Comparing General Mathematics Units 1 & 2 between the VCE and Australian Curriculum
Do you think they're the same...in general?
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 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):
Consumer arithmetic, including applications of rates and percentages, and use of spreadsheets.
Algebra and matrices, including linear and non-linear expressions, and matrices and matrix arithmetic.
Shape and measurement, including Pythagoras Theorem (sic), mensuration, and similar figures and scale factors.
Univariate data analysis and the statistical investigation process, including the statistical investigation process, making sense of data relating to a single statistical variable, and comparing data for a numerical variable across two or more groups.
Applications of trigonometry, including right- and non-right-angled trigonometry in two dimensions.
Linear equations and their graphs, including linear equations, straight-line graphs and their applications, simultaneous linear equations and their applications, and piece-wise linear graphs and step graphs.
Some differences in other states that closely follow the Australian Curriculum v8.4 (sorry NSW):
ACT: They explicitly list implementing a statistical investigation.
Qld: Reorders the topics and breaks them into 10 topics: 1 (Consumer arithmetic), 3 (Shape and measurement), 3 (Similarity and scale), 2 (Algebra and matrices), 6 (Linear equations and their graphs), 6 (Applications of linear equations and their graphs), 5 (Applications of trigonometry), 2 (Matrices), 4 (Univariate data analysis 1), 4 (Univariate data analysis 2). They remove the statistical investigation process.
SA: Reorders the topics into 1 (Investing and borrowing), 3 (Measurement), 4 (Statistical investigation), 5 (Applications of trigonometry), 6 (Linear and exponential functions and their graphs), 2 (Matrices and networks). They focus less on wages and income and more on shares and investments, and rates instead appear with Measurement and scale. They include scientific notation and significant figures, approximating areas of irregular plane shapes (including Simpson’s rule), an introduction to recursion for linear and exponential relationships (for arithmetic and geometric models), sampling and sampling methods, and connectivity and flow networks (for minimum spanning trees, shortest and longest paths, and maximum flow). They move simultaneous linear equations to stage 2 (Year 12).
Tas: Reorders the topics into 2 (Algebra and matrices), 6 (Linear equations and their graphs), 1 (Consumer arithmetic), 4 (Univariate data analysis and the statistical investigation process), 3 (Shape and measurement), and 5 (Pythagoras’ theorem and trigonometry). They include transposing formulas as part of Algebra and matrices, scatterplots and lines of best fit (by sight or regression using a calculator, and for predictions) as part of Linear equations and their graphs, volume of rainfall over an area and ratios (more broadly) as part of Shape and measurement. They explicitly list implementing a statistical investigation. They move sine and cosine rules to unit 4.
WA: Includes determining the percentage of data between a number of standard deviations from the mean, the 68-95-99.7% rule, and quantiles and probabilities for normal distributions.
The 2023 VCE units 1&2
At this point, the Victorian General Mathematics teachers are likely pointing at their screens and yelling “is that it?!” and you should definitely hold onto that feeling for units 3&4 in the next post.
Unlike VCE Mathematical Methods, VCE General Mathematics varies their areas of study and titles the subtopics.
Unit 1 includes (1) Investigating and comparing data distributions, (2) Arithmetic and geometric sequences, first-order linear recurrence relations and financial mathematics, (3) Matrices, and (4) Linear functions, graphs, equations and models.
Unit 2 includes (1) Investigating relationships between two numerical variables, (2) Graphs and networks, (3) Variation, and (4) Space, measurement and applications of trigonometry.
Let’s look at the differences.
Finance and sequences
Probably the biggest divergence of an existing topic is the financial content. The VCE General Mathematics course does not include wages and salary, government allowances and pensions, personal budgets, currency conversions, ratios for shares, and the use of spreadsheets. They do sneak in ratio, proportion, percentage, percentage change and rate, and unitary method at the end of the key knowledge, but I rarely see this actually taught or assessed, especially when following the current textbooks. Similarly, for the impact of inflation (in the AC v8.4) and the comparison of purchase options (not in the AC).
A lot of this content is relegated to VCE Foundation Mathematics, which leaves VCE General Mathematics more business or management-focused rather than consumer-focused.
Schools tend to focus on arithmetic and geometric sequences and their financial applications: simple and compound interest, and flat-rate, unit-cost, and reducing-balance depreciation (changing their names from straight-line method and declining-balance; neither of which match the Australian Tax Office’s current (primary) terminology: prime cost (straight line) and diminishing value 🤦🏻♂️). And for good reason, it’s content assessed (again) in unit 3 then further built upon. It is essentially the AC unit 3 content brought forward to unit 1. Meanwhile, in unit 1, the AC only considers interest and inflation, not depreciation (explicitly—it could theoretically be considered as part of “various contexts”).
I’m personally torn between the two approaches. I would like there to be more time and emphasis in unit 1 on reviewing ratios, rates, and percentages as these form core parts of the course going forward and are commonly not as embedded as they should be. I would also like there to be the AC content on personal finances given that General Mathematics is the predominantly taken non-calculus mathematics subject. However, I know students get a lot of benefit out of starting sequences earlier (with some schools introducing it at Year 10 for a consistent approach to simple and compound interest from then).
Linear and non-linear functions
The current study design dropped the linear and non-linear expressions content (substitution into expressions and formulae) of the AC v8.4 Algebra and matrices topic from the previous study design which used to make up part of a convenient topic that teachers and students could use to see if students will cope ok with the subject before census closed. And yet, the VCAA essentially added that topic back in as assumed knowledge in their sample plan.
Unlike the Australian Curriculum v8.4, the VCE course includes orders of magnitude and logarithmic (base-ten) scales, and direct and inverse variation, including where a variable is directly or partially proportional to the square, reciprocal, or base-10 logarithm of another variable (including linearisation) in preparation for data transformations in unit 3 (essentially making the Variation topic data transformations of the explanatory variable only with only perfectly fitting data, which I feel like is not a point made clearly enough by the study design).
The sequencing of the topics as listed in the study design mostly works growing from linear, to scatterplots and lines of good fit, to non-linear. However, I think sequences should appear after linear is reviewed given the required solving of linear equations and writing linear models for arithmetic sequences. However, this sequence doesn’t always translate to the classroom.
Matrices and networks
Spoilers for the Victorian teachers, but the AC v8.4 does not include a matrices topic in unit 3 or 4. Neither do they include matrix inverses in unit 1.
Speaking of inverse matrices, I’m still not sure if this was intentionally left in or missed, but the fact that VCE unit 1 Matrices includes using matrices to solve systems of simultaneous linear equations, but unit 4 Matrices doesn’t has baffled me since the study design was locked in. It’s not a small thing to teach well, and yet it’s discarded (likely for space for the added Leslie matrices).
The current VCE study design also brough transition matrices and steady states forward from unit 4 which I believe is meant to related to powers of matrices and using matrices for storing information, but since it wants “working with iterations”, they want matrix recurrence relations which works for the order of the topics listed in the study design, but schools don’t always follow it and get caught out by it if they teach matrices before sequences.
Similar to South Australia, the VCE General Mathematics study design also introduces networks in unit 2 (essentially the unit 3 Graphs and networks topic). This works nicely with the introduction of networks at Year/Level 10 in the AC9.0 and VC2.0. I will discuss more about graphs and networks in the unit 3&4 post.
Measurement and trigonometry
This topic has been my biggest gripe with the current VCE General Mathematics study design. With the removal of optional modules across the units, the measurement content now goes nowhere. It took seeing the AC v8.4 to realise that this is likely a combination of the VCAA checking the box that they cover it and wanting students that do not continue onto units 3&4 to have a basis of measurement and trigonometry for possible TAFE purposes. But that’s all speculation.
Here are my issues with it:
as said above, it does not lead into any of the topics in units 3 and 4.
it’s the last topic of unit 2, so if you follow the study design in order, it’ll be the last thing on students’ minds before starting unit 3, which feels pointless if they’re not going to be assessed on it in unit 3 or 4.
it includes “exact and approximate answers, scientific notation, significant figures and rounding” which the VCAA included in their added assumed knowledge topic in their sample plan at the start of the year, likely because they know that students will be expected to work with significant figure rounding (in particular) in the data analysis topics.
the majority of effort tends to be put into teaching the sine and cosine rules because students find them (fairly) the hardest part of the topic, often to the detriment to other parts of the topic, such as similarity of shapes and objects and their lengths, areas, and volumes.
My opinion is that it should either be split up like it is for the AC v8.4 (Shape and measurement in unit 1 and Applications of trigonometry in unit 2) or entirely in unit 1 if there remains no related content in units 3 and 4 so that students can keep their focus on the topics that do continue on.
Data, data, data
Ok, elephant in the room time. The AC v8.4 does not start bivariate data analysis until unit 3. Go ahead, do a quick scroll to the next section and come back. I’ll wait.
Welcome back. Let’s start with the similar univariate topic first.
For one reason or another, for the current study design the VCAA decided to include interval and ratio types of continuous numerical data in unit 1 but did not follow them through to unit 3.
A quirk that bothers me is that the study design only mentions the mode for categorical data and the mean and median for numerical data, and yet, the VCAA have assessed (in units 3&4) the mode of numerical data. I’m fine with calculating the mode or modal interval of numerical data, but it should be in the study design if they’re listing the others.
Similar to Queensland, the statistical investigation process is removed. So, students will only experience it if teachers choose to do it for the mathematical investigation or for a topic assessment (assignment or modelling task). Given the structure of application task that the VCAA wants for unit 3 (“a guided investigation of a given data set with several variables”), this puts students at a disadvantage if they do not experience something similar in unit 1 (or 2).
The VCE study design includes a preliminary bivariate topic in unit 2 for scatterplots and lines of good fit. Now, I’m all for its inclusion as I know that students can take a while to get comfortable, confident, and efficient at analysing scatterplots and regression lines and because it blends nicely from the Year/Level 10 Statistics content. However, what I’ve commonly found is that the topic becomes a mini version of the unit 3 content by including
Pearson’s product-moment correlation coefficient as it used to be in the unit 2 topic—and realistically should be—for discussing the strength of the scatterplot in a non-arbitrary way.
the least-squares regression line generally instead of the line of good fit, or as well as it. It also used to be in the unit 2 topic and is included in current textbooks, even with the formulas for the slope and y-intercept (that use the correlation coefficient).
the coefficient of determination to pre-teach it, despite a full reteach in unit 3.
This ends up bloating the topic bigger than it is meant to be, often causing it to be more rushed than necessary and requiring full reteaching in unit 3 anyway. It would be worth the VCAA considering the content of this topic more closely to what teachers would actually teach and making the appropriate space for it.
At the moment, it looks like it is intended to be a fairly light touch unit so that students have a foundation to build on in unit 3 transitioning from a line of good fit to a least-squares regression line, for example. In my opinion, it would be better to include determining residual values from the line of good fit than the least-squares regression line so that students can be taught the least-squares line after properly learning about them so they can understand what is being minimised better.
Really solid breakdown of the trig placement issue. The fact that students grind through sine and cosine rules at the end of Unit 2 only to never touch it again is bizzare curriculum design. I taught this pathway for three years and always felt like we were building toward nothing with that topic. Moving it to Unit 1 alongside the shape/measurement stuf would atleast give students breathing room before the data analysis heavy units.