This article was originally published on 12 November 2021

You’ve heard people say it. In the staffroom. At the dinner table. You may even have said it yourself: “I was rubbish at maths at school.”

Usually laughter follows, or a shrug of the shoulders. Being bad at maths doesn’t really matter, right? If you can’t read or write, that’s different, but when it comes to maths, many adults admit that they struggle without embarrassment.

But teachers know that maths matters. They know that pupils will go on to use maths every day: to work out how many minutes until the next train, to measure out their medicine doses and to make sense of statistics and graphics in the news.

Yet, there is still a disconnect between the importance that schools place on maths and how much many of us value being seen as “good” at the subject.

Why is this? When Ofsted published its research review on mathematics education in 2021, former chief inspector Amanda Spielman said that “for too many children and young people, maths is mysterious and difficult, and this has implications not just for their future attainment but also for their self-esteem”.

The review concluded that “variation in the quality of mathematics education in England is likely to be the result of the absence of systems and systems thinking, as well as possible gaps in content, instruction, rehearsal, assessment and the plans for their evolution over time”.

In other words, according to Ofsted, a lack of clarity and consistency in maths teaching could be affecting both pupils’ confidence in maths and their outcomes.

Five ‘safe research bets’ for teaching primary maths

When it comes to outcomes, the inspectorate seems to have a point. In 2019, 21 per cent of pupils left primary school below the expected standard for maths. And at secondary level, in 2021, nearly a quarter of students (23 per cent) did not pass GCSE maths.

Clearly, something is not working. But could change could be on the way?

Professor Camilla Gilmore, from the Centre for Mathematical Cognition at Loughborough University, points out that over the past 20 years or so, researchers from different fields - such as psychology and neuroscience - have become more interested in maths. While there has long been a plethora of research around early literacy and the building blocks of reading, early maths has had a lot less attention.

“There’s still not as much research on numbers as there is on language, but it is rapidly increasing. That’s really exciting but also quite challenging,” she says.

The challenge comes from the fact that more research means that teachers have to work harder to find the useful evidence, says Ruth Trundley, a primary maths adviser and Jurassic Maths hub lead. The trouble is that sweeping research reviews, such as the one conducted by Ofsted, are not always helpful here, she adds.

“We have a very bad history in education of research dictating practice without taking account of context. Often, someone will pick up a piece of research that says to do X. Everyone does X, without it really being understood,” she says.

Gilmore agrees that there is a need for caution. “There are so many different types of research which can contribute to building up theories about the development of maths. The key is then translating those theories into classroom practice,” she says.

So, which findings about early maths teaching from research should teachers trust, and how can they be brought into the classroom? Based on their expertise in maths education and research, Gilmore and Trundley suggest that there are five “safe research bets” for teaching maths at primary level, which are all demonstrably linked to later success in maths.

These five have been picked, Gilmore says, because the findings are from bodies of research over time, in different contexts, using different research methodologies. Within those bodies of research, the pair consistently found five skills that seem to be the key building blocks of effective maths education.

So, what are they?

1. Counting is a complex skill

The first skill is counting, and, more specifically, understanding cardinality. Here, Gilmore cites How Numbers Are Like the Earth (and Unlike Faces, Loitering, or Knitting) by ‪Barbara Sarnecka.

Counting, she says, involves much more than simply reciting a count sequence. Children need to understand cardinality and how counting can be used to create and compare sets. For example: a child has a set of wooden strawberries, and is asked to give a puppet two of them. To begin with, the child might grab a handful of strawberries, without counting them. After a period of time, they will realise they can count one out, and give one to the puppet, but can’t count two or three. A few months later, they may be able to count out two and hand them over, and then three and four.

Next, they will come to understand what we call “cardinality”: this means they know that once they’ve counted out three strawberries and given these to the puppet, the puppet will have a set of three. In other words, it’s the difference between knowing that the last strawberry is called “three” and knowing that there are three items in the set.

“There’s a lot of research showing this cardinality insight is really key and associated with better long-term mathematical outcomes and better maths achievement,” says Gilmore.

There are a lot of different activities that can help to instill cardinality, says Trundley.

“The activities which focus on counting out a set are so different from counting a given set,” she explains. “It’s about making sure that those opportunities are built into the classroom.”

Teachers can do this through routines around clearing up, for example. If pens and pencils belong in pots, Trundley suggests putting a label on the pot that says “five pens”, and asking the children to put five pens into the pot. When there are five pens in the pot, teachers can ask: “Have we got the whole set? Are there five pens here?”

“It’s about helping [pupils] to create the set, and asking them to recognise if the complete set is there or not,” Trundley explains. “It’s all about reinforcement, and repeating things back to children. When they count five pens, you say, ‘Yes, there are five pens altogether.’”

2. Subitising

The second safe bet is the importance of subitising, a skill that theory suggests may be the foundation for developing fluency with numbers. Gilmore points towards research by Douglas Clements, Subitizing: what is it? Why teach it?, published in 1999.

There are two different types of subitising. The first is perceptual: this is the idea that for small quantities - three for children and four for adults - we don’t need to actually count to know how many are in a set.

The second type is conceptual subitising (also known as groupitizing). This is about using perceptual subitising to grasp how many are in bigger sets: for example, seeing six items, and recognising there are three and three, which then makes six.

“Subitising is so important because it is where we start to associate number words with different quantities,” says Gilmore. “One of the theories suggests that once you can label one item as one, two items as two, three items as three, you can recognise one, two, three is the start of the count sequence.

“This is the beginning of understanding numbers, and it’s extremely powerful. If children can subitize quantities then it’s a huge benefit for them when they’re building these higher number concepts, and even going into things like addition, and how we combine smaller groups into making bigger groups of numbers.”

When it comes to embedding the skill of subitising, many adults make a simple mistake, warns Trundley: if you ask a child how many cars are there and they correctly say three, for example, many adults will then say, “Let’s count to check”.

“We need to avoid this,” says Trundley. “We can all see there are three there, the child has learned they can see three, and it needs to be celebrated. We don’t need to count it.”

Picturebooks are a great tool to support subitising, she continues. You can ask a child to find a picture for which they know how many things there are without needing to count; for example, two bears. Then ask them to find a picture for which they do need to count to know how many things there are; for example, 10 cats. Next, ask them to find a picture in which there are simply too many things to count; for example, a large crowd of people.

This helps children to realise that they can make decisions in mathematics using what they know, says Trundley. Adults need to model this skill and also consider the questions they ask.

3. Additive composition

Additive composition means knowing that numbers are composed of other numbers. To demonstrate its importance, Gilmore highlights a research review, “Understanding whole numbers”, published by Terezinha Nunes and Peter Bryant in a 2008 Nuffield Foundation report, Key Understandings in Mathematics Learning.

Additive composition is necessary for children to be able to understand numbers in an ordered sequence and to master addition and subtraction.

The following activity is often used in research: a child is given coins to buy something in a shop. If they’ve got seven pennies and a five-pence piece, and they want to buy something that is 6p, they can hand over six of the pennies. But the point of understanding comes when the child realises that a five-pence piece has the same value as five pennies, so they could also buy something for 6p using a five-pence piece and a penny.

“Research that has looked at children’s understanding and mathematical skills in later life has shown it’s this reasoning and understanding, this additive composition, that is a strong predictor of later performance,” Gilmore says.

In the classroom, teachers could use physical objects to embed this, says Trundley.

For example, you could arrange four paper dots as you’d see them on a dice, and then place one dot above. Ask the children what they see, and encourage them to see the four, and the one, which together make five. You could repeat this with three dots on the left and three dots on the right. Ask the children, “What’s the number overall?”

“It’s quite a simple thing, and it’s about seeing the small numbers within, and giving [pupils] opportunities to build on their subitizing. They can’t immediately recognise there are six items, but they can see there are three on the left and three on the right, so how many are there altogether?,” Trundley explains.

4. Equivalence

The fourth safe research bet is the importance of equivalence: understanding what the equals sign means. Gilmore points towards research published by the University of Cambridge in Cambridge Mathematics’ Espresso (issue 34), entitled “The equal sign”.

The reason why this is so important, she explains, is that too many children see the equals sign as an operator that tells you to “put the answer here”. This misconception makes it more difficult for them to grasp more advanced mathematics further down the line.

“As you can imagine, if children don’t have a good relational understanding - knowing [that] the two sides of the equal sign are the same value - then moving into things like algebra is a minefield,” Gilmore says.

Traditionally, most problems in textbooks and worksheets are presented in the style “2 + 2 = 4”. If the equals sign is only ever seen after the operation and before the answer, then it’s not surprising that children associate “=” with “the answer goes here”. Instead, teachers should vary the structure of their questions, moving the equals sign around, suggests Trundley. For example, try 5 = 2 + 3, or 3 + 3 = 4 + 2.

Again, using physical items, such as balance scales, can help here. However, language is also really important, says Trundley: “Rather than saying ‘two plus three makes five’, encourage the use of language such as ‘two plus three is the same as five’. Try to use richer language that really reflects relational pattern, rather than an operational pattern.”

There’s also an argument against using symbols too soon.

“There’s quite a disconnect between symbols and children’s understanding of mathematical situations. They can solve problems, contextualise and manipulate numbers out loud but, written down, they don’t understand those symbols as representation of the problem they’ve just solved,” Trundley explains.

Teachers should explore mathematical concepts in lots of contexts first - with blocks, with scales, with physical items - before teaching the relevant symbols. As pupils progress in their maths learning, teachers should return to these physical objects when introducing a new concept - fractions and percentages, for example.

5. Spatial skills

The final “safe bet” to take away from research is how critical spatial skills and spatial reasoning are in both early and later mathematical development.

Gilmore says that “Spatial ability as a gateway to Stem success”, published by Emily Farran (2019) and featured in Impact, the journal of the Chartered College of Teaching (issue 6), contains a wealth of evidence that spatial skills are related to understanding of mathematics and maths achievement more generally.

Spatial reasoning is a huge area. It includes the language of position (where something is, or how far away it is); which direction something may be moving in; which way up it is; how different shapes fit together and how they can move or rotate; the perspective of different viewpoints; size and distance; and appearance, scaling and navigating.

“Spatial skills are so complex; there are so many components,” Gilmore explains. “Researchers are at a point where there are theories about the different ways we think about spatial skills and categorise them.

“But I don’t think that matters so much. It’s more about knowing that all these different components are related to maths skills, they really support mathematical thinking and reasoning, and that’s why they are so important.”

It sounds complicated - but actually there are really simple play-based activities that teachers can do to foster these skills, Trundley says, such as encouraging children to play with puzzles, blocks and board games.

Trundley also recommends the Early Childhood Maths Group’s Spatial Reasoning Trajectory, and encourages teachers to make the most of giant play-based activities like number lines. Here, teachers should encourage children to physically move along the line, counting the steps as they go, and talk to them about the journey they’ve taken, asking things like, “How far have you gone? How many steps did you take to get here?”

Another activity could be playing with model characters in a setting like a doll’s house and asking the children: “Which objects can this character see from where they are? And how does it differ from another character?”

Encouraging children to draw diagrams is also useful for supporting spatial skills. For example, if a child has been riding a tricycle, rather than asking them to draw a picture of themselves doing this, ask them to draw a diagram of the route they took across the playground, Trundley suggests.

Embed these five skills throughout the day

The 2021 Ofsted research review notes that “there is no singular way of achieving high-quality education in [maths]”. And while Gilmore and Trundley agree, they do suggest that if teachers understand these five building blocks and how to embed them in their practice, they will have a better chance of teaching maths well.

At the same time, however, the pair point out that if these findings are to have a real impact, teachers need to apply them beyond dedicated maths lessons.

“Maths is around us all the time, it shouldn’t be about, ‘Right, I’m going to do a maths activity, tick that off and now carry on with the day,’” Gilmore explains. “All of these five skills can be embedded throughout the school day, using resources teachers will already have in their classrooms. That’s what will make the real difference.”

Kate Parker was schools and colleges content producer at Tes

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