Janine Simpson is NZCER's Kaituki Ratonga Aromatawai | Team Lead Assessment Services. She has worked in primary education as a classroom teacher, school leader, and mathematics facilitator.
Learning maths is difficult at the best of times - but when it comes to rational numbers, there is an entire conceptual shift that can be a massive hurdle for students. Supporting them through this shift is at the heart of effective rational number teaching.
To begin with, students work with numbers as discrete, countable quantities. They learn that 8 is bigger than 2 because they can count it, see it, and compare it. In this context, larger numbers mean more. But rational numbers shift the logic students have relied on. Instead of reasoning only about “how many”, students must reason about relationships between quantities. Suddenly, 1/8 is smaller than 1/2, even though 8 is larger than 2. Although students encounter ideas like “half” and “sharing” in everyday life, they often interpret them through a whole number lens. Rational numbers require students to think beyond counting and reason relationally about quantities. It's a conceptual shift, a change in their thinking, that can be a big step.
Building the foundations of rational number understanding
Early in my teaching career, I thought I was giving students plenty of fraction experience. They folded paper, shaded shapes, completed worksheet tasks, and practised procedures. Probably very similar to how I was taught, if I’m honest.
But listening to how students explained their thinking, I noticed something. Many talked about the numerator and denominator as if they were two separate whole numbers. They could follow the steps, but underneath, whole-number thinking was still the driving force behind their reasoning. Even though they understood the process, the underlying concepts were still not quite there.
That realisation changed how I approached teaching fractions in the early years. Experiences alone are not enough if students do not develop a deep understanding of what fractions represent. In the early years (roughly ages 5–8), the focus needs to be on building strong part–whole and quotient understandings, not just procedures to follow. This foundation can’t be rushed.
In my experience, students need regular opportunities to:
- share real objects into equal groups and talk about what makes a fair share
- partition shapes and sets into equal parts
- see that a whole can be a single object, a region, a length, or a collection
- experience fractions as something that can be counted, measured, or shared.
Hands-on materials matter here. This might include folding paper, cutting fruit, sharing teddies, using fraction strips, marking lengths on paper strips, and measuring on number lines. These kinds of experiences help students build mental models. Talking about what they are doing, using language such as one half, equal parts, or describing how many parts are being considered, helps to shape and stabilise that understanding. Without these models and the language to describe them, fraction symbols can easily become disconnected procedures rather than meaningful ideas.
Alongside this, another important shift begins to develop. In the early years, much of mathematics is additive. Students count, combine, and separate. Rational number introduces students to multiplicative and relational ways of thinking.
Understanding that 1/4 means “one of four equal parts”, and that four of those parts make one whole, is fundamentally different from addition. It is about relationships. Language matters here. Instead of simply naming a fraction, it can help to emphasise that relationship: “It takes four of these parts to make exactly one whole.”
Unit fractions are particularly important too. If a student understands that 1/5 is one equal fifth part, then 3/5 is simply three of those parts. Just as three apples are three units of apple, three fifths are three units of one fifth. Spending time on unit fractions builds a strong foundation for later work with fractions.
One thing I realised in my own practice was that I often emphasised whole-to-part situations (“Here is one cake, let’s cut it into four equal parts”) but gave less attention to part-to-whole experiences (“If this is one quarter of the shape, what does the whole look like?”). Both directions matter if students are to reason flexibly and start to see the relationships. When we opened that two-way channel in the classroom, I saw a change in how students spoke about rational number. They were thinking more relationally and providing justifications for why parts were equal and how they knew that.
Using multiple representations to build rational number understanding
Students need to experience fractions in a range of ways if they are going to build a strong understanding of rational number. It isn’t enough for them to only see fractions in circles, pizzas, pies and cakes. When fractions live only in food, they stay as food, not mathematics. Students also need opportunities to encounter fractions as measures, quantities, locations on a number line, and relationships between amounts. True competence in rational number is about flexibility.
Students also benefit from encountering rational numbers through a range of models.
Area models, such as circles, rectangles, and paper folding, are often the starting point. They are useful for introducing part–whole ideas and helping students see how a whole can be partitioned into equal parts.
Set models shift that thinking slightly. When students are asked, “What fraction of the 12 counters are red?”, the whole is no longer a single object but a collection. This helps broaden their understanding of what a “whole” can be.
Linear models are also important. Before introducing number lines, fraction strips can help students build a sense of equal parts and relative size. Aligning and comparing strips supports the idea that fractions represent quantities. The number line then strengthens this further. Placing 3/4 between 0 and 1 highlights that a fraction has magnitude. It is a number, not just two whole numbers written on top of each other.
Classroom example: The endless line
For years, I kept an empty number line on my whiteboard stretching beyond 0. As mathematical conversations unfolded, we added fractions, decimals, benchmark numbers, drawings of materials, and real-life examples. Students began to see equivalence and size visually. Parents even contributed examples from home. That number line was one way we kept a living record of proportional reasoning developing over time. It helped make students’ thinking visible and supported them to see how ideas connected and grew.
The refreshed Mathematics and Statistics curriculum places strong emphasis on the mathematical processes of representing, reasoning, and connecting. When students work across multiple models, they are learning to represent ideas in different ways, connect concepts across contexts, and reason about relationships. These processes are central to developing deep, transferable mathematical understanding.
Watch your language when teaching rational number!
And of course, we can’t talk about rational number without talking about mathematical language.
Early in my teaching I often used phrases like “three out of four” because they felt accessible for students. At first, they seemed helpful, but over time I began to see their limits and that they in fact had a short ‘shelf life’! When students encountered improper fractions, that language no longer fits.
Language such as “three one-fourths” or “three of four equal parts” keeps the focus on unit fractions and equal partitioning. This supports understanding fractions as quantities that can be composed and decomposed, including fractions greater than one.
Decimal place value introduces another layer of complexity. Although decimals build on whole-number place value, students must learn to extend their understanding of the base-ten system in new ways. Students who have not yet consolidated whole-number place value often try to apply whole-number rules to decimals.
I remember students insisting that 0.75 was larger than 0.8 because 75 is larger than 8. It made sense to them. They were applying what they already knew about whole numbers. Moments like this show how students are still making sense of new ideas. They also highlight the importance of strong place value foundations before expecting fluency with decimals.
A connected approach in the primary years
At times, aspects of rational number have been taught separately. Fractions in one block, decimals in another, percentages later. When those connections are not made clear, students often struggle to see how the ideas relate.
In the classroom, this means making those links visible. Seeing tenths represented as both fractions and decimals. Noticing that ten tenths make one whole. Counting forwards and backwards in fractional amounts. Connecting sharing, measurement, division, and ratio situations. Giving students opportunities to break wholes apart, put them back together, and explain how they know the parts are equal.
Without these experiences, rational number can feel disconnected. With them, it starts to make sense. It’s these connections that help students move beyond seeing fractions as rules to follow, and towards understanding them as meaningful relationships.
Misunderstandings such as treating fractions as two separate whole numbers or comparing decimals digit by digit are often only visible when we pay close attention to students’ reasoning. Some of my recent work has involved contributing to the development of NZCER’s new rational number assessment, designed to generate insights into students’ understanding. It helps teachers see where students are confident, where misunderstandings may be emerging, and where further support is needed, particularly across the connections between fractions, decimals, and percentages. These insights can help teachers respond more deliberately to students’ reasoning and support stronger conceptual understanding over time.
Rational numbers are challenging because they require students to rethink what numbers mean and how quantities relate. With time, multiple representations, rich discussion, and carefully connected experiences, students can move beyond memorising procedures and begin to see fractions, decimals, and percentages as meaningful mathematical ideas. We’re about to trial NZCER’s new rational number assessment and look forward to making it available to you all soon. Watch this space!
About the author
Janine Simpson has worked in primary education as a classroom teacher, school leader, and mathematics facilitator. She has also served as a national lead facilitator, supporting schools across Aotearoa to strengthen mathematics teaching and learning during the recent curriculum updates. Through this work, she has partnered closely with teachers to deepen their understanding of students’ mathematical thinking and strengthen instructional practice.
Now working with NZCER, Janine brings both classroom insight and system-level experience to her work in assessment development. Her particular interest lies in how assessment can meaningfully inform teaching practices, especially in complex areas such as rational number. As part of this work, she has supported the development of NZCER’s new rational number assessment, aimed at helping teachers better understand students’ thinking and plan responsive instruction.