Effective mathematics teaching requires deep subject knowledge combined with the pedagogical content knowledge of how to make that content accessible, engaging, and meaningful. Standard 2 asks teachers to continually develop both mathematical understanding and a repertoire of teaching strategies — including how to sequence concepts, anticipate misconceptions, use representations, and connect mathematics to learners' lives.
Volume: Cylinders, Pyramids & Compound Shapes
This lesson demonstrates pedagogical content knowledge across several dimensions: the sequencing from prisms (prior knowledge) → cylinders → pyramids → compound shapes follows a deliberate conceptual build. Animated SVG drawings show how 3D shapes are constructed rather than presenting them statically. The ⅓ relationship between pyramid and prism volumes is made visually explicit. A common-mistakes slide addresses the most frequent error (using diameter instead of radius) directly.
The lesson uses a chilli-level system to differentiate by challenge while keeping all learners working on the same topic. This avoids the deficit framing of ability grouping while still extending confident learners.
Direct application of a single formula. Values are straightforward. Builds fluency and confidence with the method.
Multi-step problems. Requires choosing between formulae and managing unit conversions or rearrangement.
Compound shapes requiring decomposition and synthesis. Non-routine contexts. Connects to Codebreaker extension task.
The Grading Algorithm — Statistical Enquiry
Uses the 2020 UK A-level algorithm scandal as a real-world hook for statistical enquiry: forming good statistical questions, classifying variable types (categorical vs numerical, discrete vs continuous), and evaluating primary vs secondary data sources. Opens with the whakatauki "Mā te huruhuru ka rere te manu." 23 slides across three parts, with quickfire rounds and a structured exit ticket.
Three further interactive lessons demonstrate content knowledge taught through genuine curriculum contexts rather than surface decoration: symmetry and transformation geometry through tukutuku panels, perimeter/area/Pythagoras through wharenui design, and bearings/trigonometry through waka hourua navigation. Full lessons on Standard 1.
30+ fully interactive extension lessons on the NCEA Level 1 algebra pathway, each with four difficulty tiers and an infinite randomised question generator, so content knowledge can be pitched precisely to each learner's level. Full evidence on Standard 3 and Standard 6.
Four anonymised student work samples with teacher feedback, covering algebra (Codebreaker), volume, symmetry/tukutuku, and statistical questions. Samples show a range of achievement levels and the feedback cycle in practice.
📄 View Work Samples- How do I demonstrate my mathematical knowledge confidently in the classroom?
- What teaching strategies do I use to make difficult concepts accessible?
- How do I use technology and visual representations to deepen understanding?
- How do I connect the curriculum to contexts that are meaningful to my learners?
Teaching cylinders and pyramids — what I know and what I'm still learning to teach
I feel confident in my own mathematical understanding of volume — the content itself is well within my knowledge. The challenge, I am finding, is not knowing the mathematics but knowing how to make it land for learners who are encountering it for the first time.
When I first taught this topic, I explained the formula V = πr²h correctly but several students struggled with the move from the general prism formula V = A × l to the cylinder as a special case. I had not anticipated that gap. In revising the lesson, I restructured the opening to spend more time on that conceptual bridge — slide 2 in the lesson above was added specifically because of that experience.
The differentiation through chilli levels was something I was nervous about. I worried students would opt for Medium even when they were capable of more. In practice, the opposite happened — students were honest about their level and several moved up mid-task. I think the framing as spice rather than ability helped remove the stigma.
What I am still developing: my ability to respond to unexpected misconceptions in the moment. During this lesson one student calculated volume using the circumference instead of the radius. I addressed it individually but fumbled my explanation slightly. I want to build a better mental library of these misconceptions and clear responses to them — this is what I understand as the heart of pedagogical content knowledge.