**Ruth Berger** (department head)

The department offers two majors: mathematics and mathematics/statistics.

Mathematics is the study of numbers, measurements, patterns, shapes, equations, relations, functions, change, symmetry, structure, sets and operations; the modeling of physical phenomena to better understand and predict nature; the development of theorems from accepted axioms through logical proof. It is abstract and applied, theoretical and experimental. Mathematics is perhaps the oldest academic discipline, yet mathematics is the primary language and theoretical foundation of modern technology. It is an extremely versatile major. Mathematics majors are encouraged to explore applications of mathematics in other disciplines, and it is a popular second major for students pursuing advanced degrees.

Statistics is the science of reasoning from uncertain empirical data. Statisticians build mathematical models to solve problems in business, the natural sciences and the social sciences. The intent of the mathematics/statistics major is to provide adequate preparation to attend graduate school in statistics or to pursue a career such as actuary, data scientist, statistical analyst, etc.

**Requirements for majors:**

**Mathematics major.** MATH 220, MATH 240, and MATH 253; CS 150 or CS 160; MATH 215, MATH 322, or MATH 327; with a minimum of eight courses (32 credits) in mathematics numbered 200 or above, including at least three courses (12 credits) in mathematics numbered 300 or above. Writing requirement completed with MATH 220 or MATH 240. (No more than two of MATH 215, MATH 271, MATH 322, MATH 327, MATH 328 can count toward the mathematics major.)

**Mathematics/Statistics major.** MATH 220, MATH 240, and MATH 253; CS 150 or CS 160; MATH 271, MATH 322, MATH 327, and MATH 328; MATH 454 or DS 320. Writing requirement completed with MATH 220 or MATH 240. (A student may not major in both mathematics and mathematics/statistics.)

**Mathematics minor.** At least five courses (20 credits) in mathematics, including MATH 220, MATH 240 and three additional courses, of which two are numbered 200 or above.

Suggested electives for majors planning careers in the following areas:

- Graduate study in mathematics: MATH 351, MATH 454, MATH 456, MATH 471, MATH 472.
- Graduate study in statistics: A mathematics/statistics major with both MATH 454 and DS 320. Consider additional courses in computer science, data science, or a science discipline.
- Secondary education: MATH 360, MATH 365, MATH 471. See education department for secondary education minor requirements.
- Actuarial science: A mathematics/statistics major, ECON 247, ECON 248, MGT 353, MGT 365, and more courses in computer science or data science.
- Computer or data science: MATH 327, MATH 328, MATH 471.
- Management (accounting, economics): MATH 327, MATH 454.
- Science: MATH 327, MATH 328, MATH 351, MATH 452, MATH 453, MATH 454, MATH 456.
- Applied statistics: A mathematics/statistics major with DS 320 and other courses in computer science and data science.

NOTE: Students earning a C or below in MATH 220 or MATH 240 are advised not to take 300+ level courses.

View program learning goals for an explanation of learning outcomes in Mathematics and Mathematics/Statistics.

The mathematics department placement procedure uses high school records, scores on outside standardized tests, and a placement test in mathematics as a basis for a recommendation. MATH 110 and MATH 115 are designed for students who will not be taking calculus. MATH 123 is only for students who major in elementary education. Students with good algebra and trigonometry skills should begin with the traditional Calculus I course, MATH 151. Students who need calculus for their major but also need a review of algebra and trigonometry should take the MATH 140 and MATH 141 sequence. Students whose math placement suggests they require more in-depth review of algebra should consider completing MATH 100 before registering for MATH 140. Students who have completed a year of calculus in high school and perform well on the Advanced Placement A/B Exam or the calculus portion of the mathematics placement test should start in Calculus II, MATH 152. Students who perform well on the Advanced Placement B/C Exam should start in MATH 220 or MATH 240.

- A student will receive credit for MATH 151 with a score of 4 or 5 on the AP Calculus A/B Exam. A student will receive credit for MATH 151 and MATH 152 with a score of 4 or 5 on the AP Calculus B/C Exam. A student will receive credit for MATH 151 with an IB Mathematics HL score of 4-7.
- A student who places into MATH 152 will receive, upon completion of the course with a grade of C or better, Luther College credit for the preceding calculus course, MATH 151 (if credit has not already been awarded through AP or transfer credit).
- A student who places into MATH 240 may petition the head of the mathematics department, upon completion of the course with a grade of C or better, to receive Luther College credit for the preceding calculus course(s) MATH 151 and/or MATH 152 (if credit has not already been awarded through AP or transfer credit). Approval of the petition will depend upon whether the student has previously covered the content of MATH 151 and/or MATH 152.
- A student who achieves a score of 4 or 5 on the AP Statistics Exam will receive credit for MATH 115.

NOTE: AP credit for MATH 115, MATH 151, or MATH 152 satisfies the all-college requirement for quantitative perspective (QUANT).

- 2 hours

This course is focused on strengthening algebraic and quantitative skills required for success in science, economics, or business majors. By preparing students for the first semester of Calculus, this course is appropriate for those desiring an entry level college mathematics course before completing MATH 140 in the following semester. Topics include simplifying mathematical expressions, functions and graphs, solving polynomial/rational equations in one variable, exponents, quantitative reasoning and mathematical models.

- 4 hours
- Fulfills: Quantitative

Quantitative literacy plays an important role in an increasing number of professional fields, as well as in the daily decision-making of informed citizens in our changing society. This course is designed to improve students' quantitative reasoning and problem-solving skills by acquainting them with various real-world applications of mathematical reasoning, such as fair division, voting and apportionment, probability, finance, sets and logic. This course is recommended for students who wish to take a non-calculus-based mathematics class as they prepare for their lives as informed members of a larger world. Prerequisite: high school algebra.

- 4 hours
- Fulfills: Quantitative

The course uses data sets from the social and natural sciences to help students understand and interpret statistical information. Computer software is used to study data from graphical and numerical perspectives. Topics covered include descriptive statistics, correlation, linear regression, contingency tables, probability distributions, sampling methods, confidence intervals, and tests of hypotheses. This class does not count towards the mathematics major or minor or the mathematics/statistics major. Students who earn credit for BIO 256, MGT 150, MATH 215, PSYC 350, or SOC 350 may not earn credit for MATH 115. Prerequisite: high school algebra.

- 4 hours
- Fulfills: Quantitative
- Prerequisites: One year of high school algebra, one year of high school geometry, EDUC 185, Elementary Education major.

This course provides pre-service K-8 teachers a strong foundation in the mathematics content areas as described in NCTM's Principles and Standards for School Mathematics. The content standards include: Number and Operations, Algebra, Geometry, and Measurement. This course will engage students in standards-based mathematics learning to prepare them for the pedagogical practices they will learn in EDUC 325.

- 4 hours
- Fulfills: Quantitative

MATH 140 and MATH 141 cover all the material in MATH 151; Calculus I, while concurrently reviewing precalculus material. Algebraic and graphical representations of functions including: polynomial, rational, exponential, and logarithmic; techniques of solving equations and inequalities; modeling with various functions. An introduction to calculus concepts such as instantaneous rates of change, limits, derivatives, continuity, and applications of derivatives. Prerequisite: a suggested placement.

- 4 hours
- Fulfills: Quantitative
- Prerequisites: MATH 140

Continuation of topics of MATH 140, including trigonometric functions, derivatives, chain rule, Riemann sum approximation for integrals, definite integrals, antiderivatives, and applications. (Students who earn credit for MATH 141 may not earn credit for MATH 151.)

- 4 hours
- Fulfills: Quantitative

Topics include rates of change, functions, limits, continuity, derivatives, optimization, curve sketching, implicit differentiation, the mean value theorem and applications; antiderivatives, definite integrals, and integration by substitution. (Students who earn credit for MATH 151 may not earn credit for MATH 140 or MATH 141). Prerequisite: three years of high school mathematics including algebra, trigonometry, and geometry, and a suggested placement.

- 4 hours
- Fulfills: Quantitative
- Prerequisites: MATH 141 or 151 or suggested placement

Applications of the definite integral, techniques of integration, separable differential equations, series and tests for convergence, and Taylor series.

- 4 hours
- Fulfills: Quantitative
- Prerequisites: MATH 141 or MATH 151

An introduction to statistics and data analysis for math and science majors who have already taken calculus. Topics include numerical and graphical descriptions of data, regression, probability, sampling distributions, confidence intervals, and hypothesis testing. Students who earn credit for MATH 115, BIO 256, MGT 150, PSYC 350, or SOC 350 may not earn credit for MATH 215.

- 4 hours
- Fulfills: Quantitative
- Prerequisites: MATH 152

This course introduces students to standard logical operations (and, or, implies), set operations (union, intersection, complement, Cartesian product, power set), quantifiers (for every, there exists), and properties of functions. These logical foundations are used to understand and produce rigorous mathematical proofs, applying the methods of direct proof, proof by contrapositive, proof by contradiction, cases, and induction. At the end of the course these tools are used to understand cardinalities of infinite sets.

- 4 hours
- Fulfills: Quantitative
- Prerequisites: MATH 152

Theory, computation, abstraction, and application are blended in this course, giving students a sense of what being a mathematics major is all about. Assignments will include computations to practice new techniques and proofs to deepen conceptual understanding. This course starts by solving systems of linear equations, views matrices as linear transformations between Euclidean spaces of various dimensions, makes connections between algebra and geometry, and then extends the theory to more general vector spaces. Topics include matrix algebra, vector spaces and subspaces, linear independence, determinants, bases, eigenvalues, eigenvectors, orthogonality, and inner product spaces.

- 4 hours
- Fulfills: Quantitative
- Prerequisites: MATH 240

The tools of calculus are developed for real-valued functions of several variables: partial derivatives, tangent planes to surfaces, directional derivatives, gradient, maxima and minima, double and triple integrals, and change of variables. Vector-valued functions are also studied: tangent and normal vectors to curves in space, arc length, vector fields, divergence and curl. The fundamental theorem of calculus is extended to line and surface integrals, resulting in the theorems of Green, Stokes, and Gauss, which have applications to heat conduction, gravity, electricity and magnetism.

- 4 hours
- Fulfills: Quantitative
- Prerequisites: MATH 141 or MATH 151

This course will examine a variety of ways that mathematical methods can be brought to bear on biological phenomena. Topics chosen will depend on student interest, and may include food webs, disease etiology, gene regulatory networks, disease transmission dynamics, phylogeny, metabolism, natural selection in haploid and diploid populations, and others. Mathematical methods used may be drawn from graph theory, linear algebra, differential equations, or others, but no previous familiarity in these areas will be assumed.

- 4 hours
- Fulfills: Quantitative
- Prerequisites: MATH 152

Axioms and laws of probability, conditional probability, combinatorics, counting techniques, independence, discrete and continuous random variables, mathematical expectation, discrete probability distributions, continuous probability distributions, functions of random variables, joint probability distributions and random samples, statistics and their distributions, central limit theorem, distribution of a linear combination of random variables.

- 4 hours
- Fulfills: Quantitative
- Prerequisites: MATH 271

Building on probability theory, learn the theory and foundations of statistical inference, a set of methods for drawing conclusions from data. Topics selected from sampling distributions of the mean, standard deviation and proportion, theory of estimation, methods of point estimation, hypothesis testing, large and small sample confidence intervals, Frequentist and Bayesian inference for means, proportions and variances; and distribution free procedures.

- 4 hours
- Fulfills: Quantitative
- Prerequisites: MATH 115, MATH 215, MATH 322, BIO 256, PSYC 350, SOC 350 or MGT 150

Explore methods of regression modeling, with applications in different fields of inquiry, including science, business, and the humanities. Topics selected from: Least square estimates, simple and multiple linear regression, hypothesis testing and confidence intervals for linear regression models, prediction intervals. Analysis of Variance (ANOVA), model diagnostics, multi-collinearity, influence analysis, logistic regression, tree regression, and time series analysis.

- 4 hours
- Fulfills: Quantitative
- Prerequisites: MATH 327

Statistical experimental design is a set of methods for designing and analyzing multi-factor experiments that maximizes the amount of information obtained given a set of experimental resources. Topics selected from: experimental factors, randomization, blocking, interaction effects, analysis of variance methods, fixed and random effects, repeated measures, factorial and response surface designs.

- 4 hours
- Fulfills: Quantitative
- Prerequisites: MATH 240

Differential equations is an area of theoretical and applied mathematics with a large number of important problems associated with the physical, biological, and social sciences. Analytic (separation, integration factors, and Laplace transforms), qualitative (phase and bifurcation diagrams), and numerical (Runge-Kutta) methods are developed for linear and nonlinear first- and higher-order single equations as well as linear and nonlinear systems of first-order equations. Emphasis is given to applications and extensive use of a computer algebra system.

- 4 hours
- Fulfills: Quantitative
- Prerequisites: MATH 240

Why is it so difficult to make accurate predictions about seemingly chaotic physical systems like weather? This course explores the behavior of nonlinear dynamical systems described by iterated functions. A variety of mathematical methods, including computer modeling, is used to show how small changes in initial conditions can drastically change the future behavior of the system. Topics will include periodic orbits, phase portraits, bifurcations, chaos, symbolic dynamics, fractals, Julia sets, and the Mandelbrot set. Offered alternate years.

- 4 hours
- Fulfills: Quantitative
- Prerequisites: MATH 220

In this course we will survey a wide variety of topics in combinatorics, an area of mathematics which focuses on understanding arrangements of objects, including things like permutations and combinations, but also more rigid structures like Sudoku grids. Combinatorialists are interested in questions such as how many arrangements of a particular type exist, what sorts of structure those arrangements have, and sometimes if any such arrangements exist at all. Topics in this course will include: combinations, permutations, the multiplication principle, the Binomial Theorem, the pigeonhole principle, the principle of inclusion and exclusion, derangements, Latin squares, graphs, and design theory.

- 4 hours
- Fulfills: Quantitative
- Prerequisites: MATH 220 or MATH 240

This course gives an introduction to the wide and diverse field of number theory. Topics may include: divisibility theory in the intergers, prime numbers, the Euclidean algorithm, solutions of Diophantine equations, congruences, Euler's theorem, algorithmic number theory, public key crytography, quadratic reciprocity, analytic number theory and the Riemann Hypothesis.

- 4 hours
- Fulfills: Quantitative
- Prerequisites: MATH 220 and MATH 240

This course follows the historical development of geometry, including the important question of which parallel postulate to include. This is a proof-oriented course focusing on theorems in plane Euclidean and hyperbolic geometry, with some mention of elliptic geometry. We examine the development of a lean set of axioms, (incidence, betweenness, congruence, continuity) and investigate which theorems about points and lines can be derived using them.

- 4 hours
- Fulfills: Quantitative
- Prerequisites: MATH 351

An introduction to initial and boundary value problems associated with certain linear partial differential equations (Laplace, heat and wave equations). Fourier series methods, including the study of best approximation in the mean and convergence, will be a focus. Sturm-Liouville problems and associated eigenfunctions will be included. Numerical methods, such as finite difference, finite element and finite analytic, may be introduced, including the topics of stability and convergence of numerical algorithms. Extensive use of a computer algebra system.

- 4 hours
- Fulfills: Quantitative
- Prerequisites: MATH 351.

This course will be devoted to developing mathematical methods useful in the physical sciences. Topics may include dimensional analysis and scaling, perturbation methods, calculus of variations and Hamilton's principle, boundary value problems, Green's functions, and integral equations.

- 4 hours
- Fulfills: Quantitative
- Prerequisites: MATH 220 and MATH 240

The course studies functions of a real variable and examines the foundations of calculus, with an emphasis on writing rigorous analytical proofs, and follows the historical development of analysis beginning with a rigorous axiomatic description of the real numbers and proceeding through the topology of the reals, sequences, series, limits, continuity, pointwise and uniform convergence, differentiation, Taylor series, and integration.

- 4 hours
- Fulfills: Quantitative
- Prerequisites: MATH 253

What happens when calculus is extended to functions of a complex variable? Geometry and analysis combine to produce beautiful theorems and surprising applications. Topics include complex numbers, limits and derivatives of complex functions, Cauchy-Riemann equations, harmonic functions, contour integrals, the Cauchy integral formula, Taylor and Laurent series, residues, and conformal mappings with applications in physical sciences. Offered in alternate years.

- 4 hours
- Fulfills: Quantitative
- Prerequisites: MATH 220 and MATH 240

Real numbers and integers satisfy many nice properties under addition and multiplication, but other sets behave differently: matrix multiplication and composition of functions are noncommutative operations. Which properties (associativity, commutativity, identity, inverses) are satisfied by operations on sets determine the basic algebraic structure: group, ring, or field. The internal structure (subgroups, cosets, factor groups, ideals), and operation-preserving mappings between sets, (isomorphisms, homomorphisms)are examined. Emphasis is on theory and proof, although important applications in symmetry groups, cryptography, and error-correcting codes may also be covered.

- 1, 2, or 4 hours
- Fulfills: Quantitative
- Prerequisites: MATH 471

Topics may include simple groups, Sylow theorems, divisibility in integral domains, generators and relations, field extensions, splitting fields, solvability by radicals, Galois theory, symmetry and geometric constructions. Offered on demand.

- 1, 2, or 4 hours