## Why take courses in this discipline?

Studying the mathematical sciences develops logical thinking, quantitative reasoning, and creative problem-solving skills. Mathematical techniques play an essential role in all of the sciences, as well as any subject that relies on quantitative information.

Our faculty members have strong **research careers** and provide opportunities for students to engage in research of their own. Majors **pursue careers** in many areas, including law, software engineering, actuarial analysis, teaching, and research. The curriculum also prepares students for graduate study in fields like mathematics, statistics, computer science, physics, engineering, and economics.

## How does the discipline contribute to the liberal arts?

Courses in mathematics focus on **quantitative reasoning**, **writing**, and **communication**.

## What kinds of questions are asked in this discipline?

Mathematicians investigate patterns, structure, and randomness in both abstract and applied settings. Using logical reasoning and creative problem-solving skills, mathematicians establish truth through **rigorous proof**, for which clear writing and attention to detail are essential. Here are a few questions that are asked in the field of mathematics:

- If we detect a possible pattern, how can we know that it continues forever?
- Can we classify all mathematical objects (such as numbers, geometric shapes, functions, etc.) of a certain type?
- Can we build one model that unifies and explains many different phenomena (either in the natural world or in mathematics itself)?

## How does a student get started?

**Note:** In fall 2024, Math 123 will not be taught. Instead, the department is offering a special topics class, MAT 195 - Introduction to Mathematical Practice. Some new students will be placed in this course based on the placement diagnostic they completed this summer. New students should read the placement letter sent to them in August and consult with their faculty adviser.

*Fall registration tip: Mathematics courses often fill to capacity during first-year registration. If a student does not prioritize a math course during registration, that student will likely be on a waiting list and will have to wait until the spring term to take the course. Note that this strategy is especially important for MAT 123-124; MAT 123 is a fall-only course, and MAT 124 is a spring-only course. Many courses in the Science and Social Studies Divisions have a math prerequisite.*

Mathematics placement at Grinnell College is generally done through an online diagnostic test. Incoming students will receive an email in early July with information about how to access the test, and all students must complete it by the end of July. Based on the results, the department will send each student an email in mid-August with guidance about their appropriate first mathematics course. If a student has additional questions or concerns about the placement given in that email, they should talk with a mathematics faculty member during New Student Orientation (information about the location and times for such meetings will be in the placement email).

**In mathematics, most students start in one of the following courses:**

**MAT 123-124, Functions and Differential Calculus (123, fall only)**, and **Functions and Integral Calculus (124, spring only)**, is a year-long sequence that covers the content of one semester of MAT 131 (see below for topics covered in MAT 131). The structure and pace of MAT 123-124 are different from MAT 131. In addition to the core calculus material, MAT 123-124 includes a thorough development of several precalculus topics from algebra and trigonometry. *The two semesters together cover the content of MAT 131 and serve as a prerequisite to MAT 133.*

Here are some examples of topics covered more thoroughly in MAT 123-124 than in MAT 131:

- functions y = f(x)
- the relationship between a formula for a function and its graph
- manipulation and simplification of complex algebraic expressions
- properties of exponents and logarithms
- trigonometric functions and identities

Other topics covered in MAT 123-124 are listed under MAT 131.

**MAT 131, Calculus I,** is an introductory course in calculus that moves at a fast pace. MAT 131 assumes high proficiency in high school algebra and trigonometry but does not require previous exposure to calculus. The topics covered in MAT 131 and during the year of MAT 123-124 include

- a review of algebraic, trigonometric, and exponential functions, their graphs, and their inverses
- the derivative of a function
*y = f(x)* - limits of functions
*y = f(x)* - interpretation of the derivative as an instantaneous rate of change
- computing derivatives in two ways: with the limit definition and with the power, sum, product, quotient, and chain rules
- maximization and minimization (optimization) problems using the derivative
- graphing functions by hand without the use of a graphing calculator using the derivative
- Riemann sums—to approximate the area between the graph of a function
*y =f(x)*and the*x*-axis - antiderivatives, definite integrals, and the Fundamental Theorem of Calculus
*u*-substitution to compute antiderivatives and definite integrals- applications of integration: areas, volume, and work
- inverse functions

**MAT 133, Calculus II**, is the sequel to MAT 123-124 or MAT 131. If you are comfortable with the material listed under MAT 131 and have not learned calculus for functions *z = f(x,y)* and their higher-dimensional analogs, then MAT 133 is a good starting point for you.

Topics studied in MAT 133 include

- parametrization and derivatives or parametric functions
- the geometry of 3-space: lines, planes, and their intersections
- vectors, the dot product (inner product), and orthogonality
- partial derivatives of functions
*z = f(x,y)*and their higher-dimensional analogs - gradients; directional derivatives; the Chain Rule for functions of several variables
- graphs of surfaces defined by
*z = f(x,y)* - level curves and their relation to orthogonality
- maximization/minimization (optimization) problems in higher dimensions; Lagrange multipliers
- Riemann sums for functions
*z = f(x,y)*and their higher-dimensional analogs - double- and triple-integrals (with a study/review of single-variable integration techniques)
- volumes of regions defined by inequalities.

*Notes:*

- Many other colleges use “Calculus III” or even “Calculus IV” to describe the material we study in MAT 133.
- If you have taken AP Calculus BC or its equivalent, you may have studied sequences and series, but these topics are not part of MAT 133; they are studied in MAT 220 (Differential Equations) or more theoretically in MAT 316 (Foundations of Analysis; a requirement for the major). AP Calculus BC does not cover most of the material in MAT 133.
- Only
*u-*substitution

**MAT 215, Linear Algebra**, builds on the knowledge about vectors and higher dimensions that begins in MAT 133. From a computational perspective, the subject studies systems of linear equations and matrix algebra. From a more abstract perspective, it studies linear transformations between vector spaces. This course balances the computational, conceptual, and theoretic aspects of the subject, and explores their connections. In addition to the core topics (matrices, vector spaces, eigenvectors, dimension), the course develops mathematical reasoning and writing skills.

Completion of AP Calculus BC or its equivalent is not sufficient preparation for MAT 215. If you have taken a linear algebra course that has primarily focused on computational techniques, matrix manipulation, and solving systems of linear equations, you will probably need to take MAT 215 at Grinnell if you plan to major in mathematics or take courses that have MAT 215 as a prerequisite.

## AP/IB Credit

Students with any of the following exam scores will receive credit for MAT 131:

- A score of 4 (or higher) on the AP Calculus AB exam.
- A score of 3 (or higher) on the AP Calculus BC exam.
- A score of 5 (or higher) on the IB Mathematics: Applications and Interpretation exam, or the IB Mathematics: Analysis and Approaches exam.

As mentioned in the course descriptions above, the overlap between MAT 133 and the material on the BC exam is small, so the BC exam does *not* give credit for MAT 133.

A score of 5 on the IB mathematics exam can count as MAT 123, 124, or 131.

## Courses in Mathematics

Regular 200-Level Courses

- Linear Algebra (MAT 215)
- Bridges to Advanced Mathematics (MAT 218 or MAT 222)
- Differential Equations (MAT 220)

Regular 300-Level Courses

- Foundations of Analysis (MAT 316)
- Foundations of Abstract Algebra (MAT 321)
- Advanced Topics in Analysis (MAT 317)
- Advanced Topics in Algebra (MAT 322)
- Mathematical Modeling (MAT 306)
- Numerical Analysis (MAT 313)
- Advanced Topics in Applied Mathematics (MAT 314)
- Probability and Statistics (MAT/STA 335-336)

## Structure of a Mathematics Major

The requirements of a mathematics major are summarized in the diagram **Mathematics Course Sequence**. Notice that several courses (including some required core courses) require a significant prerequisite chain.

## Off-Campus Study

Majors who are interested in taking several mathematics courses during off-campus study should consider the Budapest semester in mathematics program. Courses taken off-campus rarely count toward the major. Please consult with a faculty member in the department for more information.

## Contributions to Other Majors/Concentrations

Courses in mathematics and statistics contribute to the following majors:

**biology****chemistry****biological chemistry****computer science****economics****general science****physics****political science****psychology**

Courses in mathematics and statistics contribute to the following concentrations**:**

## Department Events and Opportunities

Opportunities to work on challenging problems are presented in the Putnam examination and the Mathematical Contest in Modeling, both of which are national mathematics competitions. Many students present talks in the mathematics and statistics student seminar. Visiting lecturers extend the curriculum beyond the classroom, as do opportunities for students to do summer research.