Year
2021Credit points
10Campus offering
No unit offerings are currently available for this unit.Prerequisites
MATH107 Introduction to Logic and Algebra
Teaching organisation
4 contact hours per week for twelve weeks or equivalent.Unit rationale, description and aim
This unit provides the history of and motivation for the study of Mathematics, in particular it focuses on the idea and techniques of proof. Building on the previous Mathematics units and by looking at the development of counting and number systems, the evolution of mathematical concepts in response to need and the development of proof and its various forms is studied.
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 - Describe the role that use played in evolving counting including Aboriginal and Torres Strait Islander and other multicultural aspects (GA1, GA5)
LO2 - Explain the development of numbers and number forms including natural numbers, integers, irrational numbers and complex numbers (GA5, GA9)
LO3 - Identify the links with Geometry and the notion of proof (GA4, GA5)
LO4 - Use the various methods of proof: construction, deduction, induction, contradiction and exhaustion (GA4, GA5, GA8)
LO5 - Prove some major results in mathematics including infinity of primes, irrationality of √2, countability of rational and uncountability of the reals (GA4, GA5, GA8)
LO6 - Explain the role of Gödel’s Incompleteness Theorem in the treatment of axiomatic systems (GA5, GA9)
LO7 - Describe the importance of conjectures in evolving mathematical thinking and identify some of these key problems in the history of mathematics (GA5, GA7, GA9).
Graduate attributes
GA1 - demonstrate respect for the dignity of each individual and for human diversity
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
GA9 - demonstrate effective communication in oral and written English language and visual media
Content
Topics may include:
- History of Counting from primitive to historical times: the natural numbers.
- Extending the numbers systems.
- Geometrical aspects and the development of proof.
- Different methods of proof.
- History of operating with numbers and notation used.
- Cardinality and notions of infinity. Countability of sets.
- Axiomatic systems and Gödel’s Incompleteness Theorem
- Famous problems in the evolution of mathematics
- The role of conjecture.
Learning and teaching strategy and rationale
There will be 4 hours of class time each week with a two hour lecture and two hour tutorial.
Assessment strategy and rationale
A range of assessment procedures will be used to meet the unit learning outcomes and develop graduate attributes consistent with University assessment requirements.
The assessment will include an individual essay, a test on methods of proof and a group presentation to the class with associated report that will specifically address the learning outcome 7 and graduate attribute GA7.
Overview of assessments
| Brief Description of Kind and Purpose of Assessment Tasks | Weighting | Learning Outcomes | Graduate Attributes |
|---|---|---|---|
Appropriate written task | 30% | 1-3 | GA1, GA4, GA5,GA8, GA9 |
Test | 40% | 4-5 | GA4, GA5, GA8 |
Presentation and Report | 30% | 5-7 | GA4, GA5, GA7, GA8, GA9 |
Representative texts and references
Cameron, M. (1983). Heritage mathematics. North Melbourne: Hargreen.
Crossley, J. N. (1987). The emergence of number (2nd ed.). World Scientific.
Hofstadter, D. (1970). Gödel Escher Bach: The eternal golden braid. New York: Basic Books.
Kline, M. (2001). Mathematical thought from ancient to modern times (v1–3) New Edition. Cambridge University Press.
Lakatos, I. (1976). Proofs and refutations: The logic of mathematical discovery. Cambridge; New York: Cambridge University Press.
Pottage, J. (1982). Geometrical investigations: Illustrating the art of discovery in the mathematical field. Reading, MA: Addison Wesley.
Smullyan, R. (2014). A beginner’s guide to mathematical logic. Dover.
Struik, D. (1987). A concise history of mathematics (4th rev. ed.). New York: Dover.