Comparing General Mathematics Units 3 & 4 between the VCE and Australian Curriculum
Generally speaking, could they be that different?
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 General Mathematics
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):
Bivariate data analysis, including the statistical investigation process, identifying and describing associations between two categorical variables, identifying and describing associations between two numerical variables, fitting a linear model to numerical data, association and causation, and the data investigation process.
Growth and decay in sequences, including the arithmetic sequence, the geometric sequence, and sequences generated by first-order linear recurrence relations.
Graphs and networks, including the definition of a graph and associated terminology, planar graphs, and paths and cycles.
Time series analysis, including describing and interpreting patterns in time series data, analysing time series data, and the data investigation process.
Loans, investments and annuities, including compound interest loans and investments, reducing balance loans (compound interest loans with periodic repayments), and annuities and perpetuities (compound interest investments with periodic payments made from the investment).
Networks and decision mathematics, including trees and minimum connector problems, project planning and scheduling using critical path analysis (CPA), flow networks, and assignment problems.
Some differences in other states that closely follow the Australian Curriculum v8.4 (sorry NSW):
Qld: Reorders the topics into 1 (Bivariate data analysis 1), 1 (Bivariate data analysis 2), 4 (Time series analysis), 2 (Growth and decay in sequences), X (Earth geometry and time zones), 5 (Loans, investments and annuities 1), 5 (Loans, investments and annuities 2), 3 (Graphs and networks), 6 (Networks and decision mathematics 1), 6 (Networks and decision mathematics 2). They include using the means, standard deviations, and correlation coefficient to determine the slope and y-intercept of the least-squares line, the present value and future value annuity formulae, ordinary annuities, angular distances on the Earth’s surface, time zone problems. They remove the statistical investigation, the general content on first-order linear recurrence relations (still include their financial applications).
SA: Reorders the topics into X (Modelling with linear relationships), X (Modelling with matrices), 1&4 (Statistical models), 2&5 (Financial models), 6 (Discrete models), X (Open topic). They include the normal distribution and the 68-95-99.7% rule, communication, dominance, and transition matrices, simultaneous linear equations (from unit 1&2), linear programming. They remove two-way tables and associations between categorical variables, non-causal explanations for an association, time series analysis, the general content on first-order linear recurrence relations (still include their financial applications).
Tas: Reorders the topics into 1&4 (statistical analysis), 2 (growth and decay in sequences), 5 (investment, loans and annuities), then either 3&6 (graphs, networks and decision mathematics) or X (trigonometry and earth geometry). They include sums of arithmetic and geometric sequences, sine and cosine rules (from unit 1&2), area formulae for triangles (from unit 1&2), great and small circle distances on the Earth’s surface, time zone problems.
WA: They explicitly include predicting from regression lines, making seasonal adjustments (where AC only fits least-squares, which WA also includes). They remove stated examples in the content points.
The 2023 VCE units 3&4
Told you Victorian teachers that should definitely hold onto that feeling of “is that it?!” for units 3&4.
Units 3&4 include
Data Analysis: Investigating data distributions, Investigating association between two variables, Investigating and modelling linear associations, and Investigating and modelling time series data,
Recursion and financial modelling: Depreciation of assets, Compound interest investments and loans, Reducing balance loans, Annuities and perpetuities, Compound interest investment with periodic and equal additions to the principal
Matrices: Matrices and their applications, Transition matrices
Networks and decision mathematics: Graphs and networks, Exploring and travelling problems, Trees and minimum connector problems, Flow problems, Shortest path problems, Matching problems, Scheduling problems and critical path analysis.
Let’s look at the differences.
Data analysis
In a pattern that we’ll see repeated, the VCAA includes a lot of content from units 1 and 2 so that they can explicitly assess it in units 3 and 4. For data analysis this includes much of the univariate content and scatterplots.
As part of the univariate content, the VCE study design includes using a logarithmic (base-ten) scale for histograms which creates a nice basis for the later data transformations, both of which are not in the AC v8.4.
They also include the normal distribution and the 68-95-99.7% rule and standardised z-scores. I’ll make a short point of it now, but I find it strange that, given the excessive use of the CAS calculator for VCE General Mathematics, they don’t instead use the CAS to determine percentages (which can round differently than the rule gives). Especially when the common approach is using a labelled diagram that has all (if not most) percentages written down for each possible region of the normal distribution for 3 standard deviations either side of the mean.
The AC v8.4 includes “identify the response variable and the explanatory variable” as part of Fitting a linear model to numerical data whereas the VCE study design its similar point as the first dot point of Investigating association between two variables before contingency (two-way) tables implying that they should be applied for any pair of categorical and numerical variables, and yet they only assess it for two numerical variables. It would be good for the VCAA to clarify this or otherwise assess it in full.
Similar to Queensland, Victoria also uses the means, standard deviations, and correlation coefficient between two numerical variables to determine the slope and y-intercept of the least-squares regression line.
Keeping in mind that there are other formulae for the slope and y-intercept, such as
where N is the number of observations and the fact that students are almost never to determine the five statistics first and then use them to determine the slope and y-intercept, it now feels like an excuse to incorporate solving equations by giving the equation of the least-squares line and asking students to determine one of the statistics or the coefficient of determination by first determining the correlation coefficient. Even scientific calculators can determine the coefficients and the above statistics, so it’s not even a remnant of non-CAS or non-graphics calculator courses.
A weakness of both the AC v8.4 and the VCE study design is their treatment of residuals. Both make broad statements about using residual plots or analysis to check the quality of fit and in the VCE case, it is only a subpoint of interpreting the slope and intercepts alongside making predictions and the coefficient of determination despite being regularly assessed in more detail. This has led to issues such as the current textbooks lumping the three together in a single exercise, instead of treating them one at a time. As a whole, it could be beneficial clarify the difference between
where e_i is the error or residual for x_i when predicting the value of y_i to see how the residual if the difference of the predicted and actual values (which could also be related back to the normal distribution as the residuals should be normally distributed with a mean of 0 constant standard deviation, but that’s a discussion for another day).
From the previous study design, the VCAA dropped the non-causal explanations for associations, likely due to the very limited questions they were asking on it. Instead, they continue to focus on the difference between observation and experimentation, mostly by testing students on whether they can spot the statement that does not imply a causation when only an association is observed (i.e., spot the not definitive statement with correct direction).
The big additional content included in the VCE study design is transforming one of the explanatory or response variables for non-linear associations such that it appears as a linear association so that the least-squares regression line can be determined enabling predictions to be made. While I am fine with its inclusion, I do think some textbooks make it into a bigger topic than it is by making it its own chapter (not just a couple of exercises for applying them, predicting with them, and deciding on which transformation is useful or optimal). I have more to say on data transformations, but I will leave that for another post.
Finally, we have time series. The AC v8.4 smooths them using “a simple moving average” but does not state if that is a mean or median smooth and whether that includes centring for even-numbered mean smoothing, which are all included in the VCE study design.
The AC specifically calculates seasonal indices using “the average percentage method”. This one statement sent me down several rabbit holes trying to decipher whether the VCE’s “their calculation using seasonal and yearly means” meant the same thing or not and all the other methods that exist for determining seasonal indices and deseasonalising time series. The issue came happened to come up on the 2025 examination, so I was conveniently able to clarify it with the VCAA that they would accept either “the average percentage method” (each seasonal value as a percent of the corresponding yearly mean averaged over the years) or “the simple averages method” (averages of seasonal values as a percent of the mean of all data/mean of seasonal averages). I have much more to say on this, but that’s another post. So, I’ll leave it by saying the latter is significantly more efficient if you’re not using a UDF.
And to round out, I will emphasise this point that I have to address frequently: cycles are not in either the AC v8.4 or the VCE study design (and haven’t been for a long time). Only trend, seasonality, irregular fluctuations, and outliers need to be considered despite what the textbooks may say. Cycles are essentially bundled up with trend. And on the point of seasonality, doing seasonal indices for days of the week does not fit the definition of seasonal data (divisions that divide up a year, a week is not a year, despite the fact that that there is a cyclical nature to the days of the week that can influence data values).
Recursion and financial modelling
Once you account for the fact that the VCE study design has already taught the arithmetic and geometric sequences content in unit 1, it is fairly similar to the AC v8.4 for these topics. A few small differences to start include: a broader approach to sequences beyond finance in the AC v8.4 that the VCE study design doesn’t include and a bit more detail into first-order linear recurrence relations and their long-term behaviour that is implied at best for the VCE study design.
The biggest difference was introduced in the 2016 study design. The VCE study design includes amortisation tables that break down compound interest investments and loans with payments on a step-by-step basis which seems like an odd thing for the AC to not include given the emphasis on using spreadsheets.
Both the AC v8.4 and the VCE study structure these topics similarly: by breaking them down into the various financial models and what should be done with each. I think this was a terrible idea that makes this topic feel much less connected than it actually is. I generally have to spend the most time getting students to understand how the various models are structurally similar, compressing the whole topic down considerably. The key knowledge and skills do a better job of describing the topic than the lengthy and repetitive dot points listed earlier that become hard to clarify the differences between. They could be improved further, but it’s much easier to understand what’s going on.
Something the VCAA can learn from Queensland is the more commonly used language used of the present value of ordinary annuities (for what the VCAA call annuities) and the future value of ordinary annuities (for what the VCAA call annuity investments), as that would help teachers find related resources more effectively. An annuity (where you are receiving payments) is still an investment, so “annuity investment” is not the best clarifier.
Something that both NSW and Queensland include is formulae for future and present values. These are developed through geometric series. So, if the VCAA were to introduce these, I would hope it is with arithmetic and geometric series being added back in after they were removed in 2016.
Matrices
Well, the simple answer is everything since the AC v8.4 does not include matrices at units 3&4, except for adjacency matrices of graphs and digraphs. So, instead I will get a few things off my mind.
Again, a lot of repeated content from unit 2.
Matrices and networks should be one topic by unit 4 given how related they are to each other. This would allow for better interoperability between the two (given how siloed questions are assessed for VCE General Mathematics), allowing for more back-and-forth work between the two. As of now, we get awkward distinctions between the two, especially for
communication and dominance matrices (which was originally in the networks module),
adjacency matrices (which use “maps” that are not graphs in the networks module),
transition diagrams and Leslie diagrams (which are weighted digraphs, for which their matrices are adjacency matrices using the weights).
I feel this is mostly a hangover from removing the optional modules where they were separate. This would create another topic the size of the data analysis unit. It could still be broken into smaller subtopics to help with breaking down the content for teaching.
Given there is no applications to diagonal and triangular matrices in unit 4, they serve little purpose. They either need an application (such as costing problems and solving systems of linear equations, respectively) or should be removed. Binary matrices are in a similar boat acting as an umbrella term to permutation, identity, and zero matrices and for the one-step communication and dominance matrices.
Similarly, inverses and determinants of matrices need a specific application in unit 4 (such as solving systems of linear equations). Currently, we are left with arbitrary matrix equations that border on solving systems of linear equations anyway and the odd question requiring a matrix recurrence relation to be run in reverse to determine an earlier state.
Rules for the element in row i column j of a matrix unfairly favour the TI-Nspire which can generate the matrix from the rule where the Casio Classpad cannot. The recent exams have also finally clarified that the rules can be non-linear, but prior to that, they were almost exclusively linear, meaning that other patterns could be abused.
The study design should explicitly state that students need to be able to determine the growth rate for Leslie matrices given that they do state equilibrium states in the case of regular transition matrices.
Networks and decision mathematics
Now that the AC has caught up, let’s compare the pair. Similar to recursion and financial mathematics, there’s a lot similar, despite differences in the listed order of content.
Surprisingly, there are some places where the AC v8.4 has content the VCE study design does not, such as simple and complete graphs, closed walks, and (semi-)Eulerian and (semi-)Hamiltonian graphs (noting that the AC v8.4 refers to semi-Eulerian trails and Eulerian trails rather than Eulerian trails and circuits, and similarly semi-Hamiltonian and Hamiltonian trails rather than Hamiltonian paths and cycles, despite defining trails, paths, closed trails, and cycles earlier).
Long story short, again, the VCE study design has more revision from unit 2. Basically everything from unit 2 is repeated. One oddity is that unit 2 includes greedy algorithms and Kruskal’s algorithm in addition to Prim’s algorithm which is the only one remaining in unit 4 (in line with the AC v8.4).
Speaking of greedy algorithms, it would be great if they were put more centrally within the networks topic as they can appear all over and would create a more interconnected topic but is not made obvious from the study design. This include Dijkstra’s algorithm, that is explicitly by trial-and-error only in the AC v8.4.
One absent content point that should be explicitly stated within the VCE study design that isn’t is the handshaking lemma for the sum of the degrees of vertices given how frequently it is assessed.
Determining the maximum flow through a network from source to sink is basically the same, which surprises me that the VCAA did not add any specific algorithms to solve them (only by inspection which allows for it). The maximum-flow minimum-cut theorem is fine, but it does not have a consistent way to confirm the minimum cut has actually been found, unlike other algorithms.
We’ll conclude with what feels like the VCAA’s favourite child: scheduling problems and critical path analysis (given how prominent it always features on exams, when other problems are left more neglected). The major difference to the AC v8.4 is the inclusion of crashing (reducing the completion time of a project by paying to make specific activity take less time). This tends to be students’ biggest frustration with the networks topic and probably the whole of General Mathematics because of how tedious it can be to ensure every consequence of crashing an activity has been considered. Given how tedious it is to input a network into technology, UDFs don’t help that much here either. I don’t think anyone would be upset if it were to go or otherwise for it to be more limited to school-assessed coursework where students have more time to deal with it than they do in examinations for only 1 or 2 marks per related part.
Concluding thoughts
The VCE General Mathematics course is a lot more content heavy than the Australian Curriculum v8.4 course. It would also benefit from a proper assumed knowledge section for both units 1&2 and units 3&4 to streamline the study design document.
Similar to the Methods comparison, I haven’t gone into wording comparison here, but there are plenty of improvement that could be made.
It is also quite business and management focused which seems conflicting with the name of “General Mathematics” (statistical analysis, modelling depreciating, loaned, and invested values, modelling population changes, and solving managerial decision problems). I think it would be good for the VCAA to make a clearer stance on which mathematics subject (General or Foundation) students should take by more clearly defining these subjects towards a business-mathematics focus (non-calculus management/employer mathematics) and a personal-mathematics focus (non-calculus consumer and employee mathematics). While I understand that the VCE Foundation Mathematics units 3&4 course is still new (as of 2023), the number of students taking the course is quite low compared to General:
2023 Foundation 1961 students, General 32 600 students
2024 Foundation 1909 students, General 30 556 students
2025 TBC
See my graphs of the VCAA’s performance in senior secondary statistics here. 2025 data won’t be released until after the 2026 NHT examinations have been sat and results released.