As a director of Secondary Teacher Option Masters Program, I have to refer to its official web page.

Please, find below a few important pieces of information about the program additional to the information that you can find there.

Detailed goals for some of the required courses

• MAT 511, Fundamental Concepts of Mathematics

  The goal of the course is to teach teacher students to understand mathematical language and express themselves in logically correct and mathematically literate written and colloquial language.

  Students should

  • be able to construct logically correct sentences using predicates, connectives and quantifiers;
  • be able to construct useful denials of propositions involving connectives and quantifiers;
  • understand and use literately terminology associated with conditional and biconditional sentences (sufficient and necessary conditions, antecedent and consequent, etc.) and understand the difference between implication in mathematics and causation in everyday life;
  • formulate and prove de Morgan's Laws;
  • understand the structure of mathematical definitions and theorems;
  • understand and use fluently various proof techniques: direct proof, proof by contraposition, proof by contradiction, proof by cases;
  • be able to identify, analyze and prevent typical logical mistakes, like affirming the consequent, denying the antecedent, etc.;
  • be able to apply the principle of mathematical induction in problem solving;
  • know the basic language of the set theory (membership, union, intersection, the empty set, complement) and be able to calculate the complement of a set defined in those terms;
  • be able to relate operations of logic and set theory (e.g, negation and complement, conjunction and intersection, etc.);
  • be able to prove simple set-theoretical identities and illustrate them by Venn diagrams;
  • be familiar with the power set and indexed family of sets;
  • understand the definition of direct product of sets;
  • be familiar with the terminology associated to binary relations (e.g., reflexive, irreflexive, symmetric, antisymmetric, and transitive relations);
  • be familiar with strict and non-strict partial orders;
  • be able to test whether a relation is an equivalence relation;
  • operate with equivalence classes and the quotient set, and understand the connection between partitions of a set and equivalence relations on the set;
  • know the definition of a function and use literately the terminology related to functions and mappings (e.g. domain, codomain, range, image, pre-image);
  • be familiar with special types of maps like identity maps, constant maps, restrictions, projections, inclusions, factor-maps, submaps, the characteristic function of a set;
  • know the definition of a metric on a set and be able to recognize a metric;
  • know the definition of injective, surjective and bijective map, and determine if a map is or is not injective, surjective, bijective;
  • know the definition of an invertible map and its reformulation, and be able to check if a map is or is not invertible;
  • be familiar with the notion of cardinality and related notions (finite and infinite sets, denumerable sets);
  • be familiar with the axiomatic construction of natural numbers and the Peano axioms, and be able to prove equivalence of principle of mathematical induction and well ordering principle;
  • use the Pigeonhole principle in proofs and problem solving;
  • be familiar with basic results about countability (e.g., the sets of natural numbers, integers, rational numbers are denumerable, the set of real numbers is uncountable);
  • be able to formulate the Cantor-Schroeder-Bernstein theorem and prove that cardinal numbers are linearly ordered.
• MAT 512, Algebra for Teachers

  The goal of the course MAT 512 Algebra for Teachers is to provide the algebraic material relevant to the everyday work of a high-school teacher both in the contents and methods of teaching, and form rigorous foundations for arithmetic and algebra of Common Core State Standards.

  Students should be able to

  • describe the classical number systems used in high school (natural numbers, integers, rational numbers, real numbers and complex numbers) and relations between them (This includes understanding of describing (or constructing) each of the systems, except the system of natural numbers, in terms of the preceding one. Real numbers are described by Dedekind cuts (or equivalent construction), complex numbers are given both in a formal algebraic way and via geometric interpretation, together with the trigonometric form, Moivre formula and complex conjugation.);
  • use various presentation systems for numbers (e.g., positional systems, aliquot fractions, etc.), convert the presentations to each other and perform the arithmetic operations in the systems;
  • convert an infinite periodic decimal fraction into an ordinary fraction and back;
  • formulate and use the basic properties of algebraic operations (associativity, commutativity, distributivity);
  • give definitions of the basic algebraic structures such as monoids, groups, rings, fields, and recognize them in specific situations;
  • give definitions of homomorphisms and isomorphisms of groups or rings and recognize whether a specific map matches these definitions;
  • give definitions of and fluently operate with the fundamental notions about divisibility of integers (divisor, common divisor, greatest common divisor, prime number, etc.);
  • formulate and prove Unique Prime Factorization Theorem, theorem about division with remainder, Euclidean algorithm;
  • find a linear presentation of the greatest common divisor by the Euclidean algorithm, by using continued fractions and by matrix method;
  • solve linear Diophantine equations;
  • perform basic operations with congruence classes modm and identify the congruence classes as elements of the residue ring Zm;
  • apply modular arithmetics (including the canonical ring homomorphisms Z to Z/m and Z/pq to Z/p) to a wide variety of problems such as divisibility criteria, control of calculations and Diophantine equations;
  • identify invertible elements in Z/m;
  • define the Euler function, formulate and prove its properties;
  • formulate and prove the Euler theorem and its corollaries (in particular, Fermat's little theorem), as well as apply it to solve congruence problems with large exponents;
  • define zero divisor in a ring, define integral domain;
  • solve linear equations in the residue rings;
  • formulate and prove the theorem about the field of quotients for an integral domain, apply it to the ring of integers;
  • prove that polynomials with coefficients in a ring form a ring;
  • give definitions of ideal, kernel of a ring homomorphism, quotient ring by an ideal;
  • interpret the simplest algebraic extension of fields as quotient rings of the polynomial ring over the field (in particular, represent in this way the field of complex numbers);
  • express a symmetric polynomial as a polynomial of elementary symmetric polynomials, use this in problem solving;
  • formulate and prove the Vieta theorem;
  • demonstrate knowledge of the historical development of algebra, including contributions from diverse cultures.
• MAT 515, Geometry for Teachers. I
  The goal of the course MAT 515 Algebra for Teachers is to provide the first part of geometric material relevant to the everyday work of a high-school teacher both in contents and methods of teaching, and form rigorous foundations for geometry of Common Core State Standards.

  Students should be able to

  • describe the basic objects of Euclidean plane geometry (points, lines, rays, segments, angles, circles, etc.) and relations between them;
  • formulate and prove the theorems about
    • vertical angles,
    • isosceles triangles and their properties,
    • congruence tests for triangles,
    • inequality between non-adjacent exterior and interior angles in a triangle,
    • relations between sides and opposite internal angles in a triangle,
    • triangle inequality and its corollaries,
    • existence and uniqueness of perpendicular to a line from a point,
    • slant and perpendicular;
  • describe the techniques of compass-and-straightedge constructions and solve using it the standard construction problems such as
    • from a point on a line erect a perpendicular to this line,
    • drop a perpendicular to a given line from a given point,
    • bisect a segment or an angle,
    • construct an angle with a prescribed side congruent to a given angle,
    • construct a triangle given segments congruent to its sides;
  • formulate the parallel postulate and its reformulations, prove the equivalence of reformulations;
  • formulate, prove and apply tests for parallel lines and converse theorems;
  • formulate and prove the theorem about the sum of interior angles in a triangle and its corollaries;
  • formulate and prove theorems about sides and diagonals in a parallelogram and parallelograms of special types (rectangles, rhombuses and squares);
  • formulate and prove theorems about midlines in triangle and trapezoid;
  • formulate and prove the theorem about inscribed angle and its corollaries and apply them to solving construction problems;
  • construct inscribed and superscribed circles of a triangle, prove their existence and uniqueness;
  • identify and prove existence of the classical concurrency points in a triangle;
  • formulate and prove theorems about the structure of plane isometries:
    • restoring an isometry from its restriction to 3 points,
    • representing an isometry as a composition of at most 3 reflections,
    • classification of isometries: reflections, translations, rotations and glide reflections;
  • calculate compositions of plane isometries and solve problems using the calculation of compositions (e.g., the Napoleon theorem);
  • formulate and prove Thales theorem;
  • formulate definition of similarity transformations and decompose any similarity transformation into isometry and homothety;
  • formulate, prove and apply similarity tests for triangles;
  • use similarity for solving construction problems;
  • use similarity for proving proportionality theorems, geometric means in right triangle, Pythagorus theorem and its corollaries, etc.;
  • give definition of inversion (reflection in a circle) and describe its properties, images of lines and circles;
  • solve geometric construction problems using inversions, formulate and prove the Steiner theorem;
  • outline the Poincare model for the hyperbolic plane;
  • demonstrate knowledge of the historical development of Euclidean and non-Euclidean geometries, including contributions from diverse cultures.
• MAT 520, Geometry for Teachers. II

  The goal of the course is to provide teacher students introduction into spacial Euclidean geometry and analytical geometry.

  Students should

  • know basic facts about mutual position of lines and planes in the space, and definitions of parallel and perpendicular lines and planes;
  • acquire basic techniques for drawing 3-dimensional objects (in particular, be fluent in drawing cross sections of polyhedra passing through given points);
  • know definitions of the angle and the distance between skew lines and be able to solve relevant problems;
  • know the fundamentals about polyhedra and be able to prove Euler's formula for a convex polyhedra;
  • be familiar with Platonic solids, be able to draw them and make models of Platonic solids;
  • know various definitions of vector and understand physical notions like displacement, velocity, acceleration, force, etc. as vectors;
  • know properties of vector addition and scalar multiplication, and use vectors for solving geometric problems;
  • understand the notion of center of mass and use it in proofs of Ceva's and Menelaus theorems and problem solving;
  • know the dot product, vector product and triple product of vectors, their properties, geometric interpretations, expression in the coordinate form and relation to quaternions;
  • understand various forms of equation for a line on a plane, plane in the space and line in the space (vector, standard, normal, parametric) and be able to choose an appropriate form depending on a problem under consideration;
  • know the basic facts about conic sections: various definitions and their equivalence, canonical equations, focal, directorial, and reflective properties of parabola, ellipse and hyperbola;
  • know the basic facts about quadratic surfaces in the 3-space: identify the surface by its equation (in principal axes), be able to draw surfaces and their cross sections by hand and using graphing software, identify the surfaces of rotations and ruled surfaces, make simple models of the surfaces;
  • be familiar with non-Cartesian coordinate systems (polar, cylindrical, spherical) and with some classical curves, e.g., Archimedean spiral, cardioid, lemniscate, etc;
  • be familiar with axiomatic and constructive approaches to the notion of area and be able to prove formulas for area of basic figures: rectangle, parallelogram, triangle, trapezoid, convex polygon, disk;
  • be familiar with the basic notions related to volume, and be able to prove formulas for the volume of prism and pyramid.
• MAT 516, Probability and Statistics for Teachers

  Students should:

  • understand basic combinatorics; be able to apply them to standard problems: e.g., coin tossing, throwing dice, drawing cards, Bose-Einstein versus Fermi-Dirac statistics.
  • be able to use Stirling's formula to evaluate expressions involving binomial or multinomial coefficients.
  • understand the basic formalism of probability theory: mutually exclusive versus independent events, inclusion-exclusion principle, random variables; be able to analyze problems in those terms.
  • understand distribution functions and densities (mass distributions); be able to calculate expectation, variance, standard deviation.
  • understand independence of random variables, and the joint and marginal distributions of two random variables; be able to calculate covariance and correlation.
  • be familiar with the basic discrete distributions: Bernoulli, binomial, Poisson, geometric, negative binomial, hypergeometric; be able to calculate their expectation and standard deviation.
  • be familiar with the basic continuous distributions: uniform, normal, exponential, gamma, beta; be able to calculate their expectation and standard deviation.
  • understand Poisson processes, and the sense in which some distributions above are limiting cases of others when some parameters are small or large; be able to use this knowledge to calculate approximations.
  • understand the concept of conditional probability and expectation; be able to compute using Bayes' formula and its many applications.
  • understand the weak and strong Law of Large Numbers, higher moments, the expectation and variance of sums of independent identically distributed random variables; be able to give at least the statement and interpretation of the Central Limit Theorem, and to calculate with it in simple contexts.
  • demonstrate knowledge of the historical development of probability and statistics, including contributions from diverse cultures.

About maintaining GPA

All graduate student must maintain GPA greater than or equal to 3.00. This means that your average grade should not less than B at the beginning of each semester.

See the complete official set of relevant rules and Graduate Student Handbook.

The calculation of GPA can be done on web. See GPA Calculator