Mathematics at the college level is a rich subject, full of interesting ideas and surprising applications. Students with real interests and abilities in mathematics or kindred subjects, such as physics or theoretical computing, should be thinking about eventually studying mathematics at this level. Their high school preparation should entail taking as much mathematics as they can and also reading books that will suggest some of the power and interest of mathematics, such as Mathematics Today by Lynn Steen or Wheels, Life and Other Mathematical Amusements by Martin Gardner. A fascinating reading list is given at the end of Metamagical Themas by Douglas Hofstadter.
Most students enroll in mathematics classes for a utilitarian purpose-they wish to prepare themselves for quantitative work in programs other than mathematics. For students who choose to study mathematics at the university, four years of high school math is immensely beneficial. It is essential to alert students to the pervasive quantitative nature of virtually all modern learning and analysis. Thus, the amount of utilitarian mathematics necessary for most fields of study increases daily.
Some students, counselors, and teachers think that the admission requirements, currently one year of geometry and one year of algebra (to be supplemented in the 1990s by a third year of math; see table 2, page 3) are sufficient preparation for college-level math. Many mathematics faculty, however, believe that an entering student who is adequately prepared is ready to take calculus. Certain courses and majors require mathematics prerequisites. For example, students who expect to complete certain scientific or technological majors should view calculus as basic preparation. The time needed to complete precalculus courses may add an extra semester or an extra year to complete those programs.
The need for math skills above the admissions level is very general; it exists everywhere except for some programs in the arts, education, and the humanities. Both colleges and specific majors within colleges may have math requirements beyond admissions levels, however. Students are responsible for learning of such requirements as soon as they begin the advising process. It is important to plan ahead to take prerequisites as necessary.
All entering freshmen at UW-Madison are placed in mathematics courses according to their demonstrated "competency levels"- a determination made on the basis of high school work and performance on the University of Wisconsin System Mathematics Placement Test. The competency level description appears at the end of this section (p. 9), showing various areas of college study, with the appropriate competency level for each. The competency levels are minimum, intermediate, and advanced. Normally, a student working conscientiously in a mathematics program would reach minimum competence after two years of high school study, intermediate after three, and advanced after four. These results have not been realized by students tested for UW-Madison. In the fall of 1986, entering freshmen who had two years of high school mathematics failed to achieve minimum competence in 60% of the cases. Students with three years of high school mathematics fell below minimum competence in 30% of the cases, at minimal competence in 28% of the cases, and at intermediate competence in only 42%.Part of this gap between the ideal and the actual may result from high school courses that do not cover the mathematical material central to college preparation. Another part of this gap is unquestionably the result of students who do not learn the central material at a level high enough for college preparation; more on this topic appears in the next section, "Learning Mathematics." Finally there is the problem of lack of retention. Students who do not take mathematics during their senior year forget significant parts of the mathematics they once knew, and they enter college at a level below the best they have attained.
Geometry, algebra/pre-calculus, and trigonometry are the core of the college preparatory program. Geometry must be studied at the level of learning associated with a college-track course; basic geometry courses are not sufficient. Algebra needs to be mastered thoroughly-students wishing to be well prepared will take two algebra courses. Precalculus refers to material such as function notation, logarithms and exponentials, and analytic geometry. Calculus itself fits into the core if taught in a version generating college credit (e.g., a course affiliated with the Advanced Placement Program of The College Board); other versions do not. A course in finite or discrete mathematics contributes to college preparation, as does the ability to use a scientific calculator. Computer literacy is helpful in a general way but does not contribute directly to work in entry-level mathematics courses.
Just as important as the lists of topics covered in college preparatory mathematics is the level of understanding attained. Students who see mathematics only as a collection of rules to be mechanically applied in stereotyped situations are not mathematically prepared. From the beginning, students should learn to apply mathematical skills in a variety of contexts. For example, after studying quadratics they should solve 4-x2=16x as readily as . At a somewhat later stage, they should see that solving is done by solving quadratics. From penetrating these thin disguises, they should progress by separating a problem into simpler pieces, finding a way to deal with the pieces, and putting the results together to solve the original problem. For example, given three points P, Q, and R in the plane, the student must determine whether there is a point equidistant from all three. Looking just at the points equidistant from P and Q, we note that they must lie on the perpendicular bisector of segment PQ. Now looking at the points equidistant from Q and R, we find they lie on the perpendicular bisector of QR. The two bisectors either intersect at a single point or are parallel. The student finds the solution to be that there is a unique point equidistant from P, Q, and R unless these three points lie on a line. Being prepared in geometry means being able to do this kind of analysis.
The problems in the paragraph above are not ones for which students should be prepared specifically. Rather, they are examples of a very large variety of problems accessible without specific preparation to students whose learning has been at the desired level. Of course, the best way for students to reach this level is for them to do a large variety of problems consistently throughout their high school mathematics work. Being prepared in mathematics also means knowing what kinds of problems you can solve and when you have sufficient data to determine an answer. This kind of general understanding is essential for "problem solving" and "word problems." Finally, students maintain their mathematical learning best if they have acquired it in a logical structure; such a logical structure permits them to retain the many details necessary to use mathematics successfully.
We return to an earlier point: students who enter the university with minimal mathematical preparation are at a serious disadvantage in choosing and completing some majors. This disadvantage, unfortunately, goes unrecognized at first because math skills are often a "hidden prerequisite." Many programs that have no stated mathematics requirements do require courses that make crucial use of math skills. In the social sciences, such requirements do require courses that make crucial use of math skills. In the social sciences, such requirements will be in the area of statistics (often under a guise of "Quantitative Methods"). The chemistry courses required for biological and health science majors have "hidden" math prerequisites, with intermediate and advanced courses requiring two semesters of calculus. Alert and well-advised students will take such needs into account in planning their freshman year. They will recognize that the alternatives are to limit the field of possible college majors to those for which they are already prepared, or to begin at once to get the necessary preparation, even if this adds a semester or two to their college careers. Good mathematical preparation is like an automobile; it is not necessary for survival in college, but it makes students much more mobile in their choice of college majors. The list below illustrates the math competencies students will need for completing programs in various areas at UW-Madison. Definitions of competency follow the table.
General Field or Area Level of Mathematics Required Agriculture Advanced Business, Economics Advanced (will need calculus) Education Intermediate (depending on program) Family Resources Intermediate (some programs, advanced) Life Sciences, Health Sciences and Intermediate or advanced depending on professions program; typically in preparation for college chemistry Math Sciences, Physical Sciences, Advanced (will need, calculus) Engineering Social Studies, Social Work Intermediate or advanced, depending on program; typically in preparation for college statistics
Note: Students may meet these levels during the course of other undergraduate work, but doing so detracts from work in the program itself.
From algebra and arithmetic:
|an understanding of the axioms that underlie arithmetic, the decimal system and its use in calculation, and the definition and elementary properties of rational numbers;
|basic algebraic skills, including special products, factoring, positive integral exponents and the manipulation of algebraic fractions;
|setting up and solving linear equations and inequalities.
|axioms, theorems, and proofs of theorems covering straight lines, triangles, and circles;
|graphing of linear equations and the solutions and geometric significance of systems of two linear equations, mensuration (area and volume) formulas for common two- and three-dimensional figures.
The topics of level 1, together with
|setting up and solving quadratic equations and inequalities;
|complex numbers, rational exponents, progressions;
|graphing of circles and quadratic polynomials;
|definition and elementary properties of logarithms .
Algebra: The topics of levels 1 and 2, together with:
|algebra of polynomial and rational functions;
|the function concept, theory of polynomial equations, including the remainder and factor theorems;
|solution of simultaneous linear equations;
|equations and graphs of lines and circles;
|infinite geometric progressions;
|mathematical induction and the binomial theorem.
Trigonometry. The topics of levels 1 and 2, together with:
|the function concept;
|trigonometric functions of real numbers, together with their basic properties and graphs;
|trigonometric equations and identities;
|geometric significance of the trigonometric function and elementary applications;
|trigonometric form of complex numbers and DeMoivre's Theorem.