# Mathematics Modules in the BA in Education (Thurles)

Each module carries 6 ECTS credits.

## Year 1 — Autumn

### Elementary Number Theory

Number theory is a foundational branch of mathematics. This module gives the student a solid grounding in the subject.

#### PREREQUISITE MODULE(S):

None

#### OBJECTIVES:

Number theory is a foundational branch of mathematics. This module gives the student a solid grounding in the subject.

#### LEARNING OUTCOMES:

Students who successfully complete this module should be able to:

- 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 is the first part of a two-semester course on differentiation and integration of functions depending on one real variable. It covers an area that is both a core part of every mathematical education, and fundamental for further studies in analysis or applied mathematics.

#### PREREQUISITE MODULE(S):

None

#### OBJECTIVES:

This module is the first part of a two-semester course on differentiation and integration of functions depending on one real variable. It covers an area that is both a core part of every mathematical education, and fundamental for further studies in analysis or applied mathematics.

#### LEARNING OUTCOMES:

Students who successfully complete this module should be able to:

- 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 for the study of other areas of mathematics.

#### PREREQUISITE MODULE(S):

None

#### OBJECTIVES:

Geometry is a core part of mathematical education, because it provides a paradigm of rigorous mathematical reasoning. This module equips students with basic knowledge and skills of euclidean geometry. It thereby prepares the student for the study of other areas of mathematics.

#### LEARNING OUTCOMES:

Students who successfully complete this module should be able to:

- 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

In this module vector spaces and matrices are explored with particular reference to the development of computational techniques and application skills.

#### PREREQUISITE MODULE(S):

None

#### OBJECTIVES:

To present an exploration of vector spaces and matrices with particular reference to the development of computational techniques and application skills.

#### LEARNING OUTCOMES:

Students who successfully complete this module should be able to:

- solve systems of linear equations using row reduction techniques and matrix operations;
- understand the basic structural properties of vector spaces and inner product spaces and be able, for example, to determine bases and orthogonal bases for subspaces of a given vector space;
- understand linear mappings and their relation to matrices and be able to calculate basic invariants related to such mappings;
- 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, classical adjoint, matrix inversion.
- Games of strategy; matrix games, applications to optimal decision making.
- Linear equations; methods of solution, Gaussian elimination, use of determinants, pplications to electrical networks, traffic flow.
- Linear programming; graphical methods, simplex method, applications in management science.
- Linear mappings; kernel and image, vector space isomorphisms, space of linear mappings. Linear transformations; matrix representation, change of basis.
- 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 one real variable. It covers an area that is both a core part of every mathematical education, and fundamental for further studies in analysis or applied mathematics.

#### PREREQUISITE MODULE(S):

MHP4763

#### OBJECTIVES:

This module is the second part of a two-semester course on differentiation and integration of functions depending one real variable. It covers an area that is both a core part of every mathematical education, and fundamental for further studies in analysis or applied mathematics.

#### LEARNING OUTCOMES:

Students who successfully complete this module should be able to:

- 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 (e.g. find maximum/minimum, compute area);
- 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 to include an introduction to differential equations and applications.

#### PREREQUISITE MODULE(S):

MHP4764

#### OBJECTIVES:

To develop further the knowledge of functions from functions of a single variable to functions of several variables. The notion of derivative of a function is extended to higher dimensions. Tools are provided which are needed to handle problems with several parameters and to understand the behaviour of functions of several variables.

#### LEARNING OUTCOMES:

Students who successfully complete this module should be able to:

- 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, Cartesian coordinates, alternative coordinate systems.
- Vectors and the geometry of space: coordinate axes systems, dot product, cross product, equation of a plane, comparing and contrasting lines in the plane and in 3-dimensional space, parametric curves in 3-dimensional space.
- Partial differentiation: functions in 2 and 3 variables, limits and continuity, partial derivatives, Clairautâ€™s Theorem, tangent plane and normal line to a surface, directional deerivatives.
- Maxima and minima of functions of several variables: finding maxima and minima of functions without and with constraints, classifying critical points using the Second Derivative Test, Lagrange Multipliers.

#### 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

In this module algebraic structures of groups, rings and fields are studied, and understanding of their central importance in modern mathematics and of their relevance to engineering and science is fostered.

#### PREREQUISITE MODULE(S):

MHP4731

#### OBJECTIVES:

To present the algebraic structures of groups, rings and fields in order to foster an understanding of their central importance in modern mathematics and of their relevance to engineering and science.

#### LEARNING OUTCOMES:

Students who successfully complete this module should be able to:

- 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. Groups of small order; isomorphism.
- 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 which will foster understanding of concepts and development of expertise in their applications.

#### PREREQUISITE MODULE(S):

None

#### OBJECTIVES:

The purpose of this module is to familiarise students with the laws of probability.
The main theorems of the first section of the module cover independence of events,
mutually exclusive events and Bayes' Theorem.
Using the laws of probability, the students will then be introduced to the
different types of discrete probability distributions,
namely Bernoulli, Binomial, Poisson and Geometric.

The next section will introduce the students to probability distributions in the
continuous case, namely, Normal, Exponential and Chi Square.
The course will then continue to apply the knowledge of probability to
statistical inference. Students will be introduced to sampling theory and
how to estimate population parameters from sample data.
The main statistics to be introduced in this section are estimators for the
population mean, estimators for the population proportion and
goodness of fit tests using the chi-square distribution.
The theory and application of Regression Analysis will also be introduced.

#### LEARNING OUTCOMES:

Students who successfully complete this module should be able to:

- represent sample data graphically in an appropriate way and to correctly use summary statistics such as measures of central tendency and dispersion;
- demonstrate an understanding of probability, random variables, probability distributions;
- demonstrate an understanding of expected value and variance of a random variable both discrete and continuous;
- 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; apply these estimators to test hypothese.

#### 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, p̂-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

Within this capstone module, students will be introduced to a computer algebra system in which numerical and symbolic calculations can be carried out. The main concepts of most mathematics modules taught in this programme will be dealt with from a computational perspective.

#### PREREQUISITE MODULE(S):

MHP4731, MHP4763, MHP4764

#### OBJECTIVES:

Computers are useful and indispensable tools to carry out calculations in mathematics. Within this capstone module, students will be introduced to a computer algebra system in which numerical and symbolic calculations can be carried out. The main concepts of most mathematics modules taught in this programme will be dealt with from a computational perspective.

#### LEARNING OUTCOMES:

Students who successfully complete this module should be able to:

- 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 such as number theory, linear algebra, calculus of one or several variables and statistics.

#### 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 is a core part of mathematical education. It equips students with sound knowledge of euclidean geometry and introduces them to non-euclidean geometry.

#### PREREQUISITE MODULE(S):

MHP4713

#### OBJECTIVES:

Geometry is a core part of mathematical education, because it provides a paradigm of rigorous mathematical reasoning. This module equips students with sound knowledge of euclidean geometry and introduces them to non-euclidean geometry.

#### LEARNING OUTCOMES:

Students who successfully complete this module should be able to:

- 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.