"Great Expectations" Mathematics Grant

Recommendations for a Successful First Year of Post-Secondary Mathematics

I. Introduction

II. Literature Review

III. Statement of the Problem

Developing the Report
Philosophy of the Report
Relationship to South Carolina Mathematics Framework
Organization of the Report

IV. Definition of Success in Post-Secondary Mathematics Learning

Definition of Success
Using the Definition
Implications of the Definition

V. Content Recommendations

Success and Content
Specific Content Concerns

General Concerns

Actions to Implement

VI. Technology Recommendations

Reasons for Using Technology
Implementation of Technology in Teaching
Best and Worst Practices in Using Technology
Evolution of Teaching Mathematics with Technology

VII. Teaching Recommendations

Success and Teaching
Specific Teaching Concerns

Differences between high school and post-secondary courses
High school preparation and transfer of information
Best and worst practices of individual instructors/educational institutions

Actions to Implement

Reducing impact of differences between high school and post-secondary courses
High school preparation and transfer of information
Best and worst practices of individual instructors/educational institutions

VIII. External Factors Impacting Student Learning

Success and External Factors
Specific Factors of Concern
Actions to Implement

IX. Teacher Preparation

Pre-Service Training
In-Service Training

X. Raising the Standards

XI. Appendices

Algebra Skills Recommendations
Trigonometry Skills Recommendations
Geometry Skills Recommendations
Summaries of Recommended Readings

I. Introduction

Grant specifications

In 1996 the South Carolina General Assembly passed Act 359, which established goals for higher education in the state. A critical component of this process was to establish the ability of the South Carolina educational community to prepare its students for success in subsequent levels of academic study. For this purpose, the Commission on Higher Education established the "Great Expectations" Project in five areas, one of which was mathematics. In the context of mathematics, the primary activities sponsored by this project are 1) the definition of the knowledge, skills, and abilities that students should possess in order to be ready for college in the area of mathematics and 2) the dissemination of this information to the State Board of Education and the State Department of Education, the Commission on Higher Education, parents, and especially to high school teachers in South Carolina.

The information in this report was formed through the diligent work of a committee composed of members from all parts of secondary and post-secondary mathematics education along with representatives from the business community. The report itself provides the major conduit for the information to reach the teachers and parents of prospective post-secondary students and provides the recommendations to the various state agencies. The primary focus of this report is to delineate the results of the comprehensive study of the knowledge, skills, and abilities high school students need to succeed as freshmen in post-secondary institutions; this includes a definition of "minimum academic expectations for prospective post-secondary students". In addition, this report provides suggestions for the inclusion of the recommendations of the report into professional development activities for pre-service and in-service training in mathematics.

The final outcome of the grant will be to make copies of this report available to students, parents, teachers, guidance counselors, instructors of mathematics, the State Board of Education, the State Department of Education, and the Commission on Higher Education. In order to be more helpful, the report has retained its focus on implementation rather than philosophy. Since this emphasis on functionality prevails, it is the belief of the committee that the details of the recommendations will be given the utmost consideration and first priority for implementation within the various constituencies of the educational community. The success of our South Carolina students depends on the utmost attention to their needs.

II. Literature Review

The discussion of the effectiveness of mathematics education in the United States has a long and varied history. Given recent performances by American elementary and secondary students on national and international tests, the concern is strong and the debates are heated at this time. Adding to the process are the "new" techniques, styles of teaching, and logistics being explored in the classroom – for example, group work, various technologies, inclusion of statistics in the curricula, and block scheduling. For a deeper understanding of the state, national, and international context of this report, the committee recommends the examination of the following documents:

1. Everybody Counts
2. Challenge of Numbers
3. Reshaping School Mathematics
4. State of Mathematics Achievement
5. TIMMS
6. South Carolina Mathematics Framework, Columbia, South Carolina: South Carolina Department of Education, 1993.
7. Precollege Preparation for College Mathematics: A Survey of South Carolina Faculty, by J. Christopher Tisdale, III, Danny W. Turner, and Gary T. Brooks, Department of Mathematics, Winthrop University, January, 1998.
8. What Matters In College? , by Alexander W. Astin, San Francisco: Jossey-Bass, 1993; and What Matters in College?: Four Critical Years Revisited, by Alexander W. Astin, San Francisco: Jossey-Bass Publishers, 1997.
9. Counting on You: Actions Supporting Mathematics Teaching Standards, Mathematics Sciences Education Board, Washington, D.C.: National Academy Press, 1991.

These documents add perspective to this report and represent the national understanding of the issues in mathematics education as well as issues within the state. Summaries of a selection of these works are found in the appendices of this report. Most of them can be found in any university library.

III. Statement of the Problem

Specific Details of the Process

Broadly stated, the problem addressed was the sometimes difficult transition of the secondary student into the post-secondary educational system in mathematics. In order to make this transition more orderly and more successful, the Commission on Higher Education placed the problem before a committee composed of experienced educators and representatives from the business community. The final outcome is to be recommendations for improving the first year college mathematics experience of South Carolina students; these recommendations are to be widely distributed throughout the state.

Developing the Report

The committee of mathematics educators and representatives from the business community contributing to this report included representatives from high school, two- and four-year colleges, technical colleges, and comprehensive and research universities. The membership of the committee is as follows:

Mary B. Martin, Chair, Winthrop University

Roger Allen, Francis Marion University

Eddie Brown, Burkett CPAs

Charles Cleaver, The Citadel

Lin Dearing, Clemson University

Ron Goolsby, Winthrop University

Hugh Haynsworth, College of Charleston

John Long, Midlands Technical College

Mary Ellen O’Leary, University of S. Carolina, Columbia

Julia Robbins, Rock Hill School District III

Suzie Schembri, Northwestern High School, Rock Hill

Wade Sherard, Furman University

Chris Tisdale, Winthrop University

Jane Upshaw, American Management Association

Keith Wilks, Rock Hill High School, Rock Hill

The committee met at various times, each time addressing different aspects of the report. Between meetings, the results from previous meetings were compiled and reviewed with additional topics and adjustments proposed for future meetings. This report, along with summaries compiled for parents and students, is being disseminated widely throughout the educational community in South Carolina. There is also a version of this report posted on the "Web" at http://www.winthrop.edu/mathsuccess and is thus readily available to students, parents, guidance counselors, and teachers.

Philosophy of the Report

The first priority of the committee was to determine the audience and the parameters of the report. The obvious segment of the post-secondary population to consider is the traditional 17-18 year old, full-time college freshmen; this is the most homogeneous portion of the population. Additional portions of the population include part-time students, older students and students having special learning disabilities. Since mathematics is such a carefully structured system of information, most of the recommendations within this report will apply equally to the different segments of the post-secondary education population. Accordingly, unless otherwise noted, the report makes recommendations which should be applied to students entering technical colleges, two and four year colleges, and comprehensive and research universities. In the case where additional specific recommendations can be made for a specific subgroup, these have been noted. In most cases, these special subgroups have needs which apply to all of their educational experiences and are not specific to their mathematics experience. It is the scope of other reports and other information sources to provide general educational recommendations in these instances.

This report includes within the definition of post-secondary education attendance at technical colleges, two-year or four-year colleges, comprehensive universities, and research universities. The experiences at a technical college and at a research university are not identical, nor should they be. On the other hand, the mathematics learned at a technical college – which is of a collegiate level – generates transfer credit; therefore, the transferability to collegiate course credit determines a natural division between secondary and post-secondary mathematical material. Generally speaking, material which will not transfer as college credit is considered secondary material even though it may be taught in a post-secondary setting. (There are of course exceptions to this rule.) There are two points in the report where this distinction is important.

First, within the definition of success it is assumed that the goal for the student is to be prepared for post-secondary mathematics at the chosen post-secondary institution. For example, if a student chooses to go to a technical college, the student should still be prepared to take a course not offered in the high school curriculum. While the opportunity to repeat high school material at the post-secondary level needs to be available, the student who actually does this is not considered to have made a successful transition to post-secondary mathematics within the context of this report. Nevertheless, this student can benefit from the recommendations within this report as can the educational community responsible for meeting the needs of this type of student.

Second, the section recommending specific content for courses depends on a definition of where post-secondary mathematics begins. The consensus of the committee as it represents the mathematical community is the basis for these recommendations; it includes the assumption (a nationally valid assumption) that true post-secondary mathematics is post-algebra mathematics.

In the context of this report, post-secondary includes two- and four-year colleges, technical colleges and comprehensive and research universities. In most instances, "college" will be used instead of "post-secondary" in order to facilitate clarity. The few cases where the distinction between college/university learning and technical or two-year colleges needs to be made are clearly identified as such.

Throughout the process, the goal of the committee has been to construct a document that emphasizes implementation. While this report may be "non-philosophical", it is a highly focused document. The importance of this report, and its potential impact, will rely entirely upon how many and which of the recommendations are implemented. Therefore, the emphasis and focus of this report is the development of implementable actions which will address the transition from high school to college mathematics. Nevertheless, for the record, it is the committee’s assumption that the educational community as a whole ascribes to the following philosophies:

• Mathematics is culturally valuable.
• Mathematics is important in its own right as a discipline.
• Mathematics is a critical tool of the sciences.
• Mathematics can be exciting, fascinating and enjoyable to teachers and students alike.
• An understanding of mathematics is part of being an educated human being.
• It is important to instill an appreciation of mathematics into our students.
• At this time, society does not appreciate nor value teaching and teachers, yet educators have to work within this context.

These philosophies, and their explication, have been eloquently stated in many of the reports referenced in the Literature Review. Consequently, this report will not philosophize further regarding the nature of mathematics, its beauty, its importance, nor its relevance. We will simply repeat that for this report to succeed in fostering change, the members of the mathematics community must stand behind as many of these recommendations as possible and lobby strongly anywhere and everywhere to attain the needed changes.

Relationship to South Carolina Mathematics Framework

The purpose of this report is to provide specific recommendations for implementation in order to improve the transition from secondary mathematics to college mathematics in South Carolina. As such, the report is not meant to replace other pedagogical or curricular documents nor other reports documenting the state of mathematics education. In particular, the recommendations of the South Carolina Framework for Mathematics still are critical to the development of mathematics education in the state. The South Carolina Framework for Mathematics presents curricular goals for high school graduates in South Carolina, whether or not they will attend college. The task of this report is to refine the achievement levels in order to ensure success in college mathematics. In both instances, one necessary condition to improve mathematics education within the state is to undertake a commitment Chapter 7 of the Framework and the principles expressed there.

Organization of the Report

The report naturally divides into three groupings – a definition of success in post-secondary mathematics, three sections of specific concerns and recommended actions dealing with the effective teaching and learning of mathematics, and a section recommending an interface between this report and the training of teachers. Different constituencies will be reading this report for different purposes. Certainly, all the sections can be useful; however, each section will appeal strongly to a particular subgroup of persons. Therefore, the committee recommends the following:

• Students will most benefit from reading Sections IV, V, and VII.
• Parents will most benefit from reading Sections II, IV, VI, VII, and IX.
• Teachers will most benefit from reading Sections I-IX.
• Guidance counselors will most benefit from reading Sections II, IV, VI and VII.
• Administrators will most benefit from reading Sections III, IV, and VI - IX.

Anyone who has time to read the entire report is encouraged to do so; except for certain sections of material related to content recommendations (Section V), the recommendations here do not require extensive mathematical background.

IV. Definition of Success in Post-Secondary Mathematics Learning

Contextual Definitions of Success

Any definition of student success in mathematics must be meaningful in the context of the individual student’s needs and goals. This implies that the definition must apply to any student, any major, and any choice of institution of higher learning. The overwhelming consideration is that mathematical skill, or the lack thereof, should not be the limiting factor in choice of major or career. For these reasons, we have established the following definition for a successful first year of post-secondary mathematics education.

Definition of Success

A student is successful in mathematics in the first year of college education if the following three conditions are met:

1. The student is prepared for the first college mathematics required by the student’s choice of major or interest.
2. The student completes each mathematics course attempted during the freshman year with a grade of C or better.
3. The student is able to transfer the mathematical knowledge gained in the mathematics course(s) into other courses, particularly into subsequent courses required by the major.

These conditions formulate a minimal definition of success in first-year college mathematics. It should be remembered that minimal success as a freshman does not automatically guarantee on-going success in the remainder of the post-secondary education.

Using the Definition

This definition of success deals simultaneously with background content, the ability to function at the collegiate level, and the ability to transfer knowledge within the context of the career choices of the student. To be useful, the definition must be applied in each of these three areas. For this reason, we must keep the following information in mind when making choices about mathematics courses and study.

• Calculus preparation makes available the widest choice of career possibilities. That is, there are more options open if you are prepared for the study of calculus. Most of the science majors and substantive business options are closed to students with lesser preparation.
• The traditional four courses of high school mathematics — algebra I and II, geometry, and pre-calculus — are the minimal course work necessary to prepare a student for calculus and are likewise known to be the best predictors of future success in post-secondary mathematics.
• The student who must take pre-calculus in college, after having taken it in high school, is not considered successful under this definition. That is not to say that the needs of this student should not be met; rather, the system and/or the student has failed prior to the start of the first year of college mathematics if pre-calculus material must be repeated. If the student was not able to take pre-calculus in high school, then successful completion of the pre-calculus course as a post-secondary course along with the completion of another relevant mathematics course could be considered a successful first year of mathematics.
• The factors impinging upon the completion of a college mathematics course include study skills, test-taking skills, and other external factors. These factors cannot be ignored prior to college nor during college.
• Non-mathematics courses which use mathematics, such as economics, physics, psychology, etc., require the student use mathematics outside the context of a particular mathematics text. This is a higher level of mathematics understanding which needs to be encouraged in high school curricula and continued in college curricula.

Non-traditional students and students who "get off-track" can re-enter the mathematics coursework at the appropriate level for them and continue in the correct sequence of courses -- adjusting this "definition of success" to meet their status and circumstances. On the other hand, the educational community has as its mission the success of as many of its students as possible. Therefore, the educational community needs to take this definition of success very seriously and work towards keeping as many students as possible, especially traditional students, within the boundaries of a successful transition from secondary to college mathematics.

The overwhelming use for this definition of success and likewise for this report is to provide the participants in the educational process with the best understanding of the factors for making successful transitions between secondary and college learning in mathematics. In all cases, the goal is to make available current thinking on the standards for mathematics education in South Carolina and to make students, parents, and teachers aware of the consequences of individual educational decisions. These standards for success provide a basis for the improvement of mathematics education in the state.

Implications of the Definition

The cumulative nature of science in general, and mathematics in particular, makes it necessary to plan ahead when making course decisions; it also stretches out the consequences of poor decisions into the future.

• The decisions made in early middle school regarding pre-algebra can determine whether or not a student can take calculus in high school which in turn influences which university admissions criteria will be met. It is possible to change to a stronger option later, but this involves extra time and extraordinary pressures.
• The study skills and test-taking skills necessary to succeed in college courses are developed through high school experiences.
• Businesses today are looking for two major attributes in employees: the ability to learn new methods and processes and the ability to work with others. These attributes require that a person be able to go through the learning process independently and at the same time communicate information effectively by transferring information into a new setting.
• The ability to learn new methods and processes and the ability to work with others are improved by new pedagogical methods and can also improve the understanding of the basic skills of mathematics; in no case do they replace the goal of a mathematics course – that students be able to individually perform mathematical processes.
• Although it is preferred to state benchmarks in positive terms, it is sometimes helpful to give examples of negative criteria. Individually, a student is not successful if the student believes that the mathematics taken was not and will not be used in the career or major area of study. A student is not successful if his/her graduation is imperiled by a senior year grade in a general education course in mathematics. In these cases, however, it necessary to evaluate on a case by case basis to determine if this is a failure of the education community or a failure on the part of the student.

The next sections of this report refer to recommendations for background/content in mathematics and teaching practices as well as recommendations for addressing external factors which impact learning. Combined, these sections represent the specific recommendations for implementing the definition of success in first-year college mathematics. Recommended strategies include actions which encourage the following:

• Advising the student early regarding how to keep open the most choices for areas of study and for careers.
• Emphasis on the conceptual without compromising skills.
• Activities which encourage the transfer of material between processes, classes, or areas of study.

Once again, it should be remembered that these recommendations are for a definition which is the minimal definition of success in first-year college mathematics. Minimal success as a freshman does not automatically guarantee on-going success in the remainder of the college career.

V. Content Recommendations

Specific Knowledge Requirements

Success and Content

If one reviews the definition of success, it is clear that each portion of the definition revolves critically around the assumption that content is being conveyed and evaluated in the classroom and then used in that and other classrooms. As has been so often said, mathematics is not a spectator sport. Each person in the process – each instructor and each student – must be an active participant in doing mathematics. Therefore the whole process centers on the content being taught, the selection of topics, the order in which they are presented, the pedagogical techniques used to teach, and the evaluation of the transfer of the information.

In recent times, there has been a tacit deemphasis on skill in favor of an emphasis on group learning, technology, and problem solving. While no one denies the need for students to understand problem solving, technology, and how to work together, it is equally clear that no one can claim that businesses want to hire people who have insufficient technical mathematics skills. While maintaining an appropriate amount of technology and group activities, it must be accepted that specific skills and conceptual understanding of the underlying principles cannot be sacrificed to pedagogical technique, assessment, technological issues, or other desirable aspects.

The connections between mathematics and career choices are extensive. If a science, computer science, mathematics, or engineering major is the student’s choice, then the following broad capabilities must be attained before reaching the college level:

• Strong algebra and trigonometric skills
• Conceptual understanding of algebra, analytic geometry, and trigonometry
• Ability to translate English sentences into mathematical notation
• Graphing by hand, graphing with technology, and the interpretation of graphs
• Reasoning skills and the application of logic to understanding mathematics
• Basic technological skills -- calculators primarily
• Exposure to "real-life" type numerical examples

These broadly defined skills parallel the skills recommended in the South Carolina Framework for Mathematics, but are extended to provide the depth necessary for success in freshman calculus. Once again, students, parents, teachers, and guidance counselors need to understand that not preparing for calculus does in fact close the door on a significant number of college majors and therefore narrows choices of careers.

If a "non-science" major is the student’s choice, that is a non-calculus based major, then the following broad capabilities must be attained before reaching the post-secondary level:

• Strong arithmetic and algebra skills
• Ability to translate English sentences into mathematical notation
• Reasoning skills and the application of logic to understanding mathematics
• Basic technological skills -- calculators primarily
• Exposure to "real-life" type numerical examples

Ultimately, it is important to remember that while specific content and skills are necessary and increases the probability of success, they cannot replace a conceptual understanding of the mathematics principles underlying the various skills and techniques.

Specific Content Concerns

The content /mathematical abilities needed to succeed in the first year of college mathematics education should be considered for calculus-based majors and non-calculus-based majors separately. Once again, it should be understood that preparation for calculus is a better preparation for any major and therefore provides the student with the most choices.

General Concerns

There are characteristics of mathematical understanding which are found in all levels of mathematics. These characteristics are generally clustered around overall understanding of the concepts and the analysis of mathematical processes. They all address the ability to make mathematics useful in a variety of settings. Points of specific concern are:

• Approximation skills -- primarily knowing when the answer is reasonable – should be taught in a meaningful fashion rather than in a contrived manner. Estimation and an understanding of its value are important skills.
• "Monkey mathematics" – where the instructor works two or three problems and then the students work ten more – is far too prevalent. While practice is necessary and cannot be replaced, the student must be encouraged to develop a conceptual understanding of the purpose of all the practice.
• Students must be able to read and use a formula. Formulas must be understood as more than a formal process without meaning. Instead, functions and formulas should be understood as inducing a description, a model, an algorithm or a process.
• The ability to compare – numbers, functions, and processes – is central to mathematics. For example, the ability to solve inequalities – and to realize that this is more than just a manipulation process – is needs to be more fully developed.
• The ability to see a process as useful in a variety of settings has historically been an impetus for the development of mathematics. This is what gives mathematics its power –all students eventually ask "What is this good for?" – and yet also lends to the difficulty of teaching mathematics. Courses should include the flavor of the historical development of concepts and their subsequent uses. This will lead students to a greater appreciation of the scope of mathematics.
• Mathematics instructors need to emphasize the connections between mathematical concepts and between mathematics and other disciplines.
• Minimal facility in current material almost guarantees failure in future material. For comparison purposes, if you can barely read for comprehension for yourself, you will fail miserably at reading a passage out loud for a presentation. The next step always assumes an ease with previous steps. Students, parents, and instructors all need to understand that mastering material is the only acceptable option in mathematics. If you must write out each algebraic step in detail in every problem (for comfort or for necessity), that is a sure predictor of failure in a calculus class.
  

  Summarizing, the basic areas of expertise needed for calculus preparation are arithmetic, algebra, trigonometry, geometry, use of technology, ability to work with definitions and abstract properties. These skills position a student to succeed in college mathematics regardless of the choice of major.

Actions to Implement

Within the appendices, there are course-specific recommendations for material and competencies to include in secondary mathematics. These are technical (and therefore can be found in the appendices) and can be very useful to teachers and students. In addition, there are some non-course specific content recommendations that stand out in their applicability to the general instruction of mathematics. Of generally equal importance, the following recommendations encourage the development of appropriate mathematical thinking:

1. All instructors should use technology to support the development of the concepts of their courses.
2. Preparing for and taking comprehensive exams should be part of every mathematics course; this properly prepares a student for taking exams at the college level. Allowing exemptions from exams does not prepare students for college classes.
3. Courses should be ordered carefully in order to encourage continuing development of concepts and the maintenance of previously mastered skills. In particular, it is especially important to place geometry later in the sequence of secondary mathematics courses and to have Algebra II taken immediately after Algebra I. This material needs to be fully addressed before adding statistics or other material to the curriculum.
4. Testing needs to support the teaching process in addition to measuring performance; we are not teaching in order to test but rather to encourage learning. Consequently, tests should not lie at either extreme: containing solely straight copies of homework material or containing new material only tangentially or basically connected to material presented in class.
5. Exercises which require repetitive, boring, undirected calculations in the name of "discovery learning" are inappropriate in most cases. They consume incredible amounts of time and students rarely "get the point". "Discovery learning" requires careful construction to be successful.
6. Each course should be designed to specifically and consciously use units to help clarify the problem and to help the students understand the value of mathematics in widely differing contexts.

For more course-specific recommendations concerning mathematical content, the reader is referred to the appropriate appendix.

VI. Technology Recommendations

Technology and the Learning of Mathematics

Professional teachers and mathematicians often differ about the exact extent of the use of technology in the teaching of mathematics. Included in the debate are whether or not to use it, when to use it, which technology to use, how to teach with it, and at which levels to use technology. Furthermore, technology is leading to developments in the teaching of mathematics which are still "works in progress" and are not ready to be fully evaluated. Nevertheless, there is a consensus in a large portion of the mathematical community regarding the use of technology and its position in mathematics education.

Reasons for Using Technology

In general, technology includes the various levels of available technology: calculators, CBL’s, computers, and laptops. The particular technology changes depending on resources and the level of coursework. There are a few guiding principles – like computers should not be used as fancy calculators; primarily, the determination of the technology depends upon the material being taught and the resources available. The strongest reason to use technology, of any sort, occurs upon those occasions when it can drive the introduction/discovery of new material.

The experience of mathematics educators lead us to believe that technology helps promote the following:

• Understanding the concepts of function and functional behavior;
• Support of the development of algebraic concepts;
• Visualization of solutions and functions;
• Understanding estimation and its applications;
• Development of intuition and pattern recognition;
• Confirmation of algebraic solutions
• Understanding technology’s failure to solve certain problems;
• Use of and appreciation of scientific notation;
• Evolution of a vision of mathematics as a laboratory science, which allows student group projects, written reports, and an interdisciplinary understanding of mathematics and other areas of study;
• Consideration of more realistic problems;
• Attention to applications even when complicated algebraic computations are involved
• Visual understanding of mathematics in general and functions in particular;
• Improvement in communication skills;
• Increased attention of students;
• Understanding of the reality of mathematics and the applications of mathematics.
• Use of a broader selection of functions, less meaningless memorization, and more emphasis on intellectual understanding.

Implementation of Technology in Teaching

Ultimately, technology is a tool for teaching. Accordingly, it can be used effectively or inappropriately. An unfortunate trend in mathematics education is that in individual cases, the technology is introduced without an understanding of the time needed to learn to make the process effective. The inclusion of technology into an existing course is the equivalent of starting an entirely new course preparation; nothing stays the same if the implementation is done correctly. The following aspects of including technology in a course need to be carefully considered by instructors and administrators alike:

• Making testing work with technology
• Determining the type of technology as well as the make of technology to be used.
• Determining the amount of technology to buy.
• Considering which topics should be taught using technology, and which should not.
• Adjusting for the change in time expenditure which occurs when technology is introduced.
• Constructing strategies designed to be course-specific.

There are courses where the technology is easily included, such as differential equations and differential calculus; other courses do not make the inclusion as obvious, for example algebra I and abstract algebra. In each case, there is a common body of considerations that can be used as initial guidelines.

Best and Worst Practices in Using Technology

In each course, it is helpful to consider the following actions when designing the introduction of technology:

1. Limits: When considering the goals of the course, include appropriate levels of technology and yet do not force the technology to appear in an "unnatural" or contrived manner. Also, technology has its limits. For example, different "makes" and even different models of technology include different hierarchies in the order of operations and in the handling of fractional exponents (as in graphing ).
2. Core Material: For each course, there is a core of material which may or may not be introduced with technology but which should always be known and understood independent of technology. For example, recognition of basic graphs and the production of basic graphs should be possible without technology. For example, in an algebra class, a student should be able to produce and recognize the graphs of the following functions without the use of technology: and
3. Algorithmic: Although technology is excellent at performing algorithms repeatedly, students should not be given work which induces in them the belief that mathematics is best undertaken with the "brain turned off and the calculator turned on".
4. Keystrokes: Keystrokes or other explanations of how to use a particular technology should be kept at a minimum, especially as the course level increases. Instead, the explanation of the steps in the process should be included.
5. Written Work: The use of technology does not eliminate the need to write mathematics and to communicate it. An emphasis on the communication of the construction of the answer is vital. Consequently, "handwork" must be included in all problems. This means the inclusion of the mathematical sense of what the student did to solve the problem and why; normally, it does not include lists of keystrokes.
6. Holistic approach: For technology use to be successful and to ensure a more successful transition to post-secondary mathematics work, the presentation of mathematical work, on the part of the student and the teacher, needs to include the use of problem-solving/mental analysis, written and oral communication, and the appropriate level of technology use.
7. A Tool: In all mathematics classes, it must be remembered that the technology is a tool, not the purpose of the course. Mathematical understanding is the sole guiding principle, not teaching to the technology's capabilities.

Examples of topics in precalculus or algebra which benefit greatly from the use of technology include:

• Higher degree polynomials
• Range
• Intersection of curves
• Non-standard angles in trigonometry
• Exponential functions
• Visualization of roots and factoring
• Solutions of inequalities

Other topics do not lend themselves to technology nearly as well. The challenge is to understand the difference. Ultimately, we must understand that there is a strong need for both the traditional and technological skills.

Evolution of Teaching Mathematics with Technology

The use of technology in the teaching of mathematics is an evolutionary event. While there are certain areas which are helped by the use of technology, there are other areas where the pedagogical benefit are not as clear. Layered upon this are the changes in curriculum, logistics and resources brought about by the use of technology in mathematics teaching. Finally and ultimately, we must answer the questions: Does technology devalue mathematics as a discipline? Is it possible that the beauty of mathematics and its logic become less appreciated?

With regard to the pedagogical questions, we must remember that technology must be introduced and developed with care. An interesting observation is that the really good mathematics students use technology the best and at the same time, the least often. If not developed properly, technology can hamper learning instead of enhancing it.

The changes in curriculum, logistics and resources brought about by the use of technology in mathematics teaching require both a more efficient use of resources – time, energy, and money – as well as more resources. Mathematics can now be viewed as a laboratory science with all the concomitant resource expenditures and lab preparations. This requires that teachers share information through in-house workshops and worksheet/activity sharing. At the same time, teachers must be allowed access to more external workshops, support technology persons, subscriptions, software purchase, planning time, released time for curriculum overhaul, and current technologies.

Finally, as a community, students, parents and teachers must continue to realistically evaluate and assess the benefits of technology. It would be a real shame to enable students to be more capable mathematically while at the same time decreasing their interest in and understanding of the beauty of mathematics.

VII. Teaching Recommendations

Best and Worst Practices for High School and Post-Secondary Teaching

Success and Teaching

While the content of the course is the primary focus of the educational process, and the goal of the process is to have students master content, the teacher is the major instrument of the transfer of that knowledge. Although it seems obvious, it is becoming more and more clear that it is necessary to state that teachers of mathematics must be appropriately trained for the level of course they are teaching. Content cannot be taught if it is not mastered by the instructor. Once again, this is a minimal requirement upon the educational community and does not guarantee success; the other pedagogical factors of this section then lead to the successful teaching of mathematics. Each of the aspects of the definition of success – proper background, emphasis on content and learning in the current course, and transfer to other courses and disciplines – impact on the teaching of mathematics.

There are various ways to organize the consideration of the teaching practices and the effects on learning mathematics. This section focuses on separating experiential factors, factors concerning the preparation of students, and factors stemming from the practices of individual instructors/educational institutions. Namely, of concern here are:

• Impediments to learning arising from the differences between the high school and post-secondary environment (experiential factors)
• Importance of a common high school preparation and the transfer of the appropriate content and skills ( preparation factors)
• Best and worst teaching practices of individual instructors/educational institutions

Within this structure, it is important to emphasize the fact that each of the different constituencies in the learning process must be aware of these factors and must address change within each area. It is far too easy for each member of the educational community – students, parents, teachers and administrators – to point fingers at each other rather than recognizing that each member must be fully and appropriately participating in the total process.

Specific Teaching Concerns

Differences between high school and post-secondary courses

Of all disciplines, mathematics has the most structured, continuous flow of concepts and ideas; therefore, it is always amazing that the high school and college classroom experiences can be so different. Primarily, understanding and coping with these differences are up to the student; in most cases, this is appropriate. Due to a variety of reasons, including funding, student maturity, tradition, and market demand, students need to anticipate differences in the college classroom: larger class sizes in introductory/freshman college courses, a pace of presentation that is twice as fast as the high school pace, a higher level of expectations/standards, and a difference in testing patterns. These differences are appropriate to the maturity of the student and the material being presented.

Students are used to having 35 hours a week of structured study and learning time in high school for two semesters; the structure of college, for equivalent work, is one semester of 12 hours a week of structured study and learning time. This means that college faculty, who are preparing material and lectures and grading twice as fast, are expecting students to structure an additional 23 hours per week of studying and learning time on their own. This factor alone is completely missed by students in their planning and is a primary reason for freshman college problems. As a consequence, both inside and outside the classroom, the students are expected to assume significantly more responsibility for their learning. Facts which need to be emphasized repeatedly to students and which deal with these expectations are the following:

1. In post-secondary classes, one can expect larger lecture style classes with a less personal atmosphere.
2. Normally, there are no review days in college classes before or after tests.
3. Comprehensive exams are standard in college and provide a distinctly different challenge from chapter exams. Parents need to understand that comprehensive finals which are fully used in the calculation of grades help prepare the student for college.

High school preparation and transfer of information

Although the high schools deal with a lot of factors over which they have no control in the learning process, they do have some control over the students’ schedules and course content for the time they are in the classroom. Issues which arise in the general education structure and especially impact the learning of mathematics are:

1. Block scheduling: The evidence of the negative impact of block scheduling on the teaching of mathematics is building. In particular,
   
   
• there are fewer minutes devoted to each course (up to 900 fewer minutes),
• the compression of time in which topics are introduced and developed is harmful,
• the amount of daily homework required, and its questionable efficacy, is a heavy volume compared to benefits,
• textbooks are written for 180 days of instruction.
  
  

With less time in class, and that time less effective, teachers report that they are able to teach significantly fewer topics. Algebra I and Advanced Placement calculus are especially disadvantaged by block scheduling, with only 2/3 of the course material being covered in some instances.



2. Timing: Time lapses between mathematics courses are extremely detrimental in high school, in college, and in the transition between the two.
3. Social Passing: Pressure on high school teachers to pass most/all of their students is detrimental in any discipline which requires a continuing progression of ideas.
4. Reading Comprehension: Reading skills inadequate to understand or work with modeling problems hinder the learning of mathematics and the ability to connect mathematics to other disciplines. In particular, it makes "word problems", the analytic and application oriented problems, impossible to handle.
5. Excessive testing: Excessive time spent on testing to address "accountability" or "assessment" issues removes additional time from the learning/teaching process. Most of the testing does not reinforce the current content being studied and therefore does nothing to further the individual student’s learning. This is especially true of tests in a multiple choice format; research shows that multiple choice tests in fact hinder retention and impede affective learning (synthesis and analysis).

Best and worst practices of individual instructors/educational institutions

1. Teacher Preparation: Teachers should be teaching courses which they are trained to teach. Teachers teaching out of their area, even for a short period of time, do irreparable damage to the learning of the students in that class.
2. Teacher Attitude: The teacher must display a positive and enthusiastic attitude towards mathematics and teaching. Additionally, he/she must provide an exceptionally supportive position towards the students.
3. Compressed Courses: Long days of heavily compressed material, such as regularly occurs in Summer School, can impede retention and comprehension of mathematics. Mathematics requires a deliberate pace, with sufficient time to establish a base of knowledge before building upon it.
4. Class size: Large classes, with impersonal environments and an over-dependence on multiple choice testing, hinder learning.
5. Teaching Practices: Wherever you stand on the "traditional-reform" teaching spectrum, there are certain "Worst Practices" which occur all too often due to circumstances or institutional policy. Some of these negative practices can be characterized:
   
   
• Wasting valuable class time (for example, long undirected activity sessions, and spending inordinate amounts of time reviewing previously covered material.)
• Insufficient analysis of hands-on activities to determine student learning;
• Exclusive use of technology to visualize the most basic processes and functions; i.e., some things ought to be remembered;
• Inappropriate use of partial credit: too little discourages students while too much gives students a false sense of the requirements and of their level of knowledge;
• Presentation and explanation of material from a single perspective.

Actions to Implement

Reducing impact of differences between high school and post-secondary courses

1. Comprehensive Exams: Comprehensive exams must be given in high school. This improves high school as well as college performance in mathematics. The committee recommends that a comprehensive exam, with a weight of 25% of the course grade, be given at the end of every high school mathematics course. The following benefits arise from this practice:
   
   
• Enables students to connect and integrate the topics of the course.
• Holds students accountable for all of the material of the course.
• Prepares students for the types of tests they will encounter both in college and in their career.
• Encourages basic study skills: note-taking, good organization, and test-taking skills.
• Encourages an in-depth review of the course – a valuable "second look" in the context of the course as a whole.
• Allows student to see the "big picture" – major concepts and unifying themes.
  
  
2. Office Hours: Students in college need to be encouraged to take advantage of office hours and other resources provided by the post-secondary institution.
3. Study groups: Students need to set up their own study groups and meet with them once a week outside class hours.
4. Independent Review: Students need to design their own review strategies, with the help of the instructor during office hours if necessary, to prepare for tests. It is also the student’s responsibility to learn from graded work individually or in discussion with the instructor outside of class.

High school preparation and transfer of information

1. Block scheduling: School districts must consider creative scheduling solutions that will address these concerns; in particular, AP calculus should be scheduled for a full year. Additionally, appropriate textbooks, when they become available, must be adopted to accommodate the longer sessions with less total time available.
2. Timing of Courses: We recommend that mathematics be taken all four years of high school and that college students take mathematics in their first semester to ensure continuity of learning and the maximum retention of high school material.
3. Course Scheduling: We recommend that the four-year sequence be Algebra I, Algebra II, Geometry and Pre-Calculus, in that order. Furthermore a five-year sequence, ending with AP Calculus, is a tremendous advantage toward post-secondary success with mathematics.
4. False learning: There are standard mistakes made throughout algebra and calculus which are not being corrected. The classic problems arise in the following areas:
   
   
• Lack of estimation skills and a sense of whether an answer is correct or not
• No competence with fractions; can only do arithmetic with decimals on a calculator
• Excessive written calculations which could and should be done mentally
• Insufficient explanation of the process used when writing down a solution
• Canceling terms instead of using factorization
• 
• Solving inequalities without proper attention to negatives, positives, and absolute value.

Best and worst practices of individual instructors/educational institutions

1. Teacher Preparation: High school mathematics teachers should have the equivalent course work for a major in mathematics at the Bachelor of Science or Bachelor of Arts level. In post-secondary settings, no teacher should be teaching who does not have the above credentials plus hours towards a masters in mathematics.
2. Teacher Assignments: Unprepared teachers arise from courses being assigned at the last minute. This practice should be kept at a minimum if it cannot be eliminated completely.
3. Teacher Attitude: The teacher must have working conditions and preparation time which allows a positive attitude towards mathematics and towards the students. When the proper support is provided by the institution, then the teacher can begin to affect the student attitudes.
4. Summer School: Students should be strongly discouraged from taking remedial or repeat coursework in the summer in high school and in the first two years of college.
5. Class size: A high school class size of less than 25 is appropriate; larger classes hinder learning. Although there is no universal standard applied in college courses regarding class size, the class size needs to allow for effective instruction at the particular institution.
6. Teaching Practices: Negative-teaching practices can be overcome through appropriate preparation and training. Teachers who follow the best practices often exhibit the following attributes:
   
   
• Clear teacher knowledge and competence
• Visible teacher enthusiasm and excitement
• Appropriate notification of teaching assignments
• Teaching assignments made which are appropriate to teacher’s educational training
• Technology used to enhance concept development
• Homework and teaching emphasize the numeric, graphic, and analytic characteristics of the material
• Good pacing of material – correct balance to between "covering" the necessary material and responding to student needs; avoiding a rush to catch up towards the end
• Good rapport and interactions with students – enough flexibility;
• Use of manipulatives to enhance concepts rather than as time-consuming, distracting gimmicks
• Develop high but realistic expectations of the students
• Assist students in setting realistic goals for themselves
• Committed to continuing professional development, especially in content
• Design teacher-directed lessons with active involvement by both student and teacher for the full class period.
  
  
7. Middle School: There should be a statewide implementation of a middle school certification in mathematics with identification of an appropriate body of knowledge.

VIII. External Factors Impacting Student Learning

Skills and Time Commitment Recommendations

Success and External Factors

Societal attitudes, individual student attitudes, non-classroom activities and learning skills all impact upon a student’s success in mathematics at the college level. This impact upon learning is especially evident in mathematics and science because these disciplines demand intense concentration, connections to previous material, and a structured daily study plan outside the classroom. Because mathematics is the area where students often receive their first learning "set-back", mathematics learning suffers additionally from a variety of attitude obstacles. The crucial need is to create a better climate for learning mathematics; this involves students, parents, instructors, and society in general. Every discipline would benefit from a more supportive structure external to the actual teaching process; the learning of mathematics will not even be possible if the structure does not improve. Primary areas to address are:

• Support of mathematics instruction and learning by the essential societal support systems, such as school boards and administrators, parents and guardians, business and industry, elected officials, and the media.
• Creation of a positive atmosphere for learning mathematics
• Improvement of student motivation and attitudes towards mathematics
• Attention to learning skills
• Establishment of higher, more demanding expectations
• Adjustment of homework to increase effectiveness
• Recognition that students must put education before part-time work in after-school jobs
• Encouragement of more parental/home involvement that is supported by teachers and school
• Improvement of advising/counseling of students

How a student handles the external demands often is determined before he/she enters college. The crucial need is to provide an effective climate for learning mathematics during the pre-college experience that will build a foundation for later work: this involves students, parents, educators, public leaders, and society in general.

Specific Factors of Concern

1. Societal Attitudes: Today’s students study mathematics in an environment with societal attitudes that are often indifferent and/or hostile to the learning of mathematics. For some people, mathematics is revered and feared, and for others, it and those who study it are ridiculed. Poor performance in mathematics is socially acceptable. These public perceptions encourage low performance expectations in mathematics instead of the high expectations that are needed.
2. Student Attitudes and Motivation: All too many students enter college with poor attitudes and limited motivation for learning additional mathematics. There are many reasons given for this result: unfavorable prior school experiences, inadequate prior achievement, authoritarian instructional model, view of mathematics as an unending list of rules and procedures to be memorized, attitudes of teachers, etc. These students usually elect to take as little mathematics as possible, thus restricting or eliminating for them careers in most technological and scientific areas.
3. Learning Skills: Do students entering South Carolina colleges possess the appropriate study, listening, and test-taking skills for successful performance in college mathematics? According to a recent survey of their mathematics instructors, many students were found to be deficient. The finding is reinforced by case studies comparing students in Japan, Germany, and the United States as part of the Third International Mathematics and Science Study (TIMSS). German and Japanese students were found to spend a significant amount of time in a variety of after-school academic activities related to their schoolwork. American students, on the other hand, do very little homework – a fact that was identified by both parents and teachers. United States teachers also add that
   
   

   "many students seem uncertain about what studying entails, demonstrate a limited repertoire of strategies for studying, and are not prepared to do academic work other than short assignments outside of class."(KAPPAN, March, 1998, p. 529).

Could the inadequate experience of our students with homework and studying be related to over-involvement in after-school jobs and/or extracurricular activities?

Actions to Implement

Public Policy Leaders

• School boards, state school officials, the Legislature, and the Governor must demonstrate their visible support for a high quality program of mathematics education through their actions. Significant change in the public perception of mathematics without this support is unlikely.

Parental / Home Involvement

• Parents or guardians must continuously monitor and take an interest in the educational progress of their children, communicating with the school and teachers as appropriate and supporting the educational endeavor over children’s complaints.

Educator Involvement

• Educators from classroom teachers to administrators to school boards must set higher standards and more demanding expectations for the teaching and learning of mathematics at all levels.
• Educators must give accurate, honest advice to students concerning the role of mathematics in their continuing education and in their career choices.

Student Motivation and Attitudes

• Teachers need to present mathematics as exciting and interesting.
• The mathematics curriculum should connect mathematics to the real world and include topics that are relevant to the needs and interests of the student.
• Teachers, using representatives from business, industry and government, should demonstrate how mathematics is a key which opens doors to many different careers.
• Teacher attitudes toward mathematics affect student attitudes. Teachers should be aware of the positive role model that they can set for students.

Study / Learning Skills, including Homework

• From middle school grades on, students need to have continuous instruction in
  
  
• learning how to take good notes
• learning how to study for and take a comprehensive examination
• learning to read a mathematics textbook with comprehension and understanding
• learning how to evaluate their own work
• learning to communicate with mathematics orally and in writing
• developing organizational skills and learning how to manage time in mathematical work
  
  
• The quality of homework assignments needs to be improved. Homework should emphasize not only the development of concepts and skills, but also the ability to synthesize and integrate these concepts and skills and to use them with understanding.

Proper Priorities

• Schoolwork shall take priority over employment in after-school jobs or extra-curricular activities.

IX. Teacher Preparation

Pre-Service and In-service Training Recommendations

The state of mathematics education is as fluid as ever in history. A combination of new pedagogical approaches, access to inexpensive technology, and the need to have a more mathematically literate population has caused each aspect of mathematics education to be questioned repeatedly. Everything from the physical layout of classrooms to specific content questions is being questioned, examined, changed and then re-questioned. Mathematics is no longer the "cheap" science, with no laboratory equipment needs and little on-going change or development to challenge instructors. . Chapter 7 of the South Carolina Framework for Mathematics is a good place to start to look at these issues; it has been virtually ignored in the educational community. In this context, the in-service and pre-service issues become more and more critical. It is now terribly easy to become out-of-date, both with respect to pedagogical issues and with respect to content. Consequently, the approaches to training and re-training teachers of mathematics are now even more critical issues.

Pre-Service Training

Placement of new teachers into the workforce is at a critical juncture. The interest in teaching as a profession is extremely low due to working conditions and low pay. This is exacerbated in the sciences by the many job opportunities available, and the low competition for these jobs, in the business community. There are a sizable number of teachers of mathematics leaving the job at the end of their first semester of teaching. A combination of improved working conditions and more access to a support structure can improve the experiences of both new teachers and continuing teachers.

Middle school education is an especially critical area of concern. The need for early decisions and entry into algebra described earlier is also putting pressure on middle school teachers’ training. The lack of specific standards for middle school science and mathematics training, when the curricula in the sciences and mathematics at the middle school level is getting more and more technical, is severely harming our students’ ability to achieve on the international level. This is obvious from the latest TIMMS studies. Specific curriculum requirements, with a substantial increase in expectations, need to be established as part of a certification for middle school teachers in the state of South Carolina.

All of these issues combine to make a series of changes in training and educational practices necessary. The steps which need to be taken, some of them on a school-by-school basis, include training issues and workplace issues:

1. Add middle school certification with courses restricted to potential middle school teachers; courses need to directly address content and specific mathematics education methods.
2. Establish middle school training and certification standards to include basic probability and statistics, linear systems, geometry, and calculus (single variable), using appropriate technology. These are necessary for the content currently needed in middle school mathematics.
3. Ensure all teachers of mathematics meet the National Council of Teachers of Mathematics (NCTM) standards for qualifications.
4. Include more mathematics content in the certification programs at the university level, even if this is at the expense of some educational coursework.
5. Provide experienced teachers of excellence to act as mentors for new teachers. This needs to be on-going for their first two years and demands time from both the new teacher and the mentor. There should be one course released time for the new teacher and for the mentor.
6. Ensure that new teachers get reasonable teaching assignments, instead of a schedule made up of all the worst courses.
7. Reduce non-teaching responsibilities of mathematics teachers by hiring aids to do non-content related, non-teaching related duties. The goal is to match more closely the norms in countries with better math/science educational records.
8. Reduce class time interruptions – education must come first above all other activities. These interruptions are destructive when student concentration is a necessity.
9. Continue emphasizing graduate teaching assistant training in mathematics education and pedagogy at the colleges and universities; often these GTA’s are teaching potential teachers.
10. Provide a better balance – that is more mathematics and fewer "general" education courses – during the pre-service training. Additionally, make more of the education courses specifically applicable to mathematics and science teachers.
11. Educate students to expect continued training (after hired) in content, technology and pedagogy due to the continuously changing nature of mathematics education.

In-Service Training

As mentioned above, the need for on-going education for teachers already in place in schools is critical. The training that is available is often minimal, in-house, and not connected to changes in content. Most teachers are very eager to stay up-to-date and implement changes in their profession; there simply is no time, no money, and few opportunities to allow them to maximize their expertise. Training must be provided that is more accessible in terms of time and money, more convenient, and more focused on mathematical content.

The most effective and efficient method would be to implement the actions recommended below, with a special emphasis on Advanced Placement style or National Science Foundation style training institutes for summer training. The institutes should be designed so that each one involves teachers from around the state; this would incorporate a potential for uniformity among and interaction between teachers from different school districts and from different parts of the state. The institutes should be fully funded and of a reasonable length so as to allow the most coverage of material.

Additional recommendations follow:

1. Increase workshops, especially mathematical content workshops and workshops which are offered by individuals external to the school district.
2. Continue to educate teachers to expect continued training (after hired) in content, technology and pedagogy due to the continuously changing nature of mathematics education.
3. Establish systematic retraining to include on-going innovations and exposure to the material from sources such as this report and the National Council of Teachers of Mathematics (NCTM) standards. (Note: This can be partially achieved by increasing paid attendance, including paying for leave days, at NCTM meetings and other similar professional meetings.)
4. Develop Advanced Placement style institutes which would add on content for middles school in at least the four areas of concern to middle school mathematics education; requirements should be in place which would ensure that completion of the institute training would occur within 4 years of starting middle school teaching and would weave course content across boundaries.
5. Include integrated content, methodology and technology in training courses.
6. Encourage middle school teachers to "sit-in" on high school courses; provide periodic released time or appropriate scheduling on a rotational basis to make this feasible.
7. Include in institutes: required attendance, training in learning/study skills, on-line features for follow-up, review of algebra, award of a certificate of (extended) renewal, and evaluation on an individual basis.
8. Reeducate the public and the educational community to realize that at this time "certified" does not necessarily mean qualified to teach a particular subject; bring these concepts back in alignment and eliminate open certification at the state level.
9. Continue to emphasize standards and requirements regarding mathematical content.
10. Provide opportunities for teachers to stay current with the changing issues and practices in mathematics education.

X. Raising the Standards

Future goals for success

Implementing these goals will require action on the part of all constituencies of the educational system in South Carolina. This can be not be a grass-roots action or a top-down action; it must be both. This report makes recommendations which requires change at each point of contact in the educational process: parent/student, student/teacher, teacher/parent, teacher/administration, teacher/college, etc. To change mathematics education in South Carolina will require:

• A change in expectations for learning mathematics, both internal and external to the educational community
• The participation of every constituency in South Carolina (business, government, parents, students, teachers, school boards, etc. ) through changing their own expectations of teaching and learning mathematics and through individual actions supporting mathematics education
• Each group to act as an advocate for change.

Attempts to implement all of the changes in this report will of course fall short of "perfection". The standards must be reviewed continually in the context of desirable educational practices and then repeatedly reset to make sure the students of South Carolina are ready for their life-long work and learning. For the meantime, the two primary goals of mathematics education in South Carolina must be:

• To increase the depth and intensity of mathematics and the interest and ability of South Carolina students to the point that remediation for college entrance or for collegiate success is no longer an issue in higher education for traditional freshman students
• To ensure that mathematical skill, or the lack thereof, is not the limiting factor in choice of major or career for South Carolina students.

Mathematics education is vital to the functioning of a community, state or nation for all the reasons which make education in general important to these political bodies:

• Maintains the culture of the community
• Allows the community to function as viable economic entity
• Maintains an educated citizenry, one of the principles of democratic government
• Provides a basis for life-long learning as the requirements for an individual to be productive.

Mathematics provides a specialized arena in which to achieve these goals which is particularly necessary as we enter the 21^st century. Mathematics specifically enhances thinking and technological skills:

• Allows an educated workforce which will attract businesses to the community
• Supports a higher standard of living by allowing individuals to qualify for higher paying jobs
• Enhances the ability to learn in the workplace, especially abstract concepts
• Enhances the ability to manipulate abstract concepts and to make abstract constructs in any area of knowledge or interest
• Provides a technological basis for the growth in science and technology in the 21^st century

If South Carolina is to maintain a competitive position in the nation and in the world, the issues of mathematics education must be addressed thoroughly, carefully, and continuously. This report provides a template for beginning the process.

XI. Appendices

Specific Content Recommendations

Algebra Skills Recommendations

The following skills/facts should be contained in the curriculum of the algebra sequence and should be included in practice work given to the students:

1. Addition of fractions
   
   
1. 
2. 
   
3. Long division of polynomials
4. Addition of polynomials
   
   
2. Linear Functions and Straight Lines
   
   
1. Definition of function
2. Determination of the equation of a line given two distinct points. Equation should be checked using the two given points.
3. Applications of b.
   
   
1. Conversion of degrees Celsius to degrees Fahrenheit.
2. Conversion of miles per hour to feet per second.
   
   
4. Unit conversion
5. Writing the equation of a line in slope-intercept and in point-slope form.
6. Slopes and equations of perpendicular and parallel lines.
7. Slope of a line as a constant rate with units.
8. Solution of systems of linear equations and verification of the solution .
9. Properties of < and .
10. Solution of the following linear equation and linear inequalities.
    
    
1. 
2. 
3. 
4. 
5. 
   
   
11. Solution of a linear equality by solving and graphing the line .
12. Domains of functions of the form , , , .
13. Linear word problems.
14. Application to slope to increasing, decreasing, and horizontal lines.
    
    
3. Quadratics
   
   
1. Identification of the variable for a quadratic of a single variable
   
   
1. 
2. 
   
   
2. Identification of coefficients
   
   
1. 
2. 
   
   
3. Solutions to by the quadratic equation and verification by hand of these solutions.
4. Graph of .
5. Solutions of the following inequalities:
   
   
1. 
2. 
3. 
4. 
   
   
6. The discriminant and its applications to previous 3 problem types.
7. Completing the square to find the absolute extrema.
8. Find the midpoint of the zeros and use it to find the absolute extrema.
9. Word problems which involve finding absolute extrema.
10. Application of the discriminant to determine whether a quadratic is factorable or prime over the reals.
11. Application of the discriminant to factoring when the discriminant is the square of a rational number.
12. Factoring as , where and .
    
    
4. Functions
   
   
1. Definition of function
2. Notations for functions
   
   
1. 
2. 
3. 
   
   
3. Composition of functions: if and , then .
4. The difference quotient: .
5. Definition of the polynomial
6. Definition and construction of rational functions
7. Definition of and , where a>0, a
8. Definition of trigonometric functions
9. Domains of polynomials, rational functions, ,, and where p (x) is a polynomial
10. Graphs of lines, quadratics, rational functions, and algebraic functions
11. Solving inequalities from graphs
12. Finding domains from graphs
13. The following facts about the real numbers:
    
    
1. for all provided and
2. for all provided and
3. has no solutions if
4. for all in the domain of
5. for all in the domain of
6. has no solutions
   
   
5. Solving and Checking Equations
   
   
1. Find the domain of the rational function
2. Solve the equation by solving and checking these candidates for membership in the domain.
3. Solve the equation by solving and checking these candidates for membership in. the domain
4. Solve by solving .
5. Mathematical identities
   
   
1. provided that and
2. only when
3. 
4. for all in the domain of
5. for all in the domain of
6. provided that and
7. provided that and
   
   
6. Reduction of difficulty (on appropriate domains)
   
   
1. 
2. 
3. 
   
   
7. Solve the following equations and check by hand
   
   
1. 
2. 
3. 
4. 
5. (no solutions!)
   
   
6. Factoring polynomials over the Real numbers
   
   
1. is a prime as a polynomial for all constants
2. is prime as a polynomial
3. is prime as a polynomial if
4. is prime a perfect square if
5. factors as a product of polynomials if is the square of a rational number
6. factors using the quadratic formula
7. always factors into a product of linear polynomials and prime quadratic polynomials
8. Factor a polynomial using the remainder theorem and long division
9. If with , then
10. Factorizations of special polynomials:
    
    
1. 
2. 
3. 
   
   
11. If and , then
    
    
7. Word (Modeling) Problems
   
   
1. Unit conversions using the following facts
   
   
1. 1 mile = 5280 feet
2. 1 mile = 1760 yards
3. 1 yard =3 feet
4. and
5. 1 inch=2.54 centimeters
6. 1 pound453.5 grams
7. 60 mph=33 feet per second
8. 1 kilometer .61 miles
9. 2 radians =
   
   
2. Changing units by mimicking multiplication and division
3. Finding absolute extrema of quadratics
4. Linear rate problems
5. Simple and compound interest
6. Continuous growth and/or decay
   
   
8. Solving Procedures
   
   
1. Polynomials by the following procedure:
   
   
1. 
2. 
3. Factor if possible to
   
   
2. Rational functions by the following procedures:
   
   
1. 
2. 
3. 
4. 
5. Exclude values of the independent variable which make the denominator zero
6. Solve by factoring
7. Check candidates for solutions in the original equation, be aware of step v
   
   
3. Algebraic equations by the following procedures
   
   
1. by solving for the most difficult operation using +,-,×,÷.
2. by applying inverse of the most difficult operation to both sides of the equation.
3. by rewriting left side as and following above procedures.
   
   
4. Logarithmic equations of the form by the following procedures
   
   
1. 
2. 
3. Use the properties of logarithms:
4. Use the properties of logarithms to compress:
5. Apply inverse to both sides:
6. Solve
   
   
9. Given a graph of a function , determine the following:
   
   
1. The domain of
2. Solve
3. Solve
4. Solve
5. Solve
6. Solve
7. Determine the domains of and
   
   
10. Miscellaneous
    
    
1. Midpoint formula
2. Distance formula

Trigonometry Skills Recommendations

The following skills/facts should be contained in the trigonometry curriculum and should be included in practice work given to the students:

1. The definitions of the six trigonometric functions, using both right triangles and the unit circle.
2. Basic knowledge of the number such as:
   
   
1. The circumference of the circle divided by the diameter is .
2. The number is not 3.1415 nor is it , but is approximated by these.
   
   
3. The definition of angular measurements both in radians and degrees.
4. Convert angular measurement between radians and degrees:
   
   
1. A strong understanding of the angular measurement in radians.
2. The length of the arc subtended by an angle on the unit circle.
3. 1 radian57 degrees
   
   
5. Numerical approximations (without a calculator)
   
   
1. 
2. 
   
   
6. Values of the trigonometric functions for the standard angles which are multiples of 30 degrees and 45 degrees, without using a calculator.
7. The period of each of the trigonometric functions.
8. Given a zero of the function or , with , and the period, produce all the zeros of the function without using a calculator.
9. Solve the equations and for zeros in a given interval.
10. Domains for all six trigonometric functions.
11. Solve and on an interval.
12. Understand that has no solutions; similarly for .
13. Know the graphs of the six basic trigonometric functions.
14. Solve the following inequalities from the graph of the corresponding function:
    
    
1. ,,, and
2. ,,, and
3. ,,, and
   
   
15. Be able to quickly state and easily use the following identities:
    
    
1. Unit Circle Identities
   
   
1. 
2. 
3. 
   
   
2. Basic Division Identities
   
   
1. 
2. 
3. 
4. 
   
   
3. Even and Odd Identities
   
   
1. 
2. 
   
   
4. Euler Identities
   
   
1. 
2. 
   
   
5. Half-angle and Double-angle Identities
   
   
1. 
2. 
   
   
6. Miscellaneous Identities
   
   
1. 
2. 
   
   
7. Law of Cosines: