• Success and Content
• Specific Content Concerns

General Concerns

• Actions to Implement

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.

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