 # Student Learning Goals

Mathematics is the language of science. In Galileo’s words:

Philosophy is written in this grand book, the universe, which stands continually open to our gaze. But the book cannot be understood unless one first learns to comprehend the language and read the characters in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures, without which it is impossible to understand a single word of it. Without those, one is wandering in a dark labyrinth.

Mathematics majors learn the internal workings of this language, its central concepts and their interconnections. These involve structures going far beyond the geometric figures to which Galileo refers. Majors also learn to use mathematical concepts to formulate, analyze, and solve real-world problems. Their training in rigorous thought and creative problem-solving is valuable not just in science, but in all walks of life.

### Learning Goals for Mathematics Majors

The Mathematics Department offers three majors:

(1) the Major in Mathematics,

(2) the Major in Mathematics with a Teaching Concentration, and

(3) the Major in Applied Mathematics.

Most of the learning goals for these majors are common to all three.

General skills

By the time of graduation, majors should have acquired the following knowledge and skills:

1. Analytical skills
1. An understanding of the basic rules of logic.
2. The ability to distinguish a coherent argument from a fallacious one, both in mathematical reasoning and in everyday life.
3. An understanding of the role of axioms or assumptions.
4. The ability to abstract general principles from examples.
2. Problem-solving and modeling skills (important for all, but especially for majors in Applied Mathematics)
1. The ability to recognize which real-world problems are subject to mathematical reasoning.
2. The ability to make vague ideas precise by representing them in mathematical notation, when appropriate.
3. Techniques for solving problems expressed in mathematical notation.
3. Communication skills
1. The ability to formulate a mathematical statement precisely.
2. The ability to write a coherent proof.
3. The ability to present a mathematical argument verbally.
4. Majors in Mathematics with a Teaching Concentration should acquire familiarity with techniques for explaining K-12 mathematics in an accessible and mathematically correct manner.
1. Sufficient experience in mathematical language and foundational material to be well-prepared to extend one’s mathematical knowledge further through independent reading.
2. Exposure to and successful experience in solving mathematical problems presenting substantial intellectual challenge.

Remarks:

• The skills listed above are not to be acquired solely while a major. Instead, incoming majors are expected to have many of the above skills at some level already from their K-12 education (e.g., some understanding of axioms and proofs from high school geometry). This is essential, because mathematics is largely a cumulative subject.
• The skills above are generally not taught directly. Instead they are acquired by students through experience, in the context of studying specific mathematical topics, such as those in the following section.

Subject-specific knowledge

Graduating mathematics majors should have learned the basics of “mathematical culture” in the following areas:

 (CA) Calculus and analysis: For real analysis, construction of the real numbers, rigorous derivation of basic calculus facts (e.g., Mean Value Theorem, Taylor’s Theorem, Fundamental Theorem of Calculus, etc.), computing limits and integrals. Multivariable calculus, including Lagrange multipliers, Green’s Theorem, Divergence Theorem, Stokes’ Theorem. For complex analysis, derivation of Cauchy-Riemann equations, Cauchy’s Theorem, residues for contour integrals, Liouville’s Theorem and its application to the Fundamental Theorem of Algebra. Courses: Math 1A, 1B, 53, 104, 105, 185 (LA) Linear algebra: Matrices and linear transformations, vector spaces, determinants, eigenvectors and eigenvalues, characterizations of invertible matrices, inner products, normal forms, and applications to linear differential equations, Fourier analysis, linear programming. Courses: Math 54, 110, 118, 170 (ANC) Algebra, number theory, combinatorics: Permutations and their connection to determinants, the Euclidean algorithm, Fermat’s Little Theorem and Euler’s generalization, factorization of polynomials over C and R, impossibility of certain straightedge-and-compass constructions. Basic theory of groups, rings and fields. Elementary enumeration methods, generating functions, discrete probability theory, etc. Courses: Math 55, 113, 114, 115, 116, 172

In addition, math majors take elective courses from among the following subject areas:

 (GT) Geometry and topology: Basic theory of curves and surfaces, Gauss and mean curvature, isoperimetric inequality, Gauss-Bonnet Theorem. Topological invariants. Elementary algebraic geometry. Courses: Math 130, 140, 141, 142, 143 (AM) Applied mathematics and modeling: Modeling physical and other phenomena with ordinary and partial differential equations. Approximation and optimization techniques, linear programming, game theory. Numerical/computational methods. Courses: Math 118, 121A, 121B, 123, 126, 127, 128A, 128B, 170, 189 (L) Logic: Syntax and semantics for formal languages, Gödel’s Completeness theorem. Computable functions, the unsolvability of the halting problem, Gödel’s incompleteness theorems. Transfinite cardinals, transfinite induction, and the Axiom of Choice. Courses: Math 125A, 135, 136

 (TC) Majors in Mathematics with a Teaching Concentration are expected to acquire a professional level of mastery in elementary arithmetic, geometry, and algebra. Courses: Math 151, 152, 153 (AM) Majors in Applied Mathematics must in addition complete a three course elective cluster concerning some advanced application of mathematics.

### Curriculum Map

The following tables show how both sets of learning goals are addressed in the mathematics curriculum.

General Skills

Most of the goals relating to general skills are addressed in all courses. For example, the ability to formulate precise mathematical statements and reason logically with them is a key skill used and taught in all courses. However, some skills are more naturally learned or re-inforced, in certain courses. Some connections are indicated in the following table.

 General Skill Most Relevant Courses (1) Analytical Skills (a) An understanding of the basic rules of logic. 55 (b) Distinguishing cogent arguments from fallacious ones. 55 (c) Understanding the role of axioms. 55, 125 (d) Abstracting general principles from examples. 1A, 1B, 53, 54, 55 (2) Problem Solving and Modeling Skills (a) Recognizing real-world problems suitable for mathematical reasoning. 1A, 1B, 53, 54, 128 (b) Making vague ideas precise using mathematics. 1A, 1B, 53, 54, 128 (c) Problem-solving techniques. 1A, 1B, 53, 54, 128 (3) Communication Skills (a) Ability to formulate a mathematical statement precisely. 55, 104, 113 (b) Ability to write a coherent proof. 55, 104, 113 (c) Ability to present a mathematical argument verbally. 1A, 1B, 53, 54, 55 (d) Majors in Mathematics with a Teaching Concentration should acquire familiarity with techniques for explaining K-12 mathematics in an accessible and mathematically correct manner. 151, 152, 153

Once again, however, most of the general skills are learned and used in all courses.

Subject-specific knowledge

The subject-specific knowledge is more easily connected to particular courses. This is done in the following table.

 Topics Most Relevant Courses (1) Calculus and Analysis 1A, 1B, 53, 104, 105, 185 (2) Linear Algebra 54, 110, 118, 170 (3) Algebra, Number Theory, Combinatorics 55, 113, 114, 115, 116, 172 (4) Geometry and Topology 130, 140, 141, 142, 143 (5) Applied Mathematics and Modeling 118, 121A, 121B, 123, 126, 127, 128A, 128B, 170, 189 (6) Logic 125A, 135, 136 (7) Teaching Concentration 151, 152, 153