It has now been seven years since the release of the current HSC Mathematics syllabuses in 2017, and the new 2024 syllabus is on the horizon.
Prior to the introduction of the 2017 syllabus, Dr Bill Pender, a highly respected figure in Mathematics education and the author of widely-used Cambridge Mathematics textbooks, offered insightful observations and recommendations regarding the syllabus structure and content at the time. His letters to NESA can be readily found online, but I have uploaded it for convenience sake below.
In this post, I will be reflecting on the content changes in the upcoming 2024 syllabus, my own teaching experience with the outgoing 2017 syllabus, as well as the contents of this letter.
Copy-Paste Job
In his letter, Dr Pender pointed out that parts of the syllabus appeared to be copy-pasted from other sources, leading to inconsistencies. I noticed this too, especially with the discrepancies between the glossary definition of a function and the syllabus definition, which I’ve discussed in a previous post. It is therefore encouraging to see that the upcoming 2024 syllabus has made progress to eliminate these copy-paste inconsistencies that affected the 2017 version.
I’m particularly pleased to see the syllabus beginning to use foundational set theory to explain relations and functions. While this approach may be long overdue in high school mathematics, it is a necessary shift that will support students pursuing further studies at university, where set-theoretic definitions are prominent and necessary. Introducing this now will help bridge the gap between high school and higher-level mathematical study, providing students with a clearer, more rigorous understanding.
We have a while to go before we start defining sequences of real numbers as functions \( t: \mathbb{N} \rightarrow \mathbb{R}\), or random variables as functions \( X: \Omega \rightarrow \mathbb{R} \), or even using \(f: X \rightarrow Y\) notation, but we are beginning to head in the right direction.
A consequence of a more set theoretic foundation I’d like to see in the future of mathematics education, after several more syllabus revisions, is the correct definition of a vector as an element of a vector space, rather than as objects with direction and length. However, I now digress from the main focus of this post.
Another criticism of the 2017 syllabus, from teachers as well as in the letter, was that it contained too much vague language like “understand” or “simple identities.” The 2024 update has done a lot of work to clarify these vagaries.
Exclusion of Euclidean Geometry
One significant change introduced in the 2017 syllabus was the removal of Euclidean Geometry proofs, a shift from previous syllabuses. While speaking with teachers who have HSC marking experience, I found a general consensus that this made marking simpler. However, I don’t believe that simplifying marking alone justifies removing this content.
In his letter, Dr Pender warned that removing Euclidean Geometry would strip away fundamental geometric concepts that are crucial for developing an intuitive grasp of mathematics. He also emphasised that geometry underpins several essential areas, including trigonometry, coordinate geometry, and graphing functions, all of which are vital to the logical progression into calculus.
From my experience teaching the 2017 syllabus, I found that omitting these proofs didn’t seem to significantly impact students’ understanding of these interconnected topics. Since students still encounter geometry in Stages 4 and 5, they do enter the HSC course with foundational knowledge, making this adjustment manageable.
One argument for retaining Euclidean Geometry proofs is that they could enhance students’ understanding of structured proof writing, particularly in ordering assumptions and logical implications. However, the Proof topic in Extension II now addresses this rigorously. In terms of preserving geometrical thinking, which remains central to mathematical study, the Vectors topics in Extension I and II provide an alternative platform for developing this reasoning.
With experience in teaching the 2017 syllabus and an eye toward the 2024 implementation, I find I don’t mind the removal of Euclidean Geometry as much as I initially thought.
Statistics
In his letter, Dr Pender questioned the inclusion of descriptive statistics and correlation, viewing them as overly simplistic and unsuitable for an advanced mathematics course. He argued that these topics rely too heavily on ‘black-box formulae,’ which detracts from the mathematical understanding required for higher-level study and doesn’t contribute to the rigour needed for university-level mathematics.
The draft 2024 syllabus initially aligned with Dr Pender’s perspective by omitting descriptive and bivariate statistics. However, following the ‘Have Your Say’ consultation, these topics were reintroduced. Their inclusion appears as a set of disjointed and hastily added dot points, lacking the coherence and integration evident in other sections of the 2024 syllabus. This inconsistency is concerning, as it disrupts the otherwise cohesive structure of the syllabus.
It’s disappointing to see these points reinstated after being removed from the draft. Personally, when teaching this content, I plan to defer the introduction of random variables until Year 12 when they are properly contextualised. The current Year 11 dot point on random variables feels disconnected, likely cut and reinserted from an earlier draft without consideration for the logical flow in the final syllabus.
Continuity
In reviewing teaching materials and assessments from various schools, I’ve observed that misunderstandings around continuity are common. One example was an assessment task that required students to label \( y = \frac{1}{x} \) as a discontinuous function, which is incorrect since it is continuous at all points within its domain.
Dr Pender’s remark that “continuous functions are not the sort of thing to teach at school, and serve no purpose in the three courses” was accurate; confusion persists, as ongoing professional discussions among teachers about limits and continuity reveal similar misconceptions. Perhaps this is a reason why the draft dot point on limit calculations was removed entirely.
The 2024 syllabus takes a positive step by clarifying that continuity should be considered only at specific points, rather than classifying entire functions as continuous or discontinuous . This clarification should reduce the need to categorise functions like \(y = \frac{1}{x}\) as continuous or not, and only focus on specific points within its domain.
However, a problematic dot point remains:
Define a discontinuity of a function informally as a point where the function is not continuous.
This definition works well for identifying a discontinuity at \(x=0\) for the piecewise function:
\[f(x) = \begin{cases} x+1 & \text{if } x \geq 0 \\ x-1 & \text{if } x < 0 \end{cases}\]
but creates ambiguity for functions like \( y = \log(x) \). For example, would we consider \( x = -1 \) as a point of discontinuity for \( y = \log(x) \)? What about at \( x= 0\)?
Without introducing topological concepts like the closure of sets within \(\mathbb{R}\) (with the standard topology) – specifically the closure of a function’s domain – such cases are difficult to clarify. The absence of this conceptual framework limits our ability to discuss discontinuity meaningfully in edge cases like these.
Perhaps this dot point requires revision, or removal.
Prescription of Technology
Perhaps due to his longstanding teaching context, Dr Pender recommended removing technology from certain dot points in the syllabus. As someone who benefits greatly from technology, especially graphical tools like Desmos and Python scripting, I disagree with this recommendation.
I remember when I was a student, technology was just beginning to make its way into classrooms. I was one of the few students in my maths class in Year 11/12 to bring my laptop and place it on my desk. Having access to the built-in Grapher software on my Mac allowed me to visualise mathematical problems more easily and verify my intuition — this is a powerful tool in developing students’ understanding of mathematics!
The retention of technology in the syllabus has enhanced students’ understanding of certain concepts. In classroom teaching, the quick and easy access to Desmos for graphing functions, without the time spent drawing axes and plotting points manually, has accelerated discussions on core topics, such as identifying domains and ranges from graphs.
We do live in the 21st century, where technology use is ubiquitous, and our syllabus does well to stay current with these advancements.
However, it is also important to remember that seeing something visually with a computer is not equivalent to a proof. When a concept requires rigorous justification, it is critical to present a proof that is coherent, logical, and precise. It’s all about balance.
Syllabus Committee
Dr. Pender recommended withdrawing the drafts and reconstituting the syllabus with input from leading mathematics academics and experienced teachers. He suggested that the syllabus committee include experts capable of refining the content to align with the logical structures and precision required in advanced mathematics.
Although the call to withdraw the syllabus was, in my opinion, a bit extreme, the call for expert academics and experienced teachers to develop the syllabus was sound. Indeed, I believe that is what has occurred in the 2024 syllabus’ writing process.
The Technical Advisory Group included such people, some of whom I am friends with, and I am glad that they had the opportunity to give valuable input to the writing of the 2024 syllabus. I commend everyone’s efforts in the writing process.
Although the syllabus is not perfect, as highlighted in some of my previous points, it is nonetheless a step in the right direction toward good progress. Another commendable change includes separating the concept of Sequences and Series from the title of Financial Mathematics, and I am sure there are other parts worth noting too.
As we implement and teach this syllabus, I am certain we as teachers will identify parts that resonate well with us and our students, and other parts that may not. Through these experiences, we can help shape the next revision.