"Great Expectations" Mathematics Report

Executive Summary

Mathematics provides a specialized arena in which to achieve the developmental and economic goals which are particularly necessary at the beginning of the 21^st century. Mathematics specifically enhances thinking and technological skills. This in turn:

• 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. The report for which this is an executive summary provides a template for beginning the process.

The purpose of this report is to provide specific recommendations for implementation in order to improve the transition of students in South Carolina from secondary mathematics to college mathematics. Together with the South Carolina Mathematics Framework, which provides the content underpinnings for graduation from high school, these recommendations set the goals for mathematics education in South Carolina for the 21^st century. The overwhelming consideration is that mathematical skill, or the lack thereof, should not be the limiting factor in the choice of major or career.

Often tradition, inertia, the fundamentals of the budgeting and political process, and lack of resources get in the way of the ability to establish a successful learning environment for students. While it is useless to assign blame, it is vital to enumerate the attributes necessary to improve the learning climate. It is the responsibility of all stakeholders to establish these attributes in the educational system.

Differences in the high school and college environment, insufficient high school preparation in both content and general learning skills, and the teaching practices of the high school and college can each enhance or detract from the student’s learning. From the standpoint of the high school, college teacher, and parents or other advisors, the following principles need to be considered:

• 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 1800 fewer minutes or 30 hours) and the compression of time in which topics are introduced and developed is harmful. 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.
  
  
• The recommended four-year sequence of high school mathematics is Algebra I, Algebra II, Geometry and Precalculus, in that order.
  
  
• Time lapses between mathematics courses are extremely detrimental in high school, in college, and in the transition between the two.
  
  
• 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.
  
  
• 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.
  
  
• There should be a statewide implementation of a middle school certification in mathematics with identification of an appropriate body of knowledge.
  
  
• Long days of heavily compressed material, such as regularly occurs in summer school, can impede retention and comprehension of mathematics.
  
  
• Pressure on high school teachers to pass most/all of their students is detrimental in any discipline which requires a continuing progression of ideas.
  
  
• Reading skills inadequate to understand or work with modeling problems hinder the learning of mathematics and the ability to connect mathematics to other disciplines.
  
  
• Excessive time spent on testing to address "accountability" or "assessment" issues removes additional time from the learning/teaching process.
  
  
• Teaching with technology is necessary for the future success and understanding of the student.
  
  
• Teaching successfully with technology is still an evolving process; teachers need to be aware of current thinking and techniques. Teachers need to experiment and try different techniques; while this is happening, they need to be encouraged and supported financially the school administrators and parents.
  
  
• Technology strategies must be designed on a course-by-course basis; exercises which require repetitive, boring, undirected calculations in the name of "discovery learning" are inappropriate in most cases.
  
  
• A holistic approach to material – involving problem solving/mental analysis, written and oral communication and the use of appropriate technology – should be included in all courses.
  
  
• A computer is not needed to do arithmetic; keep the technology appropriate to the task.

Teacher preparation is more crucial than ever due to the changing nature of mathematics instruction. The 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. Adding to the strain is the fact that placement of new teachers in the workforce is at a critical juncture. The interest in teaching mathematics as a profession is extremely low due to working conditions, low pay, and lack of adequate equipment and supplies. 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. If South Carolina is to have a properly educated population, she must first spend the time, energy and resources to recruit and train an exceptional teaching faculty at all levels. This includes addressing the following issues:

• Establish middle school training and certification standards to include basic probability and statistics, number systems and algebraic structures, geometry and measurement, and calculus (single variable), using appropriate technology.
  
  
• Develop Advanced Placement style institutes which would provide the requisite content for under-prepared middle school teachers and allow instruction to achieve the levels described above. 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.
  
  
• Ensure all teachers of mathematics meet the National Council of Teachers of Mathematics (NCTM) standards for qualifications, even if such credentials are technically not required for state certification.
  
  
• Continue to emphasize standards and requirements regarding mathematical content.
  
  
• 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 one for the mentor.
  
  
• Ensure that new teachers get reasonable teaching assignments, instead of a schedule made up of the worst courses.
  
  
• Reduce class time interruptions – education must come first above all other activities.
  
  
• 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.)

From the post-secondary student’s perspective, 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 course 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.

Ensuring this requires that the student, parents, and educational community lay the foundations to support the following principles:

• Calculus preparation makes available the widest choice of career possibilities. Students who do not prepare for calculus can be successful in their post-secondary mathematics career if they prepare as if they were going to take calculus.
  
  
• The traditional four courses of high school mathematics – Algebra I and II, geometry, and precalculus – are the minimal course work necessary to prepare a student for calculus and are the best predictors of future success in post-secondary mathematics.
  
  
• The cumulative nature of science in general and mathematics in particular force early decisions upon students; the decisions made in early middle school regarding pre-algebra can determine a student’s college mathematics experience.
  
  
• The study skills and test-taking skills necessary to succeed in college courses are developed through high school experiences.
  
  
• Early advising of the student regarding how to keep open the most choices for areas of study and for careers is essential.

For the student, success is closely linked to the content and the learning experience in the high school mathematics class. As has been said so often, mathematics is not a spectator sport. Each person involved – student, parent and teacher – must be an active participant in the process. For this reason, it is important for all concerned to aid the student in attaining the following broad capabilities before reaching the college classroom:

• 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 interpreting of graphs
• Reasoning skills and the application of logic to understanding mathematics
• Basic technological skills -- calculators primarily
• Exposure to "real-life" type numerical examples

All individuals involved in the educational process need to remember that minimal facility in current material almost guarantees failure in future material in mathematics.

From the perspective of the teacher, every teacher knows that there are considerations beyond the simple question of which material to cover. In addition, mathematics instructors should attempt to include the following in standards in their plans for teaching:

• Mathematics instructors need to emphasize the connections between mathematical concepts and between mathematics and other disciplines.
  
  
• All instructors should use technology to support the development of the concepts of their courses; it supports theoretical understanding, visualization of solutions and functions, and the development of mathematical intuition.
  
  
• 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.

Today’s students study mathematics in an environment with societal attitudes that are often indifferent and/or hostile to the learning of mathematics. Poor performance in mathematics is socially acceptable. All too many students enter college with poor attitudes and limited motivation for learning additional mathematics. These students usually elect to take as little mathematics as possible, thus restricting or eliminating careers for them in most technological and scientific areas. In addition, learning skills often have been grossly neglected throughout the middle school and high school experiences. According to a recent survey of college mathematics instructors from across the state of South Carolina, many of their freshman mathematics students have insufficient learning skills – note-taking, test-taking, and homework practices. Actions to correct these factors are the hardest to devise and the hardest to implement. As first steps, the following would help improve the setting in which the learning of mathematics occurs:

• 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.
  
  
• Teacher Attitudes: Teachers, using representatives from business, industry and government, should demonstrate how mathematics is a key which opens doors to many different careers. All teacher attitudes toward mathematics affect student attitudes, whether they are actually teaching mathematics or not. 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
  
  
• Proper Student Priorities: Schoolwork must take priority over employment in after-school jobs or extra-curricular activities. Students must accept responsibility for their own learning.

Implementing these actions in South Carolina will require effort on the part of all constituencies of the educational system. This can 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.

Report Information:

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. The Commission on Higher Education established the "Great Expectations" project; the resulting report provides the basis for this executive summary. A complete copy of the report, including technical content and recommendations, can be found on the Internet by accessing www.winthrop.edu/mathsuccess. Copies may be found in libraries, both high school and college, across the state. A copy can be obtained by contacting:

Department of Mathematics
Winthrop University
Rock Hill, SC 29732
803-323-2175

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.

Committee: