MATH221 - Applications of Mathematics

Year

Credit points

Campus offering

Prerequisites

MATH104 Differential and Integral Calculus OR MATH107 Introduction to Logic and Algebra

Incompatible

MATH215

Unit rationale, description and aim

Most, so-called, applications of mathematics that students have seen are quite simplistic or contrived. By considering some, fairly sophisticated, examples, actual applications of mathematics are introduced.

This unit investigates various applications of mathematics in the everyday world. Topics covered will be chosen from first peoples knowledge, financial mathematics, matrix games, error correcting codes, difference equations, fractals or related topics.

The aim of this unit is to provide students with contemporary examples of how mathematics is used in the world today. Students will be able to talk knowledgably about those examples and be able to connect the mathematics they have done to those applications.

Learning outcomes

To successfully complete this unit you will be able to demonstrate you have achieved the learning outcomes (LO) detailed in the below table.

Each outcome is informed by a number of graduate capabilities (GC) to ensure your work in this, and every unit, is part of a larger goal of graduating from ACU with the attributes of insight, empathy, imagination and impact.

Explore the graduate capabilities.

On successful completion of this unit, students should be able to:

LO1 - Solve problems involving some contemporary applications of mathematics to the real world (GA3, GA4, GA5)

LO2 - Make informed decisions based on the results of mathematical arguments and/or calculations (GA3, GA5, GA8, GA10)

LO3 - Work as a member of a team to investigate and explain a contemporary application of mathematics (GA3, GA4, GA5, GA7)

LO4 - Discuss the connections between different areas of mathematics (GA5, GA8)

LO5 - Effectively use appropriate techniques and technologies in solving problems in mathematics (GA5, GA7, GA10)

Graduate attributes

GA3 - apply ethical perspectives in informed decision making

GA4 - think critically and reflectively 

GA5 - demonstrate values, knowledge, skills and attitudes appropriate to the discipline and/or profession 

GA7 - work both autonomously and collaboratively 

GA8 - locate, organise, analyse, synthesise and evaluate information 

GA10 - utilise information and communication and other relevant technologies effectively.

Content

Topics will be chosen from applications of mathematics that impact directly upon everyday life, examples are given below and model the required detail and level of difficulty that should be attained. Ideally 3 topics will be covered each time offered. One topic with clear ethical implications must be included, such as Financial Mathematics or Matrix Games or another suitable topic.

Financial Mathematics

1. Review of Simple and Compound Interest. Depreciation
2. Annuities
3. Investment options: net present value and internal rate of return in theory and using technology

Discrete Dynamical Systems

1. Systems and models
2. Applications
3. Stability or Stochastic Processes

Error Correction

1. Introduction to transmission errors, error detection and error correction.
2. Reed-Muller codes and perfect codes.
3. Applications and uses of error correction.

Matrix Games

1. Simple two-person zero-sum games.
2. Solving matrix games.
3. The prisoner’s dilemma and other ethical issues.

Information Theory and Compression

1. Outline of information, entropy and uncertainty.
2. Redundancy in languages and unicity length.
3. Compression: Huffman and Lempel-Ziv coding.

Learning and teaching strategy and rationale

As is common in Mathematics a variety of Active Learning approaches promote the best acquisition of skills and understanding. Lectures will typically be structured to provide explanations of the material to be covered, along with examples of the applications of that material, as well as formal and informal tasks for students that will reinforce that learning. Splitting the taught content of the lectures into short topics will ensure increased student engagement with the material, as well as providing opportunities for students to consolidate their learning before new material is covered. This allows students to learn the skills and then build understanding, competence and confidence via (ideally, face-to-face) tutorials involving cooperative groups, peer review and other relevant strategies. In all cases this should be supported using available online technology.

This unit will normally include the equivalent of 24 hours of lectures (typically 2 hours per week for 12 weeks) together with 24 hours attendance mode tutorials. Lectures will also be recorded and, where possible or required, students may have access to an online tutorial.

150 hours in total with a normal expectation of 48 hours of directed study and the total contact hours should not exceed 48 hours. The balance of the hours becoming private study.

Assessment strategy and rationale

To successfully complete an undergraduate Mathematics sequence, students need an understanding of a variety of basic Mathematical topics and an ability to apply that understanding to a variety of problems. To succeed at problem solving in Mathematics, students must have these skills at their fingertips and be able to recall them and choose an appropriate approach under some pressure. The assessment strategy chosen, while traditional, tests and supports student learning by providing opportunities to develop and test their problem-solving skills through the unit.

The examination components ensure that students have fully integrated the learning and can bring a variety of strategies to bear under pressure.

Typing of Mathematical notation either requires a significant investment of time, or knowledge of advanced Mathematical typesetting software. Neither of those skills are suitable for an undergraduate course in Mathematics. Consequently, assignments, tests and examinations are typically handwritten and either submitted as hardcopy or scanned and submitted to a dropbox, rather than through Turnitin. Students may choose to type their responses and submit them electronically, but this is not a requirement of the unit.

The planned flexibility of this unit extends to the group-work assessment task. With guidance from the Lecturer-in-Charge, the teams of students will be able to complete a significant project on a chosen application of mathematics in the world. This could involve investigations of first peoples’ knowledge or simply be a contemporary application. Students would be encouraged to also select from topics that are not being explicitly covered in any particular year’s offering. In this way, students will meet a very wide range of applications that are of importance in the world. Aspects extensively covered in other units in the Mathematics sequence would be excluded, such as cryptography, statistics and mechanics.

Overview of assessments

Representative texts and references

Abas, S. J., & Salman, A. (1995). Symmetries of Islamic geometrical patterns. New Jersey: World Scientific.

Banks, R. B. (2013). Towing icebergs, falling dominoes and other adventures in applied mathematics. Princeton: Princeton University Press.

Barnsley, M. F. (2012). Fractals everywhere. New York: Dover.

Bruen, A. (2021). Cryptography, information theory and error correction: A handbook for the 21st century. 2nd Edition. Hoboken, NJ: Wiley-Blackwell.

Hill, R. (1988). A first course in coding theory. New York: Oxford University Press.

Körner, T. W. (2015). The pleasures of counting. New York: Cambridge University Press.

Mandelbrot, B. B. (1983). The fractal geometry of nature. New York: W. H. Freeman.

Osborne, M. J., & Rubinstein, A. (1994). A course in game theory. Cambridge, MA: MIT Press.

Owen, G. (1995). Game Theory (3rd ed.). Boston: Academic Press.

Roberts, F. S. (1976). Discrete mathematical models, with applications to social, biological, and environmental problems. Englewood Cliffs, N.J.: Prentice-Hall. 1976

Sandefur, J. T. (1990). Discrete dynamical systems: theory and applications. New York: Oxford University Press.

Stewart, I. (2008). Nature’s numbers: The unreal reality of mathematics. Orion Publishing Co.