A quick thought experiment – imagine asking students to solve the problem ½ + ¼. How do they respond? How do they get to those responses?
To many, this will look like a simple problem using familiar figures. But findings from the 2022 NMSSA study (now Curriculum Insights) shows that only about one third of Year 8 students answer this correctly.
Where did they land? Many students gave the response 2/6—suggesting that maybe students are combining numerators and denominators mechanically, instead of thinking about halves and quarters as quantities related to a whole.
This isn’t simply a case of students not being able to add fractions, it’s a sign that rational number thinking is much more than recalling a set of steps.
What is rational number learning?
In the refreshed New Zealand Curriculum, mathematics is framed as the study of relationships between quantities, space, and time, and using these relationships to make sense of the world. Rational number learning sits right at the heart of this idea.
Rational numbers describe relationships between quantities, not just counts of things. Unlike whole numbers, they allow us to express how much of something there is in relation to a whole or to another quantity. These relationships are often multiplicative as well as additive. Fractions, decimals, and percentages are all different ways of representing rational numbers.
For example, although they look quite different, 1/2, 0.5, and 50% all describe the same amount relative to a whole. Rational number learning involves recognising these equivalences and understanding what the numbers mean, rather than focusing only on how they are written. It is about seeing beyond surface features to the underlying relationships the numbers represent.
Rational number learning is more than a set of topics. It is about what numbers represent, how they connect to one another, and how quantities interrelate. This includes:
- understanding numbers as relationships
- using and connecting representations, such as area models and number lines
- moving flexibly between fractions, decimals, and percentages
- reasoning about proportion, comparison, and change
Why is rational number so hard to teach and learn?
Rational number is one of the most challenging areas of primary school mathematics. When taking it on, learners are asked to move beyond whole number thinking and make sense of multiplicative and proportional relationships.
Many of students’ existing intuitions, that work so well with whole numbers, do not transfer cleanly. Consider how adding or multiplying rational numbers does not always result in a “bigger” number, and quantities that look quite different (like the aforementioned 0.5, 1/2, and 50%) can represent the same amount.
Teaching students rational number takes time and care. Students require the chance to work with diagrams, materials, and numbers lines. They need to connect these representations to different symbols and pieces of language. Students then need to understand what these symbols indicate about the quantities they represent.
That’s why something that seems simple on the surface, like adding ½ and ¼, can become a roadblock in a student’s learning. When students struggle with questions like the above, it might not mean a lack of ability or effort. Instead, it might be a reflection of how well they can make the connections between the symbols, representations and meaning. If we can make those connections visible, make them explicit for students, it can go a long way to supporting students’ progress with rational numbers.
Why do rational numbers matter?
Rational number understanding underpins so much of what comes down the line in mathematics: proportional reasoning, algebra, measurement, data, and everyday numeracy. When students struggle with rational number, difficulties often persist across later learning.
Essentially, rational number isn’t an area that is started, understood and completed. It is central to so much of the mathematics in the New Zealand curriculum that it is an ongoing process of learning relationships throughout a child’s educational journey.
Questions for kaiako to consider about rational number
As teachers and leaders review programmes and consider rational number, it may be worth asking the following questions:
- How well does your maths programme build understanding of procedures and understanding of rational quantities?
- Will your students have support to explore the relationships between fractions, decimals and percentages?
- Will the assessment tasks students undertake show their reasoning and meaning-making skills
- How will rational number learning sow the seeds for future learning about proportional and algebraic concepts?
Looking ahead
At NZCER, this view of rational number as a connected learning domain is shaping work to develop a computer-adaptive assessment focused on rational number learning.
The assessment will locate student achievement in Rational Number on a described measurement progression, providing clear information about where students are in their learning and what may come next.
Watch this space – there is plenty more to come.