# List of Courses

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## EDCS 624 School Mathematics Curriculum (3 credits)

Mathematics in the Schools: Geometry and Measurement provides an overview of the following: K-12 mathematics content, curricula, pedagogy, standards, trends and issues, theory, and research. This course is the third of six graduate level courses intended to strengthen participants’ knowledge of K-12 school mathematics and to develop their strength as a facilitator of standards-based classroom instruction, assessment, and content support.

## EDCS 640M Ethnomathematics and STEM Education (3 credits)

In an effort to address issues of equitable and quality education, culturally responsive science, technology, engineering, and mathematics (STEM) pedagogy is explored across the State of Hawai‘i and Pacific based on the Mathematics Common Core State Standards (CCSS) and Next Generation Science Standards (NGSS). This course brings together research and community-based educational materials and organizations including the University of Hawai‘i System, Hawai‘i State Department of Education, Pacific American Foundation, Pacific Resources for Education and Learning, Hawai‘i P-20 Partnerships for Education, and the Polynesian Voyaging Society.

Ethnomathematics encourages the investigation of how STEM continues to be adapted by people around the world. Intellectual merit lies in assessing the effects of using culture-based STEM curriculum units to increase the engagement and enhance the learning of diverse, under-represented populations in alignment with CCSS and NGSS. The broader impact is that culture-based ethnomathematics practices will be recovered and integrated in STEM curriculum that enriches the global pedagogical base. In this course, we discover experiential learning inside and outside of the classroom through promising practices that are relevant, contextualized, and sustainable.

## EDCS 653D Mathematics in the Schools: Geometry and Measurement (3 credits)

Mathematics in the Schools: Geometry and Measurement is an analysis of research relating to teaching and learning mathematics, and allows students to apply research findings to classroom procedures. Through geometry and measurement standards and principles, we appraise recent curricular trends and critically examine assumptions underlying proposed changes in K-12 schools.

## MATH 100 Survey of Mathematics (3 credits)

Math 100 includes a variety of selected mathematical topics designed to acquaint students with examples of mathematical reasoning. The topics included in a given section or academic term are chosen by the instructor to demonstrate the beauty and power of mathematics from applied, symbolic, and abstract standpoints. Techniques, models, examples, and applications are drawn from several of the following areas: algebra, business mathematics, calculus, functions, geometry, graph theory, linear programming, logic, matrices, number systems, number theory, probability, sequences, set theory, statistics, and trigonometry. Math 100 is not intended as, and does not qualify as, a prerequisite for advanced mathematics courses.

Upon successful completion of Math 100, the student will be able to apply critical thinking, including rules of logical sequence, to problem-solving. The student will have a clearer understanding and insight into algebraic and geometric concepts and techniques and into the power of mathematics and symbolic reasoning. Specifically, the student will be able to carry out the following:

- Model applied problems symbolically and perform manipulations on the symbols within an appropriately selected mathematical or logical formal system;
- Distinguish between a rigorous proof and a conjecture;
- Author an elementary proof;
- Correctly select, then appropriately apply, formal rules or algorithms to solve numeric, symbolic, graphical, and/or applied problems;
- Assess the reasonableness of, then appropriately communicate, the solutions to problems.

## MATH 103 College Algebra (3 credits)

This course follows the elementary algebra sequence and will prepare students for pre-calculus, statistics, or other courses requiring algebraic, geometric or symbolic thinking and deduction. Students will apply algebraic and geometric techniques to solve problems, including simplifying, factoring, and/or solving radical expressions; linear, quadratic, absolute value, and literal equations; and working with inequalities, complex numbers, quadratic systems, logarithms, and introductory functions and graphs.

Upon successful completion of Math 103, the student will be prepared with the skills and knowledge necessary to succeed in pre-calculus courses (such as Math 135, 140, etc.), as well as introductory statistics, financial, and economic courses. The student will be able to apply critical thinking, including rules of logical sequence, to problem-solving. The student will have a clearer understanding and insight into algebraic and geometric concepts and techniques and into the power of mathematics and symbolic reasoning. Specifically, the student will be able to carry out the following algebraic and geometric tasks:

- Use algebraic symbols to solve problems, including but not limited to the following topics: solving equations and inequalities and systems of equations in 2 or 3 unknowns, factoring polynomials, performing arithmetic operations on rational expressions and radicals, quadratic equations, roots of equations, etc. at a greater complexity than elementary algebra.
- Demonstrate proficiency in writing and explaining mathematical expressions.
- Interpret equations geometrically and obtain the equations for lines and circles.
- Use algebraic and geometric techniques to analyze and solve applied problems.
- Utilize introductory functions and draw their graphs.
- Utilize logarithms and solve basic logarithmic and exponential equations.
- Demonstrate ability to solve systems of linear and 2nd degree equations and inequalities.

## MATH 111/112 Math for Elementary Teachers I/II (3 credits)

The series Math 111 and Math 112 are designed for prospective elementary school teachers, to enhance their mathematical skills, and to prepare them to take the teaching-methods course. Topics covered include: understanding, communicating, and representing mathematical ideas, problem solving, and reasoning. Operations are performed on sets, natural numbers, integers, fractions, real numbers, and functions, including properties and patterns of these operations. Connections are made to other parts of mathematics, pedagogy, and applications.

This course focuses on mathematical content for teaching, and includes not only topics, ideas, skills and procedures in specific mathematical domains, but also the mathematical thinking and reasoning involved in the mathematical tasks that teachers do. Upon successful completion of this course, the student will be able to carry out the following:

- Possess adequate knowledge and a flexible understanding of the mathematics necessary for teaching students in grades K-6, and the ability to use that knowledge.
- Understand the central features of an adequate mathematical explanation and be able to provide such explanations.
- Be able to interpret, evaluate, and respond to the ideas, explanations, solutions and methods of others.
- Be able to identify and analyze student errors.
- Be able to choose and use multiple representations (verbal, symbolic, visual, etc.), examine correspondences and equivalences among representations, and make sense of representations different from your own.
- Be able to adequately communicate mathematical ideas both in writing and orally, in a clear convincing, and accurate way and make use of appropriate representations when applicable.

## MATH 135 Precalculus I: Elementary Functions (3 credits)

Math 135 – Pre-Calculus: Elementary Functions includes a variety of selected mathematical topics designed to acquaint students with a functional approach to algebra, including polynomial, exponential, and logarithmic functions; higher degree equations; inequalities; sequences; the binomial theorem; and partial fractions. This course is recommended for students pursuing further studies in business, economics, mathematics, and/or science-related fields.

Upon successful completion of Math 135 – Pre-Calculus: Elementary Functions, the student will be able to apply critical thinking, including rules of logical sequence, to problem-solving. The student will have a clearer understanding and insight into algebraic and geometric concepts and techniques and into the power of mathematics and symbolic reasoning. Specifically, the student will be able to carry out the following:

- Proficiently solve equations and inequalities, including those involving radical, exponential, and logarithmic expressions.
- Examine, combine, and find the inverses of functions.
- Use properties to construct graphs of relations and functions in the Cartesian plane, including conic sections.
- Evaluate the properties of polynomial functions and their graphs, including symmetry, intercepts, and zeros with their multiplicities.
- Analyze the properties of rational functions and their graphs, including domain, range, asymptotes, and intercepts.
- Analyze the properties of exponential and logarithmic functions and their graphs, including domain, range, and asymptotes.
- Model and solve various applications problems related to the studied relations and functions.

## MATH 140 II Trigonometry and Analytic Geometry (3 credits)

Math 140 – Pre-Calculus: Trigonometry and Analytic Geometry is a study of the properties and graphs of trigonometric, circular, and inverse functions; solution of triangles; identities; solutions of trigonometric equations; conic sections; polar coordinates; and parametric equations. This course is recommended for students pursuing further studies in business, economics, mathematics, and/or science-related fields.

Upon successful completion of Math 140 – Pre-Calculus: Trigonometry and Analytic Geometry, the student will be able to apply critical thinking, including rules of logical sequence, to problem-solving. The student will have a clearer understanding and insight into algebraic and geometric concepts and techniques and into the power of mathematics and symbolic reasoning. Specifically, the student will be able to carry out the following:

- Calculate, understand, and apply the trigonometric ratios of acute angles.
- Understand and evaluate the trigonometric functions of variables expressed in general angle and radian units.
- Analyze and graph trigonometric functions.
- Solve application problems involving trigonometric concepts, including right triangles, arc length, area, and angular speed.
- Understand the derivation of, and be able to apply, the essential trigonometric identities.
- Utilize the essential trigonometric identities to simplify expressions and prove further identities.
- Analyze, graph, understand, and use the inverse trigonometric functions.
- Derive and apply the Law of Sines and Law of Cosines.
- Identify and be able to use further trigonometric topics, including vectors in the plane, parametric equations, and polar coordinates.
- Understand the properties of parabolas, ellipses, and hyperbolas.
- Solve various application problems related to trigonometry and the conic sections.

## MATH 241 Calculus I (4 credits)

Math 241 – Calculus I is a study of limits, continuity, and derivatives. Topics covered include: computations of derivatives — sum, product, and quotient formulas; implicit differentiation; the chain-rule; mean-value theorem; Simpson’s Rule, definite integrals; and the Fundamental Theorem of Calculus. This is a rigorous study of algebraic and trigonometric functions with the analysis of their derivatives. Linear approximation and Newton's method are studied, along with applications of derivatives to maximum-minimum problems and related rate problems. Applications of definite integrals are used to compute area, volume, arc lengths, and surface areas. Upon successful completion of Math 241, a student should:

- Understand and be able to compute limits.
- Understand continuity and be able to identify continuous and discontinuous functions.
- Understand the derivative as a generalization of the rate of change of a linear function.
- Understand the derivation of differentiation formulas (sum, difference, product, quotient, power, etc.).
- Be able to compute derivatives of various functions using the derived differentiation formulas.
- Understand and be able to use the concepts of increasing and decreasing functions, relative and absolute minimums and maximums, and the mean value theorem.
- Be able to solve various application problems involving differentiation, including related rates and minimum/maximum problems.
- Understand and be able to calculate antiderivatives.
- Understand the concept of the definite integral.
- Understand and be able to use the Fundamental Theorem of Calculus.
- Be able to use substitution to calculate integrals.
- Be able to solve various application problems involving integration, including area between curves and volumes using the disk, washer, and shell methods.

## MATH 242 Calculus II (4 credits)

Math 242 – Calculus II is the second course in the calculus sequence. The course extends differentiation and integration to inverse trigonometric, logarithmic, and exponential functions, and covers such topics as basic techniques of integration, improper integrals, Taylor's series of functions and their applications and differential equations. Upon successful completion of Math 242, a student should be able to:

- Compute derivatives and integrals of exponential, logarithmic, and inverse trigonometric functions.
- Apply limit theorems to solve indeterminate forms when possible.
- Apply integration techniques such as integration by parts, trigonometric substitution, and partial fractions.
- Compute improper integrals.
- Compute arc length and surface area for a revolution.
- Understand properties of, and be able to apply theorems and tests to, sequences and series, such as the integral test, comparison tests, alternating series, absolute and conditional convergence, the root test, and the ratio test.
- Understand properties of, and be able to compute coefficients of, power series and related functions.
- Model and solve various application problems.

## MATH 243 Calculus III (3 credits)

Math 243 – Calculus III is the third course in the calculus sequence for STEM (science, technology, engineering, mathematics) majors. The course covers vector algebra and geometry, vector-valued functions and motion in space, polar coordinates, differentiation in several variables, and optimization. Course content is articulated with the other University of Hawai‘i four-year campuses to ensure the depth, breadth and relevance of the course content and coverage. The articulation of the course allows students to take the next mathematics course in the sequence, and ensures that the student’s level of competency after the completion of the course is consistent throughout the system. Upon successful completion of Math 243, a student should:

- Understand and compute vectors and the geometry of space including the three dimensional coordinate system, vectors, dot product, cross product, lines and planes in space, and quadric surfaces.
- Understand properties of and be able to apply polar coordinates such as graphing in polar coordinates and area and curve length.
- Model and solve vector–valued functions and motion in space including derivatives, integrals of vector functions, arc length in space, curvature of a curve, tangential and normal components of acceleration, velocity and acceleration in polar coordinates, and Kepler’s laws.
- Compute partial derivatives such as functions of several variables, limits and continuity in higher dimensions, partial derivatives, chain rule, directional derivatives and gradient vectors, tangent planes and differentials, extreme values and saddle points, Lagrange multipliers, and Taylor’s formula for two variables.

## MATH 244 Calculus IV (3 credits)

Math 244 – Calculus IV is the fourth course in the calculus sequence for STEM (science, technology, engineering, mathematics) majors. Mathematics is the basic language for STEM fields. Understanding the language, the basic ideas and results, and the computational techniques of calculus is prerequisite to more advanced learning. Course content is articulated with the other University of Hawai‘i four-year campuses to ensure the depth, breadth and relevance of the course content and coverage. The articulation of the course allows students to take the next mathematics course in the sequence, and ensures that the student’s level of competency after the completion of the course is consistent throughout the system.

The course covers multiple integrals, integration in vector fields, line integrals and Green’s Theorem, surface integrals, and Stokes’ and Gauss’ Theorems. Upon successful completion of Math 244 – Calculus IV, the student will be able to apply critical thinking, including rules of logical sequence to problem-solving. The student will have a clearer understanding and insight into concepts and techniques the exhibit the power of mathematics and symbolic reasoning. Specifically, the student will be able to carry out problems, analyze ideas, and conduct proofs in the following two main areas:

- Compute multiple integrals including double and iterated integrals over rectangles, double integrals over general regions, area by double integrations, double integrals in polar form, triple integrals in rectangular coordinates, moments and centers of mass, triple integrals in cylindrical and spherical coordinates, and substitutions in multiple integrals.
- Apply and understand integration in vector fields such as line integrals, vector fields, work, circulation and flux, path independence, potential functions, conservative fields, Green’s Theorem in the plane, surfaces and area, surface integrals and flux, Stokes’ Theorem, and Divergence Theorem and a unified theory.

## MATH 311 Introduction to Linear Algebra (WI, 3 credits)

Linear algebra is one of the basic and foundational topics in mathematics that every mathematics major needs to understand. It is one of the early courses in which the student is exposed to mathematics at an abstract level. Thus, the student will learn to read and understand mathematical definitions, and write mathematical proofs. The quality of writing is an integral part of the overall course grade. The student may be asked to revise assignments multiple times before receiving credit. A minimum of 16 pages is required, including formal writing appropriate to the discipline. Students may receive credit for only one of Math 307 and 311. Upon successful completion of Math 311 – Introduction to Linear Algebra, the student will be able to:

- Carry out problems in linear equations and matrices such as systems of linear equations, matrix operations and Rn, and matrix transformations.
- Analyze ideas and solve linear systems such as row–echelon form of a matrix, elementary operations and matrices, and similar matrices.
- Compute and understand determinants including definition and properties, cofactor expansion, computation of inverses, and applications.
- Model real vector spaces including the definition and examples of vector spaces, subspaces, span and linear independence, basis and dimension, homogeneous systems, coordinates and isomorphisms, and rank.
- Apply inner product spaces such as definition of inner product space, geometry (length, angles, orthogonality, projections), Gram-Schmidt orthogonalization, and orthogonal complements.
- Compute and understand linear transformations and matrices including definition and examples, kernel and range of a linear transformation, matrix of linear transformation, vector space of linear transformation, and similarity.
- Conduct proofs of eigen values and eigen vectors including diagonalization of similar matrices and diagonalization of symmetric matrices.

## MATH 321 Introduction to Advanced Mathematics (WI, 3 credits)

Advanced mathematics is one of the basic and foundational topics in mathematics that every mathematics major needs to understand, including a formal introduction to the concepts of logic, finite and infinite sets, functions, methods of proof and axiomatic systems. The student will learn to read and understand mathematical definitions, and write mathematical proofs. The quality of writing is an integral part of the overall course grade. The student may be asked to revise assignments multiple times before receiving credit. A minimum of 16 pages is required, including formal writing appropriate to the discipline. In this course students are expected to work on the sometimes difficult transition from a computational approach to mathematics to an abstract and conceptual one. Upon successful completion of Math 321 – Introduction to Advanced Mathematics, the student will be able to:

- Develop and write direct proofs, proofs by contradiction, and proofs by induction.
- Formulate definitions and give examples and counterexamples.
- Understand the axiomatic approach to simple mathematical systems.
- Understand and model propositional logic.
- Compute basic number theory, including divisibility and modular arithmetic.
- Compute basic operations of set theory, functions on sets, and relations.

## MATH 327 History of Mathematics (3 credits)

In Math 327 – History of Mathematics, the student will be able to apply critical thinking, including rules of logical sequence to problem-solving centered on the beauty, power, clarity and precision of formal systems. The history of mathematics may be of interest to any student with a strong interest in mathematics and the appropriate background. It is particularly recommended for students wishing to teach secondary mathematics. This course is a historical development of mathematical techniques and ideas, including the inter-relationships of mathematics and sciences. Highlights include: Euclidean geometry and number theory including classical constructions, history of calculus, foundations for analysis, polynomial equations, and set theory and logic. Upon successful completion of Math 327 – History of Mathematics, the student will be able to:

- Solve problems using historical techniques, including ancient mathematics such as Egyptian fractions, and more modern mathematics such as solving cubic equations with Cardano’s formula.
- Describe the Calculus of Newton and Leibniz, and explain the subtle flaws in their development that are repaired by modern ǫ, δ definitions.
- State Euclid’s fifth postulate, explain why mathematicians spent centuries trying to prove it, state its negation, and explain how the negation leads to non-Euclidean geometries.
- Perform some classical constructions with a straightedge and compass, and describe the history of the three “impossible constructions.”
- Describe the lives of some important figures in the history of mathematics, including knowing when they lived, what they are famous for, and some important biographical facts.
- Explain what an axiom system is, what it means for an axiom system to be consistent, what Goedel’s work tells us about our ability to prove consistency, and how mathematicians’ views of axiom systems have evolved since the time of Euclid.
- Historically important mathematicians and scientists will be interwoven with studying the mathematics they developed.

## MATH 351 Foundation of Euclidean Geometry (WI, 3 credits)

This course covers axiomatic Euclidean geometry and an introduction to the axiomatic method, with an emphasis on writing instruction. Course content is articulated with the other University of Hawai‘i four-year campuses to ensure the depth, breadth and relevance of the course content and coverage. The articulation of the course allows students to take the next mathematics course in the sequence, and ensures that the student’s level of competency after the completion of the course is consistent throughout the system. Specifically, the student will be able to carry out problems, analyze ideas, and conduct proofs in the following areas:

- History with emphasis on the parallel postulate.
- Logic with emphasis on quantifications, implications and methods of proof. In particular, direct proof, proof by contradiction and proof by cases should be covered. Intimately related to these is the understanding of how to use and formulate definitions, as well as give examples and counterexamples.
- The axiomatic method with the notions of independence and consistency as well as the concept of models and isomorphism of models being covered.
- Incidence geometry in its most rudimentary form.
- Euclidean geometry of the plane with emphasis on the ideas of parallelism, congruence, and linear and angle measurement.

## MATH 371 Elementary Probability Theory (3 credits)

This course covers sets, discrete sample spaces, problems in combinatorial probability, random variables, mathematical expectations, classical distributions, and applications. Course content is articulated with the other University of Hawai‘i four-year campuses to ensure the depth, breadth and relevance of the course content and coverage. The articulation of the course allows students to take the next mathematics course in the sequence, and ensures that the student’s level of competency after the completion of the course is consistent throughout the system. Specifically, the student will be able to carry out problems, analyze ideas, and conduct proofs in the following areas:

- Definition of probability space, examples, and the simple theorems.
- Counting techniques such as the multiplication principle, permutations, permutations with repetition, combinations, and the binomial theorem.
- Conditional probability and independence.
- Random variables.
- Distributions such as the binomial and hypergeometric.
- Expectations.

## MATH 373 Elementary Statistics (3 credits)

This course covers estimation, tests of significance, and the concept of power. Course content is articulated with the other University of Hawai‘i four-year campuses to ensure the depth, breadth and relevance of the course content and coverage. The articulation of the course allows students to take the next mathematics course in the sequence, and ensures that the student’s level of competency after the completion of the course is consistent throughout the system. Specifically, the student will be able to carry out problems, analyze ideas, and conduct proofs in the following areas:

- Descriptive statistics: plots, population vs. sample, sample mean, variance, standard deviation, quantiles, and Chebyshev’s theorem.
- Review of probability: probabilities, sample spaces and events, rules of probability, and random variables.
- Distributions: expectation and variance, discrete distributions (uniform, geometric, binomial, Poisson), continuous distributions (exponential, normal), and reading statistics tables.
- Sampling distributions and the Central Limit Theorem.
- Large sample estimation: types and comparison of estimators, sampling distributions for means/proportions and their use in large sample estimation, and sample size.
- Large sample testing: components of a test, significance and power, p-values, and large-sample tests for means/proportions.
- Small sample inference: t-distribution, applications to small sample estimation and testing, χ2 and F-distributions, and applications to inference about variances.
- Regression: least squares, correlation coefficient, and inference.
- ANOVA: Experimental design, completely randomized design.
- χ2 Tests: Multinomial distributions, contingency tables, and goodness-of-fit.
- Nonparametrics.

## MATH 411 Linear Algebra (3 credits)

This course covers vector spaces over arbitrary fields, minimal polynomials, invariant subspaces, canonical forms of matrices, unitary and Hermitian matrices, and quadratic forms. Course content is articulated with the other University of Hawai‘i four-year campuses to ensure the depth, breadth and relevance of the course content and coverage. The articulation of the course allows students to take the next mathematics course in the sequence, and ensures that the student’s level of competency after the completion of the course is consistent throughout the system. Specifically, the student will be able to carry out problems, analyze ideas, and conduct proofs in the following areas:

- Vector spaces and linear transformations.
- Dual spaces and the transpose of a linear transformation.
- Minimal polynomials, invariant subspaces, and canonical forms.
- Euclidean and Hermitian vector spaces.
- Applications.
- Linear groups (optional).
- Tensor products (optional).

## MATH 412 Introduction to Abstract Algebra I (WI, 3 credits)

This course is an introduction to basic algebraic structures. Topics include: groups, finite groups, abelian groups, rings, integral domains, fields, factorization, polynomial rings, field extensions, and quotient fields with an emphasis on writing instruction. These topics are covered in the year sequence Math 412–413. Course content is articulated with the other University of Hawai‘i four-year campuses to ensure the depth, breadth and relevance of the course content and coverage. The articulation of the course allows students to take the next mathematics course in the sequence, and ensures that the student’s level of competency after the completion of the course is consistent throughout the system. Specifically, the student will be able to carry out problems, analyze ideas, and conduct proofs in the following areas:

- Group Theory: definition and examples, permutation groups, Cayley representation theorem, normal subgroups, quotient groups, direct products, isomorphism theorems, Abelian groups and the classification of finite abelian groups, and Sylow theorems.
- Ring Theory: definition and examples, polynomial rings, ideals and quotient rings, modules, direct products, Euclidean domains, principal ideal domains, unique factorization, and other domains.
- Field Theory: definition and examples, characteristics, vector spaces, quotient fields, construction of finite fields, field extensions, and Galois Theory.

## MATH 413 Introduction to Abstract Algebra II (WI, 3 credits)

This course is an introduction to basic algebraic structures. Topics include: groups, finite groups, abelian groups, rings, integral domains, fields, factorization, polynomial rings, field extensions, and quotient fields with an emphasis on writing instruction. These topics are covered in the year sequence Math 412–413. Course content is articulated with the other University of Hawai‘i four-year campuses to ensure the depth, breadth and relevance of the course content and coverage. The articulation of the course allows students to take the next mathematics course in the sequence, and ensures that the student’s level of competency after the completion of the course is consistent throughout the system. Specifically, the student will be able to carry out problems, analyze ideas, and conduct proofs in the following areas:

- Group Theory: definition and examples, permutation groups, Cayley representation theorem, normal subgroups, quotient groups, direct products, isomorphism theorems, Abelian groups and the classification of finite abelian groups, and Sylow theorems.
- Ring Theory: definition and examples, polynomial rings, ideals and quotient rings, modules, direct products, Euclidean domains, principal ideal domains, unique factorization, and other domains.
- Field Theory: definition and examples, characteristics, vector spaces, quotient fields, construction of finite fields, field extensions, and Galois Theory.

## MATH 480 Senior Seminar (1 credit)

This course is a seminar for senior mathematics majors, including an introduction to methods of research. A significant portion of class time is dedicated to the instruction and critique of oral presentations. All students must give the equivalent of three presentations. CR/NC only.