# College of Letters & Science

### Mathematics

Requirements for the Major

Honors in the Major

Teacher Preparation

Mathematics Certificate

General Levels of Competence in Mathematics

Placement

Satisfaction of General Education Requirements

Notes on Courses

Courses

207 Van Vleck Hall, 480 Lincoln Drive, Madison, WI 53706; 608-263-6374; undergrad_program@math.wisc.edu; www.math.wisc.edu

Professors Angenent, Assadi, Bolotin, Boston, Caldararu, Craciun, Denissov, Ellenberg, Feldman, Gong, Jin, Kiselev, Lempp, Mari-Beffa, Millar, A. Miller, Oh, Seeger, Seppalainen, Smith, Terwilliger, Viaclovsky, Waleffe, Yang; Associate Professors J. Miller, Mitchell, Paul, Thiffeault, Zlatos; Assistant Professors Anderson, Arinkin, Dymarz, Gurevich, Kent, LaVictoire, M. Matchett Wood, P. Matchett Wood, Maxim, Miller, Roch, Spagnolie, Stechman, Stoval, Street, Valko, Wang, Yin

Undergraduate advisor: see department's undergraduate program page.

Faculty diversity liaison: Professor Julie Mitchell

Mathematics is classified with both the humanities and the sciences. Its position among the humanities is based on the study of mathematics as one of the liberal arts for more than two thousand years. Still an expanding subject, mathematics offers more new and challenging frontiers than at any time in its long history—with many new fields, requiring new techniques and ideas for exploration.

The place of mathematics among the sciences is well founded. The natural sciences have invariably turned to mathematics for techniques needed to explore the consequences of scientific theories. In the last few decades social scientists have increasingly found higher mathematics of value in their training and research.

Although job opportunities have varied with the changes in the national economy, in recent years graduating math majors have obtained employment in a variety of jobs in business, industry, and governmental agencies and also have obtained teaching positions at the secondary school level (such teaching positions normally require teaching certification). Others have continued their education at the graduate level in mathematics and other fields. Departments in a variety of fields which use mathematics, including some in the social and biological sciences as well as in engineering and the physical sciences, are interested in attracting math majors into their graduate programs. Math Ph.D.'s obtain academic positions at the college and university level and nonacademic positions entailing consulting and research. The math major requirements are flexible enough to allow preparation for various goals.

Students interested in mathematics should also consider related programs offered by the computer sciences department; the statistics department; the Wisconsin School of Business; the School of Education; and the major in applied mathematics, engineering and physics.

#### Requirements for the Major

A detailed description of the mathematics major program is in the *Guidebook for Undergraduate Math Majors*.

**Acceptance.** To be accepted as a major in mathematics a student must complete Math 221, 222, and 234 (or equivalent sequence) with a grade point average of 2.5 or better in this sequence. However, a higher grade point average is advisable. Soon after completing Math 234, math majors should have their math advisors complete the Major Declaration Form. Majors are provided with math advisor information through the mathematics undergraduate program coordinator in 207 Van Vleck Hall.

The student chooses one of two options, or Honors in the Major. Since both nonhonors options (Option I and Option II) allow considerable flexibility, students should plan their programs with the advice of their math advisors. Indeed, those following Option II must have their programs formally approved by their mathematics advisors. Option II emphasizes the applications of mathematics, and those following Option I are also encouraged to take courses in other departments which involve the application of mathematics.

All students must complete the L&S requirement of at least **15 credits of upper-level work in the major **completed in residence. All math courses above 306 (except for Math 425) count toward this requirement.

#### Option I

Seven mathematics courses numbered above 306, excluding 320, 425 and 490.

These seven courses must include:

(1) 340 or 341 or 375 (341 is strongly recommended), and

(2) Three courses numbered above 500 including at least two of the following: 521, 541, 551.

The seven courses may not include more than one course from the Linear Algebra group (340, 341, 375), and not more than one course from the Differential Equations group (319, 376), and not more than one course from the Probability group (309, Stat 311, 331, 431). Moreover, students who have taken math 275 cannot include math 421 as one of their seven courses.

Students are strongly recommended to take either 341 or 421 before advancing into courses numbered above 500.

It is recommended that **students preparing for graduate work in mathematics **satisfy (a), (b), and (c) below.

(a) 341, 521, 522, 541, 542

(b) 551 or 561

(c) At least two other courses at the 500 level or higher.

Students who plan to enter a mathematics Ph.D. program should acquire a **reading knowledge of at least one foreign language **as early as possible. For mathematics study, the most useful languages are French, German, and Russian.

#### Option II

(For students interested in a particular area of application)

(a)Four courses in some area of application of mathematics, including at least three courses at the intermediate or advanced level, selected with the approval of the student’s mathematics adviser;

(b) Six mathematics courses numbered above 306, excluding 425 and 490, selected with the approval of the student’s mathematics adviser, including one course from the Linear Algebra group (320, 340, 341, 376) and two courses numbered above 500. The six courses may not include more than one course from the Linear Algebra group (320, 340, 341, 375), and not more than one course from the Differential Equations group (319, 320, 376), and not more than one course from the Probability group (309, Stat 311, 331, 431). Moreover, students who have taken math 275 cannot include math 421 as one of their six courses.

No courses may be used to fulfill both (a) and (b). Approval of a program under Option II will be required before a significant part of the program is completed and changes in the approved program will require prior consent of the mathematics advisor.

Sample programs for Option II are in the *Guidebook for Undergraduate Math Majors**. *The areas of application in these sample programs include: computer sciences, chemistry, physics, statistics, actuarial mathematics, finance, meteorology, economics, ecology, genetics, forestry, biostatistics, bioinformatics, systems biology, structural biology, business, and engineering.

#### Honors in the Major

Honors majors must successfully complete with grades of B or better the following mathematics courses: 521H, 522H, 541H, 542H, or their equivalents, and at least two more courses above 500 (usually 551 will be one of the courses) chosen in consultation with the honors advisor. They must also complete a "capstone" project consisting of Math 681–682 (Senior Honors Thesis), or a sequence of two mathematics courses numbered above 700, chosen with the approval of the honors advisor. Further, the student should request candidacy for the honors program from the honors advisor by the start of the junior year. At least one of the two sequences 521–522, 541–542 must be completed by the end of the student's junior year.

Before choosing the honors thesis (681–682) option, the student must consult with the honors advisor and a faculty thesis advisor to prepare for and complete a suitable thesis project; the student who elects to complete the alternative sequence of graduate-level courses should give a substantial report on his/her progress in the sequence. The student must complete the above program with a cumulative GPA in the major of at least 3.3. In addition, a collegewide cumulative GPA of at least 3.3 in all courses taken at UW–Madison at the time of graduation is necessary to earn any Honors degree in L&S. For information on research opportunities for undergraduates, see Honors Program in Math on the department website.

Students should check with the department honors advisor at least once a year, to report progress and to seek guidance about planning Honors in the Major curriculum that reflects their special interests.

**William Lowell Putnam Competition.** This is an annual international mathematics competition for undergraduates, based on originality and cleverness rather than sophisticated mathematical knowledge. Students interested in preparation for this examination may join the Putnam Club. The organizer of the club selects a team to represent UW–Madison. For more information, see Putnam Club on the department's website.

#### Teacher Preparation

Students interested in teaching mathematics at the primary or secondary level should consult the School of Education section in this catalog.

#### Mathematics Certificate

The department also offers a mathematics certificate. Most undergraduate and Special students are eligible for this program. To earn a mathematics certificate, an overall grade point average of 2.0 is required.

Certificate students must complete the following requirements:

- At least 25 credits of Mathematics courses numbered above 200 and 13 of those have to be in residence.
- Of the minimum 25 credits of required mathematics course work, at least 12 credits must be courses numbered above 306.
- Of the minimum 25 credits of required mathematics course work, at least 3 credits must be courses numbered above 400 (excluding 425 and 490).

When declaring for the certificate, students must see the undergraduate program coordinator, (207 Van Vleck Hall) for adviser information. For details, see Mathematics Certificate on the department website.

#### General Levels of Competence in Mathematics

Three years of mathematics preparation in high school (algebra, geometry, and a third year unit in algebra, trigonometry, analytic geometry, or calculus) satisfies the minimum requirement in mathematics for admission to UW–Madison. The Department of Mathematics strongly recommends that students take four years of mathematics preparation at the high school level. Students with only three years of mathematics preparation will be at a competitive disadvantage to other students for admission.

Students who wish to choose a major that requires calculus (e.g., the physical and biological sciences, business, economics, engineering, some majors in agricultural and life sciences) will be at a disadvantage in college without a rigorous college-preparatory mathematics sequence in high school. Such a sequence should emphasize both understanding and problem-solving in algebra, geometry, and trigonometry, and should include substantial work in algebraic manipulation and equation-solving **without the use of calculators**, algebraic and geometric proofs, mathematical modeling, trigonometric manipulation and equations, hand-graphing of functions, and 3-dimensional geometry.

Admission to one's first mathematics course at UW–Madison is based on the mathematics placement exam. For a sample of the types of questions that appear on the placement exam, see this link on the department website. It is strongly suggested that students look at this collection of mathematics problems for an indication of the skills, knowledge, and understanding expected from a rigorous high school mathematics curriculum.

Four levels of pre-university competence are specified below. (Levels of competence, except superior, are measured by the placement examinations described below.) Prospective students of mathematics, science, and engineering should achieve advanced mathematical competence (Levels 3a and 3b) before coming to the university so that they may enroll in Math 221 at the start of the freshman year. Students with only minimum and intermediate mathematical competence are strongly advised to remove this deficiency by independent study through the UW–Extension or by enrolling for the summer session preceding the freshman year.

Students must be able to apply all of the listed competencies in problem solving situations, and to select and combine techniques appropriate to the problem.

**1. Minimum mathematical competence.** 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; from geometry: axioms, theorems, and proofs of theorems concerning straight lines, triangles, and circles; graphing of linear equations and interpretation of systems of two linear equations; measurement formulas for the perimeter, circumference, area, and volume of common two- and three-dimensional figures.

**2. Intermediate mathematical competence.** The competencies of Level 1, together with: equations, laws of rational exponents, and radicals; additional topics in factoring; zero product rule; setting up and solving quadratic equations; complex numbers; algebra of polynomials and rational expressions; setting up and solving simultaneous linear equations and inequalities; graphing, including linear and quadratic polynomials; definition and application of absolute value and of scientific notation; definition and elementary properties of logarithms.

**3a. Advanced mathematical competence—algebra.** The competencies of Levels 1 and 2, together with: functions: definition, domain, range, algebraic combinations, composition, inverse, symmetries, translations, graphs; theory of polynomial equations, including the remainder and factor theorems; solution of simultaneous linear equations; equivalent and partially equivalent equations and systems of equations; equations solvable by linear and quadratic techniques; exponential and logarithmic functions, equations, and inequalities; nonlinear inequalities; analytic geometry of conic sections; representation of plane curves and regions by equations or inequalities; sequences, sums, and series, including arithmetic and geometric sequences and series; mathematical induction.

**3b. Advanced mathematical competence—trigonometry.** The competencies of Levels 1 and 2, together with: functions: definition, domain, range, algebraic combinations, composition, inverse, and graphs; trigonometric functions of a real number including their basic properties and graphs; trigonometric equations and identities; geometric significance of the trigonometric functions and elementary applications; polar form of complex numbers and DeMoivre's Theorem.

**4. Superior mathematical competence.** Some high schools find it possible to offer additional topics to able, well-prepared students who have already achieved advanced mathematical competence. For example, courses in probability and statistics, analytic geometry, number theory, calculus, or discrete mathematics are suitable for these high school students.

**Advanced placement credit. **A student who has completed a substantial calculus course in high school may earn university degree credit for Math 221 or for 221 and 222 by one of the following: (1) the College Board calculus advanced placement exam; (2) the math department calculus credit by examination exam; or (3) advanced placement tests given in conjunction with calculus courses at certain Wisconsin high schools.

#### Placement

Each entering student (freshman or transfer student not having transfer credit for a specific UW mathematics course indicating student's placement) who will take Mathematics 95, 101, 112, 113, 114, 141, 171, 210, 211, 221, or 275 is required to take the placement examinations in mathematics before enrolling in any of these courses. Placement in a course is not guaranteed on the basis of the high school record; placement in the course appropriate to the student's needs and competence will be made by the Department of Mathematics on the basis of placement scores. Transfer credit does not necessarily "place" a student.

**Expected levels and courses.** A student with three years of high school mathematics may possibly achieve Intermediate Competence and be placed in Math 112, 113, 114, or 171. A student with four years of high school mathematics may possibly achieve Advanced Competence and be placed in Math 210, 211, 221, or 275. Students with three or four years of high school mathematics who do not achieve these competencies will be placed in lower level courses. These achievements are not guaranteed since it depends on the quality of the courses taken in high school.

**Lower-level mathematics**. Students may be placed in Math 95 if the ACT (or SAT) math scores and the mathematics placements scores are low. Math 95 does not carry degree credit. L&S students whose mathematics placement test scores place them in Math 95 must complete that course or a higher-level mathematics course by the time the 30th degree credit is earned.

**Enrollment information**. The enrollment system checks for the appropriate math placement scores and the course prerequisites for the following Math courses: 95, 101, 112, 113, 114, 141, 171, 210, 211, and 221. The system will also check for the prerequisites and appropriate student classification for Math 130, 131 and 132. In addition, students will need math department authorization to enroll in Math 275, 276, 298, 375, or 376.

**Department policy regarding a D grade.** A student should repeat on a refresher basis a math course in which a grade of D is earned and which serves either as a prerequisite to another course or which satisfies a given requirement. A grade of D commonly signifies some achievement but usually denotes a weak foundation upon which to build subsequent course work. **Students should be aware of the policy that the new grade will not average into the major GPA. See L&S Policy regarding Refresher Work.**

#### Satisfaction of General Education Requirements

The course Math 141 or any 3-credit mathematics course numbered 112 or above is sufficient to satisfy the Quantitative Reasoning–Part A General Education requirement. Students may be exempted from Part A by high school course work or placement tests. Any 3-credit mathematics course numbered 200 or above is sufficient to satisfy the Quantitative Reasoning–Part B General Education requirement.

**Students who first matriculated at a college or university before May 20, 1996, **should consult their DARS report and major advisor for information regarding L&S requirements for Basic Composition and English Proficiency in the Major.

#### Notes on Courses

Detailed course descriptions can be found in the math department course guide.This additional information is provided on the math department's Course Description page.

##### LOWER-LEVEL MATH

Note that Math 95 does not carry degree credit but counts 3 credits for determining fees and a student's semester study load. L&S students whose mathematics placement test scores place them in Math 95 must complete that course or a higher-level mathematics course by the time the 30th degree credit is earned.

##### ELEMENTARY

**Note on Math 130, 131, 132, 135, 136 and 138:** Courses 130, 131, 132,135, 136 and 138 are designed for future teachers and are open only to students in the School of Education and in the School of Human Ecology. These courses may not be used for satisfaction of degree requirements within the College of Letters and Science.

##### INTERMEDIATE AND ADVANCED

**Calculus Sequences.** The Math 221–222 sequence is the first two semesters of the standard three-semester calculus sequence, completed with 234, which is normally required for all higher level math courses and should be taken by those preparing for major study in mathematics, the physical sciences, computer sciences, or engineering. It is also recommended for students in the social and life sciences who may want a more substantial introduction to calculus than is offered in the Math 211–213 sequence. Note that some biological sciences and economics programs require Math 221–222. The Math 211–213 sequence does not prepare the student for higher-level mathematics courses and does not provide adequate math background for some courses in related fields. Transferring from the 211–213 sequence into the 221–222–234 sequence is usually quite awkward. Math 211 is not adequate preparation for Math 222 and students may not earn credit for both Math 211 and 221; students may not earn full degree credit for both Math 213 and 222 or for both 213 and 234.

The sequence Math 171–217 is offered to provide a single sequence integrating the pre-calculus material of Math 114 with the content of Math 221. The honors calculus sequence, Math 275–276, is offered to provide a more rigorous presentation of standard calculus topics covered in Math 221–222.

**Pre-Business Mathematics Requirements.** The requirements vary within the Wisconsin School of Business. Interested students should obtain more detailed information from the business school.

**Courses that count toward the 15 credits of upper-level work in the major:** Mathematics courses numbered above 306 (except for Math 425) taken in residence count toward this requirement.

**Honors Courses.** In advanced mathematics courses, honors sections or sections in which honors credit is available normally will be offered in the following rotation: Sem I: 341!, 521!, 541!; Sem II: 341!, 522!, 542!. Math 681 and 682 are taken for honors credit. In addition, students may enroll for honors credit in most 4XX, 5XX and 6XX level courses, as denoted with "%," but he/she will need to reach an agreement with the instructor as to which extra work the student will be asked to complete*. *A graduate course will automatically carry honors credit.

**Admission to honors courses.** In order to be admitted to an honors section or to enroll for honors credit a student must have a 3.5 average in previous mathematics courses numbered 221 and above. A student is neither required to be in the honors program nor to be honors major in order to enroll in an honors section except for Math 681 and 682. Students will need math department authorization to enter the sequence Math 275, 276, 375, 376.