Mathematics education in Australia

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Mathematics education in Australia

Mathematics education in Australia varies significantly between states and territories, particularly at the upper secondary level. Each region operates its own education system and curriculum, which results in differences in what is taught and how it is assessed. Additionally, some schools offer the International Baccalaureate (IB) program, providing an internationally recognized alternative.

## Secondary

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### New South Wales

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#### Higher School Certificate

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The Higher School Certificate (HSC) in NSW contains a number of mathematics courses catering for a range of abilities. There are four courses offered by the NSW Education Standards Authority (NESA) for HSC Study:[1]

* Mathematics Standard 1: This course offers students a practical and relevant approach to mathematics, tailored to equip them with essential skills for navigating everyday challenges and future endeavours. Built upon a foundation laid in previous years, the course delves into various facets of mathematics, ranging from algebraic concepts to statistical analysis. In the Year 11 curriculum, students engage with fundamental topics such as algebra, measurement, financial mathematics, and statistical analysis. They explore concepts like linear relationships, measurement applications, financial management, and statistical data analysis, gaining insights into real-world applications and problem-solving techniques. Moving into the Year 12 course, students deepen their understanding by delving into additional topics such as networks. Throughout both years, emphasis is placed on fostering numeracy, building confidence, and making mathematics meaningful through practical applications. Students are encouraged to develop their skills in mathematical modelling, using these models to address present and future challenges they may encounter in various contexts, from personal finance to workplace scenarios. By the end of the course, students emerge with a solid mathematical foundation that not only serves them well in their immediate academic pursuits but also prepares them for the demands of the workforce and further training opportunities.

* Mathematics Standard 2: This course is designed to further enhance students' mathematical skills beyond Stage, the course extends students' proficiency in mathematics while steering clear of the in-depth complexities of higher-level mathematics like calculus. Students explore a variety of mathematical concepts across four main topics: algebra, measurement, financial mathematics, and statistical analysis, supplemented by the addition of networks. Within these topics, students engage with subjects like non-right-angled trigonometry, rates and ratios, investments and loans, annuities, bivariate data analysis, and critical path analysis. Through hands-on activities and practical applications, students develop a deeper understanding of mathematical concepts and their relevance in contemporary contexts. The course equips students with the analytical skills necessary to tackle complex problems and make informed decisions.

* Mathematics Advanced: This course is focused on calculus-based principles, this course challenges students to develop critical thinking skills by exploring problems through observation, reflection, and reasoning. Students embark on a comprehensive exploration of mathematical concepts across five main topics in the Year 11 curriculum, including functions, trigonometric functions, calculus, exponential and logarithmic functions, and statistical analysis. The course further extends students' proficiency across four topics: functions, trigonometric functions, calculus, financial mathematics, and statistical analysis. Through topics like graphing techniques, differential calculus, integral calculus, and modelling financial situations, students refine their mathematical skills and explore advanced concepts in depth.

* Mathematics Extension 1 (Must be studied concurrently with Mathematics Advanced): This course is designed to expand students' mathematical horizons and deepen their understanding of advanced mathematical concepts. This course provides students with opportunities to delve into rigorous mathematical arguments, proofs, and models, fostering their ability to communicate mathematical ideas concisely and precisely. Students embark on a journey to develop thorough knowledge and skills in various mathematical domains across four main topics in the Year 11 curriculum: functions, trigonometric functions, calculus, and combinatorics. Through topics such as polynomials, further trigonometric identities, rates of change, and working with combinatorics. Continuing into the Year 12 course, Mathematics Extension 1 further challenges students with topics such as proof by mathematical induction, vectors, trigonometric equations, further calculus skills, and statistical analysis. Through exploring topics like introduction to vectors, applications of calculus, and the binomial distribution.

* Mathematics Extension 2 This course is an advanced course that offers students the opportunity to delve into the fundamental ideas of algebra and calculus while developing strong mathematical manipulative skills. In the course, students engage with five main topics in the Year 12 curriculum: proof, vectors, complex numbers, calculus, and mechanics. Through topics such as the nature of proof, further proof by mathematical induction, further work with vectors, introduction to complex numbers, further integration, and applications of calculus to mechanics. While NSW Mathematics curricula does not include matrix theory nor group theory, the geometric properties of complex numbers alludes to both of these. The former is hinted at in the multiplicative properties of complex numbers, as students are required to plot the products, sums and quotients of complex numbers on the Argand plane. While group theory is not explicitly mentioned, roots of unity and cyclic groups are extensively studied. With their newfound familiarity with complex numbers, the fundamental theorem of algebra can now be formally stated. Students are now able to exploit this closure to solve even more polynomial equations. Recursive integral sequences, integration by parts and partial fraction decomposition techniques allow the solution to a wider class of problems. Projectile motion is studied in the kinematics module, which surpasses the depth of study found in HSC physics. This course synergizes with HSC Physics, as students are able to apply this knowledge in their Physics exams to arrive at more elegant and efficient solutions. The parametrisation of lines, circles and parabolas in Mathematics Extension 1 is further developed to the entire family of conics, including degenerate cases. Students are exposed to rectangular hyperbolas, however hyperbolic trigonometric functions are not included. Despite this students are expected to adapt to novel material, such as proving properties of the catenary via its expression in exponential functions.

The defining feature of content progression from Mathematics Advanced through to Extension 2 Mathematics is the level of mathematical maturity expected of students. In higher courses, students have exposure to a greater breadth and depth of techniques, and are expected to synthesize knowledge from seemingly disparate topics. In Mathematics Advanced exams students may be asked to apply familiar techniques to unfamiliar contexts, such as being given an identity through which they must solve a problem. Further mathematical maturity is vital to success in Extension 2 exams, as assessment focuses on both conceptual understanding and computational abilities.

The difficulty in HSC final exam questions generally increases throughout the course of the paper. In one Extension 2 HSC examination, the final question provides students with a series of prompts and smaller questions, which culminate in a proof of the Basel Problem.

### Victoria

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#### Victorian Certificate of Education

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The Victorian Certificate of Education (VCE) mathematics subjects are designed to cater for the varying abilities and aptitudes of Victorian students. There are four courses offered for VCE study:[2]

* Foundation Mathematics: Provide for the continuing mathematical development of students with respect to problems encountered in practical contexts in everyday life at home, in the community, at work and in study.[3]
* General Mathematics: Provide for the study of non-calculus and discrete mathematics topics. They are designed to be widely accessible and provide preparation for general employment, business or further study, in particular where data analysis, recursion and financial modelling, networks and matrices are important.[3]
* Mathematical Methods: Provide for the study of simple elementary functions, transformations and combinations of these functions, algebra, calculus, probability and statistics, and their applications in a variety of practical and theoretical contexts. They also provide background for further study in, for example, science, technology, engineering and mathematics (STEM), humanities, economics and medicine.[3]
* Specialist Mathematics: Provide for the study of various mathematical structures, reasoning and proof. The areas of study in Units 3 and 4 extend content from Mathematical Methods Units 3 and 4 to include rational and other quotient functions as well as other advanced mathematics topics such as logic and proof, complex numbers, vectors, differential equations, kinematics, and statistical inference. They also provide background for advanced studies in mathematics and other STEM fields. Study of Specialist Mathematics Units 3 and 4 assumes concurrent study or previous completion of Mathematical Methods.[3]

### Queensland

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In 2019 Queensland implemented a new QCE (Queensland Certificate of Education) system which included new syllabi for each of the senior mathematics subjects. Current senior mathematics general subjects are: General Mathematics, Mathematical Methods, Specialist Mathematics (listed in order of increasing complexity). There is also an applied subject called Essential Mathematics and a short course called Numeracy.

Up until the end of 2019 students in Queensland were able to study: Maths A, Maths B, and Maths C.

#### Mathematics A

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Maths A covers more practical topics than Maths B and C, but it is still OP eligible. There are considerably fewer algebraic concepts in this subject, and it is suitable for students who either struggled with mathematics in Year 10, or who do not require a knowledge of abstract mathematics in the future. Maths A is designed to help students to develop an appreciation of the value of Mathematics to humanity. Students learn how mathematical concepts may be applied to a variety of life situations including business and recreational activities. The skills encountered are relevant to a vast array of careers (trade, technical, business etc.). Assessments in the subject include both formative and summative written tests, assignments and practical work. It is assessed in the categories: Knowledge & Procedures (KAPS); Modelling & Problem Solving (MAPS); Communication & Justification (CAJ). Although Maths A is not a pre-requisite subject, but it is sufficient for entrance to many tertiary courses.[4]

The course is divided into four semesters. The skills learned in each semester are as follows:

Semester 1 (Year 11/Form 5):

* Data Analysis
* Managing Money
* Applied Geometry
* Linking 2 and 3 Dimensions

Semester 2 (Year 11/Form 5):

* Land Measurement
* Applied Geometry
* Statistics
* Managing Money

Semester 3 (Year 12/Form 6):

* Managing Money
* Land Measurement
* Data Analysis
* Operations Research

Semester 4 (Year 12/Form 6):

* Statistics
* Land Measurement
* Navigation
* and an elective topic on Data

#### Mathematics B

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Maths B is considerably more theoretical than Maths A, requiring advanced algebra skills to successfully complete. It is a common prerequisite for science and engineering courses at Queensland Universities. Maths B (in some schools) can be studied at the same time with either Maths A or Maths C, but not both. Maths B gives students an understanding of the methods and principles of mathematics and the ability to apply them in everyday situations and in purely mathematical contexts; the capacity to model actual situations and deduce properties from the model; an interest and ability in framing and testing mathematical hypotheses; the ability to express and communicate any results obtained; some knowledge of the history of mathematics; encouragement to think independently and creatively. Assessments are similar as those of Maths A, which includes both formative (Semester 1) and summative (Semesters 2,3 and 4) written tests, assignments and post-assignment tests. It is also assessed in the three categories Knowledge & Procedures (KAP); Modelling & Problem Solving (MAP); Communication & Justification (CAJ). Maths B is a pre-requisite for any tertiary course which deals with or uses math and/or science.[5] According to the Queensland Studies Authority, in 2010, 93% of students who studied Maths B were OP eligible.

The course is divided into four (4) semesters. The skills learned each semester are as follows:

Semester 1 (Year 11/Form 5):

* Functions (Linear, Quadratic, Absolute Value)
* Periodic Functions (Trigonometry, Sin/Cosine Functions)
* Applied Statistics (Mean, Median, Mode, Lie Factor)
* Applied Statistics 2 (Linear/Quadratic Regression, Residual Plots)

Semester 2 (Year 11/Form 5):

* Exploring Data / Statistics
* Indices and Logarithms/ Exponential Functions
* Limits and Differential Calculus 1

Semester 3 (Year 12/Form 6):

* Exponential and Log Functions
* Optimization Using Derivatives
* Integration
* Integral Calculus

Semester 4 (Year 12/Form 6):

* Applied Statistical Analysis
* Integration
* Differential Calculus 2
* Optimisation (Other Methods)

#### Mathematics C

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Maths C extends the topics taught in Maths B, and covers additional pure-maths topics (including complex numbers, matrices, vectors, further calculus and number theory). Although not necessarily more difficult, it must be studied in conjunction with Maths B. Maths C gives the students an understanding of the methods and principles of mathematics and the ability to apply them in everyday situations and in purely mathematical contexts; the capacity to model actual situations and deduce properties from the model; an interest and ability in framing and testing mathematical hypotheses; the ability to express and communicate any results obtained; some knowledge of the history of mathematics; encouragement to think independently and creatively. Assessments are in the same as the other two courses, formative and summative written tests, assignments and practical work. The student is assessed in the areas of Knowledge & Procedures (KAPS); Modelling & Problem Solving (MAPS); Communication & Justification (CAJ). Maths C can be a pre-requisite to tertiary courses with a heavy maths/science basis. Some skills learned in Maths C would be found in business and economics degrees.[6]

The course is divided into four (4) semesters. The areas learned are in the following:

Semester 1 (Year 11/Form 5):

* Real and Complex Numbers
* Matrices
* Vectors
* Groups
* Structures & Patterns

Semester 2 (Year 11/Form 5):

* Applications of Matrices
* Vectors
* Real and Complex Numbers
* Dynamics
* Structures and Patterns

Semester 3 (Year 12/Form 6):

* Structures and Patterns
* Real and Complex Numbers
* Matrices
* Periodic Functions
* Calculus
* Option I & II

Semester 4 (Year 12/Form 6):

* Vectors
* Calculus
* Dynamics
* Vectors
* Option I & II

### Western Australia

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New WACE mathematics courses were introduced for Year 11 students in 2015 to replace previous mathematics courses and being the Western Australian course in line with the Australian Curriculum.

The new WACE mathematics courses consist of four units. Each unit is studied over one semester. Therefore, Unit 1 & 2 is studied in Year 11, and Unit 3 & 4 is studied in Year 12.

The new WACE mathematics courses are:[7]

* Mathematics Preliminary General[8]
* Mathematics Foundation General[9][10]
* Mathematics Essential General[11][12]
* Mathematics Applications ATAR[13][14]
* Mathematics Methods ATAR[15][16]
* Mathematics Specialist ATAR[17][18]

ATAR mathematics courses are for university-bound students, whereas general courses are for non-ATAR students.

Syllabus information is available from the School Curriculum and Standards Authority (SCSA) website.

### South Australia

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In South Australia the mathematics courses are split into six[clarification needed] levels:

* Numeracy for Work and Community Life (up to and including Stage 1)[19]
* Essential Mathematics/Mathematical Pathways[20]
* General Mathematics/Mathematical Applications[21]
* Mathematical Methods/Mathematical Studies[22]
* Specialist Mathematics — more advanced topics that complement and are taken concurrently with Mathematical Studies[23]

## Tertiary

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See list of mathematical science schools in Australia for the university qualifications offered in the mathematical sciences in Australia.

## References

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1. ^ "Course descriptions for Mathematics Stage 6". NSW Education Standards Authority. Retrieved 10 February 2024.
2. ^ VCE Mathematics Study Design Units 1-4: 2006-2009 (PDF)
3. ^ a b c d "VCE Mathematics Study Design 2023-2027".
4. ^ Mathematics A syllabus Archived 2007-12-11 at the Wayback Machine (PDF)
5. ^ Mathematics B syllabus Archived 2007-12-11 at the Wayback Machine (PDF)
6. ^ Mathematics C syllabus Archived 2007-11-11 at the Wayback Machine (PDF)
7. ^ "WACE 15-16 - Mathematics". scsa.wa.edu.au.
8. ^ "Archived copy" (PDF). Archived from the original (PDF) on 2014-12-26. Retrieved 2014-12-26.`{{cite web}}`: CS1 maint: archived copy as title (link)
9. ^ "Archived copy" (PDF). Archived from the original (PDF) on 2014-12-26. Retrieved 2014-12-26.`{{cite web}}`: CS1 maint: archived copy as title (link)
10. ^ "Archived copy" (PDF). Archived from the original (PDF) on 2014-12-26. Retrieved 2014-12-26.`{{cite web}}`: CS1 maint: archived copy as title (link)
11. ^ "Archived copy" (PDF). Archived from the original (PDF) on 2014-12-26. Retrieved 2014-12-26.`{{cite web}}`: CS1 maint: archived copy as title (link)
12. ^ "Archived copy" (PDF). Archived from the original (PDF) on 2014-12-26. Retrieved 2014-12-26.`{{cite web}}`: CS1 maint: archived copy as title (link)
13. ^ "Archived copy" (PDF). Archived from the original (PDF) on 2014-12-26. Retrieved 2014-12-26.`{{cite web}}`: CS1 maint: archived copy as title (link)
14. ^ "Archived copy" (PDF). Archived from the original (PDF) on 2014-12-26. Retrieved 2014-12-26.`{{cite web}}`: CS1 maint: archived copy as title (link) (PDF)
15. ^ "Archived copy" (PDF). Archived from the original (PDF) on 2014-12-26. Retrieved 2014-12-26.`{{cite web}}`: CS1 maint: archived copy as title (link)
16. ^ "Archived copy" (PDF). Archived from the original (PDF) on 2014-12-26. Retrieved 2014-12-26.`{{cite web}}`: CS1 maint: archived copy as title (link)
17. ^ "Archived copy" (PDF). Archived from the original (PDF) on 2014-12-26. Retrieved 2014-12-26.`{{cite web}}`: CS1 maint: archived copy as title (link)
18. ^ "Archived copy" (PDF). Archived from the original (PDF) on 2014-12-26. Retrieved 2014-12-26.`{{cite web}}`: CS1 maint: archived copy as title (link)
19. ^ "Numeracy for Work and Community Life". sace.sa.edu.au. South Australian Certificate of Education. Retrieved 25 September 2014.
20. ^ "Mathematics Pathways". South Australian Certificate of Education. Retrieved 25 September 2014.
21. ^ Mathematical Applications syllabus Archived 2007-08-31 at the Wayback Machine (PDF)
22. ^ Mathematical Methods syllabus Archived 2007-08-31 at the Wayback Machine (PDF)
23. ^ Specialist Mathematics syllabus Archived 2007-08-31 at the Wayback Machine (PDF)

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