MATH 600Introduction to Mathematical Software for Teachers0.25 units
Purpose: The purpose of the Mathematical Software Course is to:
- expose the students to technical tools that professional mathematicians use,
- aid students in developing their mathematical abilities,
- provide a means of mathematical communication in an online environment,
- separate the requirement of mastering the use of mathematical software tools from the core course content,
- increase each student's understanding of her or his future course material.
Content:
The software presented in the Mathematical Software course will enhance each student's communication, presentation, visualization, and problem-solving skills. Proposed software will include, but will not be limited to:
GeoGebra:
GeoGebra is an interactive geometry software package which enables geometric constructions and algebraic manipulations. Constructions can be made with points, vectors, segments, lines, polygons, conic sections, and functions. All elements can be changed dynamically afterwards. GeoGebra has the ability to use numerical values or variables in designating, vectors and points. It can be used to make conjectures and prove algebraic and geometric theorems. Examples of this type of activity will be included.
Maple:
Maple is a powerful computer algebra system. It is a general-purpose program that can be used fro most mathematical work. There is extensive support for numerical computations, to specified precision, as well as symbolic computation and visualization. Maple is programmable. Custom user interfaces can also be created.
Students will be introduced to Maple using examples that will be similar in nature and scope to those that they will expect to see in later courses.
LaTeX:
LaTeX is a document markup language that can be used for creating sophisticated typeset mathematical documents. LaTeX is widely used by professional mathematicians, scientists and engineers. More recently, LaTeX has been used as the language of online scientific communication for individuals in a variety of disciplines.
LaTeX will be integrated with both Maple and GeoGebra to produce documents that include both static and dynamic mathematical diagrams. The course will also take a brief look at the history of mathematical communication and its impact on the development of the subject.
MATH 630Foundations of Probability and Statistics0.50 units
This course will explore the basic properties of probability focusing on both discrete and continuous random variable, and investigate how statistics and statistical models can be used to solve empirical problems. Students will also be introduced to sophisticated Statistical software. Topics include: Laws of probability, discrete and continuous random variable, probability distributions, mean, variance, generating functions, Markov chains, empirical problem solving, statistical graphics, statistical models, estimation and hypothesis testing, use of a statistical package.
This course will explore the foundations of differential calculus, the role of rigor in mathematics, and the use of sophisticated mathematical software. Topics include: A brief primer on logic and proof, axiom of choice and other ideas from set theory, convergence of sequences and the various forms of the completeness axiom for R, detailed study of limits, continuity and the Intermediate Value Theorem, fundamentals of differentiation and the importance of linear approximation, role of the Mean Value Theorem, the nature and existence of extrema, Taylor's Theorem and polynomial approximation, MAPLE as a tool for discovery.
This course explores the foundations of integral calculus and the use of series in approximating the basic functions of mathematics. Topics include: Understanding the Riemann Integral and its flaws, the idea of Lebesque, the geometric meaning of the Riemann-Stieltjes integral, the Fundamental Theorem of Calculus, numerical integration, numerical series, uniform convergence of functions and the extraordinary nature of power series, Fourier Series.
MATH 650Mathematical Modeling with Differential and Difference Equations0.50 units
Solving and interpreting differential and difference equations motivated by a variety of systems from the physical and social sciences. Analytical solutions of standard linear and non-linear equations of first and second order; phase portrait analysis; linearization of non-linear systems in the plane. Numerically solving initial-value and boundary-value problems using mathematical software.
This course is designed to allow the student to discover fundamental facts about geometry through the interactive use of mathematical software. Possible topics include: An introduction to affine, projective and non-Euclidean geometry, conic sections in projective geometry, inversion in circles, the Theorems of Desargues, Pappas and Pascal.
MATH 670Mathematical Connections: Real World Problems in Mathematics I0.25 units
This course is intended to give the student insight to an important area of mathematics and how it connects with problems in the real world. Topics: The course is one of four similar courses that will consist of either one six week module, or two three week modules each introducing a separate though possibly related area of mathematics. The emphasis will be on how the mathematics is used in a real world context.
MATH 671Mathematical Connections: Real World Problems in Mathematics II0.25 units
This course is intended to give the student insight to an important area of mathematics and how it connects with problems in the real world. Topics: The course is one of four similar courses that will consist of either one six week module, or two three week modules each introducing a separate though possibly related area of mathematics. The emphasis will be on how the mathematics is used in a real world context.
MATH 672Mathematicsl Connections: Real World Problems in Mathematics III0.25 units
This course is intended to give the student insight to an important area of mathematics and how it connects with problems in the real world. Topics: The course is one of four similar courses that will consist of either one six week module, or two three week modules each introducing a separate though possibly related area of mathematics. The emphasis will be on how the mathematics is used in a real world context.
MATH 673Mathematical Connections: Real World Problems in Mathematics IV0.25 units
This course is intended to give the student insight to an important area of mathematics and how it connects with problems in the real world. Topics: The course is one of four similar courses that will consist of either one six week module, or two three week modules each introducing a separate though possibly related area of mathematics. The emphasis will be on how the mathematics is used in a real world context.
We explore the who, where, when and why of some of the most important ideas in mathematics. Topics include: William T. Tutte and Decryption, Ancient Mediterranean and Mideastern Mathematics, Euclid, Archimedes, Estimating Pi, Al Khwarizmi, Durer and the Renaissance, Descartes and Analytic Geometry, Kepler and Planetary Motion.
MATH 681Problem Solving and Mathematical Discovery0.50 units
This course aims to develop the student's mathematical problem solving ability. Common heuristics such as problem modification, patterning, contradiction arguments and exploiting symmetry will be examined. A wide range of challenging problems from various branches of mathematics will provide the medium through which these important principles and broad strategies are experienced.
MATH 690Summer Workshop for Teachers of Mathematics0.25 units
This intense 3-day workshop focuses on the integration of problem solving and technology into the curriculum and enrichment activities. The Workshop is suitable for teachers from all over the world.
MATH 692Reading, Writing and Discovering Proofs0.25 units
Objectives: To develop the vocabulary, techniques and analytical skills associated with reading and writing proofs, and to gain practice in formulating conjectures and discovering proofs. Emphasis will be placed on understanding logical structures, recognition and command over common proof techniques, and precision in language.
Topics Include: rules of formal logic, truth tables, role of definitions, implications, sets, existential and universal quantifiers, negation and counter-example, proofs by contradiction, proofs using the contrapositive, proofs of uniqueness and induction.
MATH 699Master of Mathematics for Teachers Capstone0.50 units
The capstone course is designed to give students an opportunity to showcase the knowledge they have gained and to provide a forum for bringing that knowledge into their own classroom. As part of this course students will design a mini-course on an approved subject in mathematics.