# Mathematics

Mathematics

510.430.2226

mathematics@mills.edu

In view of the pervasive roles that quantitative analysis plays throughout our society, a basic familiarity with the disciplines of mathematics has become an integral part of a liberal arts education. As a college for women, Mills recognizes the importance of encouraging women to study mathematics, and of providing them with the high-quality instruction they need to succeed in these disciplines. Encouraging mathematical literacy is part of the College's continued effort to increase the analytical competence of its women graduates.

Mathematics is an excellent field both for lifetime intellectual interest and for career preparation. Women are becoming increasingly prominent in the field. Recent presidents of both the American Mathematical Society and the Mathematical Association of America have been women. Mathematics also serves as an excellent basis for business, finance, engineering, sciences, teaching, actuarial work, and fields that need highly developed analytical skills, such as law.

Small, interactively taught classes provide students with an ideal environment for learning mathematics. The cross-registration program with UC Berkeley enables outstanding students to take advantage of a wide range of mathematics courses not usually available at a small college.

Note: The basic calculus sequence (MATH 047 Calculus I–MATH 048 Calculus II) begins in the fall. Students who need additional preparation before taking calculus should enroll in MATH 003 Pre-Calculus along with MATH 003L Pre-Calculus Workshop in the spring before beginning MATH 047 Calculus I the following fall. To determine which initial course is appropriate, the student should take the self-placement quizzes offered by the department and consult with mathematics advisors. Students who plan to do further work in mathematics, science, or engineering are advised to continue the calculus sequence by taking MATH 050 Linear Algebra and MATH 049 Multivariable Calculus.

Before declaring a major in mathematics, a student must have completed MATH 047 Calculus I, MATH 048 Calculus II, and MATH 050 Linear Algebra. The grade in each of these courses should be at least a B-. Some exceptions may be allowed upon the recommendation of the department. Students required to declare a major before completing these courses may provisionally declare the mathematics major. The provisional declaration will be revoked if the student does not earn at least a B- in MATH 047 Calculus I, MATH 048 Calculus II, and MATH 050 Linear Algebra. Proficiency in basic logical and problem-solving skills, as determined by the instructor, is required for enrollment in advanced courses.

## Program Goals

- Develop analytical skills and logical reasoning.
- Develop ability to communicate mathematical thoughts in a clear and coherent fashion.
- Develop problem-solving skills, and in particular, develop the ability to handle problems that require multiple steps for their solution.
- Improve quantitative skills.
- Learn to apply a major mathematical theory to solve non-trivial problems in other scientific domains.

**MATH 001: Introduction to Mathematics (4 Credits)**

Mathematical reasoning and problem solving. Emphasis on building mathematical intuition and analytical skills via simplification of problems and inductive discovery methods. Topics are selected from logic, number theory, set theory, geometry, probability, statistics, and graph theory.

**Note(s):** Intended for students with little technical background who wish to acquire a mathematical perspective or prepare for a more advanced course such as MATH 003 or MATH 004. Those taking MATH 001 in preparation for MATH 004 must enroll concurrently in MATH 003L. Also suitable for candidates for teaching credentials.

**Meets the following Gen Ed requirements:** Quantitative and Computational Reasoning

**General Education Goals:**

Quan. & Comp. Reasoning

- Translate problems into the language of mathematics and computer science, and solve them by using mathematical and computational methods and tools (Introduced)
- Students will be able to translate real-life problems and problems from other disciplines into equations, inequalities, graphs, charts, and other mathematical constructs. For example, students will formulate questions about opinion polls and sample averages in terms of chance models involving coin tosses or boxes of tickets, and then apply appropriate equations for standard error in order to answer them; students will also be able to determine when it is appropriate to describe a relationship between variables with a linear equation and how to compute and interpret a correlation coefficient.
- Understand the structure and development of logical arguments (Introduced)
- Students will detect logical pitfalls, such as faulty or incomplete argumentation, conclusions based on false assumptions, and misapplications of statistical methods. Examples of such logical pitfalls include the regression fallacy, Simpson’s paradox, confounding variables, and non-response bias. Through learning indirect methods of argumentation involving reducio ad absurdum, students will be able to explain how an alternative hypothesis can be justified by rejecting a null hypothesis. Students will be able to use data to support or refute persuasive arguments and explain their reasoning. In particular, students will use the concepts of confidence intervals and statistical significance to determine whether an observation could be explained by chance variation.
- Understand the algebraic and graphical properties of important mathematical functions and use these functions in concrete applications (Introduced)
- Students will use charts, graphs, diagrams, and other visual forms to represent functions. Important examples of graphical representations of functions include the normal curve, time series plots, histograms, and regression lines. Students will use functions to predict dependent variables from independent variables and to decide when such predictions are meaningful.
- Understand and apply the fundamental ideas of probability and statistics (Introduced)
- Students will apply the concepts of independence, correlation, random sampling, confidence intervals, and statistical significance, to problems involving medical trials, lotteries and other games of chance, opinion polls, and the census.

**MATH 001L: Intro to Mathematics Workshop (0 Credits)**

**MATH 003: Pre-Calculus (3 Credits)**

A streamlined course designed to prepare students for the calculus sequence (MATH 047–048). Properties and graphs of elementary functions. Emphasis on developing conceptual understanding and problem-solving skills.

**Note(s):** Concurrently, students must enroll in a pre-calculus lab, MATH 003L, designed to strengthen their algebraic skills.

**Meets the following Core requirements:** Quantitative Literacy

**Meets the following Gen Ed requirements:** Quantitative and Computational Reasoning

**Core Goals:**

Quantitative Literacy

- Interpretation: Students will have the ability to explain information presented in mathematical and computational forms. (Introduced)
- Students will explain theorems, problems, and solutions in their own words.
- Students will interpret the graph of a function.
- Representation: Students will be able to convert information into mathematical and computational forms analytically and/or using computational tools. (Introduced)
- Students will identify patterns and use them to structure problem-solving techniques.
- Students will translate real-life problems and problems from other disciplines into mathematical statements and use mathematical methods and specific problem-solving skills to solve these problems in complex, multi-step arguments.
- Analysis: Students will be able to draw appropriate conclusions using mathematical or computational reasoning and understand the limits of such conclusions and the assumptions on which they are based. (Introduced)
- Students will understand a general mathematical argument and then replicate the argument in a specific example.
- Students will understand a mathematical hypothesis, condition, or definition and check that it is satisfied in a specific example.
- Students will construct rigorous logical arguments, using a variety of proof methods and logical reasoning.
- Communication: Students will be able to communicate quantitative ideas in the languages of mathematics, computer science, or quantitative social sciences and will be able to utilize quantitative information in support of an argument. (Introduced)
- Students will verbalize mathematical thoughts clearly.
- Students will communicate in an organized and clear manner their solutions to problems.
- Students will justify their written solutions to problems, using a variety of methods for logical reasoning.

**General Education Goals:**

Quan. & Comp. Reasoning

- Translate problems into the language of mathematics and computer science, and solve them by using mathematical and computational methods and tools (Introduced)
- Problem-solving techniques for a variety of situations will be introduced and applied by the students, including the use of absolute values, the triangle inequality, coordinate transformations of functions such as translations, reflections, and stretchings, and the derivation of trigonometric identities. For instance, students will prove (in a variety of examples) why particular functions fail to meet the criteria for continuity; they will develop the ability to recognize when a limit expression is indeterminate and how to correctly transform it into an expression where the limit exists.
- Understand the structure and development of logical arguments (Introduced)
- Students will analyze in depth mathematical statements about the real number system dealing with sets of ordered pairs, relations, and functions. Using logical reasoning, students will begin to derive defining equations for relations and functions. Examples of logical reasoning will include direct proof and proof by contradiction, and the use of examples and counterexamples when deriving solutions to relevant problems.
- Understand the algebraic and graphical properties of important mathematical functions and use these functions in concrete applications (Introduced)
- Students will correctly manipulate inequalities, algebraic expressions, and equations in one (or more) variables. They will also correctly calculate simple derivatives and use them to find maxima and minima of certain functions and on which intervals these functions are increasing or decreasing. Students will then apply this information to correctly sketch the graph of the function.
- Students will develop quantitative skills and apply them to graphing functions. In particular, students will be able to work with one dimensional sets, intervals, and subsets of these on the real line; in particular, they will compare them with the representations by characteristic functions for those subsets of the real line. Students will be able to draw graphs of functions in two dimensions and overlay them with representations of the relational converses (e. g., when a relation is one-one, the students will identify its converse as an inverse function).
- Students will be able to graphically recognize the convergence or divergence of certain sequences.

**MATH 003L: Pre-Calculus Workshop (2 Credits)**

**MATH 004: Discrete Mathematics I (4 Credits)**

The Discrete Mathematics I–II sequence studies the fundamental mathematical ideas that are used in various disciplines of computer science. Emphasis is placed on problem-solving techniques. Topics are selected from: logic, Boolean algebra, proof techniques such as mathematical induction and proof by contradiction, sums, sets, and the Halting Problem.

**Note(s):** Prerequisites: Strong background in high school mathematics and consent of instructor, or MATH 001 or MATH 003 and their associate workshop MATH 003L.

**Meets the following Core requirements:** Quantitative Literacy

**Meets the following Gen Ed requirements:** Quantitative and Computational Reasoning

**Program Goals:**

Computer Science Program Goals

- Understand and apply the mathematical concepts underlying computer science. (Introduced, Practiced)
- Develop ability to solve computational problems such as finding patterns in texts, parsing a programming language, or computing algebraic functions, using automata.
- Students apply math concepts such as Boolean algebra (in computer Architecture), set theory (in Databases), formal proofs, graph theory (computer networks), finite-state automata (computer networks). Students understand system of linear equality, sensitivity analysis, duality theory - underlying the Simplex algorithm.
- Students learn and write direct and indirect proofs and develop ability to show the correctness of operations performed in algorithms.
- Students develop and apply logic reasoning skills to model computational systems.

Interdisc Computer Science Program Goals

- Understand and apply the mathematical concepts underlying computer science. (Introduced, Practiced)
- Write direct and indirect proofs; develop the ability to show the correctness of operations performed in algorithms.

Data Science Program Goals

- Develop capacity to learn new analysis methods and tools. (Introduced)
- Develop a rigorous computational foundation for practical programming.
- Develop a rigorous mathematical foundation of calculus, linear algebra, and statistics.

Interdisc Computer Science Program Goals

- Understand and apply the mathematical concepts underlying computer science. (Introduced, Practiced)
- Understand and show properties of sets and be able to apply them to computational systems such as databases.
- Develop and apply logical reasoning skills to model computational systems such as programs and circuits.

Data Science Program Goals

- Effectively communicate data science-related information to a variety of audiences. (Introduced)
- Communicate technical materials in written, visual, and oral forms.

**Core Goals:**

Quantitative Literacy

- Interpretation: Students will have the ability to explain information presented in mathematical and computational forms. (Introduced)
- Write direct and indirect proofs and develop ability to show the correctness of mathematical statements.
- Representation: Students will be able to convert information into mathematical and computational forms analytically and/or using computational tools. (Introduced)
- " Translate real-world problems into the abstract language of mathematics."
- Analysis: Students will be able to draw appropriate conclusions using mathematical or computational reasoning and understand the limits of such conclusions and the assumptions on which they are based. (Introduced)
- Understand the essential ingredients of various mathematical methods and develop the ability to recognize situations in which these methods are applicable.
- Identify patterns and use them to structure problem-solving techniques.
- Recognize situations in which mathematical methods are applicable and draw conclusions in a structured, deductive manner.
- Communication: Students will be able to communicate quantitative ideas in the languages of mathematics, computer science, or quantitative social sciences and will be able to utilize quantitative information in support of an argument. (Introduced)
- Express mathematical concepts in a clear and organized fashion, both written and orally.

**General Education Goals:**

Quan. & Comp. Reasoning

- Translate problems into the language of mathematics and computer science, and solve them by using mathematical and computational methods and tools (Introduced, Practiced)
- Understand the essential ingredients of various mathematical methods and develop the ability to recognize situations in which these methods are applicable.
- Identify patterns and use them to structure problem-solving techniques.
- Understand the structure and development of logical arguments (Introduced, Practiced)
- Follow an abstract mathematical argument.
- Apply logic to understand and analyze mathematical arguments.

**MATH 006: Discrete Mathematics II (4 Credits)**

This course is a continuation of Discrete Mathematics I. Topics are selected from: functions, counting methods, probability, statistics, recursion, recurrence relations, graph theory, matrices, regular expressions, finite state automata, grammars, and languages.

**Prerequisite(s):** MATH 004

**Note(s):** Prerequisite of MATH 004 may be waived with instructor consent.

**Meets the following Core requirements:** Quantitative Literacy

**Meets the following Gen Ed requirements:** Quantitative and Computational Reasoning

**Program Goals:**

Mathematics Program Goals

- Develop analytical skills and logical reasoning. (Introduced)
- Students will construct rigorous logical arguments, using a variety of proof methods and logical reasoning.
- Students will read deeply and critique proofs and solutions to problems.
- Students will justify their solutions to problems, using a variety of methods for logical reasoning.
- Students will understand a mathematical hypothesis, condition, or definition and check that it is satisfied in a specific example.
- Students will understand a general mathematical argument and then replicate the argument in a specific example.
- Develop analytical skills and logical reasoning. (Introduced)
- Evaluate whether a theorem or formula is valid in a specific example and draw conclusions from the theorem or formula concerning that example.
- Develop analytical skills and logical reasoning. (Introduced)
- Students will analyze in depth mathematical statements.
- Develop analytical skills and logical reasoning. (Introduced)
- Apply logic to understand and analyze mathematical arguments.
- Understand a mathematical hypothesis, condition, or definition and check that it is satisfied in a specific example.
- Develop ability to communicate mathematical thoughts in a clear and coherent fashion. (Introduced)
- Students will read deeply and critique each other's solutions in a professional manner typical for mathematical seminars and talks.
- Students will communicate in an organized, rigorous, and clear manner their solutions to problems via oral participation in class and workshop and in written form on exams and homework assignments.
- Students will clearly verbalize mathematical thoughts.
- Students will translate between mathematical symbols and English words.
- Develop problem-solving skills, and in particular, develop the ability to handle problems that require multiple steps for their solution. (Introduced)
- Students will decompose a problem into smaller problems.
- Students will reduce a new problem to previously solved problems.
- Students will identify patterns and use them to structure problem-solving techniques.
- Students will analyze, contrast, and compare different approaches to solving a problem.
- Students will translate real-life problems and/or problems from other disciplines into mathematical statements and use mathematical methods and specific problem-solving skills to solve these problems in complex, multi-step arguments.

Interdisc Computer Science Program Goals

- Understand and apply the mathematical concepts underlying computer science. (Practiced)
- Model sequential circuits as finite-state automata to deduce the functions and capabilities of such circuits.
- Represent networks (computer, communication, transportation) as graphs and use graph theory to determine properties, such as degree of connectivity, of such networks.
- Apply recurrence relations to derive the running time of recursive algorithms.

Computer Science Program Goals

- Understand and apply the mathematical concepts underlying computer science. (Practiced)
- Students develop and apply logic reasoning skills to model computational systems.
- Students learn and write direct and indirect proofs and develop ability to show the correctness of operations performed in algorithms.
- Students apply math concepts such as Boolean algebra (in computer Architecture), set theory (in Databases), formal proofs, graph theory (computer networks), finite-state automata (computer networks). Students understand system of linear equality, sensitivity analysis, duality theory - underlying the Simplex algorithm.
- Students apply recurrence relations to derive the running time of recursive algorithms, and appreciate the close connection between mathematical induction and recursion (e.g. mathematical induction is used to prove the correctness of recursive algorithms).
- Develop ability to solve computational problems such as finding patterns in texts, parsing a programming language, or computing algebraic functions, using automata.

Mathematics Program Goals

- Develop problem-solving skills, and in particular, develop the ability to handle problems that require multiple steps for their solution. (Introduced)
- Write complete and coherent mathematical sentences.
- Translate mathematical symbols into English words and vice versa.
- Verbalize mathematical thoughts clearly.
- Improve quantitative skills. (Introduced, Practiced)
- Students will determine/calculate various function and objects characteristics within the context of the problem and mathematical area.
- Students will correctly calculate the derivatives and integrals of a variety of functions and use the results in the proofs of mathematical theorems and as part of solving relevant real analysis problems.
- Students will operate correctly on mathematical objects for theoretical and computational purposes.
- Improve quantitative skills. (Practiced)
- Decompose a problem into smaller problems.
- Improve quantitative skills. (Introduced, Practiced)
- Students will compute the value of a function at a specified input.
- Improve quantitative skills. (Practiced)
- Analyze, contrast, and compare different approaches to solving a problem.
- Improve quantitative skills. (Introduced, Practiced)
- Students will solve linear, quadratic, differential, and other equations.
- Students will correctly manipulate algebraic expressions, equations and systems of equations in one or more variables.
- Students will simplify answers after taking derivatives or integrating.
- Improve quantitative skills. (Practiced)
- Reduce a new problem to previously solved problems.

**Core Goals:**

Quantitative Literacy

- Interpretation: Students will have the ability to explain information presented in mathematical and computational forms. (Practiced)
- Write direct and indirect proofs and develop ability to show the correctness of mathematical statements.
- Representation: Students will be able to convert information into mathematical and computational forms analytically and/or using computational tools. (Practiced)
- Translate real-world problems into the abstract language of mathematics.
- Analysis: Students will be able to draw appropriate conclusions using mathematical or computational reasoning and understand the limits of such conclusions and the assumptions on which they are based. (Practiced)
- Understand the essential ingredients of various mathematical methods and develop the ability to recognize situations in which these methods are applicable.
- Identify patterns and use them to structure problem-solving techniques.
- Recognize situations in which mathematical methods are applicable and draw conclusions in a structured, deductive manner.
- Communication: Students will be able to communicate quantitative ideas in the languages of mathematics, computer science, or quantitative social sciences and will be able to utilize quantitative information in support of an argument. (Practiced)
- Express mathematical concepts in a clear and organized fashion, both written and orally.

**General Education Goals:**

Quan. & Comp. Reasoning

- Translate problems into the language of mathematics and computer science, and solve them by using mathematical and computational methods and tools (Practiced)
- Identify patterns and use them to structure problem-solving techniques.
- Understand the structure and development of logical arguments (Practiced)
- Follow an abstract mathematical argument.
- Apply logic to understand and analyze mathematical arguments.
- Understand the algebraic and graphical properties of important mathematical functions and use these functions in concrete applications (Introduced)
- Understand and demonstrate one-to-one and onto properties of a variety of functions with application to computability.
- Use the transition diagram of finite-state automaton modeling a sequential circuit to find the language accepted by the automaton.
- Understand and apply the fundamental ideas of probability and statistics (Introduced)
- Apply counting techniques to solve probabilistic problems.

**MATH 047: Calculus I (3 Credits)**

Calculus I & II and Multivariable Calculus (MATH 049) are designed to build a solid foundation in calculus. Topics in Calculus I include: limits; continuity; derivatives; techniques for differentiation; linearization and differentials; the Mean Value Theorem; interpretations of derivatives in geometry and science; extreme values of functions, with applications to graphing and optimization problems in economics, life sciences, and physics; and an introduction to integrals.

**Note(s):** Concurrently with MATH 047, students must enroll in a calculus workshop, MATH 047L. Prerequisite: MATH 003 or high school equivalent.

**Meets the following Core requirements:** Quantitative Literacy

**Meets the following Gen Ed requirements:** Quantitative and Computational Reasoning

**Program Goals:**

Mathematics Program Goals

- Develop analytical skills and logical reasoning. (Introduced)
- Students will read deeply and critique proofs and solutions to problems.
- Students will construct rigorous logical arguments, using a variety of proof methods and logical reasoning.
- Develop analytical skills and logical reasoning. (Introduced)
- Students will justify their solutions to problems, using a variety of methods for logical reasoning. (I,P) Students will construct rigorous logical arguments, using a variety of proof methods and logical reasoning. (P,M)
- Students will understand a general mathematical argument and then replicate the argument in a specific example. (I) Students will understand a mathematical hypothesis, condition, or definition and check that it is satisfied in a specific example. (I,P) Students will analyze in depth mathematical statements. (P,M)
- Develop analytical skills and logical reasoning. (Introduced)
- Students will justify their solutions to problems, using a variety of methods for logical reasoning.
- Students will analyze in depth mathematical statements.
- Students will understand a general mathematical argument and then replicate the argument in a specific example.
- Students will understand a mathematical hypothesis, condition, or definition and check that it is satisfied in a specific example.
- Develop ability to communicate mathematical thoughts in a clear and coherent fashion. (Introduced)
- Students will communicate in an organized, rigorous, and clear manner their solutions to problems via oral participation in class and workshop and in written form on exams and homework assignments.
- Students will read deeply and critique each other's solutions in a professional manner typical for mathematical seminars and talks.
- Develop ability to communicate mathematical thoughts in a clear and coherent fashion. (Introduced)
- Students will communicate in an organized, rigorous, and clear manner their solutions to problems via oral participation in class and workshop and in written form on exams and homework assignments.
- Students will clearly verbalize mathematical thoughts.
- Develop ability to communicate mathematical thoughts in a clear and coherent fashion. (Introduced)
- Students will clearly verbalize mathematical thoughts.
- Students will translate between mathematical symbols and English words.
- Develop ability to communicate mathematical thoughts in a clear and coherent fashion. (Introduced)
- Students will translate between mathematical symbols and English words.
- Develop problem-solving skills, and in particular, develop the ability to handle problems that require multiple steps for their solution. (Introduced)
- Students will decompose a problem into smaller problems.
- Students will reduce a new problem to previously solved problems.
- Students will identify patterns and use them to structure problem-solving techniques.
- Students will translate real-life problems and/or problems from other disciplines into mathematical statements and use mathematical methods and specific problem-solving skills to solve these problems in complex, multi-step arguments.
- Students will analyze, contrast, and compare different approaches to solving a problem.

Data Science Program Goals

- Develop capacity to learn new analysis methods and tools. (Introduced)
- Develop a rigorous mathematical foundation of calculus, linear algebra, and statistics.

Mathematics Program Goals

- Students will decompose a problem into smaller problems. (I,P) Students will reduce a new problem to previously solved problems. (I,P)
- Students will identify patterns and use them to structure problem-solving techniques. (I,P) Students will analyze, contrast, and compare different approaches to solving a problem. (I,P,M)
- Improve quantitative skills. (Introduced, Practiced)
- Students will compute the value of a function at a specified input. (I) Students will simplify answers after taking derivatives or integrating. (I,P) Students will correctly manipulate algebraic expressions, equations and systems of equations in one or more variables. (I,P,M)
- Students will solve linear, quadratic, differential, and other equations. (I,P) Students will operate correctly on mathematical objects for theoretical and computational purposes. (I,P,M)
- Students will correctly calculate the derivatives and integrals of a variety of functions and use the results in the proofs of mathematical theorems and as part of solving relevant real analysis problems. (P,M) Students will determine/calculate various function and objects characteristics within the context of the problem and mathematical area. (I,P,M)
- Improve quantitative skills. (Introduced, Practiced)
- Students will compute the value of a function at a specified input.
- Students will simplify answers after taking derivatives or integrating.
- Students will correctly manipulate algebraic expressions, equations and systems of equations in one or more variables.
- Students will solve linear, quadratic, differential, and other equations.
- Students will operate correctly on mathematical objects for theoretical and computational purposes.
- Students will determine/calculate various function and objects characteristics within the context of the problem and mathematical area.
- Students will correctly calculate the derivatives and integrals of a variety of functions and use the results in the proofs of mathematical theorems and as part of solving relevant real analysis problems.

Data Science Program Goals

- Interpret data analysis outcomes. (Introduced)
- Interpret data, extract meaningful information, and assess findings.
- Effectively communicate data science-related information to a variety of audiences. (Introduced)
- Communicate technical materials in written, visual, and oral forms.

**Core Goals:**

Quantitative Literacy

- Interpretation: Students will have the ability to explain information presented in mathematical and computational forms. (Introduced)
- Students will explain theorems, problems, and solutions in their own words.
- Students will interpret the graph of a function.
- Representation: Students will be able to convert information into mathematical and computational forms analytically and/or using computational tools. (Introduced)
- Students will identify patterns and use them to structure problem-solving techniques.
- Students will translate real-life problems and problems from other disciplines into mathematical statements and use mathematical methods and specific problem-solving skills to solve these problems in complex, multi-step arguments.
- Students will apply differential and integral calculus to solve problems in other sciences.
- Analysis: Students will be able to draw appropriate conclusions using mathematical or computational reasoning and understand the limits of such conclusions and the assumptions on which they are based. (Introduced)
- Communication: Students will be able to communicate quantitative ideas in the languages of mathematics, computer science, or quantitative social sciences and will be able to utilize quantitative information in support of an argument. (Introduced)
- Students will verbalize mathematical thoughts clearly.
- Students will communicate in an organized and clear manner their solutions to problems.
- Students will justify their written solutions to problems, using a variety of methods for logical reasoning.

**General Education Goals:**

Quan. & Comp. Reasoning

- Translate problems into the language of mathematics and computer science, and solve them by using mathematical and computational methods and tools (Introduced)
- Students will understand the properties of various mathematical methods and recognize opportunities to apply them.
- Students will identify patterns and use them to structure problem-solving techniques.
- Understand the structure and development of logical arguments (Introduced)
- Students will follow an abstract mathematical argument.
- Students will adapt mathematical arguments modeled in class.
- Students will refresh their understanding of trigonometric functions and their use in differential calculus.
- Students will learn about exponential and logarithmic functions and their role in differential calculus.

**MATH 047L: Calculus I Workshop (2 Credits)**

**Program Goals:**

Data Science Program Goals

- Develop capacity to learn new analysis methods and tools. (Introduced)
- Develop a rigorous mathematical foundation of calculus, linear algebra, and statistics.
- Interpret data analysis outcomes. (Introduced)
- Interpret data, extract meaningful information, and assess findings.
- Effectively communicate data science-related information to a variety of audiences. (Introduced)
- Communicate technical materials in written, visual, and oral forms.

**MATH 047LX: Calculus I Lab (0 Credits)**

**MATH 048: Calculus II (3 Credits)**

This course is a continuation of Calculus I. Topics include: the notion of integral; the Fundamental Theorem of Calculus; techniques of integration, including substitution and integration by parts; numerical integration; concrete interpretations of the integral in geometry and science; applications of the integral to problems of measurement and of physics; improper integrals; infinite series and tests of convergence; the algebra and calculus of power series; and Taylor series approximations.

**Prerequisite(s):** MATH 047

**Note(s):** Students must enroll in a calculus workshop MATH 048L

**Meets the following Core requirements:** Quantitative Literacy

**Meets the following Gen Ed requirements:** Quantitative and Computational Reasoning

**Program Goals:**

Mathematics Program Goals

- Develop analytical skills and logical reasoning. (Introduced)
- Students will read deeply and critique proofs and solutions to problems.
- Develop analytical skills and logical reasoning. (Introduced)
- Students will justify their solutions to problems, using a variety of methods for logical reasoning. (I,P) Students will construct rigorous logical arguments, using a variety of proof methods and logical reasoning. (P,M)
- Students will understand a general mathematical argument and then replicate the argument in a specific example. (I) Students will understand a mathematical hypothesis, condition, or definition and check that it is satisfied in a specific example. (I,P) Students will analyze in depth mathematical statements. (P,M)
- Develop analytical skills and logical reasoning. (Introduced)
- Students will justify their solutions to problems, using a variety of methods for logical reasoning.
- Students will analyze in depth mathematical statements.
- Develop ability to communicate mathematical thoughts in a clear and coherent fashion. (Introduced)
- Students will read deeply and critique each other's solutions in a professional manner typical for mathematical seminars and talks.
- Develop ability to communicate mathematical thoughts in a clear and coherent fashion. (Introduced)
- Students will clearly verbalize mathematical thoughts.
- Develop ability to communicate mathematical thoughts in a clear and coherent fashion. (Introduced)
- Students will clearly verbalize mathematical thoughts.
- Students will translate between mathematical symbols and English words.
- Develop ability to communicate mathematical thoughts in a clear and coherent fashion. (Introduced)
- Students will translate between mathematical symbols and English words.
- Students will decompose a problem into smaller problems.
- Students will reduce a new problem to previously solved problems.
- Students will identify patterns and use them to structure problem-solving techniques.
- Students will translate real-life problems and/or problems from other disciplines into mathematical statements and use mathematical methods and specific problem-solving skills to solve these problems in complex, multi-step arguments.
- Students will analyze, contrast, and compare different approaches to solving a problem.

Data Science Program Goals

- Develop capacity to learn new analysis methods and tools. (Introduced)
- Develop a rigorous mathematical foundation of calculus, linear algebra, and statistics.

Mathematics Program Goals

- Students will decompose a problem into smaller problems. (I,P) Students will reduce a new problem to previously solved problems. (I,P)
- Students will identify patterns and use them to structure problem-solving techniques. (I,P) Students will analyze, contrast, and compare different approaches to solving a problem. (I,P,M)
- Improve quantitative skills. (Introduced, Practiced)
- Students will compute the value of a function at a specified input. (I) Students will simplify answers after taking derivatives or integrating. (I,P) Students will correctly manipulate algebraic expressions, equations and systems of equations in one or more variables. (I,P,M)
- Students will solve linear, quadratic, differential, and other equations. (I,P) Students will operate correctly on mathematical objects for theoretical and computational purposes. (I,P,M)
- Students will correctly calculate the derivatives and integrals of a variety of functions and use the results in the proofs of mathematical theorems and as part of solving relevant real analysis problems. (P,M) Students will determine/calculate various function and objects characteristics within the context of the problem and mathematical area. (I,P,M)
- Improve quantitative skills. (Introduced, Practiced)
- Students will compute the value of a function at a specified input.
- Students will simplify answers after taking derivatives or integrating.
- Students will correctly manipulate algebraic expressions, equations and systems of equations in one or more variables.
- Students will solve linear, quadratic, differential, and other equations.
- Students will operate correctly on mathematical objects for theoretical and computational purposes.
- Students will determine/calculate various function and objects characteristics within the context of the problem and mathematical area.
- Students will correctly calculate the derivatives and integrals of a variety of functions and use the results in the proofs of mathematical theorems and as part of solving relevant real analysis problems.

Data Science Program Goals

- Interpret data analysis outcomes. (Introduced)
- Interpret data, extract meaningful information, and assess findings.
- Effectively communicate data science-related information to a variety of audiences. (Introduced)
- Communicate technical materials in written, visual, and oral forms.

**Core Goals:**

Quantitative Literacy

- Students will explain theorems, problems, and solutions in their own words.
- Students will interpret the graph of a function.
- Students will identify patterns and use them to structure problem-solving techniques.
- Students will translate real-life problems and problems from other disciplines into mathematical statements and use mathematical methods and specific problem-solving skills to solve these problems in complex, multi-step arguments.
- Students will apply differential and integral calculus to solve problems in other sciences.
- Students will verbalize mathematical thoughts clearly.
- Students will communicate in an organized and clear manner their solutions to problems.
- Students will justify their written solutions to problems, using a variety of methods for logical reasoning.

**General Education Goals:**

Quan. & Comp. Reasoning

- Students will understand the properties of various mathematical methods and recognize opportunities to apply them.
- Students will identify patterns and use them to structure problem-solving techniques.
- Understand the structure and development of logical arguments (Introduced)
- Students will follow an abstract mathematical argument.
- Students will adapt mathematical arguments modeled in class.
- Understand the algebraic and graphical properties of important mathematical functions and use these functions in concrete applications (Practiced)
- Students will refresh their understanding of trigonometric and exponential functions and their use in integral calculus.

**MATH 048L: Calculus II Workshop (2 Credits)**

**Program Goals:**

Data Science Program Goals

- Develop capacity to learn new analysis methods and tools. (Introduced)
- Develop a rigorous mathematical foundation of calculus, linear algebra, and statistics.
- Interpret data analysis outcomes. (Introduced)
- Interpret data, extract meaningful information, and assess findings.
- Effectively communicate data science-related information to a variety of audiences. (Introduced)
- Communicate technical materials in written, visual, and oral forms.

**MATH 049: Multivariable Calculus (4 Credits)**

The theory of calculus in higher dimensional spaces. Vector functions and scalar functions of several variables. The notions of derivative and integral appropriate to such functions. In particular, partial derivatives, gradient, multiple integration, extrema, and applications of these notions. Line and surface integrals, Green's Theorem, and Stoke's Theorem.

**Prerequisite(s):** MATH 048

**Meets the following Core requirements:** Quantitative Literacy

**Meets the following Gen Ed requirements:** Quantitative and Computational Reasoning

**Program Goals:**

Mathematics Program Goals

- Develop analytical skills and logical reasoning. (Introduced)
- Develop analytical skills and logical reasoning. (Introduced)
- Students will analyze in depth mathematical statements about functions of several variables, their partial derivatives and various types of integrals.
- Students will justify their solutions to problems, using a variety of methods for logical reasoning. Examples include direct reasoning or reasoning by contradiction; correct use of examples and counterexamples, as well as applying many problem-solving techniques within discussion of theory and solutions to relevant problems.
- Develop analytical skills and logical reasoning. (Introduced)
- Students will read deeply and critique proofs and solutions to problems.
- Students will justify their solutions to problems, using a variety of methods for logical reasoning.
- Students will analyze in depth mathematical statements.
- Develop ability to communicate mathematical thoughts in a clear and coherent fashion. (Introduced, Practiced)
- Students will be able to comfortably work in 2- and 3-dimensional coordinate systems and to visually represent in them functions of several variables and illustrate theorems of multivariable calculus, e.g., Green's and Stokes' Theorem.
- Students will be able to draw graphical representations of vectors and vector functions, surfaces and solids, dot- and cross-products, vector fields and various types of integrals, for the purposes of comparing and deriving properties of basic multivariable calculus objects and solving problems about them.
- Develop ability to communicate mathematical thoughts in a clear and coherent fashion. (Introduced, Practiced)
- Students will translate between mathematical symbols and English words.
- Develop ability to communicate mathematical thoughts in a clear and coherent fashion. (Introduced, Practiced)
- Students will be able to work in polar coordinates to represent functions and calculate certain integrals.
- Students will clearly verbalize mathematical thoughts.
- Students will clearly verbalize mathematical thoughts.
- Develop problem-solving skills, and in particular, develop the ability to handle problems that require multiple steps for their solution. (Introduced, Practiced)
- Students will decompose a problem into smaller problems.
- Students will reduce a new problem to previously solved problems.
- Students will reduce a new problem to previously solved problems.
- Students will analyze, contrast, and compare different approaches to solving a problem.
- Develop problem-solving skills, and in particular, develop the ability to handle problems that require multiple steps for their solution. (Introduced, Practiced)
- Students will communicate in an organized and clear manner their solutions to multivariable calculus problems via oral participation in class, written and oral presentation in workshops, and written form on exams and homework assignments.

Data Science Program Goals

- Develop capacity to learn new analysis methods and tools. (Mastered)
- Develop a rigorous mathematical foundation of calculus, linear algebra, and statistics.

Mathematics Program Goals

- Develop problem-solving skills, and in particular, develop the ability to handle problems that require multiple steps for their solution. (Introduced, Practiced)
- Students will combine mathematical correctness with good presentation skills.
- Students will read deeply and critique proofs and solutions to problems in multivariable calculus.
- Improve quantitative skills. (Practiced, Mastered)
- Students will operate correctly on mathematical objects for theoretical and computational purposes.
- Students will compute the value of a function at a specified input.
- Students will simplify answers after taking derivatives or integrating.
- Students will solve linear, quadratic, differential, and other equations.
- Improve quantitative skills. (Practiced, Mastered)
- Students will be able to translate a number of real-life problems and problems from other disciplines into statements on multivariable functions and use multivariable calculus methods and specific problem-solving skills to solve these problems in complex, multi-step arguments. Classic examples involve problems on arc length, curvature, velocity and acceleration, calculation of double and iterated integrals, and others.
- Learn to apply a major mathematical theory to solve non-trivial problems in other scientific domains. (Introduced)
- Students will translate real-world problems into abstract mathematical models.
- Students will abstract, analyze, and draw analogies between objects from other areas and mathematical constructs and theories.
- Students will learn and use problem-solving techniques to solve problems from other areas.
- Students will apply differential and integral calculus to solve problems in other sciences.
- Students will build mathematical models to solve real-world decision problems arising from a wide range of domains.
- Learn to apply a major mathematical theory to solve non-trivial problems in other scientific domains. (Introduced)
- Students will correctly manipulate algebraic expressions, equations and systems of equations in one or more variables, especially in the applications of Lagrange multipliers.
- Students will operate correctly on multivariable functions, in order to calculate their derivatives and integrals, and to work with vector fields and other concepts for theoretical and calculational purposes.

**Core Goals:**

Quantitative Literacy

- Interpretation: Students will have the ability to explain information presented in mathematical and computational forms. (Practiced)
- Students will explain theorems, problems, and solutions in their own words.
- Students will interpret the graph of a multivariable function.
- Representation: Students will be able to convert information into mathematical and computational forms analytically and/or using computational tools. (Practiced)
- Students will identify patterns and use them to structure problem-solving techniques.
- Students will apply differential and integral calculus to solve problems in other sciences.
- Analysis: Students will be able to draw appropriate conclusions using mathematical or computational reasoning and understand the limits of such conclusions and the assumptions on which they are based. (Practiced)
- Communication: Students will be able to communicate quantitative ideas in the languages of mathematics, computer science, or quantitative social sciences and will be able to utilize quantitative information in support of an argument. (Practiced)
- Students will verbalize mathematical thoughts clearly.
- Students will communicate in an organized and clear manner their solutions to problems.

**General Education Goals:**

Quan. & Comp. Reasoning

- Translate problems into the language of mathematics and computer science, and solve them by using mathematical and computational methods and tools (Practiced)
- Students will connect the theory of multivariable calculus with many other academic disciplines, industry, and everyday life. The problem-solving techniques of the class will be used by the students to solve optimization problems in economics, as well as problems in physics, astronomy, chemistry, and other areas.
- Students will operate correctly on multivariable functions, in order to calculate their derivatives and integrals, and to work with vector fields and other concepts for theoretical and calculational purposes.
- Students will correctly manipulate algebraic expressions, equations and systems of equations in one or more variables, especially in the applications of Lagrange multipliers.
- Understand the structure and development of logical arguments (Introduced)
- Students will analyze in depth mathematical statements about functions of several variables, their partial derivatives and various types of integrals.
- Students will justify their solutions to problems, using a variety of methods for logical reasoning. Examples include direct reasoning or reasoning by contradiction; correct use of examples and counterexamples, as well as applying many problem-solving techniques within discussion of theory and solutions to relevant problems.
- Understand the algebraic and graphical properties of important mathematical functions and use these functions in concrete applications (Practiced)
- Students will be able to draw graphical representations of vectors and vector functions, surfaces and solids, dot- and cross-products, vector fields and various types of integrals, for the purposes of comparing and deriving properties of basic multivariable calculus objects and solving problems about them.
- Students will be able to comfortably work in 2- and 3-dimensional coordinate systems and to visually represent in them functions of several variables and illustrate theorems of multivariable calculus, e.g., Green's and Stokes' Theorem.

**MATH 050: Linear Algebra (4 Credits)**

Matrix algebra and determinants, and the theory of vector spaces, including: the notion of subspace, independence, basis and dimension, linear transformations, and eigenvalues and eigenvectors. Applications to geometry, systems of linear equations, and the theory of approximations are given.

**Prerequisite(s):** MATH 047 and MATH 048

**Note(s):** Prerequisites of MATH 047 and MATH 048 may be waived by instructor consent.

**Meets the following Core requirements:** Quantitative Literacy

**Meets the following Gen Ed requirements:** Quantitative and Computational Reasoning

**Program Goals:**

Mathematics Program Goals

- Develop analytical skills and logical reasoning. (Introduced)
- Develop analytical skills and logical reasoning. (Introduced)
- Students will justify their solutions to problems, using a variety of methods for logical reasoning. Examples include direct reasoning or reasoning by contradiction (in specialized examples: also reasoning by mathematical induction); correct use of examples and counterexamples, as well as applying many problem-solving techniques within discussion of theory and solutions to relevant problems.
- Students will analyze in depth mathematical statements about systems of equations, matrices, linear transformations, and vector spaces.
- Develop analytical skills and logical reasoning. (Introduced)
- Students will read deeply and critique proofs and solutions to problems.
- Students will justify their solutions to problems, using a variety of methods for logical reasoning.
- Students will analyze in depth mathematical statements.
- Develop ability to communicate mathematical thoughts in a clear and coherent fashion. (Practiced)
- Students will be able to comfortably work in 2- and 3-dimensional coordinate systems and to visually represent in them the action of linear transformations. Students will construct diagrams representing linear transformations between vector spaces, and to overlay such diagrams with important features of the involved linear algebra objects, e.g., subspaces, images and kernels of linear transformations, injectivity and surjectivity, as well as linear independence, bases, isomorphisms of vector spaces, eigenvalues and eigenvectors, and many more.
- Students will be able to draw graphical representations of vectors, vector spaces, and subspaces, and of dot- and cross-products, for the purposes of comparing and deriving properties of basic linear algebra objects and solving problems about them.

Data Science Program Goals

- Select, apply, and evaluate models to devise solutions to data science tasks. (Introduced)
- Identify and classify relevant variables for data science tasks.

Mathematics Program Goals

- Develop ability to communicate mathematical thoughts in a clear and coherent fashion. (Practiced)
- As possible extra topics, students will be able to graphically represent dynamical systems and depict the corresponding recursive sequences, as well as to visualize least-square approximations and data fitting.
- Students will translate between mathematical symbols and English words.
- Students will clearly verbalize mathematical thoughts.
- Students will decompose a problem into smaller problems.
- Students will reduce a new problem to previously solved problems.
- Students will identify patterns and use them to structure problem-solving techniques.
- Students will analyze, contrast, and compare different approaches to solving a problem.

Data Science Program Goals

- Develop capacity to learn new analysis methods and tools. (Introduced, Practiced)
- Develop a rigorous mathematical foundation of calculus, linear algebra, and statistics.
- Develop a rigorous computational foundation for practical programming.

Mathematics Program Goals

- Develop problem-solving skills, and in particular, develop the ability to handle problems that require multiple steps for their solution. (Practiced)
- Students will communicate in an organized and clear manner their solutions to linear algebra problems via oral participation in class, written and oral presentation in workshops, and written form on exams and homework assignments.
- Students will combine mathematical correctness with good presentation skills.
- Students will read deeply and critique proofs and solutions to problems in linear algebra.
- Improve quantitative skills. (Practiced, Mastered)
- Students will simplify answers after taking derivatives or integrating.
- Students will solve linear, quadratic, differential, and other equations.
- Students will operate correctly on mathematical objects for theoretical and computational purposes.
- Improve quantitative skills. (Practiced)
- Students will be able to translate a number of real-life problems and problems from other disciplines into statements on matrices, linear transformations, and vector spaces, and use linear algebra methods and specific problem-solving skills to solve these problems in complex, multi-step arguments. Classic examples involve optimization problems and least-square approximations, systems of equations from economics/scheduling/traffic or bio/eco-systems, the Fibonacci sequence, dynamical systems, and many more.

Data Science Program Goals

- Interpret data analysis outcomes. (Introduced)
- Interpret data, extract meaningful information, and assess findings.

Mathematics Program Goals

- Improve quantitative skills. (Practiced, Mastered)
- Students will compute the value of a function at a specified input.
- Learn to apply a major mathematical theory to solve non-trivial problems in other scientific domains. (Introduced)
- Students will build mathematical models to solve real-world decision problems arising from a wide range of domains.

Data Science Program Goals

- Effectively communicate data science-related information to a variety of audiences. (Introduced)
- Communicate technical materials in written, visual, and oral forms.

Mathematics Program Goals

- Students will abstract, analyze, and draw analogies between objects from other areas and mathematical constructs and theories.
- Students will translate real-world problems into abstract mathematical models.
- Students will learn and use problem-solving techniques to solve problems from other areas.
- Students will apply differential and integral calculus to solve problems in other sciences.
- Learn to apply a major mathematical theory to solve non-trivial problems in other scientific domains. (Practiced, Mastered)
- Students will correctly manipulate algebraic expressions, equations and systems of linear equations in many variables.
- Students will operate correctly on matrices, to calculate their determinants, inverses, characteristic polynomials and other important features in a variety of ways, to find efficiently their eigenvalues and eigenvectors, and to work with linear transformations for theoretical and calculational purposes.
- In more advanced/bonus examples, students will correctly calculate and use derivatives and integrals within classic examples of vector spaces and isomorphisms between vector spaces, and as part of solving relevant linear algebra problems.

**Core Goals:**

Quantitative Literacy

- Interpretation: Students will have the ability to explain information presented in mathematical and computational forms. (Practiced)
- Students will explain theorems, problems, and solutions in their own words.
- Students will interpret solutions to systems of equations in the context of real-world problems.
- Representation: Students will be able to convert information into mathematical and computational forms analytically and/or using computational tools. (Practiced)
- Students will identify patterns and use them to structure problem-solving techniques.
- Students will utilize systems of linear equations to solve problems in other sciences.
- Analysis: Students will be able to draw appropriate conclusions using mathematical or computational reasoning and understand the limits of such conclusions and the assumptions on which they are based. (Practiced)
- Communication: Students will be able to communicate quantitative ideas in the languages of mathematics, computer science, or quantitative social sciences and will be able to utilize quantitative information in support of an argument. (Practiced)
- Students will verbalize mathematical thoughts clearly.
- Students will communicate in an organized and clear manner their solutions to problems.

**General Education Goals:**

Quan. & Comp. Reasoning

- Translate problems into the language of mathematics and computer science, and solve them by using mathematical and computational methods and tools (Practiced)
- Student will connect the theory of linear algebra with many other academic disciplines, industry, and to everyday life. The problem-solving techniques of the class will be used by the students to solve optimization problems from economics and transportation/scheduling, from dynamical systems in chaos theory and the theory of recursive sequences, and many other areas. The methods of linear algebra which will be mastered and used for such applications are numerous; e.g., Gauss-Jordan elimination and reduction of matrices, use of determinants, minors and Laplace expansions, Cramer's rule, and others.
- Students will correctly manipulate algebraic expressions, equations, and systems of linear equations in many variables. Students will operate correctly on matrices, to calculate their determinants, inverses, characteristic polynomials, and other important features in a variety of ways, to find efficiently their eigenvalues and eigenvectors, and to work with linear transformations for theoretical and calculational purposes.
- In more advanced/bonus examples, students will correctly calculate and use derivatives and integrals within classic examples of vector spaces and isomorphisms between vector spaces, and as part of solving relevant linear algebra problems.
- Understand the structure and development of logical arguments (Practiced)
- Students will analyze in depth mathematical statements about the systems of equations, matrices, linear transformations, and vector spaces.
- Students will justify their solutions to problems, using a variety of methods for logical reasoning. Examples include direct reasoning or reasoning by contradiction (in specialized examples: also reasoning by mathematical induction); correct use of examples and counterexamples, as well as applying many problem-solving techniques within discussion of theory and solutions to relevant problems.
- Understand the algebraic and graphical properties of important mathematical functions and use these functions in concrete applications (Practiced)
- Students will be able to draw graphical representations of vectors, vector spaces, subspaces, and of dot- and cross-products, for the purposes of comparing and deriving properties of basic linear algebra objects and solving problems about them.
- Students will be able to comfortably work in 2- and 3-dimensional coordinate systems and to visually represent in them the action of linear transformations. Students will construct diagrams representing linear transformations between vector spaces, and to overlay such diagrams with important features of the involved linear algebra objects, e.g., subspaces, images and kernels of linear transformations, injectivity and surjectivity, as well as linear independence, bases, isomorphisms of vector spaces, eigenvalues and eigenvectors, and many more.
- As possible extra topics, students will be able to graphically represent dynamical systems and depict the corresponding recursive sequences, as well as to visualize least-square approximations and data fitting.

**MATH 080: Topics in Mathematics (0.25-1.25 Credits)**

**MATH 102: Probability and Statistics (4 Credits)**

An introduction to the concepts and applications of probability and statistics, with a strong foundation in theory as well as practice, including the possible use of technology. Topics include fundamentals of probability, random variables, distributions, expected values, special distributions, sampling, tests of significance, statistical inference, regression, and correlation.

**Prerequisite(s):** MATH 047 and MATH 048

**Note(s):** The prerequisite of MATH 048 may be taken concurrently, at the discretion of the instructor.

**Meets the following Gen Ed requirements:** Quantitative and Computational Reasoning

**Program Goals:**

Mathematics Program Goals

- Develop analytical skills and logical reasoning. (Practiced)
- Students will analyze in depth statements about probability and statistics.
- Students will justify their solutions to problems, using a variety of methods for logical reasoning. Examples include reasoning directly or reasoning by contradiction; correct use of examples and counterexamples; and application of statistical techniques along with critical evaluation of the appropriateness of their use.
- Develop analytical skills and logical reasoning. (Introduced, Practiced)
- Students will analyze in depth mathematical statements.
- Students will justify their solutions to problems, using a variety of methods for logical reasoning.
- Students will read deeply and critique proofs and solutions to problems.

Data Science Program Goals

- Select, apply, and evaluate models to devise solutions to data science tasks. (Introduced)
- Choose and apply tools and methodologies to solve data science tasks.
- Identify and classify relevant variables for data science tasks.
- Assess the methods, tools, and models used in the analysis.

Mathematics Program Goals

- Students will translate between mathematical symbols and English words.
- Students will clearly verbalize mathematical thoughts.
- Develop problem-solving skills, and in particular, develop the ability to handle problems that require multiple steps for their solution. (Practiced)
- Students will decompose a problem into smaller problems.
- Students will reduce a new problem to previously solved problems.

Data Science Program Goals

- Develop capacity to learn new analysis methods and tools. (Introduced, Practiced)
- Develop a rigorous mathematical foundation of calculus, linear algebra, and statistics.

Mathematics Program Goals

- Develop problem-solving skills, and in particular, develop the ability to handle problems that require multiple steps for their solution. (Practiced)
- Students will combine mathematical and statistical correctness with good presentation skills.

Data Science Program Goals

- Develop capacity to learn new analysis methods and tools. (Introduced, Practiced)
- Develop a rigorous computational foundation for practical programming.

Mathematics Program Goals

- Students will identify patterns and use them to structure problem-solving techniques.
- Students will analyze, contrast, and compare different approaches to solving a problem.
- Improve quantitative skills. (Practiced)
- Students will be able to translate a number of real-life problems and problems from other disciplines into the language of probability and statistics, determine which statistical techniques apply to the problem at hand, and decide what assumptions are needed to justify their use.
- Students will apply specific mathematical and statistical techniques to solve problems in complex, multi-step arguments. Classic examples involve calculation of confidence intervals, estimation of parameters from data, tests of hypotheses, and calculation of probability and expected value.
- Students will use statistical software to perform multi-step computations.
- Improve quantitative skills. (Introduced, Practiced)
- Students will operate correctly on mathematical objects for theoretical and computational purposes.

Data Science Program Goals

- Interpret data analysis outcomes. (Practiced)
- Interpret data, extract meaningful information, and assess findings.
- Evaluate the limitations of data science findings.

Mathematics Program Goals

- Improve quantitative skills. (Introduced, Practiced)
- Students will solve linear, quadratic, differential, and other equations.
- Students will simplify answers after taking derivatives or integrating.
- Students will compute the value of a function at a specified input.
- Learn to apply a major mathematical theory to solve non-trivial problems in other scientific domains. (Practiced)
- Students will translate real-world problems into abstract mathematical models.
- Students will learn and use problem-solving techniques to solve problems from other areas.
- Students will apply differential and integral calculus to solve problems in other sciences.
- Learn to apply a major mathematical theory to solve non-trivial problems in other scientific domains. (Introduced, Practiced)
- Students will correctly calculate and use derivatives and integrals of a variety of functions.
- Students will correctly manipulate algebraic expressions, equations, and systems of equations in one or more variables.
- Learn to apply a major mathematical theory to solve non-trivial problems in other scientific domains. (Practiced)
- Students will build mathematical models to solve real-world decision problems arising from a wide range of domains.
- Students will abstract, analyze, and draw analogies between objects from other areas and mathematical constructs and theories.

**General Education Goals:**

Quan. & Comp. Reasoning

- Students will be able to translate a number of real-life problems and problems from other disciplines into the language of probability and statistics, determine which statistical techniques apply to the problem at hand, and decide what assumptions are needed to justify their use.
- Students will apply specific mathematical and statistical techniques to solve problems in complex, multi-step arguments. Classic examples involve calculation of confidence intervals, estimation of parameters from data, tests of hypotheses, and calculation of probability and expected value.
- Students will connect the theories of probability and statistical inference with many other academic disciplines, industry, and everyday life. Probability and statistical techniques will be used to solve problems, analyze data, and test hypotheses in medicine, psychology, business, biology, chemistry, engineering, and many other areas. Students will use statistical software to analyze data using methods and techniques learned in class.
- Understand the structure and development of logical arguments (Introduced)
- Students will analyze in depth mathematical statements about random variables, probability distributions, parameters, and hypotheses.
- Students will justify their solutions to problems, using a variety of methods for logical reasoning. Examples include direct reasoning or reasoning by contradiction; correct use of examples and counterexamples; and application of statistical techniques along with critical evaluation of the appropriateness of their use.
- Students will become familiar with important probability distributions such as binomial, normal, lognormal, and Poisson distributions. Students will be able to draw probability density curves for these distributions and compute fundamental properties such as means and variances.
- Students will be able to determine which of the standard probability distributions are plausible models for real-world data and estimate parameters from data.
- Understand and apply the fundamental ideas of probability and statistics (Mastered)
- Students will understand and apply basic concepts of probability, such as independence, conditional probability, and counting techniques.
- Students will use properties of discrete and continuous random variables, including expected value and variance, to solve problems.
- Students will understand the principles of estimation of parameters and use them to compute point estimates and confidence intervals. Students will understand concepts of hypothesis testing, including null and alternative hypotheses, test statistics, and p-values. Students will apply tests of hypotheses appropriately to analyze real data.

**MATH 104: Differential Equations (4 Credits)**

Ordinary differential equations of first and second order as well as systems of such equations. More general techniques for finding solutions are developed gradually. Applications to physical and social sciences.

**Prerequisite(s):** MATH 050 and MATH 048

**Meets the following Gen Ed requirements:** Quantitative and Computational Reasoning

**Program Goals:**

Mathematics Program Goals

- Develop analytical skills and logical reasoning. (Practiced)
- Students will analyze in depth mathematical statements about differential equations and systems of such.
- Students will rigorously justify their solutions to problems, using a variety of methods for logical reasoning and proof. Examples include direct proofs or proofs by contradiction or by mathematical induction; correct use of examples and counterexamples, as well as applying many problem-solving techniques within discussion of theory and solutions to relevant problems.
- Develop analytical skills and logical reasoning. (Introduced, Practiced)
- Students will analyze in depth mathematical statements.
- Students will justify their solutions to problems, using a variety of methods for logical reasoning.
- Students will read deeply and critique proofs and solutions to problems.
- Develop ability to communicate mathematical thoughts in a clear and coherent fashion. (Practiced)
- Students will translate between mathematical symbols and English words.

Data Science Program Goals

- Select, apply, and evaluate models to devise solutions to data science tasks. (Introduced)
- Assess the methods, tools, and models used in the analysis.

Mathematics Program Goals

- Develop ability to communicate mathematical thoughts in a clear and coherent fashion. (Practiced)

Data Science Program Goals

- Select, apply, and evaluate models to devise solutions to data science tasks. (Introduced)
- Choose and apply tools and methodologies to solve data science tasks.

Mathematics Program Goals

- Develop ability to communicate mathematical thoughts in a clear and coherent fashion. (Practiced)
- Students will clearly verbalize mathematical thoughts.

Data Science Program Goals

- Develop capacity to learn new analysis methods and tools. (Mastered)
- Develop a rigorous mathematical foundation of calculus, linear algebra, and statistics.

Mathematics Program Goals

- Develop problem-solving skills, and in particular, develop the ability to handle problems that require multiple steps for their solution. (Practiced, Mastered)
- Students will identify patterns and use them to structure problem-solving techniques.
- Students will analyze, contrast, and compare different approaches to solving a problem.
- Students will combine mathematical correctness with good presentation skills.
- Students will read deeply and critique proofs and solutions to problems in differential equations.
- Develop problem-solving skills, and in particular, develop the ability to handle problems that require multiple steps for their solution. (Practiced, Mastered)
- Students will reduce a new problem to previously solved problems.
- Students will decompose a problem into smaller problems.
- Students will communicate in an organized, rigorous, and clear manner their solutions to differential problems via oral participation in class, written and oral presentation in workshops, and written form on exams and homework assignments.
- Improve quantitative skills. (Practiced, Mastered)
- Students will operate correctly on mathematical objects for theoretical and computational purposes.
- Students will solve linear, quadratic, differential, and other equations.
- Students will simplify answers after taking derivatives or integrating.
- Students will compute the value of a function at a specified input.
- Improve quantitative skills. (Mastered)
- Students will be able to translate a number of real-life problems and problems from other disciplines into the language of differential equations, and use differential equations methods and specific problem-solving skills to solve these problems in complex, multi-step arguments. Classic examples involve problems on reduction of pollution, growth of biological populations, inflation and house-pricing, and others, to be solved with a variety of problem-solving techniques such as diagonalizing and Jordanizing matrices from linear algebra, undetermined coefficients and variation of parameters, trigonometric examples, power series solutions and recursive sequences, and others.
- Learn to apply a major mathematical theory to solve non-trivial problems in other scientific domains. (Introduced, Practiced, Mastered)
- Students will translate real-world problems into abstract mathematical models.
- Students will learn and use problem-solving techniques to solve problems from other areas.
- Students will apply differential and integral calculus to solve problems in other sciences.
- Learn to apply a major mathematical theory to solve non-trivial problems in other scientific domains. (Practiced, Mastered)
- Students will correctly calculate and use derivatives and integrals of a variety of functions.

Data Science Program Goals

- Effectively communicate data science-related information to a variety of audiences. (Practiced)
- Communicate technical materials in written, visual, and oral forms.

Mathematics Program Goals

- Learn to apply a major mathematical theory to solve non-trivial problems in other scientific domains. (Practiced, Mastered)
- Students will correctly manipulate algebraic expressions, equations and systems of equations in one variable.
- Students will operate correctly on matrices for the purposes of diagonalizing or Jordanizing them within solving systems of differential equations.
- Learn to apply a major mathematical theory to solve non-trivial problems in other scientific domains. (Introduced, Practiced, Mastered)

**General Education Goals:**

Quan. & Comp. Reasoning

- Students will be able to translate a number of real-life problems and problems from other disciplines into the language of differential equations, and use differential equations methods and specific problem-solving skills to solve these problems in complex, multi-step arguments. Classic examples involve problems on reduction of pollution, growth of biological populations, inflation and house-pricing, and others, to be solved with a variety of problem-solving techniques such as diagonalizing and Jordanizing matrices from linear algebra, undetermined coefficients and variation of parameters, trigonometric examples, power series solutions and recursive sequences, and others.
- Students will connect the theory of differential equations with many other academic disciplines, industry, and everyday life. The problem-solving techniques of the class will be used by the students to solve optimization and equilibrium problems from economics, chemistry, biology and physics, and other areas.
- Understand the structure and development of logical arguments (Practiced)
- Students will analyze in depth mathematical statements about differential equations and systems of such.
- Students will rigorously justify their solutions to problems, using a variety of methods for logical reasoning and proof. Examples include direct proofs or proofs by contradiction or by mathematical induction; correct use of examples and counterexamples, as well as applying many problem-solving techniques within discussion of theory and solutions to relevant problems.
- Students will read deeply and critique proofs and solutions to problems in differential equations.
- Students will apply their Calculus knowledge about functions (such as their growth, derivatives, integrals, and others) to intelligently guess and rigorously prove solutions to systems of differential equations.

**MATH 108: Mathematical Modeling (4 Credits)**

A mathematical model is a description of a real-world system using mathematical concepts and language. This course is an introduction to the basics of mathematical modeling emphasizing model construction, analysis and application. Using examples from a variety of fields such as physics, biology, chemistry, economics, and sociology, students will learn how to develop and use mathematical models of real-world systems.

**Prerequisite(s):** MATH 048 and MATH 050

**Program Goals:**

Mathematics Program Goals

- Develop analytical skills and logical reasoning. (Introduced, Practiced)
- Students will understand a general mathematical argument and then replicate the argument in a specific example. Students will understand a mathematical hypothesis, condition, or definition and check that it is satisfied in a specific example. Students will analyze in depth mathematical statements.
- Students will justify their solutions to problems, using a variety of methods for logical reasoning. Students will construct rigorous logical arguments, using a variety of proof methods and logical reasoning.
- Students will read deeply and critique proofs and solutions to problems.
- Develop ability to communicate mathematical thoughts in a clear and coherent fashion. (Practiced)
- Students will translate between mathematical symbols and English words.
- Students will clearly verbalize mathematical thoughts.
- Develop problem-solving skills, and in particular, develop the ability to handle problems that require multiple steps for their solution. (Practiced, Mastered)
- Students will identify patterns and use them to structure problem-solving techniques. Students will analyze, contrast, and compare different approaches to solving a problem.
- Students will decompose a problem into smaller problems. Students will reduce a new problem to previously solved problems.
- Improve quantitative skills. (Practiced, Mastered)
- Students will correctly calculate the derivatives and integrals of a variety of functions and use the results in the proofs of mathematical theorems and as part of solving relevant real analysis problems. Students will determine/calculate various function and objects characteristics within the context of the problem and mathematical area.
- Students will solve linear, quadratic, differential, and other equations. Students will operate correctly on mathematical objects for theoretical and computational purposes.
- Students will compute the value of a function at a specified input. Students will simplify answers after taking derivatives or integrating. Students will correctly manipulate algebraic expressions, equations and systems of equations in one or more variables.
- Learn to apply a major mathematical theory to solve non-trivial problems in other scientific domains. (Introduced, Practiced, Mastered)
- Students will apply differential and integral calculus to solve problems in other sciences. Students will learn and use problem-solving techniques to solve problems from other areas.
- Students will translate real-world problems into abstract mathematical models. Students will abstract, analyze, and draw analogies between objects from other areas and mathematical constructs and theories.

**MATH 127: Linear Optimization (4 Credits)**

Introduction to linear optimization, optimizing a linear function subject to a set of linear constraints, emphasizing quantitative modeling, methodology, and the underlying mathematical structures and geometrical ideas. Topics include problem formulation, simplex method, sensitivity analysis, and duality theory.

**Prerequisite(s):** MATH 004 or MATH 047 or ECON 081

**Note(s):** Concurrent workshop required. Prerequisite of MATH 004 or MATH 047 or ECON 081 may be waived by instructor.

**Meets the following Gen Ed requirements:** Quantitative and Computational Reasoning

**Crosslisted with:** CS 127

**Program Goals:**

Mathematics Program Goals

- Develop analytical skills and logical reasoning. (Practiced)
- Students will justify their solutions to problems, using a variety of methods for logical reasoning.
- Students will read deeply and critique proofs and solutions to problems.
- Develop analytical skills and logical reasoning. (Practiced)
- Apply logic in order to understand and analyze mathematical arguments.
- Develop analytical skills and logical reasoning. (Practiced)
- Develop analytical skills and logical reasoning. (Practiced)
- Evaluate alternative solutions, and analyze trade-offs among competing resources.
- Develop skills to formulate decision problems: the steps involved in this process include specifying objectives, identifying decision variables, and quantifying resource constraints.
- Develop analytical skills and logical reasoning. (Practiced)
- Students will analyze in depth mathematical statements.
- Develop ability to communicate mathematical thoughts in a clear and coherent fashion. (Practiced)
- Students will translate between mathematical symbols and English words.
- Students will clearly verbalize mathematical thoughts.
- Develop problem-solving skills, and in particular, develop the ability to handle problems that require multiple steps for their solution. (Mastered)
- Students will decompose a problem into smaller problems.
- Students will reduce a new problem to previously solved problems.
- Students will identify patterns and use them to structure problem-solving techniques.
- Students will analyze, contrast, and compare different approaches to solving a problem.
- Verbalize mathematical thoughts clearly.
- Write complete and coherent mathematical sentences.
- Formulate real-world decision problems as linear optimization problems, and be able to explain and validate the mathematical model.
- Improve quantitative skills. (Practiced, Mastered)
- Students will solve linear, quadratic, differential, and other equations.
- Students will compute the value of a function at a specified input.
- Students will operate correctly on mathematical objects for theoretical and computational purposes.
- Students will simplify answers after taking derivatives or integrating.
- Improve quantitative skills. (Mastered)
- Decompose a problem into smaller problems.
- Reduce a new problem to previously solved problems.
- Analyze, contrast, and compare different approaches to solving a problem.
- Learn to apply a major mathematical theory to solve non-trivial problems in other scientific domains. (Mastered)
- Students will translate real-world problems into abstract mathematical models.
- Perform operations on systems of linear equations involving multiple variables.
- Solve systems of linear equations using the Gauss-Jordan method.
- Learn to apply a major mathematical theory to solve non-trivial problems in other scientific domains. (Mastered)
- Students will learn and use problem-solving techniques to solve problems from other areas.
- Students will apply differential and integral calculus to solve problems in other sciences.

**General Education Goals:**

Quan. & Comp. Reasoning

- Translate problems into the language of mathematics and computer science, and solve them by using mathematical and computational methods and tools (Mastered)
- Recognize and formulate real-world decision problems as linear optimization problems. Examples of real-world problems are drawn from finance, marketing, production planning, transportation planning, telecommunications, operations management, system design, and games.
- Find optimal solutions to linear optimization problems using the Simplex algorithm.
- Perform sensitivity analysis – a technique for systematically changing parameters to determine the effects on the optimal solution – on linear optimization problems.
- Understand the structure and development of logical arguments (Practiced)
- Follow an abstract mathematical argument.
- Apply logic to understand and analyze mathematical arguments involving systems of linear equations.
- Understand the algebraic and graphical properties of important mathematical functions and use these functions in concrete applications (Practiced, Mastered)
- Solve algebraically and graphically systems of linear inequalities in two and three dimensions, and find the optimal solution, based on a linear objective function, that satisfies all linear constraints.
- Generalize the algebraic approach from two and three dimensions to higher dimensions of systems of linear inequalities, and find the optimal solution, based on a linear objective function, that satisfies all linear constraints.

**MATH 128: Theory of Computation (4 Credits)**

An introduction to the mathematical basis for the study of computability and to the formal theory behind compiler design. Topics include the formal models of computation such as finite state automata, pushdown automata, and Turing machines; languages and grammars, such as regular languages and grammars, context-free languages and grammars, and recursively enumerable languages and grammars; and the problems that a machine can and cannot solve.

**Prerequisite(s):** MATH 006 or MATH 048

**Note(s):** Concurrent workshop required. Prerequisite of MATH 006 or MATH 048 may be waived by the instructor.

**Meets the following Gen Ed requirements:** Quantitative and Computational Reasoning

**Crosslisted with:** CS 128

**Program Goals:**

Mathematics Program Goals

- Develop analytical skills and logical reasoning. (Mastered)
- Students will justify their solutions to problems, using a variety of methods for logical reasoning.
- Students will read deeply and critique proofs and solutions to problems.
- Develop analytical skills and logical reasoning. (Mastered)
- Apply logic to understand and analyze mathematical arguments involving automata or formal languages.
- Develop analytical skills and logical reasoning. (Mastered)
- Develop analytical skills and logical reasoning. (Mastered)
- Analyze computational problems with respect to language/automata hierarchy and computability.
- Solve computational problems by creating automata or formal languages.
- Develop analytical skills and logical reasoning. (Mastered)
- Students will analyze in depth mathematical statements.
- Develop ability to communicate mathematical thoughts in a clear and coherent fashion. (Mastered)
- Students will translate between mathematical symbols and English words.
- Students will clearly verbalize mathematical thoughts.
- Develop problem-solving skills, and in particular, develop the ability to handle problems that require multiple steps for their solution. (Mastered)
- Students will decompose a problem into smaller problems.
- Students will reduce a new problem to previously solved problems.
- Students will identify patterns and use them to structure problem-solving techniques.
- Students will analyze, contrast, and compare different approaches to solving a problem.
- Develop problem-solving skills, and in particular, develop the ability to handle problems that require multiple steps for their solution. (Mastered)
- Translate mathematical symbols into English words and vice versa.
- Write complete and coherent mathematical sentences.
- Verbalize mathematical thoughts clearly.
- Improve quantitative skills. (Mastered)
- Students will solve linear, quadratic, differential, and other equations.
- Students will compute the value of a function at a specified input.
- Students will operate correctly on mathematical objects for theoretical and computational purposes.
- Students will simplify answers after taking derivatives or integrating.
- Improve quantitative skills. (Mastered)
- Develop ability to solve computational problems using automata – finite state machines, pushdown automata, and Turing machines.
- Develop ability to solve computational problems using formal languages – regular languages and context-free languages.
- Demonstrate properties of languages and apply them to solve computational problems.
- Learn to apply a major mathematical theory to solve non-trivial problems in other scientific domains. (Introduced, Practiced)
- Students will translate real-world problems into abstract mathematical models.
- Students will learn and use problem-solving techniques to solve problems from other areas.
- Students will apply differential and integral calculus to solve problems in other sciences.

**General Education Goals:**

Quan. & Comp. Reasoning

- Translate problems into the language of mathematics and computer science, and solve them by using mathematical and computational methods and tools (Mastered)
- Translate real-world problems into computational problems associated with automata and formal languages.
- Solve computational problems by creating automata or formal languages, or show that they are unsolvable by proving no such automata or formal languages exist.
- Analyze computational problems with respect to language/automata hierarchy and computability.
- Understand the structure and development of logical arguments (Mastered)
- Follow an abstract mathematical argument concerning an automaton or a formal language.
- Apply logic to understand and analyze mathematical arguments involving automata and formal languages.

**MATH 131: Introduction to Abstract Algebra (4 Credits)**

This course introduces mathematical proof techniques in the context of Abstract Algebra. Set theory, logic, equivalence relations, and proof techniques are interwoven with basic number theory and modular congruence in the integers and in polynomial rings. Other topics include criteria for reducibility and irreducibility in polynomial rings over the rational, the real, and the complex numbers; the quotient of a polynomial ring; abstract rings, subrings, and ring homomorphisms and isomorphisms; and ideals and quotient rings. Basic group theory is included as time permits.

**Prerequisite(s):** MATH 048

**Note(s):** The instructor may waive the prerequisites for exceptional students.

**Program Goals:**

Mathematics Program Goals

- Develop analytical skills and logical reasoning. (Practiced, Mastered)
- Students will understand a general mathematical argument and then replicate the argument in a specific example. Students will understand a mathematical hypothesis, condition, or definition and check that it is satisfied in a specific example. Students will analyze in depth mathematical statements.
- Students will justify their solutions to problems, using a variety of methods for logical reasoning. Students will construct rigorous logical arguments, using a variety of proof methods and logical reasoning.
- Students will read deeply and critique proofs and solutions to problems.
- Develop ability to communicate mathematical thoughts in a clear and coherent fashion. (Practiced, Mastered)
- Students will translate between mathematical symbols and English words.
- Students will clearly verbalize mathematical thoughts.
- Students will decompose a problem into smaller problems. Students will reduce a new problem to previously solved problems.
- Students will identify patterns and use them to structure problem-solving techniques. Students will analyze, contrast, and compare different approaches to solving a problem.
- Improve quantitative skills. (Practiced, Mastered)
- Students will correctly calculate the derivatives and integrals of a variety of functions and use the results in the proofs of mathematical theorems and as part of solving relevant real analysis problems. Students will determine/calculate various function and objects characteristics within the context of the problem and mathematical area.
- Students will solve linear, quadratic, differential, and other equations. Students will operate correctly on mathematical objects for theoretical and computational purposes.
- Students will compute the value of a function at a specified input. Students will simplify answers after taking derivatives or integrating. Students will correctly manipulate algebraic expressions, equations and systems of equations in one or more variables.

**MATH 132: Topics in Algebra (4 Credits)**

This course goes deeper into the study of Abstract Algebra. Topics vary from group theory (homomorphisms and isomorphisms, Lagrange's Theorem, normality, quotients, fundamental isomorphism theorems, symmetric groups, direct products, classification of finite abelian groups) to basic commutative algebra (integral domains, Euclidean domains, principal ideal domains, unique factorization domains, factorization of quadratic integers, and the field of fractions of an integral domain) to Galois theory. Additional topics as time and interest permits.

**Prerequisite(s):** MATH 131

**Program Goals:**

Mathematics Program Goals

- Develop analytical skills and logical reasoning. (Practiced, Mastered)
- Students will understand a general mathematical argument and then replicate the argument in a specific example. Students will understand a mathematical hypothesis, condition, or definition and check that it is satisfied in a specific example. Students will analyze in depth mathematical statements.
- Students will justify their solutions to problems, using a variety of methods for logical reasoning. Students will construct rigorous logical arguments, using a variety of proof methods and logical reasoning.
- Students will read deeply and critique proofs and solutions to problems.
- Develop ability to communicate mathematical thoughts in a clear and coherent fashion. (Practiced, Mastered)
- Students will translate between mathematical symbols and English words.
- Students will clearly verbalize mathematical thoughts.
- Students will decompose a problem into smaller problems. Students will reduce a new problem to previously solved problems.
- Students will identify patterns and use them to structure problem-solving techniques. Students will analyze, contrast, and compare different approaches to solving a problem.
- Improve quantitative skills. (Practiced, Mastered)
- Students will correctly calculate the derivatives and integrals of a variety of functions and use the results in the proofs of mathematical theorems and as part of solving relevant real analysis problems. Students will determine/calculate various function and objects characteristics within the context of the problem and mathematical area.
- Students will solve linear, quadratic, differential, and other equations. Students will operate correctly on mathematical objects for theoretical and computational purposes.
- Students will compute the value of a function at a specified input. Students will simplify answers after taking derivatives or integrating. Students will correctly manipulate algebraic expressions, equations and systems of equations in one or more variables.

**MATH 141: Real Analysis I (4 Credits)**

The Real Analysis sequence is a rigorous presentation of the basic concepts of real analysis, including the real number system, suprema and infima, and completeness; estimations and approximations; sequences, subsequences, and convergence; cluster points, limits of sequences, and the Bolzano-Weierstrass Theorem; Cauchy sequences; infinite series and the convergence tests; and power series.

**Prerequisite(s):** MATH 048 and MATH 131

**Program Goals:**

Mathematics Program Goals

- Develop analytical skills and logical reasoning. (Practiced)
- Students will analyze in depth mathematical statements.
- Develop analytical skills and logical reasoning. (Practiced)
- Students will construct rigorous logical arguments, using a variety of proof methods and logical reasoning. Examples include direct proof, proof by contradiction or by mathematical induction; correct use of examples and counterexamples, and of numerous problem-solving techniques within proofs of real analysis theorem and solutions to relevant problems.
- Students will analyze in depth mathematical statements about the real number system, sequences, series, and functions.
- Develop analytical skills and logical reasoning. (Practiced)
- Students will read deeply and critique proofs and solutions to problems.
- Students will justify their solutions to problems, using a variety of methods for logical reasoning.
- Develop ability to communicate mathematical thoughts in a clear and coherent fashion. (Practiced)
- Using Newton's method, students will be able to graphically explain the convergence or divergence of certain sequences.
- Students will be able to work with the real number line and to visually represent various notions related to real numbers and sequences, such as rationality, monotonicity, boundedness, supremum and infinimum, convergence, and others.
- Students will clearly verbalize mathematical thoughts.
- Students will translate between mathematical symbols and English words.
- Develop ability to communicate mathematical thoughts in a clear and coherent fashion. (Practiced)
- Students will be able to draw graphs of functions and overlay them with representations of areas of plane figures (such as rectangles, trapezoids, triangles and circles), for the purposes of comparing and deriving inequalities and other properties of sequences, series, and functions.
- Students will decompose a problem into smaller problems.
- Students will reduce a new problem to previously solved problems.
- Students will identify patterns and use them to structure problem-solving techniques.
- Students will analyze, contrast, and compare different approaches to solving a problem.
- Students will learn to combine mathematical correctness with clear presentation skills, both oral and written.
- Students will communicate in a rigorous and clear manner their solutions to real analysis problems via oral participation in class, written and oral presentation in workshops, and rigorous written form on exams and homework assignments.
- Students will learn to critique each other's solutions in workshop in a professional manner typical for mathematical seminars and talks.
- Improve quantitative skills. (Practiced, Mastered)
- Students will operate correctly on mathematical objects for theoretical and computational purposes.
- Students will simplify answers after taking derivatives or integrating.
- Improve quantitative skills. (Practiced)
- Students will translate some real-life problems and problems from other disciplines into statements on sequences, series, and functions. Typical examples include the Fibonacci sequence, nested intervals, cluster points, boundedness of sets and sequences, suprema and infima and the Completeness Property, rearrangements of conditionally convergent series, Cesaro summable series, and many more.
- Problem-solving techniques for a variety of situations will be introduced and applied by the students. Examples include the use of absolute values, triangle inequality, "sandwich" techniques, and trigonometry in proofs of inequalities; the ability to intelligently guess and rigorously prove direct formulas for sequences defined recursively or via sums; the insight to know when it is appropriate and useful to rationalize or clear denominators, or otherwise manipulate fractions in problems, and when to use the telescoping method to find the sum of a series; to correctly apply the multistep technique of Newton's method for approximating roots, or to create error terms and integrals in search of limits of sequences.
- Improve quantitative skills. (Practiced, Mastered)
- Students will solve linear, quadratic, differential, and other equations.
- Students will compute the value of a function at a specified input.
- Students will correctly calculate derivatives and integrals within the proofs of mathematical theorems and as part of solving relevant real analysis problems.
- Students will correctly manipulate inequalities and algebraic expressions and equations in one (or more) variables.

**MATH 142: Real Analysis II (4 Credits)**

Continuation of Real Analysis I. Topics include: elementary properties of functions of a single variable; local and global behavior of functions; continuity and limits; Intermediate Value Theorem; properties of continuous functions on compact intervals; Rolle's Theorem; Mean Value Theorem; l'Hospital's rule for indeterminate forms; linearization and applications to convexity; theory of Taylor polynomials; the Riemann integral; Fundamental Theorems of Calculus; improper integrals; and sequences and series of functions.

**Prerequisite(s):** MATH 141

**Program Goals:**

Mathematics Program Goals

- Develop analytical skills and logical reasoning. (Practiced)
- Students will analyze in depth mathematical statements.
- Develop analytical skills and logical reasoning. (Practiced)
- Students will construct rigorous logical arguments, using a variety of proof methods and logical reasoning. Examples include direct proof, proof by contradiction or by mathematical induction; correct use of examples and counterexamples, and of numerous problem-solving techniques within proofs of real analysis theorem and solutions to relevant problems.
- Students will analyze in depth mathematical statements about the local and global properties of functions, limits and derivatives, Taylor polynomials and Riemann integrals.
- Develop analytical skills and logical reasoning. (Practiced)
- Students will read deeply and critique proofs and solutions to problems.
- Students will justify their solutions to problems, using a variety of methods for logical reasoning.
- Develop ability to communicate mathematical thoughts in a clear and coherent fashion. (Practiced)
- Develop ability to communicate mathematical thoughts in a clear and coherent fashion. (Practiced)
- Using method of subdivision of intervals, students will be able to graphically explain the Intermediate Value Theorem. Combining the notions of first and second derivatives, students will be able to graphically depict convexity of functions.
- Students will be able to work with the real number line and to visually represent various notions related to functions, such as monotonicity, boundedness, supremum and infinimum, convergence, differentiabililty, integrability, and others.
- Develop ability to communicate mathematical thoughts in a clear and coherent fashion. (Practiced)
- Students will clearly verbalize mathematical thoughts.
- Students will translate between mathematical symbols and English words.
- Develop ability to communicate mathematical thoughts in a clear and coherent fashion. (Practiced)
- Students will be able to draw graphs of functions and overlay them with representations of areas of plane figures, for the purposes of comparing and deriving properties of integrals and of evaluating integrals, as well as proving theorems in real analysis.
- Students will decompose a problem into smaller problems.
- Students will reduce a new problem to previously solved problems.
- Students will identify patterns and use them to structure problem-solving techniques.
- Students will analyze, contrast, and compare different approaches to solving a problem.
- Students will learn to combine mathematical correctness with clear presentation skills, both oral and written.
- Students will communicate in a rigorous and clear manner their solutions to real analysis problems via oral participation in class, written and oral presentation in workshops, and rigorous written form on exams and homework assignments.
- Students will learn to critique each other's solutions in workshop in a professional manner typical for mathematical seminars and talks.
- Improve quantitative skills. (Practiced, Mastered)
- Students will operate correctly on mathematical objects for theoretical and computational purposes.
- Students will simplify answers after taking derivatives or integrating.
- Improve quantitative skills. (Practiced)
- Students will translate some real-life problems and problems from other disciplines into statements on functions and integrals. Typical examples include applications of L'Hospital's rule, Linearization, Taylor polynomials and Lagrange's remainder, Stirling's formula and the gamma function.
- Problem-solving techniques for a variety of situations will be introduced and applied by the students. Examples include the use of the limit definition to derive limit laws for functions and then use these in problems; to use the definition of integrals to derive properties of and theorems about integrals and use these in problems.
- Improve quantitative skills. (Practiced, Mastered)
- Students will solve linear, quadratic, differential, and other equations.
- Students will compute the value of a function at a specified input.
- Students will correctly calculate derivatives and integrals within the proofs of mathematical theorems and as part of solving relevant real analysis problems.
- Students will correctly manipulate inequalities and algebraic expressions and equations in one (or more) variables.

**MATH 154: Foundations of Geometry (4 Credits)**

A survey of various systems of geometry from a modern point of view, using techniques from algebra and logic. Possible topics include Euclidean geometry, non-Euclidean geometries (such as elliptic, hyperbolic, and parabolic geometry), affine geometry, projective geometry, and finite geometries.

**Prerequisite(s):** MATH 050

**Note(s):** Prerequisite of MATH 050 may be waived with instructor consent.

**Program Goals:**

Mathematics Program Goals

- Develop analytical skills and logical reasoning. (Mastered)
- Develop analytical skills and logical reasoning. (Practiced, Mastered)
- Students will analyze in depth mathematical statements.
- Students will justify their solutions to problems, using a variety of methods for logical reasoning.
- Students will read deeply and critique proofs and solutions to problems.
- Develop ability to communicate mathematical thoughts in a clear and coherent fashion. (Practiced, Mastered)
- Students will translate between mathematical symbols and English words.
- Students will clearly verbalize mathematical thoughts.
- Students will reduce a new problem to previously solved problems.
- Students will decompose a problem into smaller problems.
- Students will analyze, contrast, and compare different approaches to solving a problem.
- Students will identify patterns and use them to structure problem-solving techniques.
- Improve quantitative skills. (Mastered)
- Learn to apply a major mathematical theory to solve non-trivial problems in other scientific domains. (Practiced)
- Students will translate real-world problems into abstract mathematical models.
- Students will learn and use problem-solving techniques to solve problems from other areas.
- Students will apply differential and integral calculus to solve problems in other sciences.

**MATH 158: Topics in Topology (1-3 Credits)**

This course provides an introduction to the area of mathematics that studies geometric properties unaffected by continuous deformation. Topics may vary among point-set topology, metric spaces, compactness, surfaces, the Fundamental Group, simplicial homology, computational topology, and topological data analysis.

**Program Goals:**

Mathematics Program Goals

- Develop analytical skills and logical reasoning. (Mastered)
- Students will read deeply and critique proofs and solutions to problems.
- Develop ability to communicate mathematical thoughts in a clear and coherent fashion. (Mastered)
- Students will translate between mathematical symbols and English words.
- Students will clearly verbalize mathematical thoughts.
- Improve quantitative skills. (Mastered)
- Learn to apply a major mathematical theory to solve non-trivial problems in other scientific domains. (Mastered)
- Students will apply differential and integral calculus to solve problems in other sciences. Students will learn and use problem-solving techniques to solve problems from other areas.
- Students will translate real-world problems into abstract mathematical models. Students will abstract, analyze, and draw analogies between objects from other areas and mathematical constructs and theories.

**MATH 160: Complex Analysis (4 Credits)**

An introduction to the calculus of functions that have complex numbers as arguments and values. Topics include algebra and geometry of complex numbers; elementary functions of a complex variable; differentiation and integration of complex functions; Cauchy's Integral Theorem; Taylor's and Laurent's (infinite) series for complex functions; residues; and conformal mapping.

**Prerequisite(s):** MATH 141

**Program Goals:**

Mathematics Program Goals

- Develop analytical skills and logical reasoning. (Practiced, Mastered)
- Students will analyze in depth mathematical statements.
- Develop analytical skills and logical reasoning. (Practiced)
- Develop analytical skills and logical reasoning. (Practiced, Mastered)
- Students will justify their solutions to problems, using a variety of methods for logical reasoning.
- Students will read deeply and critique proofs and solutions to problems.
- Students will translate between mathematical symbols and English words.
- Students will clearly verbalize mathematical thoughts.
- Develop ability to communicate mathematical thoughts in a clear and coherent fashion. (Mastered)
- Students will decompose a problem into smaller problems.
- Students will identify patterns and use them to structure problem-solving techniques.
- Students will analyze, contrast, and compare different approaches to solving a problem.
- Students will reduce a new problem to previously solved problems.
- Improve quantitative skills. (Mastered)
- Students will operate correctly on mathematical objects for theoretical and computational purposes.
- Improve quantitative skills. (Mastered)
- Improve quantitative skills. (Mastered)
- Students will solve linear, quadratic, differential, and other equations.
- Students will compute the value of a function at a specified input.
- Students will simplify answers after taking derivatives or integrating.
- Students will learn and use problem-solving techniques to solve problems from other areas.
- Students will learn and use problem-solving techniques to solve problems from other areas.
- Students will apply differential and integral calculus to solve problems in other sciences.

**MATH 179: Directed Research (0.25-1 Credits)**

**MATH 180: Topics in Mathematics (4 Credits)**

Offers topics that are not offered in the regular curriculum from the following fields: algebra, algebraic geometry, algebraic logic, analysis, applied linear algebra, combinatorics, geometry, linear algebra, mathematical logic and foundations of mathematics, number theory, representation theory, and topology.

**Prerequisite(s):** MATH 047 and MATH 048

**Meets the following Gen Ed requirements:** Quantitative and Computational Reasoning

**Program Goals:**

Mathematics Program Goals

- Develop analytical skills and logical reasoning. (Practiced, Mastered)
- Students will analyze in depth mathematical statements.
- Students will justify their solutions to problems, using a variety of methods for logical reasoning.
- Students will read deeply and critique proofs and solutions to problems.
- Students will clearly verbalize mathematical thoughts.
- Students will translate between mathematical symbols and English words.
- Develop problem-solving skills, and in particular, develop the ability to handle problems that require multiple steps for their solution. (Introduced, Practiced, Mastered)
- Students will analyze, contrast, and compare different approaches to solving a problem.
- Students will decompose a problem into smaller problems.
- Students will reduce a new problem to previously solved problems.
- Students will identify patterns and use them to structure problem-solving techniques.
- Improve quantitative skills. (Introduced, Practiced, Mastered)
- Students will compute the value of a function at a specified input.
- Students will simplify answers after taking derivatives or integrating.
- Students will solve linear, quadratic, differential, and other equations.
- Students will operate correctly on mathematical objects for theoretical and computational purposes.
- Students will translate real-world problems into abstract mathematical models.
- Students will apply differential and integral calculus to solve problems in other sciences.
- Students will learn and use problem-solving techniques to solve problems from other areas.

**General Education Goals:**

Quan. & Comp. Reasoning

- Translate problems into the language of mathematics and computer science, and solve them by using mathematical and computational methods and tools (Introduced, Practiced)
- Students will be able to apply problem-solving techniques and theoretical and computational tools in solving multi-step problems in multiple areas of mathematics such as: inversion in the plane, combinatorics, abstract algebra (applications to Rubik's cube), number theory, point-mass geometry, complex numbers, invariants and monovariants, circle geometry and others.
- Understand the structure and development of logical arguments (Introduced, Practiced)
- Students will be able to develop logical arguments and construct proofs, using proof techniques such as direct proof, proof by case-analysis, mathematical induction, by contradiction, and special extra constructions.
- Students will occasionally use properties and features of functions such as monotonicity, tangent and secant slopes, concavity, rate of increase and other notions from Calculus and Pre-calculus, in order to analyse problems in several mathematical areas.
- Understand and apply the fundamental ideas of probability and statistics (Introduced)
- Students will learn basic ideas from Combinatorics, such as permutations, combinations, binomial coefficients, and others, and will apply their knowledge of summation of specific series to problems in game theory.

## Faculty

**Maia Averett**

Associate Professor of Mathematics

Department Head of Mathematics and Computer Science

CPM 200A, 510.430.2106, maverett@mills.edu

Professional interests: Algebraic topology, homotopy theory, women and mathematics

**Barbara Li Santi**

Professor of Mathematics and Computer Science

CPM Room 200B, 510.430.2247, barbara@mills.edu

Professional Interests: Artificial intelligence applications in education, cognitive science, intelligent tutoring systems, computer science education, linear algebra

**Susan S. Wang**

Professor of Computer Science

Interdisciplinary Computer Science Program Head

Sarlo Award Recipient

CPM Room 202, 510.430.2138, wang@mills.edu

Professional Interests: Design and analysis of algorithms, very large-scale integrated systems, parallel computation

## Staff

**Holly Robinson**

Faculty Administrative Assistant

CPM Room 104, 510.430.2226, holly@mills.edu