MTH0003 | 2022/3 | University of Exeter

MODULE TITLE	Exploring Mathematics	CREDIT VALUE	15
MODULE CODE	MTH0003	MODULE CONVENER	Dr Houry Melkonian (Coordinator)

DURATION: TERM	1	2	3
DURATION: WEEKS	0	11	0

Number of Students Taking Module (anticipated)	30

DESCRIPTION - summary of the module content

This module is designed to develop an understanding of the nature of mathematics and mathematical thinking. It takes you on a voyage of exploration of how mathematics was invented and developed. For millennia BC, mathematics was used for various purposes such as in the study of planets and their motions as well as in construction and trading, and since then the study of mathematics was expanded and new branches were explored. The module aims to enhance your ability to use mathematical reasoning, abstraction and logic while enjoying the process of learning and performing mathematical procedures. In its first theme, it focuses on the concept of number sets and cardinality, where notions like, ‘infinity’ are discussed and explored through examples and simple proofs. The second theme develops an understanding of the mathematical language through an illustration of symbolic representations, patterns and relationships, such as the relationship between the number of spirals in a pine cone (a natural pattern) and the famous Fibonacci sequence (a number pattern), showing how those patterns are linked to other parts of mathematics such as geometry, arithmetic and probability. This is followed by a third theme about Euclidean geometry featuring some of its intriguing theorems and properties including simple geometrical proofs as well as some special geometric shapes (e.g. Kepler triangle), it also covers some aspects of coordinate geometry and transformations.

Each theme will be covered over three weeks with scheduled weekly sessions led by the module instructor/convenor. During those sessions you will be introduced to the topic of the theme through lecture presentations, learning materials and provided exercise worksheet. In this module you will learn how to: use theories and definitions; analyse and critique mathematical claims; perform simple proofs; formulate examples and communicate results. This will be achieved by encouraging you to develop solutions for the weekly formative exercises in the class while working in small groups, as well as through effective management skills and self-directed study outside scheduled hours. You will be assessed through a combination of summative assessments: in-class tests (3 tests – a test per theme), coursework and presentation (a mini-project), and a final exam. Each in-class test will take place during the end of the third week of the subject theme; the coursework will be in the form of a mini-project consisting of two components selected from a list of given problems – components should be selected from two different topics, and the solution of a component could be presented in the form of an essay, a recorded audio/clip, a written mathematical argument, or could be a combination of those.

Students are expected to have knowledge of Principles of Pure Mathematics (MTH0001) as a co-requisite.

AIMS - intentions of the module

This module aims to develop your knowledge and understanding of core mathematical skills, including the ability to use abstract ideas; formulate accurate and rigour justifications; provide concise and logical proofs; generalise concepts; form examples that demonstrate understanding of a topic. By studying a number of topics you will become familiar with the development stages of mathematics, and you will learn how to communicate ideas and solutions. The module is designed to broaden your horizon by exploring a number of divisions of mathematics and the connections between them.

INTENDED LEARNING OUTCOMES (ILOs) (see assessment section below for how ILOs will be assessed)

Module Specific Skills and Knowledge:

1 Demonstrate a general appreciation of the development of key mathematical concepts explored in this module;

2 Reveal awareness of a selection of topics in mathematics and the connections between them;

Discipline Specific Skills and Knowledge:

3 Demonstrate a basic knowledge and understanding of fundamental concepts necessary for progression to further studies in mathematics or in other quantitative degree pathways;

4 Develop skills to reason and solve problems using abstract ideas; critique mathematical claims; test concepts by creating examples; understand the importance of logic in proofs; recognize underlying simple ideas common to many areas of mathematics;

Personal and Key Transferable/ Employment Skills and Knowledge:

5 Acquire ability to: perform symbolic representation of concepts and generalise ideas; formulate and solve problems and communicate reasoning and solutions effectively in writing and oral presentation;

6 Work in groups to solve in-depth problems effectively, and learn to analyse and evaluate solutions;

7 Demonstrate appropriate use of learning resources; Demonstrate ability to use library resources;

8 Demonstrate self-management and time management skills.

SYLLABUS PLAN - summary of the structure and academic content of the module

Theme A: Number sets (natural, integer, rational, irrational, real and complex numbers); Cardinality of sets (finite sets, countable and uncountable infinite sets, How big is infinity?); Transcendental and algebraic numbers. [3 weeks]

Theme B: Patterns - special numbers (i.e. π Archimedes’ constant, e Euler’s constant, φ Golden ratio, i=√(-1) imaginary unit); number patterns: Triangular, Fibonacci, Pascal’s triangle, Leibniz harmonic triangle, geometric and arithmetic progressions, prime and perfect numbers; connections with arithmetic, probability, algebra, geometry, etc. [3 weeks]

Theme C: Geometry – Plane Euclidean geometry and geometrical proofs: Angles and Pythagoras’ theorem; congruency and similarity (e.g. congruence triangles, similarity of triangles, fundamental theorem of similarity); Circles and Circle Theorems; Coordinate geometry: Transformations and symmetry (translation, rotation, reflection, dilations, shearing); Special shapes: Golden rectangles and triangle, Kepler Triangle, Logarithmic versus Archimedean spiral. [3 weeks]

LEARNING AND TEACHING

LEARNING ACTIVITIES AND TEACHING METHODS (given in hours of study time)

Scheduled Learning & Teaching Activities	33	Guided Independent Study	117	Placement / Study Abroad	0

DETAILS OF LEARNING ACTIVITIES AND TEACHING METHODS

Category	Hours of study time	Description
Scheduled learning & teaching activities	22	Formal lectures of new learning material
Scheduled learning & teaching activities	11	Tutorials with worked examples; individual and group support; in-class tests
Guided Independent Study	117	Lecture, worked examples and assessment preparation; further reading.

ASSESSMENT

FORMATIVE ASSESSMENT - for feedback and development purposes; does not count towards module grade

Form of Assessment	Size of Assessment (e.g. duration/length)	ILOs Assessed	Feedback Method
In-class worked examples; basic exercises per topic	10 x 1 hour	1-8	Provided solutions, exercises discussed in class, oral in-class feedback and support

SUMMATIVE ASSESSMENT (% of credit)

Coursework	50	Written Exams	50	Practical Exams	0

DETAILS OF SUMMATIVE ASSESSMENT

Form of Assessment	% of Credit	Size of Assessment (e.g. duration/length)	ILOs Assessed	Feedback Method
3 tests	3 x 10	40 mins	1-8	Annotated scripts and oral feedback with provided solutions
Coursework	15	800 words	1-8	Written comments
Presentation	5	5-10 mins during one of the seminars (end of term)	1-8	Emailed/written feedback
Written Exam	50	1½ hours	1-8	Annotated scripts/feedback sheet

DETAILS OF RE-ASSESSMENT (where required by referral or deferral)

Original Form of Assessment	Form of Re-assessment	ILOs Re-assessed	Time Scale for Re-assessment
3 tests	3 tests (3 x 10%)	1-8	August Ref/Def period
Coursework	Coursework (15%)	1-8	August Ref/Def period
Presentation	Presentation (5%)	1-8	August Ref/Def period
Written exam	Written exam (50%)	1-8	August Ref/Def period

RE-ASSESSMENT NOTES

Deferral – if you have been deferred for any assessment, you will be expected to complete relevant deferred assessments as determined by the Mitigation Committee. The mark given for re-assessment taken as a result of deferral will not be capped and will be treated as it would be if it were your first attempt at the assessment.

Referral – if you have failed the module overall (i.e. a final overall module mark of less than 40%) you will be required to undertake re-assessments as described in the table above for any of the original assessments that you failed. The mark given for a re-assessment taken as a result of referral will be capped at 40%.

RESOURCES

INDICATIVE LEARNING RESOURCES - The following list is offered as an indication of the type & level of
information that you are expected to consult. Further guidance will be provided by the Module Convener

Basic reading/Web-based and electronic resources:

• ELE – College to provide hyperlink to appropriate pages

Other resources:

• ‘Thomas' Calculus Based on the Original Work by George B. Thomas, Jr.’ by Finney, R.L, Maurice, D., Weir, M. and Giordano, F.R, 10th edition or later, Addison-Wesley, 2003, [Library]

• ‘Exploring mathematics : problem-solving and proof’ by Daniel Grieser, Springer ,2018

• ‘Fundamentals of University Mathematics’ by McGregor, C., Nimmo, J. & Stothers, W., 2nd edition, Horwood, Chichester, 2000, [Library]

• ‘How to Think Like a Mathematician: A Companion to Undergraduate Mathematics’ by Houston, K., 1st edition, Cambridge University Press, 2009, [Library]

• ‘A Concise Introduction to Pure Mathematics’ by Liebeck, M., 3rd edition, Chapman & Hall/CRC Press, 2010, [Library]

Reading list for this module:

There are currently no reading list entries found for this module.

CREDIT VALUE	15	ECTS VALUE	7.5

PRE-REQUISITE MODULES	MTH0001
CO-REQUISITE MODULES	MTH0001

NQF LEVEL (FHEQ)		AVAILABLE AS DISTANCE LEARNING	No
ORIGIN DATE	Thursday 29th July 2021	LAST REVISION DATE	Wednesday 27th October 2021

KEY WORDS SEARCH	Problem solving, critical thinking, logic, geometry, number sets, number patterns

Exploring Mathematics - 2022 entry