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NMBR140 Introduction to Mathematical Thinking

Unit rationale, description and aim

In an increasingly technological society, an understanding of mathematics is a major asset to an individual seeking to participate fully and meaningfully. Mathematical content in this unit draws from the areas of Number, Algebraic Thinking, Geometry, Measurement, Probability, Statistics and the Application of Mathematics; and ways of mathematical thinking including reasoning, problem solving and conceptual understanding. Learnings and examples from an Indigenous perspective will be used to explore the mathematical concepts.

This unit is designed for pre–service teachers who are First Peoples with an aim of presenting mathematical knowledge from indigenous, non–Western and Western perspectives.

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 - Demonstrate an appreciation of how mathematical concepts developed in indigenous and other cultures (GA1, GA4, GA9; APST 2.1)

LO2 - Demonstrate understanding and application of some elementary mathematical concepts (GA5; APST 2.1)

LO3 - Solve a variety of mathematical tasks (GA4, GA5, GA10; APST 2.1)

LO4 - Identify the structure inherent in various mathematical situations and undertake simple mathematical modelling (GA4, GA5, GA8; APST 2.1)

LO5 - Demonstrate an understanding of the interconnectedness of different mathematical topics and their application to real world contexts using a range of resources (GA5, GA8; APST 2.1)

LO6 - Communicate mathematical thinking and reasoning using mathematical language including spoken, written and visual representations (GA9, GA10; APST 2.1)

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

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

GA9 - Demonstrate effective communication in oral and written English language and visual media

GA10 - Utilise information and communication and other relevant technologies effectively


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

2.1 Demonstrate knowledge and understanding of the concepts, substance and structure of the content and teaching strategies of the teaching area.


Topics will include:

  • Numbers and Counting
  • Why did all cultures develop numbers and number systems?;
  • What is place value notation and why is it useful;
  • Computational strategies including mental, estimation and use of calculators;
  • Natural numbers, integers, fractions, factors, prime numbers, prime factorisation, divisibility tests;
  • Proportional reasoning, ratio;
  • Rational numbers and their representations in different cultures;
  • Simple operations with fractions;
  • Decimals, percentages.
  • Algebraic Thinking
  • Figural and numeral patterns leading to generalisation;
  • Numbers as counts versus numbers as objects – was algebra only invented once?;
  • Functions, variables and relationships;
  • Foundations of Geometry;
  • Shape and geometry in different cultures;
  • Shape versus geometrical figures;
  • Finding your place – the night sky and angles;
  • Angles and angle sums of polygons;
  • Foundations of Measurement
  • What is it important to measure – area, length, mass, volume and time;
  • Metric and other measurement systems;
  • Position and navigation;
  • Foundations of Probability;
  • Chance and real life – examples in many cultures and contexts;
  • Sample space and basic probability;
  • Single and multi-stage events;
  • Tree diagrams and two-way tables;
  • Representation and interpretation of data
  • Population and samples
  • Distribution: measures of centre, spread and variability
  • Presenting, summarising, analysing, interpreting data

Embedded within each topic is the application of mathematical reasoning and problem-solving drawing upon examples from First Peoples perspective.

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.


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

Technology Enhanced Learning

Lectures, online resources and access to apps to improve mathematical content knowledge will be available for students.

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. An early low-weighted assessment task is used to provide feedback to students within the first six weeks of a standard semester is used to identify students who are experiencing difficulties with numeracy and mathematics. These students will be advised to seek assistance from the Academic Skills unit.

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.

Overview of assessments

Brief Description of Kind and Purpose of Assessment TasksWeightingLearning OutcomesGraduate Attributes

Assessment Task 1

Knowledge test and development of a learning plan.




Assessment Task 2

Two investigative tasks, one of which must be a mathematical modelling task dealing with a real-world context and the other a purely mathematical investigation with use of varied resources.

The accompanying report will require a combination of written and/or oral multimedia forms of presentation and use of varied resources.


LO1, LO2, LO4, LO5, LO6

GA1, GA4, GA5, GA8, GA9, GA10

Assessment Task 3

Final Examination

Written examination: demonstrating an understanding of key mathematical content and problem-solving skills undertaken in the unit.


LO2, LO3, LO4, LO6

GA4, GA5, GA8

Representative texts and references

Required text(s)

Australian Curriculum

Australian Curriculum, Assessment and Reporting Authority (ACARA)

McLeod, G. et al (2019). Introduction to mathematical thinking (Custom ed. eBook). Melbourne: Pearson.

Recommended references

Belos, A. (2010). Alex’s adventures in numberland. London: Bloomsbury.

Belos, A. (2015). The grapes of math: How life reflects numbers and numbers reflect life. ? City: Simon & Schuster.

Booker, G. (2011). Building numeracy: Moving from diagnosis to intervention. South Melbourne, Vic: 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.

Gullberg, J. (1997). Mathematics from the birth of numbers. London: W. W, Norton.

Jacobs, H. R. (2002). Mathematics: A human endeavour: A book for those who think they don’t like the subject (3rd ed.). New York, NY: W. H. Freeman.

Reeve, R. (2010) Using mental representations of space when words are unavailable: Studies of enumeration and arithmetic in Indigenous Australia. Teaching Mathematics? Make it Count ACER Research Conference 15-17 August 2010 Melbourne.

Shryock-Boyke, K (2011) Introduction to plane geometry: Explorations and explanations. Pearson.

Velleman, D.J. (2019). How to prove it. (3rd ed.). Cambridge, UK: Cambridge University Press.

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