# Mathematics

The learning objectives of all undergraduate programs with a focus in Mathematics are based on this observation attributed to Carl Friedrich Gauss: “Mathematics is the Queen of the Sciences … She often condescends to render service to astronomy and other natural sciences, but in all relations she is entitled to the first rank.”

The courses comprising the major have been chosen to ensure the student obtains the fundamental algorithmic and computational skills, incisive thinking skills, and analytical mindset sufficient for participating in any kind of science or for becoming a Mathematician.

Learning objectives for Mathematics major are achieved through training and courses which may be categorized in three ways: **(1) Foundational Skills **courses, **(2) Intellectually Transformative** courses, and **(3) Mathematical Fluency** courses. Foundational Skills courses consist of courses taken by intra- and inter-departmental and inter-college students and are intended to give students the algorithmic and computational skills needed in any discipline regardless of major. These courses develop computational skills and begin to develop symbolic fluency and comfort with abstraction. Intellectually Transformative courses are intended to acclimate the student to the thinking styles and philosophies indigenous to theoretical Mathematics. These courses build on the foundation obtained from the Foundational Skills courses, especially on the abilities engendered by the symbolic manipulations, turning those skills into a fluency with a language useful in all disciplines but with a focus tuned more for the development sought in the final category of courses. The Mathematical Fluency courses introduce the student to the ideas from which the topics in Foundational Skills courses were borne. This training allows the student to succeed in graduate-level courses, and gives them the skills that potential employers value: dynamic problem solving skills coupled with technically precise thinking and analytical skills. Technical communication skills are also developed and tested within the guidelines of rigorous proof.

The objective of all BS degrees is, very briefly, to produce mathematicians poised for industry, or for scientific or academic research.**Student Performance in Courses that Promote Learning Objectives**

**Foundational Skills Courses**

All Mathematics majors develop mastery of foundational algorithmic skills including symbolic manipulation from three major areas: calculus, linear algebra, and differential equations. Calculus studies the mathematics of change and of motion. The skills developed in the courses MATH 1210, Calculus I), MATH 1220 (Calculus II), and MATH 2210 (Multivariable Calculus) prepare all students from the future physicists and engineers to mathematicians for the applications and manipulations of formulas indigenous to their respective disciplines that deal with change and motion. I Linear Algebra students master the solution of systems of equations and abstract the process into something applicable to all disciplines that involve any calculation whatsoever --- from optimization problems such as Linear Programs to projection problems such as Least-Squares curve fitting. Additionally, in MATH 2270 (Linear Algebra), students develop a vocabulary for matrices and a variety of their factorizations that are used in applications from computer graphics and image recognition to solving systems of equations. The course MATH 2280 (Ordinary Differential Equations) prepares students for the applying models based on change and equips with the vocabulary and basic skills needed to find solutions or to identify instances where solutions will be difficult, if not impossible to find. The mastery of the skills developed is assessed in these courses.

**Intellectually Transformative Courses**

It is at the level of Intellectually Transformative courses that the various Mathematics major specializations begin to diverge based on the needs of the courses deemed optimal with respect to the specialization. The transformative courses introduce the Mathematics major to ideas and ways of thinking engendered in the mathematical sciences. Ideas like the power of abstraction are communicated via the language of Algebra, of Sets, or Geometry. Thinking tuned with the philosophy of axiomatic systems is developed via the process of proof. The development of abstraction and proof akin to algebra is developed and assessed in MATH 4310 (Introduction to Algebraic Structures). The language of Set Theory and of Logic is developed and assessed on MATH 3310 (Discrete Math), and that of Geometry in MATH 3110 (Modern Geometry).

**Mathematical Fluency Courses**

The languages and ways of thinking developed in the Intellectually Transformative courses provide the field on which the final learning objectives are played out. In the courses aimed at producing a budding mathematician students develop a deep understanding of the algorithmic skills and computational skills from which the degree grew. Skills with using Logic, Set Theory, Algebra, and the axiomatic methods previously introduced are developed via the study the real number system and functions and operations of calculus in the courses MATH 4200 (Foundations of Analysis), MATH 5210 (Introduction to Analysis I), and MATH 5220 (Introduction to Analysis II). In fact, the theory underpinning all Foundational Skills courses is addressed in the 5000-level courses. The theory of the methods and content in MATH 2270 (Linear Algebra) is addressed in MATH 5340 (Theory of Linear Algebra), that of MATH 1210, 1220, and 2210 (Calculus sequence) and much of MATH 2280 (Differential Equations) is addressed in MATH 4200, 5210, and 5220.

Proficiency in the communication of mathematical arguments, such as proof, is developed in essentially all 5000-level mathematics courses, whence many 5000-level mathematics courses are designated as *Communications Intensive* and contribute the USU’s University Studies program.

The communication, analysis, symbolic computation, manipulation of axioms, and thinking within the confines of logic are all assessed in these courses.

**Student Performance in Courses that Promote Learning Objectives****Foundational Skills Courses**All Mathematics majors develop mastery of foundational algorithmic skills including symbolic manipulation from three major areas: calculus, linear algebra, and differential equations. Calculus studies the mathematics of change and of motion. The skills developed in the courses MATH 1210, Calculus I), MATH 1220 (Calculus II), and MATH 2210 (Multivariable Calculus) prepare all students from the future physicists and engineers to mathematicians for the applications and manipulations of formulas indigenous to their respective disciplines that deal with change and motion. I Linear Algebra students master the solution of systems of equations and abstract the process into something applicable to all disciplines that involve any calculation whatsoever --- from optimization problems such as Linear Programs to projection problems such as Least-Squares curve fitting. Additionally, in MATH 2270 (Linear Algebra), students develop a vocabulary for matrices and a variety of their factorizations that are used in applications from computer graphics and image recognition to solving systems of equations. The course MATH 2280 (Ordinary Differential Equations) prepares students for the applying models based on change and equips with the vocabulary and basic skills needed to find solutions or to identify instances where solutions will be difficult, if not impossible to find. The mastery of the skills developed is assessed in these courses.

**Intellectually Transformative Courses**It is at the level of Intellectually Transformative courses that the various Mathematics major specializations begin to diverge based on the needs of the courses deemed optimal with respect to the specialization. The transformative courses introduce the Mathematics major to ideas and ways of thinking engendered in the mathematical sciences. Ideas like the power of abstraction are communicated via the language of Algebra, of Sets, or Geometry. Thinking tuned with the philosophy of axiomatic systems is developed via the process of proof. The development of abstraction and proof akin to algebra is developed and assessed in MATH 4310 (Introduction to Algebraic Structures). The language of Set Theory and of Logic is developed and assessed on MATH 3310 (Discrete Math), and that of Geometry in MATH 3110 (Modern Geometry).

**Mathematical Fluency Courses**The languages and ways of thinking developed in the Intellectually Transformative courses provide the field on which the final learning objectives are played out. In the courses aimed at producing a budding mathematician students develop a deep understanding of the algorithmic skills and computational skills from which the degree grew. Skills with using Logic, Set Theory, Algebra, and the axiomatic methods previously introduced are developed via the study the real number system and functions and operations of calculus in the courses MATH 4200 (Foundations of Analysis), MATH 5210 (Introduction to Analysis I), and MATH 5220 (Introduction to Analysis II). In fact, the theory underpinning all Foundational Skills courses is addressed in the 5000-level courses. The theory of the methods and content in MATH 2270 (Linear Algebra) is addressed in MATH 5340 (Theory of Linear Algebra), that of MATH 1210, 1220, and 2210 (Calculus sequence) and much of MATH 2280 (Differential Equations) is addressed in MATH 4200, 5210, and 5220.

Proficiency in the communication of mathematical arguments, such as proof, is developed in essentially all 5000-level mathematics courses, whence many 5000-level mathematics courses are designated as

*Communications Intensive*and contribute the USU’s University Studies program.The communication, analysis, symbolic computation, manipulation of axioms, and thinking within the confines of logic are all assessed in these courses.

**Student Evaluations of Courses and Instructors**The standardized USU course evaluation form is conducted in all courses taught by Mathematics and Statistics faculty to allow students to evaluate both the course and the instructor.

**College of Science Interviews and Questionnaire**Each year, the Dean of the College of Science interviews a number of majors from each department in the college. In addition, every student applying for graduation in the College of Science is given a questionnaire to complete. These collectively provide information on general student satisfaction with the degree program, courses, faculty and facilities. This information is collected anonymously and then returned to the department in the summer following graduation.

**Self-Study and External Review**The Department of Mathematics and Statistics periodically conducts a Regents-mandated self-study and external review. The self-study allows the faculty to reassess the program and its direction as well as its goals and objectives. The recommendations made by the external review team are used to modify and improve the program.

**Capstone Experience and Math GRE**

Beginning in 2017-18, the Department of Mathematics and Statistics will formally organize a capstone experience for undergraduates, in order to consistently measure the broader impact of the program on our learning objectives. This capstone experience will be fulfilled by one of the following: (1) a senior thesis on a special topic within the mathematical sciences, resulting in a concise report and oral presentation; (2) preparing and taking the GRE Mathematics Subject Test; or (3) the successful completion of an internship for USU credit.

**Student success in subsequent graduate programs**

A significant proportion of our graduating students in Mathematics and Statistics pursue graduate and professional degrees. Successful admission into such programs and subsequent degree completion can also help to understand how Beginning in 2016-17, these outcomes will be tracked both retrospectively and prospectively, for use in future calibration of our undergraduate programs.

- Student Course Evaluations
- College of Science interviews and questionnaire
- Self-Study and Regents Review
- GRE Mathematics Subject Test results – starting in 2017
- Student success in subsequent graduating programs – data collection underway during 2016-17.