What I can fathom about the new National Curriculum for Mathematics
I’m spending some extra effort today figuring out the differences between the current Western Australian senior mathematics courses in secondary school, and that proposed as the National Curriculum for mathematics. Someone intending to do mathematics, engineering and the quantitative sciences should take 3AMAT, 3BMAT, 3CMAT, 3DMAT and 3(A/B/C/D)MAS. So if we do not speak of the student who is not intending to have a good level of mathematics after graduation, then there are eight units to take, each unit is a semester long and in total it would normally be contained in the last two years of secondary school. This is the current WA system, which has only been instituted recently.
The National Curriculum has four mathematics streams consisting of four units each. The weakest is “Essential Mathematics” which we would not include in our criteria as we are just considering the student who is intending to go to university. The other three are “Specialist Mathematics”, “Mathematical Methods” and “General Mathematics”. On a first glance, it seems the content of these units will be an improvement over the current WA courses. However, there is a problem in that the endemic problem of students choosing lower level mathematics subjects is likely to not change in the new curriculum. Some people have questioned the titles of these subjects, for example “Specialist Mathematics”, which some would see as a pseudonym for “Very Hard Mathematics”. Such titles may give students the illusion that Specialist Maths is much much harder than Math Methods; and this is certainly not the case on inspection of their content. So here I will summarise each subject of the WA system, adjacent to a summary of each subject in the National Curriculum. Then it might be possible to make comparisons between the two. I will only look at what could be potentially be taken at Year 12 by a student wanting to do science or engineering. I will say one more thing though, the documentation for the National Curriculum is much clearer and easier to read than the WA Curriculum (a tip for you budding bureaucrats!).
The Western Australian system
3CMAT and 3DMAT
Estimation and calculation | Solve equations and inequalities that involve algebraic fractions (up to quadratics) |
Define as the limit of as n goes to infinity | Two-variable linear programs |
The graph of | Systems of equations in three variables |
Composites of functions | Critical points of functions and the graphs of the first and second derivative |
Domain and range of a function | Using the Fundamental Theorem of Calculus |
Simplifying equations and inequalities | Algebraic deductive proofs |
Differentiation of polynomials | Rectilinear motion |
Sum, product and quotient rules for differentiation | Volumes of solids of revolution about an axis |
The derivatives of exponentials | Mensuration formulas |
Chain rule | Geometric arguments: using congruence of triangles, right-angles etc |
Second derivatives | Probability laws and calculating probabilities |
Basic use of calculus (turning points, tangent lines, etc) | Discrete, binomial, uniform and normal distributions |
Anti-differentiation and the integral of a function | Use of the Central Limit Theorem |
Definite integrals | Confidence intervals |
Integrating polynomials and exponentials | Calculation of sample sizes to obtain a certain confidence interval |
Solutions of | |
Area under curves | |
Combinations and arrangements | |
Bayes laws for probability | |
Conditional probabilities | |
Bernoulli trials and the binomial theorem | |
Uniform and normal distributions | |
Random variables | |
Mean and standard deviation |
3CMAS and 3DMAS
Vectors in 3D | Matrices |
Parallelogram rule | Algebraic properties of matrices (esp. multiplication is not commutative) |
Dot product of vectors | Singular, diagonal matrices; row and column matrices |
Cosine formula for a dot product | Determinants and inverses of 2×2 matrices |
Angles between vectors | Solve systems of up to 5 linear equations |
Parametric equation of a vector | Geometric properties of 2×2 matrices; actions on the plane |
Limit of (graphically, numerically) | Matrices as linear maps |
Differentiation of basic trig functions | Areas of shapes and the determinant |
The natural logarithm as an integral | Simple harmonic motion problems (simple second-order DEs) |
Change of base for logarithms | Growth and decay |
Integration by subsitution | More complicated integration of log and trig functions |
Derivatives of polynomials | Applying calculus to problems in kinematics and finance |
Product, quotient and chain rules | Conjecture in number theory (twin primes, Goldbach etc) |
Areas under curves | Proof by induction, de Moivre’s Theorem |
Basic separable differential equations | Algebraic properties of complex numbers |
Recurrence relations | Use complex numbers, de Moivre to establish trig identities |
Axioms and theorems | Roots of unity in the Argand plane |
Basic geometric proofs using vectors | Euler’s formula for |
Prove harder trig identities | |
Proof by exhaustion (?!!) | |
Proof by contradiction, including the “infinitude of primes” | |
Complex numbers | |
Complex numbers in polar form | |
Conjugation and its properties | |
Argand diagrams | |
Polar coordinates |
The National Curriculum
Mathematical Methods (Units 3 and 4)
Exponential functions | Two-outcome populations |
Define as the unique number such that as n goes to 0 | Survey procedures, sampling, |
Derivatives of exponentials | Bias (various kinds) |
Natural logarithm | Sample proportion |
Sum, product and quotient rules for differentiation | The sample prop as a random variable |
The second derivative, and its applications | Mean and variance of the sample prop |
Definite integrals | Standard error |
Areas under curves | Approx normal distribution of sample prop |
The Fundamental Theorem of Calculus | Confidence intervals |
Applications of integration (total change, average values, etc) | Calculation of sample sizes to obtain a certain confidence interval |
Linear equations in several variables | Critiquing voting polls (esp. in the media) |
The notion of a solution of a system of linear equations | Absolute value |
Gaussian elimination | Distances between reals |
Row reduction methods | Periodicity, symmetry, phases, asymptotes of trig functions |
Row echelon form | The graph of |
Continuous random variables | Changes in amplitude, period and phase |
Discrete and continuous variables | Solving algebraic equations involving dilation, translation and trig functions |
Probability density function | Least squares, curve fitting |
Calculating probs using integrals | Log-linear plots |
Mean, variance and standard deviation | Derivatives and anti-derivatives of trig functions |
The normal distribution | Applications of calculus to optimisation and exp. decay |
Calculating quantiles |
Specialist Mathematics (Units 1 and 2)
I’ve included these units because I was surprised by the level of content!
Basic logic (implication, converse, contrapositive) | Parametric equations of circles, parabolas, ellipses, hyperboles |
Proof by contradiction | Geometric properties of parabolas |
Symbolic logic | Basic graph theory |
Quantifiers (“for all” and “there exists”) | Isomorphism of graphs |
Deduction in Euclidean geometry | Bipartite graphs |
Infinitude of primes | Konigsberg bridge problem |
Basic properties in circle geometry | Eulerian circuits, Hamiltonian circuits |
Concurrency theorems in Euclidean geometry | Trees |
Rational numbers as terminating and recurring decimals | Spanning trees and Prim’s algorithm |
Irrational numbers | Planarity, Euler’s formula, Kuratowski’s Theorem, Platonic solids |
Proofs of irrationality of and | Double angle trig formulae |
Approximation of irrationals by rationals | Trig identities, triple angle formulae |
Imaginary numbers and complex numbers | Complicated trigonometric equations |
Conjugation and properties thereof | Periodic phenomena |
Basic algebraic properties of the complexes | Applications to simple harmonic motion, music, etc |
The Argand plane | Basic Kinematics |
Complex roots of equations | |
Recurrence relations | |
Fibonacci sequence, arithmetic sequence, geometric seq. | |
First-order linear recurrence relations and apps | |
Direct formulas for sequences | |
Matrix arithmetic | |
Determinants and inverses of 2×2 matrices | |
Solving simultaneous equations in two variables | |
Geometric interpretations of unique solution, no solution and infinitely many solutions | |
Transformations in the plane | |
Determinants vs area |
Specialist Mathematics (Units 3 and 4)
Mathematical induction and examples | Integration techniques, partial fractions |
Vector arithmetic (in more than 2 dimensions) | Integration using trig identities |
Parallel and perpendicular vectors, projection | Integration by substitution |
Vector equations, concurrence of lines | Inverse trig functions |
Geometric proofs using vectors | Derivatives of inverse trig functions |
Graph sketching | Applications of integral calculus |
Simple rational functions and their differential properties | Calculating area under and between curves |
Complex numbers and arithmetic | Volumes of solids of revolution about an axis |
de Moivre’s Theorem | Newton’s method |
Vectors in the Argand plane | Simple first-order DEs |
Roots of complex numbers, conjugate pairs | Separation of variables |
Factorisation of polynomials with real coefficients | Euler’s method for finding approx solutions to first-order DEs |
Applications to DEs and physics and biological sciences | |
Simple harmonic motion | |
Option 1: Statistical Inference | |
Option 2: Vectors and dynamics | |
Option 3: Further calculus techniques and inequalities |