Puzzles, Paradoxes and Paradigms - 2019 entry

The development of mathematics over five or so millennia has followed a pattern of puzzle and paradox, and then an enlightened paradigm shift where a new idea prevails over an old. This can be seen in the discovery of irrational numbers by the ancient Greeks grappling with the commensurate length of the side of an isosceles triangle and further seen in the creation of complex numbers by Gerolamo Cardano during the Renaissance in response to a puzzle involving the factorisation of general cubic polynomials. Recently, the puzzle-paradox-paradigm cycle is seen in the development of fractals and chaos.

This module embraces the puzzle-paradox-paradigm developmental cycle. You will undertake a guided exploration of 4 selected topics in mathematics. Each topic will begin with a short historical introduction, to understand the emergence of new paradigms in this field within their historical context. You will then be introduced to some basic examples to explain key concepts. The majority of the time will then be spent working on a set of problems. You will receive guidance on how to approach these problems through in class discussions, but are expected to be self-motivated and work largely independently.

You will develop your ability to “think mathematically”, including the ability to: formulate problems symbolically; reason and solve problems using abstract ideas and rules; ask crucial questions that cut to the core of a problem; test concepts by creating examples; unravel theorems to understand their inner workings; understand the importance of formal proof; understand the power of abstraction and generalisation; recognize underlying simple ideas and structures common to many areas of mathematics. As with other modules, it will also develop your ability to: communicate reasoning and solutions effectively in writing and oral presentation; self-management and time-management skills; broaden your use of learning resources including the use of IT.

The module content may vary from year to year, depending on the interests of the module leaders. The module content will also be co-developed by a number of 3rd year students through the module ECM3903 Mathematical Sciences Project.

The assessment for this module is entirely by in-class tests and coursework (no final exam). For each topic you will take a short in-class test, during the term, which will assess understanding of basic concepts and methods; this will consist of questions similar to those seen in worked examples and on a set of basic exercises. This should encourage you to work out solutions for all formative exercises (this can happen in small groups, but the in class assessment must be carried out individually) and hence ensure familiarity with basic concepts. For each topic, there will also be a set of medium and hard problems, and the major part of the assessment will require you to submit a portfolio consisting of solutions to a number of these problems. It will be permitted to work in groups on medium level problems, but hard level problems must be an individual effort. There will also be couple of alternatives to submitting hard problems: 1. Create a video that explains a difficult but known problem; 2. Critique a selection of mathematical YouTube videos from a given list, indicating mistakes and ambiguities, and filling in crucial missing details.

The aim is to develop knowledge and skills in a number of key areas of mathematics, alongside developing the crucial skills required to become an independently thinking mathematician. The focus will be on the process of problem solving and independent thinking, as opposed to the memorisation of facts and methods; in order to achieve this, the module will utilise ideas from the pedagogy of Problem Based Learning. By introducing you to a number of different branches of mathematics, this module will also have the effect of broadening your horizons with respect to the mathematical world.

On successful completion of this module, you should be able to:

Module Specific Skills and Knowledge:

1 Demonstrate an understanding of the key concepts and methods explored in the module and their application to particular problems;

2 Demonstrate awareness of a number of divisions of mathematics, as well as connections between topics;

Discipline Specific Skills and Knowledge:

3 Demonstrate a basic knowledge and understanding of fundamental concepts necessary for progression to successful further studies in the mathematical sciences;

4 Acquire skills to ‘think like a mathematician’: formulate problems symbolically; reason and solve problems using abstract ideas and rules; ask crucial questions that cut to the core of a problem; test concepts by creating examples; unravel theorems to understand their inner workings; understand the importance of formal proof; understand the power of abstraction and generalisation; recognize underlying simple ideas and structures common to many areas of mathematics;

Personal and Key Transferable / Employment Skills and Knowledge:

5 Reason using abstract 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/assess other solutions for problems;

7 Acquire ability for self-criticism of your work;

8 Demonstrate appropriate use of learning resources;

9 Demonstrate self-management and time management skills.

The exact content of the syllabus may vary from year to year, depending on the interests of the module leaders and those taking 3rd year projects (module ECM3903) to co-develop this material. Overall there will be 4 topics of study. Example topics:

- Naïve Thinking and Naïve Set Theory

Set theory is a fundamental and unifying concept in mathematics. One can quickly run in to paradoxes here, stemming from mistaken intuition and careless definitions – choosing the “correct” definitions and rules, or axioms, can put things right. Possible topics for study: Basic ideas from naïve set theory, including operations on infinite collections of sets; Sets on the real line; countability and infinities; exploring some properties of interesting sets, e.g. the Cantor Set (countability, density, Lebesgue measure); xiomatic set theory, Russell’s paradox, Axiom of Choice, Banach-Tarski Paradox;

- “To infinity and beyond!”

Basic concepts around infinities; bijection definition of cardinality; continuum hypothesis; cardinal and ordinal numbers. Investigating some interesting ‘phenomena’ related to infinities, e.g.: Hilbert’s infinite hotel, slaying the mathematical Hydra, the Banach-Tarski Paradox;

- Absurd Numbers

Understanding the development of the number systems, from natural numbers up to the complex numbers: their origins, construction and their sometimes reluctant acceptance into mainstream mathematics. Possible topics for study: their algebraic properties (as fields, groups etc.); a particular focus given to complex numbers, including Euler’s formula, polar form and the complex plane, applications of complex numbers;

- Ubiquitous Numbers

Understanding the origins, definitions and properties of some famous numbers in mathematics, for example: Euler’s Number e, Archimedes’ Constant π, the Golden Ratio φ (and the associated Fibonacci sequence), the Imaginary Unit i. Understanding properties such as: rational/irrational numbers, transcendental numbers, normal numbers;

- Special Functions

Understanding the origins, definitions and properties of some famous functions in mathematics, for example: Trigonometric functions, hyperbolic functions, natural logarithm, gamma function;

- Epsillon, Delta and Other Greek Letters

Delving deeper into concepts from Real Analysis that have been introduced in the Calculus and Geometry and Advanced Calculus modules. Possible topics for study: Series; Cauchy sequences; infima and suprema; limits of sequences of functions; Bolzano-Weierstrass Theorem; mean value theorems; Taylors theorem; point set topology; metric spaces; fixed point theorems and their applications;

- Symmetry and Invariance

Symmetry groups and their appearance in the real world;

- Puzzles and Games

The mathematical theory behind some popular toys, puzzles and games, e.g.: Rubik’s cube, mathematics of juggling, chess, board games (Group theory, game theory, combinatorics);

- Coding Theory

Encoding, Compression and Encryption. Mathematics behind digital encoding of information, efficient storage and transmission, error correction and secret codes (e.g. Hamming codes, RSA cryptography, the Enigma machine) and the associated mathematical concepts, e.g. groups, finite fields, vector spaces, FFT;

- Order and Chaos

Deterministic chaos: sensitive dependence on initial conditions; the logistic map. Fractals: construction of fractals; calculating lengths, areas and volumes; fractal dimension(s). Links between chaos and fractals: Logistic map bifurcations; Mandelbrot set; strange attractors.

If a module is normally assessed entirely by coursework, all referred/deferred assessments will normally be by assignment.

If a module is normally assessed by examination or examination plus coursework, referred and deferred assessment will normally be by examination. For referrals, only the examination will count, a mark of 40% being awarded if the examination is passed. For deferrals, candidates will be awarded the higher of the deferred examination mark or the deferred examination mark combined with the original coursework mark.

Basic Reading:

ELE -  http://vle.exeter.ac.uk

Reading list for this module: