

Mathematical Connections

The study of mathematics must emphasize making Connections among the various topics within mathematics, between mathematics and other disciplines, and between mathematics and "real world" situations.

The mathematics curriculum is often viewed as consisting of several discrete stands; so topics tend to be taught in isolation. Unless the learners connect ideas both among and between areas of mathematics, they learn isolated skills rather than develop the ability to recognize general principles and procedures relevant to several areas. Connecting conceptual understanding to procedures will enable learners to apply, recreate, and invent new procedures when needed. Failure to connect conceptual understanding to procedures results in a view of mathematics as an arbitrary set of rules. Learners should have many opportunities to observe and work with the interaction of mathematics with other subjects and with everyday society. Problems become meaningful when they relate to the learners' experiences. Mathematics must be integrated into contexts that give its symbols and processes practical meaning. The school environment is rich with opportunities to use mathematics in other subject areas as well as other subject area content in mathematics.

"Mathematical Connections" concentrate on enabling the learners to:

• Appreciate mathematics as an integrated whole, linking conceptual and procedural knowledge within the discipline and relating multiple representations of concepts or procedures to one another.
• Apply mathematical thinking and modeling to solve substantial problems that arise in other disciplines and curriculum areas, such as art, business, music, psychology, industrial arts, computer technology, social studies, and sciences, such as biology, chemistry, and physics.
• Use, recognize, and value the varied roles of mathematics in their lives, cultures, and society.

9-12 Mathematics Course Listing

Vocational courses which receive mathematics credit are listed below. The frameworks for these courses may be found in the State Department of Education's Technology Preparation Framework.

Advanced placement mathematics courses are listed below. Descriptions for these courses may be found in the Advanced Placement Course Descriptions provided by The College Board.

Traditional Mathematics Course Sequence

Students may receive mathematics credit for only ONE of the following: Algebra I, Mathematics for Technology II, or Integrated Mathematics I. It is strongly recommended that students who begin in the Traditional Program (Alg I, Geom, Alg II) not move to the Integrated program. Transfer students may be an exception to this recommendation.

Students may receive mathematics credit for only ONE of the following: Algebra I, Mathematics for Technology II, or Integrated Mathematics I. It is strongly recommended that students who begin in the Integrated Program (Integrated Mathematics I, II, & III) not move to the Traditional Program. Transfer students may be an exception to this recommendation.