NMBR141 - Number Development in Mathematics

Year

Credit points

Campus offering

Prerequisites

Unit rationale, description and aim

Development of participants' mathematical content knowledge is the key foci of the unit sequence.

This unit offers a historical overview of the development of number and deals with some aspects of number theory. Further, conjecturing, specialising, generalising, justifying, mathematising, proving together with argumentation problem posing, problem solving, investigation and mathematical modelling are key elements of mathematical thinking and working mathematically that are developed across the sequence of units. Historical and real world situations are pivotal.

The aim of this unit is to develop the mathematical content knowledge of pre-service teachers.

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   explain the historical development of number and the effect this development has had on different societies and cultures (GA4, GA6, GA8; APST 2.1)

LO2   demonstrate an understanding of arithmetic, including the binary operations: addition and multiplication in various number systems (GA4, GA5, GA8; APST 2.1)

LO3   estimate the size of numbers and the size of solutions to problems (GA5, GA8, GA10; APST 2.1)

LO4   solve a variety of problems involving numbers (GA5, GA6, GA10; APST 2.1).

LO5   use technology and resources that will facilitate understanding of number, number systems and aspects of number theory (GA5, GA8, GA10; APST 2.1)

Graduate attributes

GA4 - think critically and reflectively 

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

GA6 - solve problems in a variety of settings taking local and international perspectives into account

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

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

AUSTRALIAN PROFESSIONAL STANDARDS FOR TEACHERS - GRADUATE LEVEL

On successful completion of this unit, pre-service teachers should be able to:

Content

In all topics the effective use of technology will be key, allowing students to focus on the understanding, rather than the mechanics of each particular topic.

Why bother with numbers, and what are they?

• Why did we need numbers anyway? Why did all cultures develop numbers? (Note, not why do we need numbers, but why did they.)
• Types of numbers (natural, integer, rational)
• Working with numbers in the real world (revision of rationals, perceptron algorithms, etc. estimation, rounding, accuracy, number systems from other cultures, eg. Babylonians, time and angles)

Working with numbers

• Why don’t we use Roman numerals anymore? Place notation, large and small numbers. Reporting numbers (Place notation, sci not, etc, other cultures)
• Is 10 the only option? Different ways of representing numbers and conversion between them (Place value, other cultures – including geeks!)
• Field axioms — what are our choices and what are the consequences (Field axioms)

Working with whole numbers

• You can’t farm with half a sheep? Working with whole numbers and integers.
• Types of whole numbers: Primes, greatest common divisor, Euclidean algorithm
• Solving equations needing whole numbers. (Diophantine equations, incl Pythagorean triples)

Learning and teaching strategy and rationale

Teaching and learning organisation can take several forms. This could include intensive weekend classes supported by web-based tools, Intensive one week winter or summer schools supported by web-based tools or weekly face-to-face classes during semester.

Pre-service teachers may be expected to participate in online discussion and sharing via eLearning. Class resources will be available via eLearning as will access to relevant web links.

This is a 10-credit point unit and has been designed to ensure that the time needed to complete the required volume of learning to the requisite standard is approximately 150 hours in total across the semester. To achieve a passing standard in this unit, students will find it helpful to engage in the full range of learning activities and assessments utilised in this unit, as described in the learning and teaching strategy and the assessment strategy. The learning and teaching and assessment strategies include a range of approaches to support your learning such as reading, reflection, discussion, webinars, podcasts, video etc. 

Duration

This unit includes 4 contact hours per week for 12 weeks, comprising 2 hours of lectures and 2 of tutorials.

This unit will normally include the equivalent of 24 hours of lectures together with 24 hours attendance mode tutorials.

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

Assessment strategy and rationale

The assessment tasks for this unit have been designed to contribute to high quality student learning by both helping students learn (assessment for learning), and by measuring explicit evidence of their learning (assessment of learning). Assessments have been developed to meet the unit learning outcomes and develop graduate attributes consistent with University assessment requirements. The assessment tasks provide multiple opportunities (presentation, problem solving and examination) in different ways (visual, verbal and written) for students to demonstrate:

• Knowledge of content
• Application of mathematics in real world contexts
• Development, use and communication of appropriate mathematical language

Minimum Achievement Standards

The assessment tasks for this unit are designed to demonstrate achievement of each learning outcome.

In order to pass this unit, students are required to complete all required assessment tasks as per the Assessment Policy and gain an overall pass mark.

Electronic Submission, Marking and Return

Assessment tasks are submitted electronically whenever possible. Marking will include a moderation process. Assessment returns will occur within the 3 week period as per the Assessment Policy. 

Overview of assessments

Representative texts and references

Required text(s)

Australian Curriculum https://www.australiancurriculum.edu.au/

Australian Curriculum, Assessment and Reporting Authority (ACARA) www.acara.edu.au

McLeod, G. et al (2019). Introduction to mathematical thinking. Custom Edition. Pearson

Australian Curriculum Mathematics. https://www.australiancurriculum.edu.au/f-10-curriculum/mathematics/

Relevant state and territory Mathematics curriculum documents

Recommended references

Bellos, A. (2010). Alex’s adventures in numberland: Dispatches from the wonderful world of mathematics. London: Bloomsbury.

Bellos, A. (2015). Alex through the looking glass: How life reflects numbers, and numbers reflect life. London: Bloomsbury.

Burton, D. (2011). Elementary number theory (7th ed.). Boston: McGraw-Hill Higher Education.

Clapham, C., & Nicholson, J. (2014). The concise Oxford dictionary of mathematics. Oxford: Oxford University Press.

Du Sautoy, M. (2011). The number mysteries: A mathematical odyssey through everyday life (1st Palgrave Macmillan ed.). New York, NY: Palgrave Macmillan.

Joseph, G. G. (2011). The crest of the peacock: Non-European roots of mathematics (3rd ed.). Princeton, NJ: Princeton University Press.

Katz, V. (2009). A history of mathematics (3rd rev. ed.). Pearson Education.

Matthews, C., Cooper, T., & Baturo, A. (2007). Creating your own symbols: Beginning algebraic thinking within Indigenous students. In Woo, J. H., Lew, H. C., Park, K. S. & D.Y. Seo (Eds.). Proceedings of the 31st Conference of the International Group for the Psychology of Mathematics Education, Vol. 3, pp. 249-256. Seoul: PME.

Silverman, J.H. (2013). A friendly introduction to number theory (4th ed.). Boston, MA: Pearson.

Wells, D. G. (1987). The Penguin dictionary of curious and interesting numbers. Harmondsworth, UK: Penguin.