Program Overview

Mathematics Modules in the BA in Education (Thurles)

The BA in Education (Thurles) program offers a comprehensive range of mathematics modules. Each module carries 6 ECTS credits.

Year 1 -- Autumn

Elementary Number Theory

This module provides a solid grounding in number theory, a foundational branch of mathematics.

• Prerequisite Module(s): None
• Objectives: Understand the basic elements of number theory and use known algorithms to solve problems related to divisibility.
• Learning Outcomes:
  • Understand the basic elements of number theory
  • Understand notation and conventions associated with the topic
  • Use known algorithms to solve problems related to divisibility
  • Master modular arithmetic
  • Produce coherent and convincing arguments
  • Communicate solutions to problems clearly and coherently
• Module Content:
  • Representations of numbers
  • The binomial theorem
  • Mathematical induction
  • Divisibility of integers
  • Prime Numbers and The Fundamental Theorem of Arithmetic
  • Euclid's algorithm
  • Congruence
  • Linear Diophantine equations
  • Fermat's Little Theorem
  • Using congruences to solve more complex problems
  • Pythagorean Triples
• Prime Texts:
  • ORE, O. (1969) Invitation to Number Theory. Mathematical Association of America.
  • SILVERMAN, J. H. (2012) A Friendly Introduction to Number Theory, Pearson Education.
• Other Relevant Texts:
  • NIVEN, I.M., ZUCKERMAN, H.S. (1980) An introduction to the theory of numbers, Wiley.
  • DUDLEY, U. (2008) Elementary Number Theory, Dover.
  • BURN, R.P. (1997) A pathway into number theory, Cambridge University Press.
  • FORMAN, S., RASH, A. (2015) The Whole Truth About Whole Numbers, Springer.

Calculus I: Differentiation

This module covers the first part of a two-semester course on differentiation and integration of functions depending on one real variable.

• Prerequisite Module(s): None
• Objectives: Understand the concepts of differentiation and integration of functions depending on one real variable.
• Learning Outcomes:
  • Differentiate functions of one real variable
  • Demonstrate knowledge and understanding of the mathematical concepts of function, limit, continuity, and derivative
  • Use these concepts appropriately in solving problems and in discussing solutions
• Module Content:
  • Functions and graphs
  • Slope, Newton quotient, and derivative
  • Limits
  • Differentiation rules for sums, products, quotients, composite functions
  • Trigonometric functions, logarithms, exponential functions, and their derivatives
  • Continuous functions
  • Nested intervals, completeness of the real numbers
  • Intermediate Value Theorem
  • Inverse functions and their derivatives
• Prime Text:
  • LANG, S. (2002) Short Calculus. Springer.
  • FLANDERS H. (1985) Single Variable Calculus. Freeman.
• Other Relevant Texts:
  • ANTON H. (1998) Calculus. A New Horizon. Volume 1. John Wiley & Sons.
  • STRANG G. (1991) Calculus. Wellesley-Cambridge Press.

Year 1 -- Spring

Introduction to Geometry

This module equips students with basic knowledge and skills of Euclidean geometry and prepares them for the study of other areas of mathematics.

• Prerequisite Module(s): None
• Objectives: Understand the basic knowledge and skills of Euclidean geometry.
• Learning Outcomes:
  • Understand, express, and use geometric results
  • Carry out geometric constructions
  • Determine certain geometric quantities from others
  • Use coordinates to solve geometric problems analytically
• Module Content:
  • Angle, distance, length, area
  • Coordinates
  • Lines, triangles, and circles
  • Geometric constructions
  • Congruence and similarity
• Prime Texts:
  • OSTERMANN, A., WANNER, G. (2012) Geometry by Its History, Springer.
  • LANG, S., MURROW, G. (1988) Geometry, Springer.
  • GARDINER, A.D., BRADLEY, C.J. (2005) Plane Euclidean Geometry: Theory and Problems, The United Kingdom Mathematics Trust.
• Other Relevant Texts:
  • BRUMFIELD, C.F., VANCE, I.E. (1970) Algebra and Geometry for Teachers, Addison-Wesley.
  • CLARK, D.M. (2012) Euclidean Geometry: A Guided Inquiry Approach, American Mathematical Society.

Year 2 -- Autumn

Linear Algebra

This module explores vector spaces and matrices with particular reference to the development of computational techniques and application skills.

• Prerequisite Module(s): None
• Objectives: Understand vector spaces and matrices and their applications.
• Learning Outcomes:
  • Solve systems of linear equations using row reduction techniques and matrix operations
  • Understand the basic structural properties of vector spaces and inner product spaces
  • Understand linear mappings and their relation to matrices
  • Find the eigenvalues and eigenvectors of a matrix and use them in the process of diagonalization
• Module Content:
  • Vectors and vector spaces
  • Subspaces, linear independence, bases, physical applications
  • Inner products, norm and distance, orthogonality, orthonormal basis
  • Matrices, matrix operations, echelon matrices, algebra of square matrices
  • Games of strategy, matrix games, applications to optimal decision making
  • Linear equations, methods of solution, Gaussian elimination, use of determinants
  • Linear programming, graphical methods, simplex method, applications in management science
  • Linear mappings, kernel and image, vector space isomorphisms, space of linear mappings
  • Eigenvalues and eigenvectors, characteristic polynomial
• Prime Text:
  • CHENEY, W. & KINCAID, D. (2009) Linear Algebra: Theory and Applications. Jones and Bartlett.
• Other Relevant Texts:
  • ANTHONY, M. & HARVEY, M. (2012) Linear Algebra: Concepts and Methods. Cambridge University Press.

Year 2 -- Spring

Calculus II: Integration

This module is the second part of a two-semester course on differentiation and integration of functions depending on one real variable.

• Prerequisite Module(s): MHP4763
• Objectives: Understand the concepts of differentiation and integration of functions depending on one real variable.
• Learning Outcomes:
  • Integrate functions of one real variable
  • Demonstrate knowledge and understanding of the theory of differentiation and integration of such functions
  • Apply differentiation and integration techniques to solve problems
  • Express their mathematical thoughts clearly
• Module Content:
  • Maxima and minima
  • Boundedness of continuous functions on closed intervals
  • Rolle's Theorem, Mean Value Theorem
  • Increasing and decreasing functions
  • Antiderivatives, indefinite integrals, integration by parts, substitution
  • Area, Riemann sums, definite integrals
  • Least upper bound, greatest lower bound
  • Fundamental Theorem of Calculus
  • Taylor's Formula
  • Infinite series, convergence
• Prime Text:
  • LANG, S. (2002) Short Calculus. Springer.
  • FLANDERS H. (1985) Single Variable Calculus. Freeman.
• Other Relevant Texts:
  • ANTON H. (1998) Calculus. A New Horizon. John Wiley & Sons.
  • STRANG G. (1991) Calculus. Wellesley-Cambridge Press.

Year 3 -- Autumn

Multivariable Calculus

This module explores further advanced calculus through a range of topics not covered elsewhere and includes an introduction to differential equations and applications.

• Prerequisite Module(s): MHP4764
• Objectives: Develop further the knowledge of functions from functions of a single variable to functions of several variables.
• Learning Outcomes:
  • Understand how notions in the single variable calculus extend to multivariable calculus
  • Understand notation and conventions associated with the topic
  • Describe the behaviour and characteristics of functions of more than one variable
  • Graph function of two variables
  • Communicate solutions to problems clearly and coherently
• Module Content:
  • Plane curves: Cartesian equations, parametric equations, tangent lines as approximations to curves, area under a curve, surfaces of revolution
  • Vectors and the geometry of space: coordinate axes systems, dot product, cross product, equation of a plane
  • Partial differentiation: functions in 2 and 3 variables, limits and continuity, partial derivatives, Clairaut's Theorem
  • Maxima and minima of functions of several variables: finding maxima and minima of functions without and with constraints
• Prime Text:
  • STEWART, J. (1998) Calculus: Concepts and Contexts, London: Pacific Grove.
  • HASS, J., HEIL, C., WEIR, M. (2019) Thomas' Calculus: Early Transcendentals. Pearson.
• Other Relevant Texts:
  • ADAMS, R. A. (1999) Calculus: A Complete Course. Addison-Wesley.

Year 3 -- Spring

Abstract Algebra

This module studies algebraic structures of groups, rings, and fields and fosters an understanding of their central importance in modern mathematics and their relevance to engineering and science.

• Prerequisite Module(s): MHP4731
• Objectives: Present the algebraic structures of groups, rings, and fields to foster an understanding of their central importance in modern mathematics.
• Learning Outcomes:
  • Use the vocabulary, symbolism, and basic definitions used in abstract algebra
  • Apply the concepts of groups, rings, and fields to solve problems in which their use is fundamental to obtaining and understanding the solution
• Module Content:
  • Groups; axioms and examples, subgroups, mappings and symmetries, applications of symmetry groups
  • Subgroups; cosets, Lagrange's theorem
  • Binary codes; application of group codes, error correction
  • Conjugacy; normal subgroups, factor groups, homomorphism, isomorphism
  • Permutation groups; Cayley's theorem
  • Rings; axioms and examples, polynomial rings
  • Subrings; ideals, quotient rings, ring homomorphisms, isomorphisms
  • Integral domains; integers
  • Congruences; Fermat's theorem, Euler's theorem, application of Euler's theorem to public key codes
  • Fields; axioms and examples, polynomials over a field
• Prime Text:
  • LAURITZEN, N. (2003) Concrete Abstract Algebra, Cambridge University Press.
• Other Relevant Texts:
  • DURBIN, J.R. (1979) Modern Algebra, Wiley.
  • FRALEIGH, J.B. (1976) A First Course in Abstract Algebra, Addison Wesley.
  • HUMPHREYS, J.F., PREST, M.Y. (1989) Numbers, Groups and Codes, Cambridge University Press.
  • IRVING, R. (2004) Integers, Polynomials and Rings: A Course in Algebra, Springer.
  • KIM, K.H., ROUSH, F.W. (1983) Applied Abstract Algebra, Ellis Horwood.
  • LEDERMANN, W. (1973) Introduction to Group Theory, Oliver and Boyd.
  • WHITELAW, T.A. (1988) Introduction to Abstract Algebra, Blackie.

Introduction to Probability and Statistical Inference

This module provides an introduction to the theory of probability and to statistical techniques in a manner that fosters understanding of concepts and development of expertise in their applications.

• Prerequisite Module(s): None
• Objectives: Familiarize students with the laws of probability and introduce them to statistical techniques.
• Learning Outcomes:
  • Represent sample data graphically in an appropriate way and correctly use summary statistics
  • Demonstrate an understanding of probability, random variables, probability distributions
  • Demonstrate an understanding of expected value and variance of a random variable
  • Demonstrate an understanding of sampling theory and the Central Limit Theorem
  • Apply theory to the analysis of sample data in order to find estimators and their properties
• Module Content:
  • Laws of probability; mutually exclusive events; addition and multiplication rules; independent events
  • Bayes' Theorem
  • Random variables; expected value and variance of a random variable
  • Probability distributions to include Bernoulli, Binomial, Poisson, Uniform, Normal, and chi-square
  • Descriptive statistics to include mean, median, mode, standard deviation, variance, kurtosis, and skewness
  • Sampling theory
  • Hypothesis testing to include test statistics, z-test, t-test, chi-square test, F-test, p-test for population proportions
  • Correlation and regression analysis
• Prime Text:
  • MENDENHALL, BEAVER et al. (2019) Introduction to Probability and Statistics. Brooks/Cole.
• Other Relevant Texts:
  • HOGG, R., McKEAN, J., CRAIG, A. (2020) Introduction to Mathematical Statistics. Pearson.
  • HOEL, P. (1976) Elementary Statistics. Wiley.
  • FRANK, H. (1974) Introduction to Probability and Statistics. Wiley.

Year 4 -- Autumn

Computational Mathematics

This capstone module introduces students to a computer algebra system in which numerical and symbolic calculations can be carried out.

• Prerequisite Module(s): MHP4731, MHP4763, MHP4764
• Objectives: Use a computer algebra system to investigate mathematical concepts and to solve mathematical questions.
• Learning Outcomes:
  • Use a computer algebra system to investigate mathematical concepts and to solve mathematical questions
  • Solve equations numerically and symbolically
  • Use computers to study problems from various areas of undergraduate mathematics
• Module Content:
  • Introduction to a computer algebra system
  • Calculations in number theory, linear algebra, calculus, and in statistics
• Prime Text:
  • BARD, G. V. (2015) Sage for Undergraduates, American Mathematical Society.
  • KOSAN, T. (2007) SAGE for Newbies. Available online
• Other Relevant Texts:
  • STEIN, W. (2012) Sage for Power Users. Available online

Euclidean and Non-Euclidean Geometry

This module equips students with sound knowledge of Euclidean geometry and introduces them to non-Euclidean geometry.

• Prerequisite Module(s): MHP4713
• Objectives: Equip students with sound knowledge of Euclidean geometry and introduce them to non-Euclidean geometry.
• Learning Outcomes:
  • Express, justify, and establish geometric results
  • Determine geometric quantities using theorems and techniques of Euclidean geometry
  • Demonstrate understanding of geometry beyond classical Euclidean geometry
  • Carry out and describe geometric constructions
• Module Content:
  • Basic notions and theorems of Euclidean geometry
  • Geometric constructions
  • Analytic geometry
  • Transformations and symmetry
  • Vectors and dot product
  • Non-Euclidean geometry
• Prime Text:
  • BYER, O., LAZEBNIK, F., SMELTZER, D. (2010) Methods for Euclidean Geometry, MAA.
  • OSTERMANN, A., WANNER, G. (2012) Geometry by Its History, Springer.
  • STILLWELL, J. (2005) The Four Pillars of Geometry, Springer.
• Other Relevant Texts:
  • BARKER, W., HOWE, R. (2007) Continuous Symmetry, From Euclid to Klein. American Mathematical Society.
  • COXETER, H.S.M. (1961) Introduction to Geometry, John Wiley & Sons.
  • GARDINER, A.D., BRADLEY, C.J. (2005) Plane Euclidean Geometry: Theory and Problems, UKMT.