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What I can fathom about the new National Curriculum for Mathematics

May 25, 2010

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 e as the limit of (1+1/n)^n as n goes to infinity Two-variable linear programs
The graph of f(x) =a e^{(b(x-c)}+d 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 y'=ky
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 \sin(x)/x (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 e^{i\theta}
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 e as the unique number a such that (\frac{a^n-1}{n}\mapsto 1 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 y=af(b(x+c))+d
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 \sqrt{2} and log_2(5) 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
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