Comparing Mathematical Methods between the VCE and Australian Curriculum
Mathematical Methods... flip it around... Wicked Witch 🧹
Where the first post focused on structural differences between the VCE Mathematics study design and other Australian and international senior mathematics curricula, the next series of posts will look at the content differences for General Mathematics (units 1&2 and units 3&4), Mathematical Methods, and Specialist Mathematics (units 1&2 and units 3&4) between (primarily) the Australian Curriculum v8.4 and the current 2023 VCE Mathematics study design and the VCAA’s sample course plans. Where relevant, there may be references to other Australian and international curricula to make a point.
Differences with Mathematical Methods
Let’s start with arguably the most on-the-surface different subject—Mathematical Methods. The subject mostly likely to claim the title by students of being the Wicked Witch of the West.
The Australian Curriculum v8.4 units 1&2
The Australian Curriculum v8.4 has the following six topics across units 1&2 (topics 1–3 for unit 1 and topics 4–6 for unit 2):
Functions and graphs, including lines and linear relationships, review of quadratic relationships, inverse proportion, powers and polynomials, graphs of relations, and functions.
Trigonometric functions, including cosine and sine rules, circular measure and radian measure, and trigonometric functions.
Counting and probability, including combinations, language of events and sets, review of fundamentals of probability, and conditional probability and independence.
Exponential functions, including indices and the index laws, and exponential functions.
Arithmetic and geometric sequences and series, including arithmetic sequences, and geometric sequences
Introduction to differential calculus, including rates of change, the concept of the derivative, computation of derivatives, properties of derivatives, applications of derivatives, and antiderivatives.
Some differences in other states that closely follow the Australian Curriculum v8.4 (sorry NSW):
ACT: offers Mathematical Methods T and Specialist Methods T which adds several more advanced content points to the core Mathematical Methods T course but is not the Specialist Mathematics T course.
QLD: Removes the lines and linear relationships sub-topic, adds a surds sub-topic and brings down the logarithms and logarithmic topic from unit 4 and the differentiation rules sub-topic from unit 3 into unit 2, and breaks the three AC v8.4 topics per unit into five topics using the sub-topics. It also lists the suggested time for each sub-topic in hours.
WA: Reorders the topics into 3 (Counting and probability), 1 (Functions and graphs), 2 (Trigonometric functions) for unit 1 and follows the same order as AC v8.4 for unit 2. It also lists the suggested time for each topic in hours.
SA: splits the Functions and graphs topic into Functions and graphs and Polynomials (which includes the quadratics content), and places sequences and series after the introduction to differential calculus. They include statistics (including standard deviation) and normal distributions, and logarithmic functions.
Tas: closer to Vic, so see below.
The 2023 VCE study design units 1&2
The current VCE Mathematical Methods study design has the same four areas of study for each unit: (1) Functions, relations and graphs, (2) Algebra, number and structure, (3) Calculus, and (4) Data analysis, probability and statistics.
Let’s look at the differences.
Trigonometry
Where the Australian Curriculum v.8.4 has trigonometry included in unit 1, the VCE Mathematics study design positions it in unit 2. By positioning the topic in unit 1, it makes it much easier to sequence Specialist Mathematics trigonometry content without rushing through it at the start of unit 2 so that the Specialist teacher(s) don’t have to teach the Methods content as well. Where does the VCAA position it in their sample course plan? Weeks 6–10 of unit 2 (so, end of term 3-ish) and yet leads their Specialist Mathematics sample course plan with three topics that rely on understanding sine, cosine, and tangent as functions of real angles using the unit circle (Trigonometry, Transformations, and Vectors in weeks 1–9).
I believe the biggest reason for the shift is because of the increase amount of content related to polynomial functions that is included in the VCE study design for unit 1 Methods compared to the greater focus on specifically the cubic polynomial in AC v8.4 and the inclusion of the rates of change content in unit 1 from unit 2 in AC v8.4. Despite the fact that publishers tend to place that content with the rest of the introductory differential calculus content at the end of the textbook, encouraging teachers to essentially follow the AC v8.4 structure if following the textbook order, I prefer it being taught in unit 1, but I think being separated from the rest of the functions and algebra content in its own area of study makes it too standalone, rather than something that could and should be integrated throughout the study of each new function as the concepts in calculus are a core part of Mathematical Methods and so should be introduced as early as possible and reiterated throughout the units.
Additionally, the VCE study design does not include the sine and cosine rules and the included area formula for triangles that the Australian Curriculum does. While the inclusion or exclusion of the sine and cosine rules are debatable, the included area formula for triangles not being explicitly included is an oversight given that it appears on the unit 3&4 formula sheet for the examinations but is (and was in VC1.0) optional 10A content. The same goes for surface areas and volumes of cones, pyramids, and spheres, but I will look at those below.
Additionally, AC v8.4 includes the angle sum and difference identities where the Vic does not. The inclusion is helpful for proving the derivatives of trigonometric functions by first principles but otherwise does not create issues.
Sequences and series
This one surprised me the most the first time I saw that it was included in Methods in AC v8.4 rather than Specialist like it is for VCE Mathematics, despite being a core part of Calculus (both for concepts like convergence, and as discrete cases for rates of change using differences and ratios of terms and accumulation using summations).
Keeping in mind that VCE Mathematical Methods includes algorithms, such as specifically the Newton-Raphson method that uses a recurrence relation (Joseph Raphson gets credit in this house), limiting values, including the fact that students will learn integration as a limiting value of a sum (not to mention the trapezium rule being the only numerical method as rectangular approximations were removed, for the most part), without the inclusion of sequences and series (even just for arithmetic and geometric ones), students are left to incidentally learn
subscript notation for terms of a sequence and contrasting it with the use of subscripts to denote the value of a parameter a function is also in terms of,
the difference between recurrence relations and explicit nth term rules, and generating sequences using them,
convergence vs divergence of sequences and functions (such as for limits and asymptotic values),
determining summations and the relationship between integration and summation.
While there are places you can include sequences in Year 10 (such as using recurrence relations as algorithms for VC2M10A06), many Methods-bound Year 10 classes are unlikely to do that whereas General-bound Year 10 classes are more likely to, and that is unlikely to include work on summation and series.
As a part of early consideration of accumulation and limiting values, we could look at how the formulae for the surface areas of spheres and volumes of pyramids, cones, and spheres were determined without the use of integration and surfaces and solids of revolution, which then provides an opportunity to introduce these formulae that otherwise appear incidentally with optimisation problems if students were not taught them in Year 10 using optional level 10A content.
Probability
I assume that the VCAA decided to split the Data analysis, probability and statistics area of study into two sections (compared to the Australian Curriculum that has it all in unit 1) so that they could have the same areas of study in each unit and so that they could be assessed in both units (note: this dates back to pre-2016 study design).
The biggest change that’s happened to this content over the years for units 1&2 (mainly from the 2016 to 2023 study design) is the reversal of the which unit the content appears in moving the probability calculations to unit 2 and the combinations to unit 1. I have recently come around on this decision as I’ve dug into the other curricula and have been mapping out the study design myself more carefully. While bringing forward combinations to fit with expanding powers of linear expressions was favourable, I originally was opposed to it for two main reasons:
most textbooks, many schools, and even the VCAA’s sample course plan sequence them one after the other anyway (last topic of unit 1 into first topic of unit 2 with an assessment only appearing in unit 2, which feels disingenuous to their own study design), making it fundamentally one topic, so it seemed like a fairly pointless change, especially when
the VCAA left in references to probability in unit 1 despite being a specifically listed point to reintroduce in unit 2, despite unit 1 also referring to proportions which they could have used instead (i.e., “determine the proportion of all combinations that have this selection" or “determine the relative frequency of selecting this combination”).
Now, there’s two relatively quick-and-easy fixes for this:
give clearer information about assumed knowledge (how much of probability do students need to recall from 7–10, given that “the properties that probabilities for a given sample space are non-negative and the sum of these probabilities is one”1 is a key knowledge point), and
remove the references to probability in unit 1, either completely or changing them to proportions and/or relative frequencies. This would allow the content to introduce randomness and random variables, differentiate between outcomes and elements of a set, and properly introduce the biggest sore thumb in the Mathematics Methods study design.
Pseudocode
Oh, pseudocode. Maybe you were a well-intentioned inclusion to build in coding to Methods and Specialist as part of numerical and computer-based methods that are used in industry (such as for estimating probabilities), but to say it’s poorly received is a massive understatement. Let’s go through some of the biggest issues:
for whatever reason, Mathematical Methods only refers to pseudocode and algorithmic design in a key knowledge dot point in outcome 2, computational thinking and algorithms in a key knowledge and skills dot points in outcome 3, and no specific introduction is included for outcome 1, and insufficient details around what students are expected to know and do with it (do they need to read and apply provided pseudocode?, do they need to be able to write pseudocode?), yet Specialist Mathematics gets a proper introduction in unit 1,
pseudocode was brought in just as several topics were removed for having too much crossover with other subjects (looking at you forces and mechanics, I still miss you) where pseudocode is included in VCE Algorithmics, and for being too arbitrarily included (how matrix transformations held on after transition matrices were removed despite the fact that no introduction to matrix arithmetic was included in Methods is still a mystery) and now the pseudocode is commonly taught briefly and just before exams so teachers can tick the box and students can hopefully get the one multiple-choice question on it correct rather than being incorporated throughout the subjects as was likely intended,
by choosing pseudocode (likely to avoid issues with languages going out of date) and not a specific programming language that is used by mathematicians, such as Python, students cannot simply implement the code to a computer or device to run it (only by hand), they need separate teaching to convert the pseudocode to an actual programming language to then run it,
following on, the two CAS calculators most commonly used by schools (the Casio Classpad and the TI-nspire CX, given VCAA’s approved technology list) can both run Python code, and so would have been a perfect opportunity to incorporate with the technology students already have (I’ll save the discussion around the inclusion of CAS calculator for another blogpost),
lack of (mathematics) teacher knowledge and experience around pseudocode and coding in general meant a massive state-wide amount of professional development was required for the non-computer science teachers, part of which is a webpage that initially included the introduction, examples, and sample questions, which was updated last year to include where pseudocode can be used in both subjects (and then proceeded to examine none of the listed points for Methods, and instead an extremely rudimentary algorithm),
lack of student knowledge and experience around pseudocode and coding in general despite the inclusion of computational thinking and pseudocode in the Victorian Curriculum 2.0 I don’t think has yet translated into a sufficient baseline for students entering Mathematical Methods in Year 11,
the VCAA not being consistent with their own set out conventions for writing pseudocode and taking a number of years to get to a decent formatting.
As I’m mapping out the study design, I’m being intentional about introducing pseudocode and algorithms, and bringing it up with each of the suggested topics for coherence, but my Spidey-sense tells me that changes are coming for pseudocode.
Rapid-fire round of other differences
“substitute integer, simple rational and irrational numbers in exact form into expressions, including rules of functions and relations, and evaluate these by hand” and the inclusion of rational exponents being the only direct implications that students need to learn to operate with surds (otherwise the fact that exact value for trigonometry implies it indirectly)
graphs of side parabolas and circles were removed for the current study design (but their functional sections can appear as a square root function and composite function, respectively)
no scientific notation and significant figures
no areas of sectors as part of radian measure
no horizontal translations of exponential, logarithmic, and circular functions (despite what the textbooks might make you believe, but only for unit 2)
minimal emphasis on straight-line motion (only motion graphs)
inclusion of systems of equations
inclusion of inverse functions (in Specialist Mathematics for AC v8.4)
inclusion of logarithms and logarithmic functions (in unit 4 for AC v8.4) which relates to the inclusion of inverse functions but also for solving exponential equations
inclusion of central difference approximation of gradients
explicit inclusion of other limits (not just first principles for derivatives), and the concepts of continuity and differentiability
inclusion of addition and multiplication principles for counting.
The Australian Curriculum v8.4 units 3&4
The Australian Curriculum v8.4 has the following six topics across units 3&4 (topics 1–3 for unit 3 and topics 4–6 for unit 4):
Further differentiation and applications, including exponential functions, trigonometric functions, differentiation rules, and the second derivative and applications of differentiation.
Integrals, including anti-differentiation, definite integrals, fundamental theorem, and applications of integration.
Discrete random variables, including general discrete random variables, Bernoulli distributions, and binomial distributions.
The Logarithmic function, including logarithmic functions, and calculus of logarithmic functions.
Continuous random variables and the normal distribution, including general discrete random variables, general continuous random variables, and normal distributions.
Interval estimates for proportions, including random sampling, sample proportions, and confidence intervals for proportions.
Some differences in other states that closely follow the Australian Curriculum v8.4 (sorry NSW):
SA: includes standard scores, and the sample means distribution and related confidence intervals. They explicitly show the chain rule versions of derivatives. They swap the order of integrals and discrete random variables, and some of the sub-topics.
QLD: includes the trapezoidal rule for area approximations and standard scores. Since logarithmic functions have already been taught, only the calculus is included, and put into unit 3. The sine rule, cosine rule, and included area formula for triangles are also included in unit 4. They explicitly show the chain rule versions of derivatives. Again, they break up the topics into five topics per unit and list the suggested time for each topic in hours.
WA: includes integrals of the form
\(\int\frac{f'(x)}{f(x)}\mathrm{d}x\)They make some small order and wording changes. Again, they list the suggested time for each topic in hours.
The 2023 VCE study design units 3&4
The current VCE Mathematical Methods study design has the same four areas of study for each unit: (1) Functions, relations and graphs, (2) Algebra, number and structure, (3) Calculus, and (4) Data analysis, probability and statistics.
Let’s look at the differences.
Second derivatives
Starting with my current biggest bugbear. The current VCE study design does not explicitly include second derivatives. Despite that, it does include determining non-stationary points of inflection and determining maximum and minimum rates of change. Now, these are intended to be done using technology for numerical and graphical solutions, but the fact that that the Australian Curriculum v8.4 includes the second derivative which solves this issue drives me up the wall. Not only that, but it also deals with the issue that I commonly see with textbooks, that they include determining acceleration from position or displacement functions and vice versa within VCE Mathematical Methods where successive derivatives or antiderivatives are required, despite not being a part of the course.
Separation of church and state discrete and continuous random variables
A weird quirk of the current study design was the choice to finally separate unit 1 and 2 for all other subjects than Methods (which was already separated) but leave units 3&4 joined at the hip. So, from the study design alone, we cannot tell if the VCAA would suggest teaching all of area of study 4 (data analysis, probability and statistics) together or separated. However, the sample course plan does separate them…but not discrete and continuous. They put discrete (including binomial) random variables, the normal distribution, and sample proportions together before integration and then finishing the subject with continuous random variables, which is certainly a choice. While VCE General Mathematics also includes the normal distribution without the framing of continuous random variables and area as probability, I think it’s poor form to not include it for Methods (or at least have the contextual information first).
Timing of differentiation and integration
Speaking of integration, the VCAA’s sample course plan put it in weeks 23–25 of 27 weeks of teaching time (including the 4 weeks of SACs). Meanwhile, their sample course plan has integration start in week 16 for VCE Specialist Mathematics. For the less familiar, there is a lot of integration in Specialist Mathematics, and yet the VCAA’s samples course plans don’t set the two subjects up so that the Specialist teacher(s) don’t have to also teach the fundamentals of integration (antidifferentiation is known, but not integration specifically). It makes much more sense in the Australian Curriculum v8.4 where it takes place in unit 3 and sets things up nicely for two of the three topics in unit 4: logarithmic functions and continuous random variables.
And yet, they do the exact opposite with differentiation.
Most textbooks take the excessive amount of review content from the study design, reteach it to death (while sneaking in the few new non-calculus content, such as e and general solutions to trigonometric equations (I would include graphing exponential, logarithmic, and circular functions with horizontal translations, but they include that in unit 2), and then get around to all of the differential calculus content which tends to happen around week 10–12 (ish) and even later for antidifferentiation and integration.
Whereas the VCAA’s sample course plan gets right into differentiation and antidifferentiation from the start of unit 3 which I applaud. There is no reason to arbitrarily forget about calculus for a term after the (hopefully) mind altering revelation of how we can determine gradients of non-linear functions (specifically polynomial functions at that point).
The logarithmic and absolute value functions
The VCE Mathematical Methods course introduces the logarithmic function in unit 2, whereas the Australian Curriculum v8.4 introduces it in unit 4. Given that the AC v8.4 does not include inverse functions, it makes more sense to delay the introduction of the function, and ordering it so that it comes after integration means that teachers can choose to introduce the natural logarithmic function as an integrally-defined function for the area bound by the reciprocal function and the x-axis from x = 1 (not that the AC v8.4 sets it up that way).
Speaking of integrals of reciprocal functions, the AC v8.4 explicitly only considers the positive domain: no absolute value, but vague on whether ln(−x) for x < 0 is meant to be included as it is in Vic. I’d argue that both the AC and the VCE study design would benefit from including the absolute value function as a specific example of a hybrid function with only linear segments. It solves a lot of issues around domains with integrating with reciprocal functions and areas bound by hyperbolae.
Rapid-fire round of other differences
AC v8.4 only includes sum, difference, product, quotient, and composite functions for calculus purposes, whereas the VCE study design considers their graphs (quotient being the only one specific to Specialist Mathematics), in particular VCE Mathematical Methods places a large emphasis on composite functions, and their domains and ranges
similar to units 1&2, Vic includes inverse functions (Specialist Mathematics for AC v8.4)
rectangle area approximations are replaced with the trapezium rule
inclusion of the Newton-Raphson method
no explicit uniform and non-uniform random variables
inclusion of literal equations and general solutions in terms of a parameter, including for systems of equations
no increments formula
no kinematics (only where velocity or acceleration is given as a type of rate and clearly described like other rates)
inclusion of average value
AC v8.4 includes “understand the effects of linear changes of scale and origin on the mean and the standard deviation.”2 which gives a possible way to teach
E(aX + b) and E(g(x)) to introduce the formulas for determining variance and standard deviation, but is unclear given it appears after both are introduced at the end of continuous random variables. The VCE study design does not include it (included as part of VCE Specialist Mathematics).inclusion of the standard normal distribution and z-scores
no margin of error (though related questions have been asked)
Concluding thoughts
Have you ever taught VCE Mathematical Methods and thought “I swear there are things missing that should be here?”, well hopefully this blog has made you feel less gaslit about it when a lot of those things are in the Australian Curriculum but weren’t brought across. What I want most from updates to the study design are changes that makes the foci of the subject (processes and their inverses, representations of relations and functions, ratios and rates of change and accumulation, approximations and convergence, generalisation, and probability and statistics) clearer and more intwined like they actually are rather than the pseudo-splitting the study design has.
Something I haven’t focused much on in this blog is differences in wording for the same idea and things like the inclusion of examples which I may return to. But for now, I will leave that as an exercise for the reader to investigate.