# Structure of Specialist Mathematics

### Structure of Specialist Mathematics

Specialist Mathematics is structured over four units. The topics in Unit 1 broaden students’ mathematical experience and provide different scenarios for incorporating mathematical arguments and problem solving. The unit provides a blending of algebraic and geometric thinking. In this subject there is a progression of content, applications, level of sophistication and abstraction. For example, vectors in the plane are introduced in Unit 1 and then in Unit 3 they are studied for three-dimensional space. In Unit 3, the topic ‘Vectors in three dimensions’ leads to the establishment of the equations of lines and planes, and this in turn prepares students for solving simultaneous equations in three variables.

Unit 1 Unit 2 Unit 3 Unit 4
Combinatorics
Vectors in the plane
Geometry
Trigonometry
Matrices
Real and complex numbers
Complex numbers
Functions and sketching graphs
Vectors in three dimensions
Integration and applications of integration
Rates of change and differential equations
Statistical inference

#### Units

Unit 1 contains three topics that complement the content of Mathematical Methods. The proficiency strand, ‘Reasoning’, of the F–10 curriculum is continued explicitly in the topic ‘Geometry’ through a discussion of developing mathematical arguments. This topic also provides the opportunity to summarise and extend students’ studies in Euclidean Geometry, knowledge which is of great benefit in the later study of topics such as vectors and complex numbers. The topic ‘Combinatorics’ provides techniques that are very useful in many areas of mathematics, including probability and algebra. The topic ‘Vectors in the plane’ provides new perspectives on working with two-dimensional space, and serves as an introduction to techniques which can be extended to three-dimensional space in Unit 3. These three topics considerably broaden students’ mathematical experience and therefore begin an awakening to the breadth and utility of the subject. They also enable students to increase their mathematical flexibility and versatility.

Unit 2 contains three topics, ‘Trigonometry’, ‘Matrices’ and ‘Real and complex numbers’. ‘Matrices’ provides new perspectives for working with two-dimensional space, ‘Real and complex numbers’ provides a continuation of the study of numbers. The topic ‘Trigonometry’ contains techniques that are used in other topics in both this unit and Units 3 and 4. All of these topics develop students’ ability to construct mathematical arguments. The technique of proof by the principle of mathematical induction is introduced in this unit.

Unit 3 contains three topics, ‘Complex numbers’, ‘Vectors in three dimensions’, and ‘Functions and sketching graphs’. The Cartesian form of complex numbers was introduced in Unit 2, and in Unit 3 the study of complex numbers is extended to the polar form. The study of functions and techniques of calculus begun in Mathematical Methods is extended and utilised in the sketching of graphs and the solution of problems involving integration. The study of vectors begun in Unit 1, which focused on vectors in one- and two-dimensional space, is extended in Unit 3 to three-dimensional vectors, vector equations and vector calculus, with the latter building on students’ knowledge of calculus from Mathematical Methods. Cartesian and vector equations, together with equations of planes,enables students to solve geometric problems and to solve problems involving motion in three-dimensional space.

Unit 4 contains three topics: ‘Integration and applications of integration’, ‘Rates of change and differential equations’ and ‘Statistical inference’. In this unit, the study of differentiation and integration of functions is continued, and the techniques developed from this and previous topics in calculus are applied to the area of simple differential equations, in particular in biology and kinematics. These topics serve to demonstrate the applicability of the mathematics learnt throughout this course. Also in this unit, all of the students’ previous experience in statistics is drawn together in the study of the distribution of sample means. This is a topic that demonstrates the utility and power of statistics.

#### Organisation of achievement standards

The achievement standards in Mathematics have been organised into two dimensions: ‘Concepts and Techniques’ and ‘Reasoning and Communication’. These two dimensions reflect students’ understanding and skills in the study of mathematics.

Senior secondary achievement standards have been written for each Australian Curriculum senior secondary subject. The achievement standards provide an indication of typical performance at five different levels (corresponding to grades A to E) following the completion of study of senior secondary Australian Curriculum content for a pair of units. They are broad statements of understanding and skills that are best read and understood in conjunction with the relevant unit content. They are structured to reflect key dimensions of the content of the relevant learning area. They will be eventually accompanied by illustrative and annotated samples of student work/ performance/ responses.

The achievement standards will be refined empirically through an analysis of samples of student work and responses to assessment tasks: they cannot be maintained a priori without reference to actual student performance. Inferences can be drawn about the quality of student learning on the basis of observable differences in the extent, complexity, sophistication and generality of the understanding and skills typically demonstrated by students in response to well-designed assessment activities and tasks.

In the short term, achievement standards will inform assessment processes used by curriculum, assessment and certifying authorities for course offerings based on senior secondary Australian Curriculum content.

ACARA has made reference to a common syntax (as a guide, not a rule) in constructing the achievement standards across the learning areas. The common syntax that has guided development is as follows:

1. Given a specified context (as described in the curriculum content)
2. With a defined level of consistency/accuracy (the assumption that each level describes what the student does well, competently, independently, consistently)
3. Students perform a specified action (described through a verb)
4. In relation to what is valued in the curriculum (specified as the object or subject)
5. With a defined degree of sophistication, difficulty, complexity (described as an indication of quality)

Terms such as ‘analyse’ and ‘describe’ have been used to specify particular action but these can have everyday meanings that are quite general. ACARA has therefore associated these terms with specific meanings that are defined in the senior secondary achievement standards glossary and used precisely and consistently across subject areas.

#### Role of technology

It is assumed that students will be taught the Senior Secondary Australian Curriculum: Mathematics subjects with an extensive range of technological applications and techniques. If appropriately used, these have the potential to enhance the teaching and learning of mathematics. However, students also need to continue to develop skills that do not depend on technology. The ability to be able to choose when or when not to use some form of technology and to be able to work flexibly with technology are important skills in these subjects.