Luke works as a maths tutor Melbourne for high school and University students and provides one-to-one and student-organised group tutoring in person or online since 1997.

**Mathematical Development**

The development of mathematics is human’s ever-increasing endeavours of abstraction. Prehistoric people might realise that a collection of two apples and a collection of two oranges has something in common, namely quantity of their members. More complex mathematics did not appear until around 3000 BC, when the Babylonians and Egyptians began to use arithmetic, algebra and geometry for surveying, building, and observing universe. People studied the relationships among quantities, whether of magnitudes (as in geometry) or of numbers (as in arithmetic) or the generalization of these two fields (as in algebra).

Descartes developed analytic geometry (the Cartesian coordinate system) in 1637, which uses algebra to describe geometry. He invented the convention of representing variables in equations by x, y, and z, and constants by other letters, such as a, b, c and etc. These play a fundamental role in the development of the calculus by Isaac Newton and Gottfried Leibniz.

During the 19th Century, however, mathematics broadened to encompass mathematical or symbolic logic, and thus came to be regarded increasingly as the science of relations or of drawing necessary conclusions (although some see even this as too restrictive).

The discipline of mathematics now covers – in addition to the more or less standard fields of number theory, algebra, geometry, analysis (calculus), mathematical logic and set theory, and more applied mathematics such as probability theory and statistics.

**VCE, IB, and University Maths**

VCE (Victorian Certificate of Education) mathematical studies have three choices for students. Specialist Mathematics is the most difficult, followed by Mathematical Methods and then by Further Mathematics. However, the IB (International Baccalaureate) mathematics has both the standard level (SL) and the higher level (HL).

Mathematics teaching in secondary school is designed to cater for students of different abilities and interests. VCE maths methods, VCE further maths, and the IB maths (SL) are for students with varied backgrounds and abilities. The IB mathematics higher level (HL) and VCE specialist math are for students with a strong background in mathematics and competence in a range of analytical and technical skills. These students will be likely to include mathematics as a major component of university studies, either in their own right or within courses such as physics, engineering or technology.

The university mathematics is classified into more specialised courses depending of different students’ majors. BHG Tutoring focuses on algebra, trigonometry, functions, calculus, probability and statistics.

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**VCE mathematics study design**

The new mathematics study design (2016 – 2018) for the first time separated Specialist Mathematics Units 1 and 2 from General Mathematics Units 1&2. This gives three relatively independent maths subjects.

General Mathematics Units 1 and 2 provide for a range of courses of study involving non-calculus based topics for a broad range of students and may be implemented in various ways to reflect student interests in, and applications of, mathematics. They incorporate topics that provide preparation for various combinations of studies at Units 3 and 4 and cover assumed knowledge and skills for those units.

Mathematical Methods Units 1 and 2 are completely prescribed and provide an introductory study of simple elementary functions, algebra, calculus, probability and statistics and their applications in a variety of practical and theoretical contexts. They are designed as preparation for Mathematical Methods Units 3 and 4 and cover assumed knowledge and skills for those units.

Specialist Mathematics Units 1 and 2 comprise a combination of prescribed and selected non-calculus based topics and provide courses of study for students interested in advanced study of mathematics, with a focus on mathematical structure and reasoning. They incorporate topics that, in conjunction with Mathematical Methods Units 1 and 2, provide preparation for Specialist Mathematics Units 3 and 4 and cover assumed knowledge and skills for those units.

Further Mathematics Units 3 and 4:

There have been minor edits and refinements throughout. In addition:

- the General Mathematics topic Computation and practical arithmetic has replaced the topic Financial arithmetic with respect to coverage of assumed knowledge and skills for the Core area of study in Further Mathematics
- for Recursion and financial modelling students are expected to be able to apply recurrence relations from first principles for

The module Geometry and measurement has been revised to:

- remove the use of Simpson’s rule for estimating the surface area of irregular shapes
- remove the interpretation and use of contour maps
- remove the area of lune and lens and applications
- remove determination of the great circle distance between any two points on the surface of the earth using

Mathematical Methods Units 3 and 4

There have been minor edits and refinements throughout, however no significant changes have been made to the areas of study and topics, or to the outcomes and key knowledge and key skills from the consultation draft to the final revised study.

Specialist Mathematics Units 3 and 4

There have been minor edits and refinements throughout, in addition:

- graphs of regions in the complex plane have been removed
- the application of integration to problems involving the exponential distribution has been removed
- limiting equilibrium when a body is at rest has been removed
- linear combinations of random variables have been included

hypothesis testing for a population mean has been included

IB Mathematical studies standard level course, which is quite similar to VCE maths methods includes 130 hours of instruction on eight topics. Students have 20 hours to spend project work – an individual piece of work.

- Introduction to the graphic display calculator
- Number and algebra
- Sets, logic and probability
- Functions
- Geometry and trigonometry
- Statistics
- Introductory differential calculus
- Financial mathematics

The IB Diploma Programme mathematics higher level course includes 190 hours of instruction on seven topics

- Algebra
- Functions and equations
- Circular functions and trigonometry
- Matrices
- Vectors
- Statistics and probability
- Calculus

Students also have spend 40 hours in one of the following topics

- Statistics and probability
- Sets, relations and groups
- Series and differential equations
- Discrete mathematics

Students have 10 hours to develop a portfolio including two individual pieces of work based on mathematical investigation and mathematical modelling.